Commodity Procurement with Operational and Financial Instruments

1 Introduction 1.1 Motivation Increasing global competition and cost pressure force enterprises and supply chains to discover undetected cost-saving potentials. In particular, interfaces to the procurement market are a promising field for improvement. Recent developments in international trade outline the high influence of uncertain deliveries and highly volatile prices on the companies’ costs. A vital question in research, industry, and politics addresses the optimal procurement policy of raw materials due to uncertain future prices and is of highest priority for the firms’ success and even the wealth of a nation. Nowadays, commodity markets and commodity derivative markets offer transparent, fast, and efficient trade and risk sharing for raw materials and financial products on raw material prices, e.g., option contracts. This coexistence of operational and financial procurement instruments, i.e., buying opportunities on spot and derivative markets, offers an auspicious chance to optimize the procurement policy in the crucial problem of raw material procurement which is the focus of this thesis. The most basic distinction of commodities is between storable and non-storable commodities (see Pirrong (2008)). The vast majority of commodities is storable, mainly at some costs, but this class is fairly heterogeneous. Commodities can be characterized by seasonality in demand (such as crude oil or natural gas) or production (take wheat or sugar as example) or contrarily by a rather continuous production and consumption (e.g., nickel and aluminum). The recent economic developments have strongly increased the attention researchers, managers, politicians, and even private households pay on the development of raw material prices (see Der Spiegel, 24/2008.). Several raw materials, among them the most important as copper, iron ore, tin, aluminum, crude oil, wheat, soybeans, sugar, or coffee, are traded at commodity exchanges and therefore, a financial market is existent. Derivatives as futures or option contracts are available. For example, the London Metal Exchange offers future contracts for various metals such as aluminum, copper, tin, or lead with a term of 3, 15, or 27 months. As other examples, the New York Mercantile Exchange offers option contracts on copper and the Chicago Board of Trade derivatives on Ethanol,

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1 Introduction

Livestock, and plenty of others. Many fields of essential importance in daily life are affected directly by the price level of raw materials, e.g., the costs for energy, heating, or fuel as illustrated in Figure 1.1 for one of the most important commodities, crude oil. Due to a highly industrialized agriculture, prices of basic food and

Figure 1.1: Quotations of Brent Crude Oil [US-$/Barrel]

agricultural products nowadays are almost as volatile as the crude oil price. An increasing global population increases demand for agricultural products. On the other hand, soil sealing and alternative usage of cultivable land decrease available acreage which cannot fully be compensated by technological innovations. Nevertheless, prices of agriculturals are of crucial interest for many, especially developing, countries. Figure 1.2 provides the price of wheat as representative for several key agricultural products (e.g., soybeans, coffee, orange juice) in order to illustrate the recent price development. The success of the heavy and processing industry is not only influenced by energy costs, but also by costs for main ingredients as copper, aluminum, or steel, which are usually produced and consumed rather continuously. To give an impression of the development of non-precious metal prices, Figure 1.3 maps the quotation of aluminum prices. Certainly, far not all raw materials are traded at commodity exchanges but on dedicated markets which are quite often organized as over-the-counter markets. Reasons can be found in a lack of standardization (e.g., steel), due to storage and transportation difficulties (natural gas), or due to few market participants on the demand and/or supply side (benzene).

1.1 Motivation

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Figure 1.2: Quotations of Wheat Futures [US-Cent/Bushel]

Figure 1.3: Quotations of Aluminum [US-$/Tonne]

Nowadays, more than 2.000 steel sorts (see DIN EN 10027-1) are distinguished. A single sort quite often does not have a sufficient trade volume to justify its trade at an exchange. Recently, the London Metal Exchange has made first efforts developing futures contracts for steel (see Campopiano (2007)). Nevertheless, these relatively new instruments have not yet fully been established. Steel consumers

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1 Introduction

regularly set up long-term contracts with their suppliers specifying a certain volume within a certain period or simply fixing a price for a certain period in advance. Likewise, options on pre-specified volumes are usual. Natural gas is usually transported via pipelines which provide a regionally limited availability. Although recent technological innovations have made techniques for gas liquidation efficient and workable (see Younger and Eng (2004)), the argument of storage and transportation difficulties still is sufficiently strong to prevent natural gas to be traded at exchanges. (Private) gas consumers therefore depend one unilateral price fixing of the suppliers which take the crude oil price as indicator for the price fixing. Although the worldwide trading volume of Benzene is relatively high (e.g., about 8.5 billion metric tons in West Europe in 2007 (see CMAI (2008)) only few suppliers and consumers arrange the trade mainly among each others. The need to set up a public quotation at an exchange is low as trade agents, e.g., Platts or Dewitt, publish historical quotations and established a system of average prices serving as guideline for transactions. This system is the basis for trading option contracts, similar to standardized derivatives, which require an individual agreement of both contractual partners. Summarizing, numerous commodities are traded at commodity exchanges such as the Chicago Board of Trade or the London Metal Exchanges. These exchanges offer a quotation of commodities’ spot prices and match supply with demand. An increasingly important business area are derivatives on the commodities such as futures or call and put option contracts which are comparable to derivatives offered at stock exchanges. Similar contracts are usual at dedicated over-the-counter markets. This choice offers potential of improvement for a firm’s procurement policy, but requires research considerations from an operational and a financial viewpoint. The young research field at the interface of finance and operations is increasing researchers’ attention due to undetected potential of improvement. Combing the fields of operations and finance enables to raise emersed benefits by connecting the strengths of both disciplines. To highlight the importance of combined research in operations and finance and to give guidelines for research, Birge et al. (see Birge et al. (2007), p. 355.) state: "Two basic implications of the seminal work of Modigliani and Miller [see Modigliani and Miller (1958)] are that the management of the firm’s operations can be separated from its financial management, and that hedging the variability of the cash-flows associated with the firm’s operations adds no value to the shareholders. Largely as a consequence of these fundamental conclusions, the fields of Finance and Operations Management have independently developed during the last half of the past century. While illuminating, the Modigliani-Miller theory rests

1.2 Research Questions

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on a number of simplifying assumptions, whose violations make the interface between Operations Management and Finance relevant. [...] It is now timely to compile state-of-the-art research at the interface of Finance and Operations Management [...]. Such research will capitalize on the richness of the two fields while exploiting gaps left in their interfaces [...]."

1.2 Research Questions The purpose of this thesis is multi-fold. The key question is the optimal procurement strategy of a buying agent who can match his demand just-in-time or in advance via inventories or financial option contracts. The thesis is at the interface between the disciplines of operations management and finance. The research questions contain the aspects of timing the purchase and determining the order quantity. This is studied within deterministic continuous time models using the theory of optimal control in order to isolate dynamic effects. Additionally, the aim is to answer the question how to optimally combine the procurement instruments, i.e., how to mix operational and financial instruments. This aspect is studied in two-period stochastic models under the assumption of a perfect commodity market involving arbitrage-free raw material prices in order to isolate stochastic effects. One key research question addresses the optimal manufacturing and procurement time and quantity when cost and demand parameters are deterministic and dynamic. A continuous time modelling approach is appropriate to answer this question. In detail: • Which combination of inventories and JIT-procurement is optimal? • In how far is the optimal manufacturing/procurement decision influenced by dynamic prices? • Which trade-off is responsible for the optimal manufacturing/production quantity and the timing of this decision? The question of the optimal mix of procurement instruments under uncertain future demands and prices when advance procurement via inventories, the procurement of call option contracts, and JIT-procurement is possible is answered by a two-period stochastic modelling approach. In detail, the research questions involve the following aspects. • How are the procurement instruments inventory holding, option contracts, and JIT combined optimally? Under which circumstances does one of these procurement instruments dominate an other one?

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1 Introduction

• In how far is the procurement decision influenced by the risk attitude of the decision maker? Does the quantity of advance procurement increase or decrease when decision makers are risk-averse? • In how far is the procurement decision influenced by a competitive sales market? Is the procurement decision different under the assumption of a monopolistic or duopolistic sales market? Does the mode of the duopoly, Cournot or Bertrand competition, influence this decision?

1.3 Structure and Overview The thesis is divided into three parts. Chapter 2 introduces the relevant fundamentals from financial theory and reviews the literature closely related to this thesis. Financial markets, arbitrage, and option pricing are addressed in Section 2.1, Section 2.2 introduces to the relevant aspects of commodity markets and commodity prices. Literature in the field of operations, especially single- and multi-period procurement models under risk-neutrality and risk-aversion, are presented in Section 2.3.1. Related literature from industrial organization investigating a competitive sales market is reviewed in Section 2.3.2. Section 2.3.3 investigates research at the interface of operations and finance. Deterministic manufacturing and procurement models are addressed in Chapter 3. Chapter 3.2, based on Arnold et al. (2008), analyzes an EPQ-type manufacturing problem with dynamic cost and demand parameters. Chapter 3.3, which is based on Arnold et al. (2007), addresses a related EOQ model. An investigation of stochastic models is provided in Chapter 4 of this thesis. Section 4.1 is based on Arnold and Minner (2009) and investigates the optimal procurement policy on an arbitrage-free commodity market under different optimization objectives, risk-neutrality and risk-aversion, i.e., mean-variance optimization and expected utility optimization. Section 4.2, which is based on Arnold and Minner (2008), investigates procurement strategies on an arbitrage-free commodity market when the sales market is characterized by price-sensitive demand within a monopoly (Section 4.2.3) or competition, i.e., Cournot quantity competition (Section 4.2.4) and Bertrand price competition (Section 4.2.5). Part 5 concludes the thesis. It provides a summary of the major findings and a discussion of extensions and implications for future research.

2 Fundamentals and Literature Review 2.1 Financial Fundamentals 2.1.1 Financial Markets The purpose of this section is to outline the key concepts of financial theory, financial markets and arbitrage-free pricing. Global and regional trade nowadays highly depend on the existence of markets, e.g., real estate markets, agricultural markets, or financial markets. Trade takes places at a market which constitutes binding guidelines for all participants in order to guarantee fast and efficient trades. A market is "a means by which the exchange of goods and services takes place as a result of buyers and sellers being in contact with each others" (cp. Enceclopædia Britannica (2005)). Consequently, a financial market deals with financial products such as stocks, futures, or option contracts. Financial theory usually assumes an idealistic market, the so-called perfect capital market, which is characterized as follows (cp. Ross (2008)). • No transaction costs, no taxes, and no regulations exist. • Market participants are rational, utility maximizing, and have homogeneous expectations. • Assets are perfectly divisible. • Market participants are price-taking. • All information, independent whether certain or uncertain, is costless and available for every market participant. Financial markets and pricing of contracts traded at financial markets are among the key interest in financial theory and financial research. The functions of financial markets are multilayered. Mishkin (2004) lists borrowing and lending, price determination, information aggregation, risk sharing, liquidity, and efficiency. For the purpose of this thesis, the financial market serves several of these functions, i.e., as information source for current and expected future prices, as a spot market for procurement, and in order to share risks. The

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existence of a spot market is a prerequisite for the existence of a derivative market. Derivatives are financial contracts contingent on the realization of a spot market price. In the course of this thesis, derivatives are used as instrument for procurement and hedging.

2.1.2 No-Arbitrage Arbitrage is defined as "a trading strategy that takes advantage of two or more securities being mispriced relatively to each other" (cp. Hull (2006), p. 741). Financial theory distinguishes two kinds of arbitrage, the so-called "free lunch" and the "free lottery". A free lunch provides its owner a deterministic positive profit without usage of capital whereas a free lottery provides the participation at a lottery with positive expected outcome (cp. Föllmer and Schied (2004)). Financial theory assumes markets to be free of arbitrage, i.e., no financial product providing a free lunch or a free lottery exists. There is an intuitive explanation. If two financial products providing the same future outcome are connected with different current prices, every agent interested in buying one of the two contracts would rather buy the cheaper one. On the other hand, every agent interested in selling one of the contracts would rather sell the more valuable. An increasing demand (supply) for the cheaper (the more valuable) contract will increase (decrease) the price of the respective contract until both are traded at the same price. This effect is called the "law of one price". A situation relevant for this thesis considers no-arbitrage on a commodity market. Assume a purchasing agent who has to buy a commodity which is required at t = 1. The current point in time is t = 0. In general, there are two simple strategies in order to own one unit of the commodity at t = 1 if a warehouse is available. First, procure at t = 0 and store until t = 1. The costs of this strategy are composed of procurement costs p0 (the deterministic spot price at t = 0) and holding costs h per unit and period which are assumed to be payable at t = 0. Second, wait until t = 1 and procure at the future spot price p1 which is stochastic from the view of t = 0 but will realize just before the purchase is made at t = 1. The expected value of p1 is E(p1 ). Payments are assumed to be discounted with a rate of ρ per period. The two strategies are illustrated in Figure 2.1.2. Absence of arbitrage claims equality of the net present values (NPVs) of both strategies, i.e., !

NPV1 = NPV2

⇔

!

−(p0 + h) = −ρ E(p1 ).

(2.1)

2.1 Financial Fundamentals

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Figure 2.1: Two Strategies to Procure one Unit of a Commodity

Note, that this equation states a tight connection between parameters of the stochastic price process, i.e., current and expected future prices are related via an intertemporal no-arbitrage condition. This relationship is relevant for the sections of this thesis considering procurement strategies at financial markets in Chapter 4. Equation (2.1) shows that the commodity price increases by the risk-free interest rate plus holding costs which is known as principle of risk-neutral valuation. The consequence of this principle is that (2.1) applies in a risk-neutral world and in a risk-averse world (see Hull (2006)).

2.1.3 Option Contracts and Option Pricing A derivative is a financial "instrument whose price depends on, or is derived from, the price of another asset" (see Hull (2006), p. 747). Nowadays financial markets offer a plurality of different derivatives such as options, futures, or swaps. The main focus of this thesis is on European call options on commodities. A European call option provides its owner the right (but not the obligation) to buy a specified amount of the underlying at a specified point in time (expiration date) at a fixed price f , the so-called exercise price. Likewise, a European put option provides the right to sell at a fixed exercise price. Opposed to this, American options provide the right to exercise the contract not only at a specified point in time, but during a specified period.

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Financial researchers have been working on the pricing of option contracts for many decades. The seminal works among this research are summarized in short. Black and Scholes (1973) developed the so-called Black-Scholes option pricing formula for option contracts on shares. Within a continuous-time framework, the future stock price follows a Brownian motion and has a lognormally distributed drift rate. They determine a closed-form solution of the option price of a European call option using a second-order Taylor-series expansion of the price process. Black (1976) extended the Black-Scholes approach to various derivative contracts, especially for goods involving holding costs. Holding costs may arise when the underlying is, e.g., a commodity or an exchange rate. These costs can be included in the option’s pricing formula by additional discounting with the holding cost rate. A modification of the approach of Black and Scholes (1973) for discrete time quotation of the underlying is presented in Cox et al. (1979). The so-called binomial tree model investigates a two-point distribution of future prices. The price of the underlying can move upwards above the option’s exercise price or downwards below the exercise price. The authors present a closed-form solution of the option value. The Black-Scholes formula is proven to correspond to the limiting value of the option price in the binomial tree model when the time intervals between two quotations tend to zero. Barone-Adesi and Whaley (1987) provide an approach to determine the value of option contracts on commodities and commodity futures. Both European and American call and put options are analyzed. American options can only be valued via an approximation function. However, the authors show how holding costs that arise from storing commodities can be integrated in a pricing formula for option contracts. The synopsis of this work is characterized in the following in order to present the framework for option pricing used in this thesis. Assume a spot market for a commodity with deterministic current price p0 , stochastic future price p1 and holding costs h per unit and time unit and a purchasing agent who requires one unit of the commodity at t = 1. A third strategy is to buy one option contract at price c0 at t = 0 with exercise price f . c1 denotes the future option price, E(c1 ) its expected value. If the price at t = 1 is above f , the agent exercises the option, otherwise he procures at the spot market at t = 1. This strategy guarantees the ownership of one unit of the commodity at t = 1 (as the first two strategies did) and has to result in an equal NPV in order to satisfy the law of one price. An illustration is provided in Figure 2.1.3. c0 and c1 have to satisfy c0 = ρ E(c1 )

(2.2)

2.1 Financial Fundamentals

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Figure 2.2: A Third Strategy to Procure one Unit of a Commodity

in order to guarantee arbitrage-free pricing of option contracts. Let p1 be continuously distributed between a lower bound p1 and an upper bound p1 and the density function be denoted by φ (p1 ). Then E(p1 ) =

 p1 p1

p1 φ (p1 )dp1

(2.3)

and E(c1 ) =

 f p1

0φ (p1 )dp1 +

 p1 f

(p1 − f )φ (p1 )dp1 =

 p1 f

(p1 − f )φ (p1 )dp1 (2.4)

hold which again corresponds to risk-neutral valuation. A focal interest in financial theory is to calculate the current option value c0 = ρ E(c1 ) which can be expressed in parameters known at t = 0 by equation (2.4). Note, that the state-of-the-art assumption p1 > 0 must hold as a price of zero implies an infinite profit when the price changes from zero to a positive value. The NPV of the third strategy thus is  NPV3 = −c0 − ρ

f

p1

p1 φ (p1 )dp1 +

 p1 f

 f φ (p1 )dp1

which has to equal NPV1 and thus NPV2 to satisfy the law of one price.

(2.5)

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2.2 Commodities Commodities addressed in this thesis and exemplified in Section 1 are understood in a wide context. They distinguish from other physical goods mainly by a relatively high degree of standardization, take wheat or crude oil as examples, a low degree of processing progress, for example, natural gas or benzene, and a usage as basic ingredient for trade or production. Many of these raw materials are traded at the world-wide exchanges, e.g., the Chicago Board of Trade or the London Metal Exchange or via dedicated over-the-counter markets. Geman (2005) defines commodities from the viewpoint of an economist as follows. "An economist would say that it is a consumption asset whose scarcity, whether in the form of exhausting underground reserves or depleted stocks, has a major impact on the world’s and country-specific economic development." Her textbook provides a broad overview on commodities, commodity markets, and commodity derivatives. The target of this section is to emphasize the most relevant issues of commodities necessary for the purpose of this thesis, i.e., to address the irrelevance of convenience yield in this context and to present the fundamentals of commodity prices. It refers closely to Geman (2005).

2.2.1 Convenience Yield Geman (see Geman (2005)) formulates, that the spot-forward relationship for stocks (2.6) and storable commodities (2.7), i.e., the relation between current and expected future prices, must satisfy p0 = ρ E(p1 )

(2.6)

p0 = ρ E(p1 ) − h − j

(2.7)

in case of a stock and for a commodity (the formulae provided in Geman have been adapted to the framework of time and notation used in this thesis). In the latter case, the so-called convenience yield j has to be incorporated into the intertemporal no-arbitrage consideration formulated in (2.1). This convenience yield can be defined as the advantage for the owner of a physical good but not for the holder of a derivative contract (see Kaldor (1939)). Especially within an inventory context, physical goods can provide benefits. Inventories have a value as they allow to react on unexpected demand and to reduce the cost of modifications of the production schedule and manufacturing disruptions (see Geman (2005)). From an operational point of view, these benefits can be regarded as lead time reduction and safety argument. While items

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in the inventory are immediately available, e.g., for demand satisfaction, the ownership of a derivative does not provide this advantage. This thesis addresses stylized two-period stochastic models characterized by two discrete points in time. Therefore, the physical ownership of the commodity cannot provide any advantage as no lead time reduction is possible and, consequently, the convenience yield is zero. On the other hand, a redefinition of the holding cost parameter h such that h := h + j in (2.7) allows to easily integrate a deterministic convenience yield into the spot-forward relationship formulated in (2.1).

2.2.2 Commodity Prices Commodity prices have attracted researchers’ interest for decades and led to a broad variety of modelling approaches to this question. The history of this research is outlined in Geman (2005) and Pirrong (2008). Early work on the behaviour of commodity prices is based on the so-called theory of storage. The theory of storage was developed by various authors such as Kaldor (1939), Working (1948), Brennan (1958), and Telser (1958). Commodity prices are explained by an equilibrium resulting from quantities produced and stored and the demand. The difference between the future price and the current spot price results from the level of the convenience yield, inventory holding costs, and forgone interests. The key results gained from the theory of storage are summarized as follows (cp. Geman (2005)). Higher levels of global inventories of a commodity are related to lower price volatilities. Inventories therefore have a balancing effect on prices. Higher price levels are related to higher price volatilities. This so-called inverse leverage effect exists as both factors have a negative relation to the inventory level. Another view interprets this difference as premium due to the bias of the futures price as a forecast of the future spot price. This alternative view splits a futures price into an expected risk premium and a forecast of a future spot price and was developed by Cootner (1960), Dusak (1973), Breeden (1980), and Hazuka (1984). Fama and French (1987) compare these viewpoints. The key result is a slight advantage for approaches based on the theory of storage, but a strong dependence on the type of commodity and the variance of its price. Forecast results for commodities with a low price variance are slightly better explained via convenience yield and interest rates, commodities prices with a high variance can be modeled slightly better via the bias of the futures price as a forecast of the future spot price. Later contributions describe the behaviour of commodity prices by a tendency to mean-reversion and an influence of random shocks. The feature of mean-reversion results from an adjustment of supply to demand such that prices remain approx-

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imately stable on average. This can be represented by an Ornstein-Uhlenbeck process as in Vasicek (1977). Merton (1976) developed the jump-diffusion model which is capable to describe random shocks in supply or demand on commodity markets. These cognitions have been amended by models including multiple state variables in models based on time-series analysis. Geman identifies "the convenience yield, the long-term value of mean-reversion and stochastic volatility" (see Geman (2005)) as promising candidates for state variables. Gibson and Schwartz (1990) developed a two-factor model for pricing contingent claims on the price of crude oil and provide empirical tests. The spot price of crude oil and the convenience yield are identified as the two relevant factors. While a geometric Brownian motion serves to describe the oil spot price, an OrnsteinUhlenbeck process models stochastic convenience yield. The insights are applied in order to determine the present value of various crude oil futures. A seminal article considering the theory of storage is from Deaton and Laroque (1992) who address the phenomenon of autocorrelation in commodity price time series. They formulate a stochastic dynamic programming model including inventories, current equilibrium prices, and expected future prices. This model is capable to explain in parts how risk-neutral decision makers increase autocorrelation by combined speculation and inventory holding. A related theory of commodity price fluctuations is developed by Chambers and Bailey (1996). They investigate price fluctuations of a storable commodity traded in an open market. Under the influence of random shocks to supply and/or demand a unique stationary rational expectations equilibrium is derived based on the investigation of Deaton and Laroque (1992). Three cases, independent and identically distributed demand disturbances, correlated disturbances, and periodic disturbances are distinguished. Schwartz (1997) studies models explaining the stochastic behaviour of commodity prices. He takes into account a one-factorial model assuming mean-reversion, a two-factorial model adding convenience yield, and a three-factorial model incorporating mean-reverting stochastic interest rates. An empirical investigation for crude oil, copper, and gold is provided. McLaren (1999) presents a model of oligopolistic commodity speculation. Assuming an infinite time horizon, a limited number of speculators applies noncooperative storage. He solves the model via subgame perfect equilibrium and shows that inventory levels are lower and prices more volatile than under perfect competition. Schwartz and Smith (2000) investigate a two-factorial model for commodity prices combining mean-reversion in the short term and an uncertain equilibrium

2.3 Literature

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level to which prices revert when the convenience yield is constant over time. An estimation of the long term level is gained from prices of futures contracts with long maturity. The short term variation is estimated from the difference between prices of long and short term contracts. The authors show the equivalence of their model to the model of Gibson and Schwartz (1990) considering stochastic convenience yield. Due to the broad variety of commodities, an individual analysis of each commodity is appropriate in order to reflect the specific behaviour and dominance of one of the key characteristics (mean-reversion, random shocks, convenience yield, or stochastic volatility). This detailed investigation is neither objective of this thesis, nor required as the following analysis rather assumes arbitrary price functions (Chapter 3) and arbitrary distribution functions of prices (Chapter 4) within a twoperiod context and thus doesn’t need the statement of a specific (multi-period) price process.

2.3 Literature 2.3.1 Operations Literature The contributions reviewed in the following address the sourcing aspects of procurement and production decisions, i.e., timing and allocation. The timing aspect considers the question when to procure or manufacture in order to satisfy current and future demands. The allocation aspect refers to the mode of how materials required for production are procured - using JIT-procurement, (supply chain) options, or storing inventory. In the following, continuous time and discrete time models are distinguished. Continuous Time Models Various researchers address deterministic, continuous time procurement and planning models. This research is categorized into models with linear cost functions, EOQ (Economic Order Quantity) and EPQ (Economic Production Quantity) models with linear variable costs and fixed order/setup costs, and models with convex variable costs. Ijiri and Thompson (1970) develop the so-called wheat-trading problem involving linear variable costs and no fixed costs. Within a finite planning horizon and continuous time, a speculator buys and sells a commodity (wheat) in order to optimize his assets consisting of a stock of wheat and money at the terminal point in time. The wheat quotation is deterministic and fluctuating in time. As costs

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are linear, Ijiri and Thompson identify a bang-bang property, i.e., an all-or-nothing property. Dependent on the relation between the instantaneous price change and holding costs of the commodity, it is either optimal to fill the inventory with maximal rate or to sell items from the inventory with maximal rate. EOQ- and EPQ-type models involve fixed costs. Bensoussan et al. (1974) study an EOQ control model including deterministic demand and lot-sizing in continuous time within a finite planning horizon. The authors relax for discounting and dynamic parameters besides the demand rate. The inventory is replenished by lot-sizes where fixed costs occur and a deterministic demand has to be fully satisfied. Due to the fixed costs, the model involves a non-continuous adjoint trajectory representing impulse-controls at junction points, i.e., order times. First-order conditions for the optimal order times and quantities and a simple algorithm to find the number of replenishments are provided. Luhmer (1986) formulates a dynamic optimal control EOQ problem considering ordering cost in different settings, backlogging, non-backlogging, and limited storage space combined with (non-)backlogging. The impulse maximum principle is used to derive necessary conditions for optimal solutions including jumpconditions for the state trajectory. He points out the importance of a correct assessment of inventory holding cost by a discounted cash flow approach. Giri et al. (1996) investigate an EOQ model for deteriorating items. Demand rate, inventory holding cost, ordering cost, and the deterioration rate fluctuate in time incorporating the possibility of partial backlogging. Optimal order points are characterized by first-order conditions and the optimal number of replenishments is obtained algorithmically. A numerical example and sensitivity analysis illustrate the results. Al-Fawzan and Al-Sultan (1997) develop an EPQ model with a controllable production rate to minimize total cost in two cases, with and without the possibility of shortages. They analyze effects of the manufacturing rate on manufacturing quality, inventory levels, and fixed costs. An EOQ model with partial backlogging and shortages is presented in Teng and Yang (2004). The demand rate and the production costs are deterministic and dynamic. The authors determine the optimal replenishment policy which is proven to be unique. As the function of total costs is convex in the number of replenishments, the search for the optimum can be accelerated. Teng et al. (2005) analyze EPQ models with dynamic demand and production costs. They prove the convexity of the total cost function in the number of replenishments and provide a starting point for searching the optimal replenishment number.

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Models with convex variable costs are addressed by Pekelman (1974 and 1975) and Stöppler (1985). Pekelman (1974) investigates simultaneous price and production decisions within a deterministic, continuous time model. The demand rate and the price function are modeled as dynamic parameters during a finite planning horizon. The model is of an EPQ-type including convex production costs in absence of fixed costs. A solution is obtained via optimal control theory, in particular, by using conditions for the state constraint of a nonnegative inventory level. The existence of planning horizons leads to a decomposition property exploited by an algorithm that assures an optimal and unique solution. A related optimal control problem is analyzed by Pekelman (1975). Demand rate and price function are deterministic and dynamic during the finite planning horizon. The firm maximizes the total profit by choosing the production rate when adjustment of output incurs additional cost. An algorithmic solution is possible as a property of planning and forecast horizons is identified. Stöppler (1985) analyzes an EPQ problem with cyclic demand in absence of fixed costs as optimal control model. Inventory holding costs are dynamic, manufacturing costs are represented by a twice continuously differentiable, convex function yielding a non-linear control problem. He compares a discounted and a non-discounted approach and devotes attention to the relation between the variability of demands and the optimal manufacturing policy. Stöppler shows, that increasing inventory holding costs yield an increased synchronization of production and demand and therefore, a higher variability of the production level. While continuous time models often are represented in deterministic models but allow for analytical results, discrete time models incorporate stochastic parameters. Differences and similarities between deterministic continuous and discrete time approaches are investigated in Khmelnitsky and Tzur (2004). They consider a stationary production-inventory problem with limited production capacity in discrete and continuous time. The discrete time version equals an infinite horizon capacitated dynamic lot-sizing problem. The authors develop an analogous continuous time version and find out circumstances under which the solution of the continuous problem is easier to handle than the solution of the discrete problem. Discrete Time Models Cohen and Huchzermeier (1999) review optimization models considering supply chain management under uncertainty. They emphasize the importance of risk management in a firm’s operations and finance in global supply chain management. A review of supply chain risk management is provided by Tang (2006). Besides addressing supply network design, supplier relationships, demand, product, and

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information management, he reviews literature investigating uncertain demand and uncertain supply costs. Tang outlines the investigation of uncertain supply costs and wholesale prices as underrepresented field of research. Single-Period Models Single-period, single item inventory models with uncertain future parameters are based on the Newsvendor framework (see, e.g., Silver et al. (1998) or Cachon and Terwiesch (2007)) which allows for a broad variety of modifications. The key modifications in the focus of this thesis are risk-aversion, stochastic future costs and prices, and competition in Newsvendor models. A literature overview on single period Newsvendor problems and risk-aversion is contributed by Khouja (1999). Lau (1980) investigates risk-aversion in Newsvendor models. He distinguishes a mean-standard-deviation approach, a maximization of expected utility (von Neumann-Morgenstern-utility), and a maximization of the probability to obtain a budgeted profit level. He shows that in the first case, the optimal inventory level of a risk-averse decision maker is below the solution of a risk-neutral decision maker. In the second and third case, the relation between the risk-averse and the riskneutral solution depends on the parameter values and the specific distribution of demand. Anvari (1987) applies the Newsvendor model within the framework of the capital asset pricing model. This procedure incorporates risk aspects in comparison to market risk at the financial market. The author provides a numerical comparison to a maximization of the expected profit which leads to an amount ordered far below the risk-neutral solution. Eeckhoudt et al. (1995) examine the risk-averse Newsvendor and provide various comparative statics. A maximization of the expected utility of the total wealth of a Newsvendor with some initial wealth is investigated. They consider the effect of increasing demand risk and an additional independent risk to the Newsvendor’s background wealth. Eeckhoudt et al. conclude that a risk-averse Newsvendor holds less items in the inventory than a risk-neutral Newsvendor. The optimal inventory level is increasing in the salvage value and decreasing in the variable costs. In special cases when the Newsvendor’s utility function is not characterized by decreasing absolute risk-aversion, wealthier decision makers order more items. Chen and Federgruen (2000) analyze mean-variance optimization in inventory models. They address Newsvendor problems based on profit maximization and cost minimization in combination with the profit’s or cost’s variance. In the first case, the inventory level turns out to be below the solution of the risk-neutral Newsvendor. In the latter case, the inventory is above the risk-neutral Newsvendor solution.

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19

Agrawal and Seshadri (2000) investigate a Newsvendor model under risk-aversion. The expected utility is maximized when the order level and the selling price are determined simultaneously. When a change in the selling price affects the scale of the demand distribution, the selling price (order quantity) in the solution under risk-aversion is above (below) the level under risk-neutrality. When a change in the selling price affects the location of the demand distribution, the risk-averse Newsvendor charges a selling price below the solution under risk-neutrality. In this case, the effect on the order quantity is unclear. Collins (2004) analyzes risk-averse Newsvendor models for various distribution functions of the demand and compares the solutions of optimizing expected costs and expected utility. He shows conditions under which the solution of a risk-averse and a risk-neutral decision maker are rather the same. Martínez-de-Albéniz and Simchi-Levi (2006) address operational hedging and investigate a manufacturer signing option contracts with a supplier and access to a spot market under mean-variance optimization. The manufacturer trades-off demand risk in case of excess option contracts and price risk in case of spot market procurement. The authors characterize an efficient frontier of optimal portfolios and show that their results cannot be compared to a financial hedging strategy. Wang et al. (2009) address the risk-averse Newsvendor problem for a broad class of utility functions including CARA and IARA, and some DARA functions. The authors show for a special case that the optimal order quantity is decreasing in the selling price. They prove that the decision maker orders less than an arbitrarily small quantity if the selling price is increasing in case that the price is above a certain value. Operations literature considering uncertain costs within Newsvendor models is investigated by various authors. Most of this research is done in the interface of operations and finance and presented in Section 2.3.3. Stochastic future costs in a Newsvendor model with two ordering occasions are analyzed by Gurnani and Tang (1999). Prior to a single selling season of a seasonal product, the retailer can procure facing uncertain demand. Demand forecasts can be improved exploiting market signals observed between the first and second ordering occasion. The production costs for items manufactured during the second instant are uncertain and allowed to be both higher or lower than the unit cost at the first instant. Key result is the retailer’s trade-off between a more accurate forecast and potentially higher manufacturing costs and conditions under which postponement of order decisions is optimal. Seifert et al. (2004) analyze optimal procurement strategies in presence of forward and spot markets. The authors derive a closed-form solution in case that the spot market can be used for buying and selling and compare this situation to pure

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2 Fundamentals and Literature Review

forward procurement. They show that a combination of forward and spot market procurement is beneficial, that the usage of the spot market increases the expected service level, and that the profit is more volatile if spot markets are used. Newsvendor models in connection with competition aspects are considered by Parlar (1988) and Lippman and McCardle (1997). Parlar (1988) investigates two Newsvendors with dedicated demands in competition. When demand cannot be satisfied by the dedicated Newsvendor, a fraction switches to the competitor. He investigates the situation with and without coordination and addresses irrational behaviour. Parlar proves the existence of a unique Nash solution and shows that the Newsvendors’ optimal decisions equal those in a standard Newsvendor model in case that a maximin criterion is applied. Lippman and McCardle (1997) assume that initial demand of a Newsvendor is responsible for the fraction of future demand. The model thus is characterized as allocation-reallocation problem. The authors investigate different types of splitting rules for the initial demand. They show that the sum of the inventory levels is identical in case of two competitors and the sum of expected profit tends to zero when the number of competitors is increasing. Multi-Period Models Multi-period models allow for multiple time instants to perform procurement or production. The well-known article of Wagner and Whitin (1958) contributed the dynamic version of the economic lot-sizing model. The authors developed a forward algorithm for a discrete-time, single-item problem where demands, inventory holding costs, and fixed costs are dynamic within a finite horizon. The objective is to minimize total cost composed of fixed order costs and inventory holding costs. Morris (1959) investigates purchasing policies subject to uncertain procurement prices. He provides a stochastic dynamic programming approach in order to minimize the total expected procurement costs. This solution is compared to heuristics using only the spot market (dollar averaging and speculation). Dollar averaging is a heuristic which invests every period the same amount of money in an asset. As less (more) items of this asset are procured when prices are high (low), Morris can prove that dollar averaging reduces the expected procurement costs in comparison to a JIT policy. Speculation includes buying and selling at the market and is proven to promise expected profits which are increasing in the price variability. Fabian et al. (1959) solve a discrete time procurement problem for fluctuating prices. They assume holding and shortage costs and known distribution functions for future prices and demands of the raw material, develop a dynamic programming model, and present an optimal solution. A case study exemplifies and illustrates the optimal solution.

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21

Kingsman (1969 and 1985) addresses a discrete time problem with stochastic future procurement prices distributed identically in every period. Deterministic demand in every period has to be satisfied either by procurement in advance and subsequent inventory holding or by just-in-time procurement. Due to the property of identically distributed procurement prices (in contrast to mean-reversion or a Markovian process), the probability of increasing or decreasing prices depends on the current realization, i.e., the price process is not arbitrage-free. Low (high) realizations of the procurement price close to the lower (upper) bound of the price distribution therefore are an argument for advance procurement (postponement) as the probability of increasing (decreasing) prices is high. Kingsman developed a multi-period stochastic dynamic program exploiting this property in order to determine critical prices, so-called price-breaks, indicating an optimal inventory coverage. A modification of Kingsman (1969) is studied by Kalymon (1971). He considers a price process with Markov property, convex inventory holding costs, fixed costs, and allows for shortages. Kalymon provides solutions for a finite and an infinite time horizon and focuses on computational aspects of an optimal solution. Golabi (1985) analyzes an other modification of Kingsman’s (1969) model. He additionally incorporates discounting and an infinite horizon and provides an optimal solution. Guimaraes and Kingsman (1990) develop a commodity procurement heuristic for a modified performance measure which uses a ratio of relative savings and relative price volatility. The authors show that the heuristic outperforms the model of Kingsman (1969) when applied for historical corn prices. Other Related Models Models from the field of dynamic pricing in the Newsvendor context, thus revenue management models, are loosely related to this thesis as they involve price setting aspects similar to competition models. Petruzzi and Dada (1999) provide a review of pricing problems based on the Newsvendor model when the inventory level and the selling price are determined simultaneously. Parlar and Weng (2006) investigate the pricing problem in connection with duopolistic price competition on the supply market. They compare a situation when the price setting marketing department and the operations department which determines the inventory level are not coordinated to a vertically coordinated situation in each firm. The authors show that the coordinated solution dominates the non-coordinated. A model incorporating dynamic quantity competition and economic order quantities in continuous time is considered by Transchel and Minner (2008). The two

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competing retailers are characterized by a different cost structure. The first retailer has both variable and fixed costs and inventory holding costs such that he replenishes in batches. His competitor only has variable ordering costs and thus uses a just-in-time strategy. The authors show that the first retailer (second retailer) decreases (increases) his market share during an order cycle and can act as a monopolist in certain settings.

2.3.2 Industrial Organization Literature In terms of duopolistic competition, game theory mainly deals with two forms, quantity competition as developed in Cournot (1838) and price competition as presented in Bertrand (1883). The two concepts distinguish by how selling price and corresponding demand are determined. A Cournot duopolist fixes the quantity he wants to sell and is assigned a selling price via a price-response function. A Bertrand duopolist fixes a selling price and is assigned a corresponding demand. Theory claims that - in the most simple and symmetric (textbook) form of both concepts - only Cournot duopolists are able to achieve a positive profit while in a Bertrand competition firms compete aggressively by setting prices low such that no positive profit is possible. Cournot Competition in Terms of Quantity Commitment Various researchers have analyzed the concept of Cournot in terms of commitment strategies and timing aspect. This research is usually done in an inventory, production, or capacity planning context assuming increasing costs over time. Consequently, the potential influence of a no-arbitrage assumption and a postponement strategy within a financial context is neglected. Dixit (1979) investigates a Cournot duopoly with fixed costs. An incumbent can realize an output in which a second firm’s entry is prevented. He focuses two aspects of product differentiation with distinct effects. If the established firm has an absolute advantage in demand, entry is harder for the competitor. This tendency is even intensified by a lower cross-price effect. In the basic model Dixit maintains the pre-entry quantity after entry. An extension incorporates the case where the output increases after the second firm’s entry which is made possible by setting up excess capacity before the entry. Dixit (1980) analyzes capacity investment of an established firm in order to generate market entry barriers in a Cournot duopoly. He incorporates fixed costs and variable cost of capacity within an entirely deterministic model. The established firm can build up capacity during a pre-entry period and during an eventual (post-

2.3 Literature

23

entry) competition period. Dixit points out conditions when market organization changes due to the capacity investment. If the competitor decides to enter the game, the incumbent will not install capacity left-idle in the pre-entry period. Arvan (1985) studies two-period Cournot games where the players can hold inventories. His focus is on the existence of equilibria. Circumstances under which equilibria are (a)symmetric and do (not) exist are shown. Klemperer and Meyer (1986) investigate the influence of uncertainty on price and quantity competition. Two competitors producing differentiated products simultaneously choose either a quantity to produce or a price to charge. If demand is deterministic, there exist four types of equilibria resulting from the combination of each two strategies for both firms. If, however, market demand is uncertain, firms have a preference between setting price and quantity. The key drivers for this preference are the marginal cost and the nature of demand uncertainty. A modification of the Cournot model allowing for two production periods before the market clears is presented by Saloner (1987). The competitors choose their production simultaneously in the first period. These outputs are observable before the firms fix their production quantity in the second period and enter a Cournot game. All parameters are known with certainty. Saloner’s results are composed of Stackelberg and Cournot elements where the smallest Stackelberg outcomes are Pareto optima. Pal (1991) generalizes the model of Saloner (1987). He investigates a Cournot duopoly in presence of two production periods and allows for time-varying costs. If costs fall slightly over time, multiple equilibria result. Else, the equilibrium is unique and the duopolists produce their single-period Cournot outputs in the period connected with lower production costs. Another two-period model within the field of capacity commitment is considered by Maskin (1999). During an investment period under uncertain future variable costs, an incumbent chooses a capacity level. Adjacent, the potential entrant either stays out of the market or installs his capacity. After uncertainty resolves, the incumbent acts as monopolist or competes with the entrant within a Cournot game. Maskin shows that stochastic costs or demand yield a higher capacity investment for the incumbent than necessary under certainty to keep the competitor from entering the market. Poddar and Sasaki (2002) analyze the strategic benefit of production in advance in terms of quantity commitment in a deterministic two-period duopoly. In the investment period, two firms simultaneously and independently decide on advance production levels. In the competition period, firms compete in a market characterized by a linear price-response function. Advance production is assumed to be more expensive than just-in-time production. They show that Cournot duopolists

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2 Fundamentals and Literature Review

use advance production, independently of the firms’ a priori symmetry. Poddar and Sasaki argue that in a price-setting market, advance production is disadvantageous. Models Combining Aspects from Cournot and Bertrand Competition Other researchers investigated models in order to resolve the problems arising from the different outcomes of the models of Bertrand and Cournot. The seminal work of Kreps and Scheinkman (1983) investigates quantity commitment in connection with a Bertrand game and inspired various researches to extensions. The authors consider a two-period deterministic duopoly. They outline that the solution of a duopoly depends both on the strategic variables and on the game form in which those variables are employed. During an investment period, two symmetric firms decide simultaneously on their capacity level for the production of a homogeneous good. During the subsequent competition period they enter a Bertrand game where they can produce and sell a quantity up to the capacity level decided in the investment period. The rationing rule applied first serves customers from the cheapest supplier. The key result is a unique equilibrium characterized by a Cournot outcome within the rules of a Bertrand game. This is achieved by producing the Cournot quantity in the first and charging the Cournot price in the second period. This result holds likewise if the first period decision is interpreted as capacity and production in the second period is still costly. Investment in capacity therefore serves as commitment not to charge aggressively low prices. Davidson and Deneckere (1986) investigate a duopoly model where firms decide on their capacity level before entering in a Bertrand-like competition. They strongly refer to the results of Kreps and Scheinkman (1983) which are shown to critically depend on how demand is rationed when the lower-priced firm is not able to meet demand. The authors show that the Cournot outcome is unlikely to establish. The equilibrium rather tends to be more competitive as output is higher and price is lower than in the corresponding Cournot model. The participants additionally tend to install asymmetric firm sizes and price dispersion resulting from mixed strategies they use in equilibrium. Haskel and Martin (1994) provide an empirical investigation of the findings of Kreps and Scheinkman (1983). They test the theoretical result that undercutting rivals by Bertrand pricing is only feasible if firms have the capacity to serve the market. If on the other hand firms have limited capacity, the Cournot model is more reasonable. Haskel and Martin discover a positive relationship between limited capacities and Cournot-like competition using survey data on capacity constraints and panel industry data set for the United Kingdom.

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25

Boccard and Wauthy (2000) extend the model of Kreps and Scheinkman (1983) where two symmetric firms choose capacities and then compete with prices to an oligopoly. Capacity installed in the investment period can be supplemented by additional capacities in the competition period for larger costs than in the investment period. Therefore, capacity is an imperfect commitment in terms of not to price aggressively in the second period. The production costs in the second period are the key parameter responsible for the results. When the production costs in the competition period are larger than the Cournot price, the Cournot outcome characterizes the unique subgame perfect equilibrium. Decreasing the production costs from the Cournot price towards zero, the whole range of prices, from Cournot to Bertrand, is obtained in the equilibrium. De Francesco (2003) and Boccard and Whauty (2004) correct technical mistakes found in the contribution of Boccard and Wauthy (2000). A modification of the model of Kreps and Scheinkman (1983) to price competition of more than two players is presented in Moreno and Ubeda (2006). The model is simplified by the assumption of strictly convex capacity costs, zero production costs during the competition period, and no production possibility above the capacity level installed in the investment period. The authors show that the Cournot outcome results in every pure strategy equilibrium. Anupindi and Jiang (2008) extend the model of Kreps and Scheinkman (1983) to demand uncertainty. The duopolists decide on capacity before realization of demand and on production quantity, the sales price, and deliveries after demand has realized. The authors show that price postponement can protect inflexible firms from the effect of uncertain demand whereas flexible firms can use production and delivery postponement to increase the profit under demand uncertainty. Besides literature based on the article of Kreps and Scheinkman (1983), other researchers investigate the interface of Bertrand and Cournot under different cost structures or uncertainty. Judd (1996) investigates a dynamic game where a firm chooses both output and price in order to resolve strategic limitations of the Cournot and Bertrand models. He considers differentiated products generating production and inventory holding costs and allows for simultaneous and repeated choices of output and price. Each period’s inventory level is observable, unsold outputs are stored. While constant marginal production costs yield Bertrand equilibrium prices, linear-quadratic production and inventory holding costs can yield Cournot outcomes in connection with high adjustment costs. The results are obtained via subgame perfect equilibria. Janssen and Rasmusen (2002) investigate a Bertrand model in which the number of active firms facing a monopsony is uncertain. Each firm has to decide on

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its price before the uncertainty is solved. The solution involves a mixed-strategy equilibrium in which expected profits are positive and decreasing in the number of active firms unlike in a Cournot model with similar uncertainty.

2.3.3 Literature on the Interface of Operations and Finance Several contributions in the literature developed approaches combining aspects from finance and operations. These contributions are of special relevance for Chapter 4 of this thesis addressing integrated approaches in the presence of a financial market. A predominant assumption in existing literature is due to the fact that an arbitrage-free financial market does not provide a simple potential for improvement. Therefore, these contributions assume risk-averse decision makers optimizing an expected utility or a mean-variance criterion. Van Mieghem (2003) reviews the literature on capacity management concerned with determining the sizes, types, and timing of capacity investments and adjustments under uncertainty. He studies three aspects, optimal investment in capacity by single and multiple risk-neutral decision makers in a stationary environment, dynamic adjustment of capacity, and risk-aversion in capacity investment involving hedging strategies with financial and operational instruments. Van Mieghem outlines that most of this research addresses an inventory context and emphasizes the combination of financial and operational means as promising field for improvement. Boyabatli and Toktay (2004) review literature considering operational hedging. They distinguish literature from the fields of operations, finance, and strategy and international business. They find that operational hedging strategies need a higher capital investment than financial hedging strategies but, on the other hand, provide a long-term hedge against risk. A literature review considering procurement problems in the presence of spot markets is provided by Haksöz and Seshadri (2007). The survey distinguishes contributions from operations literature and from financial and economics literature. The operational literature is mainly dealing with optimal procurement strategies when both a spot market and a derivative market are accessible. Financial and economics literature mainly focuses on the valuation aspect of supply contracts, often based on the theory of rational expectations equilibrium. Haksöz and Seshadri outline that problems incorporating financial and operational procurement instruments are widely unexplored. The emerging field of research on the interface of operations and finance can be distinguished in research addressing uncertainty in prices, uncertainty in demands, and uncertainty in prices and demands.

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27

Uncertain Prices Ritchken and Tapiero (1986) address combined procurement with financial and operational instruments of a risk-averse decision maker. Within a two-period setting, a procurement decision to satisfy a known future demand is made before the future procurement price realizes. A commodity can be procured just-in-time or in advance and stored in the inventory. Likewise, an option contract can be procured providing the right to buy a certain amount of the commodity when the price has realized. Future prices follow a log-normal distribution in order to employ the Black-Scholes option pricing formula. The objective is to optimize a utility composed of the profit and the variance from buying and selling the commodity. The key result is that both option contracts and inventories are used in an optimal procurement plan though prices are arbitrage-free. Birge (2000) shows how to incorporate risk arguments gained from financial considerations into an operational planning model. He analyzes a multistage stochastic linear program and uses the binomial option pricing model by Cox et al. (1979). The incorporation of risk is made possible by adjusting the capacity level in the planning model and assuming availability of hedging instruments. Wu et al. (2001) integrate contract and spot markets using tradable capacity options and address market and equilibria aspects. Buyers and sellers can trade contracts for advance delivery or they can sell and buy units in a spot market. The key question is the structure of the utility-optimizing portfolios of buyers and sellers consisting of spot and contract transactions, and the market equilibrium price. The authors show that the offer of a contract fee which equals variable costs is optimal for the sellers. Jammernegg and Paulitsch (2004) investigate risk-hedging supply chains when raw material prices are stochastic and demands deterministic. The authors show how short- and medium term contracts can be used to build up speculative inventories in order to hedge against price uncertainty. Risk management using financial hedging instruments in global procurement is addressed by Wildemann (2007). He outlines futures, forwards, swaps, and option contracts as promising contracts to hedge the cash flows arising from procurement activities. The contribution argues that operational risks can be transferred to the financial market. Uncertain Demands Chod et al. (2007) study the interface between finance and operations by setting up a procurement problem under demand uncertainty. Operational flexibility allows

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to exploit a pooling effect attenuating the problems arising from stochastic demand. Likewise, a derivate correlated to the demand risk is introduced serving as hedging instrument. The decision maker optimizes a concave utility-function and is allowed to chose a hedging contract offering an optimal payoff-structure. The authors show under which circumstances financial and operational instruments can be regarded as complements or substitutes. Financial hedging and postponement flexibility are substitutes. In contrast, financial hedging and product flexibility are complements in case that product demands are positively correlated and substitutes for a negative correlation. Uncertain Prices and Uncertain Demands Chung (1990) provides a valuation approach for a firm’s output decision using contingent claims. Demand and production costs are uncertain and the existence of an asset who’s price is perfectly correlated to the uncertain demand is assumed. The author finds out that the profit is decreasing in the demand volatility and the optimal output level is increasing in the interest rate. Caldentey and Haugh (2003) analyze a simultaneous optimization of the production strategy and the financial hedging strategy of a risk-averse firm optimizing a mean-variance criterion. The operating profit results from a Newsvendor model and is assumed to be correlated to the return of a financial asset. The authors distinguish a situation with complete and with incomplete information, and a situation with a dynamic hedging strategy in continuous time and a static hedge. Their numerical results show that the optimal ordering level in case of incomplete information is lower in comparison to complete information, and lower when the hedging strategy is static in comparison to a dynamic hedging strategy. Chen et al. (2004) analyze multi-period inventory models under risk-aversion and an endogenous price. They include fixed costs and use financial hedging in order to reduce risk. The authors develop base stock policies and show numerically that both the reorder level and the order-up-to level are decreasing in increasing risk-aversion in most of the numerical experiments.

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29

Gaur and Seshadri (2005) investigate hedging of inventory risk when the price of a financial contract is correlated with a product’s demand and develop an optimal hedging strategy. The key results are threefold. First, hedging increases the expected utility of a risk-averse decision maker and decreases the variance of the profit. Second, for a broad class of utility functions hedging provides an incentive to order a quantity closer to the quantity maximizing the expected profit. Third, hedging does not yield to additional investment. An optimal order quantity under commodity price and product demand uncertainty in complete markets is investigated by Tapiero (2008). The purpose is to outline how production level and financial transactions can be combined optimally. The decision maker maximizes an exponential utility function within a two-period problem and has private information about the future commodity price. Tapiero shows how to determine the optimal order quantity and provides various numerical studies.

3 Procurement with Deterministic Costs 3.1 Introduction At the beginning of many manufacturing processes, semi-finished products are manufactured from raw materials. Customer demands and raw material markets represent the interfaces of supply chains. Prices paid in procurement markets and for manufacturing can highly influence tied-up capital and therefore the total cost of final goods and company’s profit. Growing international demand even further increasing by the market entry of emerging nations recently pushed raw material prices to unknown levels (see Nestmann (2006)). Nowadays, raw material prices are responsible for an increasing portion of total procurement cost. Economic production quantities (EPQ) and economic order quantities (EOQ) address important problems in operations management. The objective in an EPQ model is to determine the optimal manufacturing quantity that minimizes total cost required to manufacture goods and hold inventories when production is performed incrementally during the manufacturing process. In an EOQ model, the objective is to determine the optimal order quantity minimizing ordering and inventory holding cost when orders are delivered in batches. The concepts of EOQ and EPQ offer a wide field of application, e.g., within a manufacturing context a company producing industrial goods in batches. Likewise, applications in procurement problems are usual. For example, a customer filling his inventory with raw materials in order to exploit dynamic changes of procurement prices. Although each concept provides interpretations both as manufacturing and procurement problem, the EPQ model is referred to as manufacturing problem (Section 3.2) and the EOQ model is referred to as procurement problem (Section 3.3). These problems incorporate fluctuating demand, manufacturing (or procurement) cost and inventory holding cost, i.e., parameters are dynamic throughout, and assume no fixed order/setup costs. The usage of optimal control theory to solve the models provides analytical insights into the problem and allows for interpretations of shadow prices. The correct assessment of capital cost on inventories and manufacturing is guaranteed by discounting. A determination of the most appropriate period-length is avoided by a continuous time planning approach.

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The investigation differs from related contributions by integrating four major issues: a discounted cash-flow approach, solution with the theory of optimal control, a combination of the three policies just-in-time-procurement (JIT), backlogging, and inventory holding, and a finite warehouse capacity. Allowing for a discounted cash-flow-oriented inventory holding cost evaluation is especially of importance if procurement prices and capital tied up in inventory vary over time. Using the modelling and solution approach of optimal control goes beyond the suggestion of a pure numerical approach and allows for analytical insights into the optimal structure of a solution and for an economic interpretation of Lagrangian multipliers and optimality conditions. Future raw material prices and manufacturing costs can have a stochastic nature, but the prices of derivative instruments like futures contracts and commodity options give a comprehension of market expectations on these prices. Deterministic analysis of the problem gives first insights into possible heuristic strategies in a stochastic environment. The decision to be made facing fluctuating raw material prices or manufacturing costs is when, i.e., at which price to purchase the commodity or manufacture the product. This section establishes a model-based approach to this question. There are manifold practical applications. E.g., consider a power supplier with a warehouse who procures natural gas from a gas company via a pipeline. The gas price is subject to long-term agreements and thus frequent and announced changes. As reaction on a future price increase, the power supplier can procure gas in advance at the pipeline’s capacity limit and store it until its usage for power generation. Section 3.2 formulates and solves a model for dynamic economic production quantities. Section 3.3 presents a corresponding model for economic order quantities. This model is a special case of the EPQ model and allows for a more detailed investigation of storage capacities and backlogging of demands. The investigation concludes in Section 3.4.

3.2 Manufacturing Model 3.2.1 Model Formulation This section introduces the required notation and assumptions and formulates an optimization model for a dynamic manufacturing problem. The objective is to minimize the NPV of total cost during the planning horizon t ∈ [0; T ] consisting of manufacturing and inventory holding costs. The variable production costs p(t) are

3.2 Manufacturing Model

33

strictly positive and represented by a twice continuously differentiable function. Dynamic demand has to be satisfied at every time instant from existing inventories or current production and is given by the continuously differentiable strictly positive function d(t). Inventories y(t) are subject to the continuously differentiable out-of-pocket holding cost function h(t). The control variable x(t) corresponds to the production rate, the state variable y(t) denotes the inventory level. The production rate has to be non-negative (x(t) ≥ 0) and respect the upper bound x(t) ≤ xmax , xmax > d(t). Inventories are non-negative (y(t) ≥ 0). Within a discounted cashflow approach, payments are discounted exponentially with the positive interest rate ν which allows for an adequate assessment of capital costs. The optimization objective is min x(t)

NPV =

 T 0

e−ν t [p(t)x(t) + h(t)y(t)] dt

(3.1)

subject to y(t) ˙ = x(t) − d(t) x(t) ≥ 0

xmax − x(t) ≥ 0

k(y(t),t) = y(t) ≥ 0 y(0) = y0 ,

y(T ) = yT .

(3.2) (3.3) (3.4) (3.5)

The inventory movement over time is represented by the differential equation (3.2). Conditions (3.3) consider the lower and upper bound on the manufacturing rate x(t). (3.4) guarantees non-negativity of the inventory level y(t). The initial inventory y(0) and the terminal inventory y(T ) are fixed by (3.5). Using an optimal control approach requires an adjoint variable represented by λ (t) in order to state the current-value Hamilton-function H (x(t), y(t), λ (t),t) = − [p(t)x(t) + h(t)y(t)] + λ (t) [x(t) − d(t)] .

(3.6)

λ (t) expresses the value of one unit in the inventory at time t. The HamiltonLagrange-function (3.7) adds Lagrangian multipliers αi (t), i = 1, 2 and β (t) associated with (3.3) and (3.4). L (x(t), y(t), λ (t),t) = H (x(t), y(t), λ (t),t) + α1 (t)x(t) + α2 (t) [xmax − x(t)] + β (t)y(t). (3.7)

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Necessary conditions are given by Pontryagin’s Maximum Principle, KarushKuhn-Tucker (KKT) conditions (see, e.g., Sethi and Thompson (2000)) and jumpconditions at junction times θ where the state constraint (3.4) is active.

Pontryagin’s Maximum Principle H(x∗ (t), y∗ (t), λ (t),t) = max

H (x(t), y(t), λ (t),t)

x(t)

(3.8)

∂L = −p(t) + λ (t) + α1 (t) − α2 (t) = 0 ∂ x(t)

(3.9)

∂L = νλ (t) + h(t) − β (t) λ˙ (t) = νλ (t) − ∂ y(t)

(3.10)

∂L = x(t) − d(t) = y(t) ˙ ∂ λ (t)

(3.11)

KKT-Conditions

α1 (t)x(t) = 0, α2 (t) [x

max

− x(t)] = 0,

β (t)y(t) = 0,

x(t) ≥ 0, x

max

α1 (t) ≥ 0

− x(t) ≥ 0,

y(t) ≥ 0,

α2 (t) ≥ 0

β (t) ≥ 0

(3.12)

Jump-Conditions

λ (θ − ) = λ (θ + ) + η (θ )

∂k ∗ (y (θ ), θ ), ∂y

(3.13)

    ∂k H x∗ (θ − ), y∗ (θ ), λ (θ − ), θ = H x∗ (θ + ), y∗ (θ ), λ (θ + ), θ − η (θ ) (y∗ (θ ), θ ), ∂t (3.14) η (θ ) ≥ 0, η (θ )k(y∗ (θ ), θ ) = 0. (3.15) Due to the fixed terminal state in (3.5), a transversality-condition is irrelevant.

3.2 Manufacturing Model

35

3.2.2 Model Solution According to (3.8), x∗ (t) maximizes the Hamilton-function. Facing a linear objective function and linear constraints, the optimal control x∗ (t) is of a bang-bang-type ⎧ ⎨

0 x (t) = undefined ⎩ xmax ∗

if if if

λ (t) < p(t) λ (t) = p(t) λ (t) > p(t).

(3.16)

If λ (t) = p(t), the optimal control is undefined, but satisfaction of demand from (3.35) and (3.12) requires x(t) = d(t), i.e., just-in-time production. In presence of state constraints (3.4) and jump-conditions (3.13)-(3.15), special attention devotes continuity of the adjoint trajectory at junction times, i.e., points in time when x∗ (t) switches between the values in (3.16). Let θ be an entry or exit time into or from the state constraint y(t) ≥ 0. Feichtinger and Hartl (1986) provide a continuity property of the adjoint function: If (3.4) is entered or left nontangentially, i.e., the control function is non-continuous, the adjoint function λ (t) is continuous at θ implying a jump-parameter of zero. The condition y(t) ≥ 0 can only be left if the control left of θ is x∗ (t) = d(t) and right of θ is x∗ (t) > d(t). From (3.16) results x∗ (t) = xmax if x∗ (t) > d(t) and therefore, the control trajectory (adjoint function) at exit times is non-continuous (continuous). The same argumentation applies for the entry time with x∗ (t) = 0 left of θ and x∗ (t) = d(t) right of θ . Summarizing, the adjoint function is continuous throughout and jump-conditions are needless. The derivation of the optimal policy is in two steps. First, all possible combinations of x(t) and y(t) are enumerated and infeasible sets excluded. Second, all sequences of the remaining sets are enumerated and infeasible sequences excluded. Feasible Combinations of Control and State Variable In order to derive optimal control and state functions, x∗ (t) from (3.16) and y(t) ≥ 0 are combined to feasible sets by enumeration. In total, an optimal x(t) can take the values zero, d(t), and xmax whereas y(t) can equal or exceed zero which leads to six possible combinations. • x∗ (t) = xmax implies y(t) > 0 due to (3.2). This case is referred to as inventory holding periods. • x∗ (t) = 0 requires y(t) > 0 due to satisfaction of demand. This set characterizes destocking periods.

36

3 Procurement with Deterministic Costs

• x∗ (t) = d(t) can be combined with y(t) = 0. This case is referred to as JIT-period. x(t) = 0 ∩ y(t) = 0 is inadmissible as demand is not satisfied and x(t) = xmax ∩ y(t) = 0 cannot hold during an interval with positive length. Let tnW J denote a point in time where the optimal policy changes from inventory holding to JIT. Si(x(t) = d(t), multaneous inventory holding (y(t) > 0, β (t) = 0) and JIT-production

α1 (t) = α2 (t) = 0) imply λ (t) = p(t) from (3.9) and λ (t) = eν t ttnW J h(u)e−ν u du +

eν (t−tn ) p(tnW J ) from (3.10) which is not satisfied during an interval with positive length (see Feichtinger and Hartl (1986)). WJ

Feasible Sequences of Policies An optimal policy consists of a sequence of the feasible sets, i.e., subsequent inventory holding, destocking, and JIT-intervals. Let N intervals occur during the planning horizon being characterized by a certain sequence of policies. The beginning of the n-th interval corresponds to the n-th change from JIT-manufacturing to inventory holding. Let tnW D denote the transition time between a period of inventory holding and a period of destocking in the n-th interval. Accordingly, tnDJ (tnDW , tnJW ) denotes transition times between periods of destocking and JIT (destocking and inventory holding, JIT and inventory holding). Assume a JIT-period with y(t) = 0 implying x(t) = d(t) and λ (t) = p(t) from (3.16). A subsequent period of destocking is not admissible as demand cannot be satisfied due to an empty inventory. Therefore, only a period of inventory holding can follow a JIT-period. This requires an interval with y(t) > 0, β (t) = 0 such that (3.10) yields λ˙ (t) = νλ (t) + h(t). (3.17) This first-order differential equation has an initial condition at the intersection of JIT-manufacturing and inventory holding, λ (tnJW ) = p(tnJW ), due to the continuity property of the adjoint function (which satisfies continuity of the Hamiltonfunction either). The solution of (3.17) and this initial condition is

λ (t) = eν t

 t tnJW

h(u)e−ν u du + eν (t−tn

JW )

p(tnJW )

(3.18)

(see, e.g., Kamien and Schwartz, 1991). This function is increasing and convex as exemplified in Figure 3.1. The left common point of λ (t) and p(t) is t JW due to the initial condition p(t JW ) = λ (t JW ). The adjoint function is affected by the procurement price p(tnJW ) at the point in time when the order is placed and by

3.2 Manufacturing Model

37

Figure 3.1: Illustration of the Adjoint Function for the Manufacturing Model

accumulated marginal inventory holding and manufacturing costs after this point in time. Further, (3.16) requires

λ (t) > p(t)

for

lim t = tnJW + ε .

ε →0+

(3.19)

The optimal policy changes at the first intersection point of λ (t) and p(t) after t JW . As λ (t) from (3.18) is valid after this intersection point and p(t) is a continuously differentiable function, λ (t) < p(t) holds right of this intersection point. Therefore, the intersection point characterizes a point in time tnW D where the optimal policy switches from inventory holding to destocking with λ (t) < p(t) for tnW D + ε (in contrast to (3.19)). At the subsequent intersection point of λ (t) and p(t), the optimal policy switches from destocking to inventory holding or JITmanufacturing. DW occurs if y(t DW ) > 0 A transition to inventory holding characterized by tn+1 n+1 DW DW ) > p(t DW ) for as in that case β (tn+1 ) = 0. λ (t) from (3.17) applies and λ (tn+1 n+1 DW + ε . In this case, t = tn+1

DW

tn+1

tnJW

(x∗ (t) − d(t))dt > 0 holds. On the other hand, a

transition to JIT-manufacturing characterized by tnDJ occurs if y(tnDJ ) = 0 implying λ (t) = p(t) for t = tnDJ + ε . Summarizing, a period of JIT-manufacturing can be followed by a period of inventory holding, inventory holding can succeed an interval of destocking or JIT, and a period of destocking can follow a period of inventory holding. The distinction between intervals where a period of inventory holding follows a period of destocking and intervals where a period of JIT-manufacturing does is as follows.

38

3 Procurement with Deterministic Costs

Assume that the sequence of optimal policies is JIT, inventory holding, destocking for each of the N intervals. The n−th such interval then requires  tW D n tnJW

(xmax − d(t)) dt =

 t DJ n tnW D

d(t)dt,

(3.20)

i.e., inventories built up during the inventory holding interval at tnJW ≤ t ≤ tnW D equal the demand during the destocking period at tnW D ≤ t ≤ tnDJ . This presupposes empty inventories before and after the inventory-holding-destocking-period (note, that periods with x(t) = d(t) ∩ y(t) < 0 are not feasible). For each interval three points in time, tnJW , tnW D , and tnDJ , have to be determined using (3.18) and the continuity properties

λ (tnW D ) = p(tnW D ) and λ (tnDJ ) = p(tnDJ )

(3.21)

with (3.20). This procedure yields subsequent intervals, each with the sequence of policies JIT, inventory holding, destocking and empty inventories during every JIT-period. JW doesn’t hold By calculating all N such intervals it is possible that tnDJ ≤ tn+1 implying a violation of the continuity property of λ (t). In these cases, (3.20) has JD the inventory doesn’t to be extended such that between intervals with tnDJ > tn+1 deplete to zero. This characterizes intervals where a period of inventory holding follows a period of destocking without JIT-manufacturing in between. Assume Mn consecutive subintervals in interval n with tnDJ > tnJD , tnDJ > tnJD , ..., m m +1 m +1 m +2 DJ JD tnm +Mn −1 > tnm +Mn . Then condition (3.20) changes to  tW D nm +Mn tnJW m

(xmax − d(t)) dt =

 t DJ nM

n

tnWmD

d(t)dt

(3.22)

ensuring that the inventory is empty at the beginning and the end of the Mn consecutive intervals. Initial and Terminal Inventories The beginning and the end of the planning horizon with y(0) = y0 and y(T ) = yT have to be treated separately. If y(0) = 0, the slope of the production costs, p(t), ˙ ˙ ≤ is responsible for the decision whether to produce JIT or with rate xmax . If p(0) ν p(0) + h(0), starting with JIT-production (t1JW > 0) is optimal as no function λ (t) > p(t) exists and no inventories can be destocked. Else, t = 0 characterizes

3.2 Manufacturing Model

39

the beginning of a period of inventory holding if (3.20) has a solution with t1JW = 0 or the optimal policy starts with JIT-manufacturing. If initial inventories are non-negative, the optimal inventory level y∗ (t) (calculated with y(0) = 0)  y∗ (t) =

t

0

(x∗ (t) − d(t)) dt

(3.23)

can be compared to an inventory level y(t) resulting from simply destocking initial inventories,  y(t) = y0 −

t

d(t)dt.

(3.24)

0

The point in time ti where the optimal policy switches from destocking initial inventories to the optimal policy in case no initial inventories are available results from equating (3.23) and (3.24) in order to satisfy the continuity property of λ (t). For 0 ≤ t < ti , x∗ (t) = 0 is optimal. Else, the optimal control is given above and the optimal inventory level y∗∗ (t) = max{y∗ (t); y(t)}

(3.25)

results. The adjoint function requires a separate treatment which additionally allows for the determination of a fictitious optimal gratis initial inventory. The continuity property implies

λ (ti − ε ) = λ (ti + ε ) for ε → 0.

(3.26)

Positive inventories left of ti yield β (t) = 0 for t < ti and the adjoint function is the solution of the differential equation (3.17) with boundary condition (3.26) which is an increasing function between t = 0 and t = ti . As λ (t) represents the value of a unit in the inventory at time t, gratis initial inventories become nocent as soon as λ (t) < 0 in the region 0 ≤ t < ti . The economic rationale is that during destocking initial inventories holding costs occur which may exceed later manufacturing costs if the inventory level is above a certain level. At the end of the planning horizon, a terminal stock y(T ) = yT is required which implies that λ (T ) is free (see Feichtinger and Hartl (1986)). t f is defined as point in time where an optimal policy starts reserving the desired terminal inventories. A distinction of two cases is necessary. First, if y(T ) = 0, t f = T results and inventory holding at t = T is not admissible as in that case positive terminal inventories result. Again, the slope of the function p(t) decides whether to destock totally or to manufacture JIT in the last interval until T . If p(T ˙ ) ≤ ν p(T ) + h(T ), JIT-manufacturing in the end of the planning

40

3 Procurement with Deterministic Costs

horizon is optimal and tNDJ < T . Else, the optimal policy ends with destocking, tNDJ = T . Second, yT > 0 results in t f < T . Assume yT = 0 and decrease t f until y∗ (t f ) +

 T tf

(xmax − d(t)) dt = y(T ).

(3.27)

If thereby new intersection points of the adjoint and the price trajectories are generated this requires an additional adjustment of the optimal policy according to (3.16). In that case, applying (3.27) only once still does not yield to the desired terminal inventory level and has to be repeated until y(T ) = yT . Solution Algorithm The optimal policy can be calculated by the following algorithm. (0) Determine all tnJW , tnW D , and tnDJ from (3.18), (3.20), and (3.21) for n = 1, 2, . . . , N. Repeat JW and combine them to connected intervals. (1) Identify intervals with tnDJ > tn+1

(2) Determine the remaining number K of intervals. Label the beginning (the DJ end) of the k−th interval as tkJW (as tk+M ). Mk corresponds to the number k −1 of connected intervals. Let k := 1. Repeat If Mk = 1, then solve (3.18), (3.20), and (3.21). Else solve (3.18), (3.21), and (3.22) Until k = K. Let n := n + 1. Until (1) yields to an empty set. (3) Incorporate initial inventories by applying (3.25) and terminal inventories by applying (3.27).

3.2 Manufacturing Model

41

Numerical Example An example illustrates the optimal policy for the parameters t ∈ [0, 19], d(t) = 1+0.1t, h(t) = 0.1, ν = 0.05, p(t) = 5+sin(t − π )+0.4t −0.002(t −5)2 , xmax = 5, y0 = yT = 0. The resulting points in time are t1JW = 0.32, t1W D = 2.31, t2DW = 7.20, t2W D = 8.96, t2DJ = 11.94, t3JW = 13.88, t3W D = 15.65, t3DJ = 17.33 and yield trajectories as shown in Figure 3.2. The total discounted costs sum up to 161.37. 10

λ(t) p(t) x(t) y(t)

8

6

4

2

0

tJW 1 0

tWD 1 2

tDW 2 4

6

tWD 2 8

10

tDJ 2

tJW 3

tWD 3

12

14

16

tDJ 3 18 time

Figure 3.2: Numerical Example for the Manufacturing Model

This example suggests to exploit periods of low manufacturing costs (with p(t) < λ (t)) to build up inventories in order to cover demand during intervals with high manufacturing costs (with p(t) > λ (t)). At t2DW , a period of inventory holding starts although inventories have not depleted to zero as the increase in production costs is sufficient enough to justify this absence of a zero-inventory-property. Interpreting as procurement model, periods with low procurement prices are useful to build up inventories. During periods with high procurement prices, demand is satisfied from inventories. Interpretation of Shadow Prices Theory of optimal control allows for an economic interpretation of shadow prices. During inventory holding and destocking periods, λ (t) is given by (3.18). Applying a first-order Taylor-series expansion and assuming h(t) := h simplifies (3.18) to

λ (t) = p(t JW ) + ν p(t JW ) + h (t − t JW ), (3.28) i.e., the adjoint trajectory is composed of the manufacturing costs at the beginning of an inventory holding period (p(t JW )) and marginal inventory holding cost

42

3 Procurement with Deterministic Costs

(ν p(t JW ) + h) from the beginning of an inventory holding period until to the current point in time. Another interpretation results from inserting (3.21) into (3.18) and analyzing the difference,

λ (t DJ ) − λ (t W D ) = ν p(t JW ) + h (t DJ − t W D ). (3.29) The increase of the value of a unit in the inventory λ (t DJ ) − λ (t W D ) during the entire inventory holding and destocking cycle equals the marginal holding costs ν p(t JW ) +h during this cycle from t W D to t DJ . The Lagrangian multipliers α1 (t), α2 (t), and β (t) can be interpreted exploiting conditions (3.9) and (3.10). Clearly, α1 (t) is zero if α2 (t) is positive and vice versa. α1 (t) is positive during intervals with x(t) = 0, i.e., during periods of destocking. Reformulation of equation (3.9) now yields to α1 (t) = p(t) − λ (t). Therefore, α1 (t) corresponds to the costs saved by destocking a marginal unit at time t rather than manufacturing it JIT. α2 (t) is positive during intervals with x(t) = xmax . In that case equation (3.9) yields α2 (t) = λ (t) − p(t) and α2 (t) equals the costs of increasing the inventory by a marginal unit rather than procuring JIT. The value of β (t) is positive if the inventory level is zero which is only possible during periods of JIT-manufacturing. (3.10) in junction with (3.16) simplifies to β (t) = ν p(t) + h(t) − p(t). ˙ This shadow price thus reflects the difference of marginal holding costs (composed of capital costs rp(t) and inventory holding costs h(t)) and marginal manufacturing costs p(t). ˙

3.3 Procurement Model The assumptions required for the procurement model deviate only slightly from those made in Section 3.2. Production costs p(t) and the production rate x(t) are interpreted as procurement costs and procurement rate. xmax = ∞ allows for batch ordering and y(t) ∈ [ymin , ymax ] allows for backlogging up to the lower bound ymin ≤ 0 and introduces a warehouse with limited storage capacity ymax . Positive inventories y(t) are subject to out-of-pocket holding cost of hw (t) per unit and time unit whereas backorders are subject to a penalty hb (t) per unit and time unit, i.e., h(t) = hw (t) if y(t) ≥ 0 and h(t) = −hb (t) if y(t) < 0. The NPV of procurement and inventory holding cost over the planning horizon subject to the inventory movement, non-negativity of the decision variable, the possibility of backorders, and constant warehouse capacity is minimized.

3.3 Procurement Model

min x(t)

43

NPV =

T

e−ν t [p(t)x(t) + h(t)y(t)]dt

0

subject to y(t) ˙ = x(t) − d(t), k1 (y(t),t) = y(t) − ymin ≥ 0, y(0) = y0 ,

x(t) ≥ 0,

k2 (y(t),t) = ymax − y(t) ≥ 0, y(T ) = 0.

(3.30) (3.31)

The following analysis concludes in a situation considering limited warehouse capacity and backlogging in Model IV (Section 3.3.4). Sections 3.3.1-3.3.3 deal with special cases in order to derive properties of an optimal solution sequentially. These special cases are the uncapacitated situation without backlogging (ymax → ∞, ymin = 0) analyzed in Model I (Section 3.3.1), the backlogging case (ymin → −∞) in Model II (Section 3.3.2), and the case of limited warehouse capacity without backlogging (ymax < ∞, ymin = 0) in Model III (Section 3.3.3). First, the Hamilton-function and the Hamilton-Lagrange-function are stated. The current-value Hamilton-function H (x(t), y(t), λ (t),t) consists of the direct procurement and inventory holding cost at t and the adjoint variable λ (t) multiplied by the inventory movement. The Hamilton-Lagrange-function adds Lagrangian multipliers α1 (t) and α2 (t) associated with the lower bounds of x(t) and y(t), respectively, and α3 (t) associated with the warehouse capacity constraint. H (x(t), y(t), λ (t),t) = −[p(t)x(t) + h(t)y(t)] + λ (t)[x(t) − d(t)] L (x(t), y(t), λ (t),t) = H (x(t), y(t), λ (t),t) + α1 (t)x(t) + α2 (t)[y(t) − ymin ] + α3 (t)[ymax − y(t)] The necessary conditions are formed by Pontryagin‘s Maximum Principle and Karush-Kuhn-Tucker (KKT) conditions which apply for all continuity points of the optimal control x∗ (t) (see, e.g., Sethi and Thompson, 2000) and jump-conditions at junction times θ where the state constraints (3.30) are active.

44

3 Procurement with Deterministic Costs

Pontryagin‘s Maximum Principle: H(x∗ (t), y∗ (t), λ (t),t) = max H(x(t), y(t), λ (t),t),

(3.32)

∂L = −p(t) + λ (t) + α1 (t) = 0, ∂ x(t)

(3.33)

∂L = rλ (t) + h(t) − α2 (t) + α3 (t), λ˙ (t) = νλ (t) − ∂ y(t)

(3.34)

x(t)

∂L = x(t) − d(t) = y(t). ˙ ∂ λ (t) Karush-Kuhn-Tucker Conditions:

α1 (t)x(t) = 0, α2 (t)[y(t) − y

min

α1 (t) ≥ 0,

x(t) ≥ 0,

] = 0,

α3 (t)[ymax − y(t)] = 0,

,

α2 (t) ≥ 0,

y(t) ≤ ymax ,

α3 (t) ≥ 0.

y(t) ≥ y

min

Jump-Conditions for i = 1, 2:

λ (θ − ) = λ (θ + ) + η (θ )

∂ ki ∗ (y (θ ), θ ), ∂y

(3.35)

  H x∗ (θ − ), y∗ (θ ), λ (θ − ), θ   ∂ ki ∗ = H x∗ (θ + ), y∗ (θ ), λ (θ + ), θ − η (θ ) (y (θ ), θ ), (3.36) ∂t

η (θ ) ≥ 0,

η (θ )ki (y∗ (θ ), θ ) = 0.

(3.37)

Note, that due to the fixed terminal state in (3.31) no transversality condition is required. According to (3.32) the optimal solution x∗ (t) maximizes the Hamiltonfunction. The target in the following is to derive properties of the optimal solution, i.e., the trajectories of control and state variable, the adjoint variable, and Lagrangian multipliers. Based on these properties, solution algorithms are provided.

3.3 Procurement Model

45

3.3.1 Model I: Uncapacitated Storage, no Backlogging First, the purchasing problem is analyzed without a warehouse capacity constraint and without the possibility of backlogging, i.e., y(t) ≥ 0 and ymax → ∞, h(t) = hw (t). Thus conditions considering limited warehouse capacity can be omitted. The maximization of the Hamilton-function yields ∗

x (t) =



0 undetermined

if λ (t) < p(t) if λ (t) = p(t).

(3.38)

Regarding the adjoint variable λ (t) as the value of a unit in inventory at time t, nothing is procured if using inventory is cheaper compared to procurement for a price p(t). In case that both alternatives have the same value, the optimal control is undetermined. The following property states that in an optimal solution only intervals of either JIT-procurement or demand satisfaction from inventory replenished in batches (impulses) occur and identifies candidate points in time for the transition between these intervals.

Property 3.1 If the marginal raw material price increase exceeds the interest on the current price plus holding cost ( p(t) ˙ > ν p(t) + hw (t)), inventory holding is preferred over JIT-procurement (y(t) > 0). If inventory is positive (y(t) > 0), the procurement rate equals zero (x(t) = 0), i.e., destocking inventories and JITprocurement do not occur simultaneously.

Proof Assume that y(t) > 0 and 0 < x(t) < ∞ hold at the same time. Then, α1 (t) = α2 (t) = 0 and λ˙ (t) = νλ (t) + hw (t). From (3.38) results p(t) = λ (t) and ˙ > ν p(t) + hw (t) holds and therefore, p(t) ˙ = λ˙ (t). However, by assumption p(t) (3.34) can only be satisfied if α2 (t) > 0 which contradicts the assumptions.  Two major results are extracted: 1) Inventory holding y(t) > 0 and procurement with rate 0 < x(t) < ∞ at the same time exclude each other which requires the construction of consecutive procurement and destocking intervals.

46

3 Procurement with Deterministic Costs

2) The condition !

p(t) ˙ = ν p(t) + hw (t) p(t ˙ + ε ) > ν p(t + ε ) + hw (t + ε ), ε → 0+

(3.39)

t ∈ [0, T ] identifies candidates for entering intervals with positive inventories. Exploiting this result, the optimal control from (3.38) can be stated when λ (t) = p(t): x(t) = d(t) applies in order to satisfy demand. However, a transition between procuring JIT and entering intervals with positive inventories requires filling the inventory with a batch (x(t) = ∞) as the Hamilton-function is linear in the control in absence of upper bounds on the control. In the following let t JD denote a point in time where an optimal policy changes from JIT-procurement to destocking. These are the candidates identified from (3.39). Further let t DJ denote a point in time where inventory depletes to zero and the policy returns back to JIT-procurement. For JIT-intervals, x(t) = d(t) and therefore λ (t) = p(t). In order to derive the complete adjoint function, analysis of its continuity at junction times θ is required. The state constraint k1 (y(t),t) = y(t) ≥ 0 is left at θ = t JD (exit time) and entered at θ = t DJ (entry time). According to Feichtinger and Hartl (1986), the adjoint variable is continuous at exit (entry) times if the control variable leaves (enters) non-tangentially, i.e., x∗ (θ ) is not continuous. Due to (3.38), x(θ ) is not continuous at θ = t JD as x∗ (t JD− ) = d(t) = x∗ (t JD+ ) = 0. Consequently for intervals with y(t) > 0, α2 (t) = 0 and λ˙ (t) = νλ (t) + hw (t) from (3.34). The solution of this first-order differential equation with initial value p(t JD ) due to the continuity property is given by

λ (t) = eν t

 t t JD

hw (u)e−ν u du + eν (t−t

JD )

p(t JD )

(3.40)

(see, e.g., Kamien and Schwartz, 1991). This function is increasing and convex and exemplified in Figure 3.3. The left common point of λ (t) and p(t) is t JD and results from the initial condition p(t JD ) = λ (t JD ). When the optimal policy switches back from inventory holding (y∗ (t) > 0, x∗ (t) = 0) to JIT (y∗ (t) = 0, x∗ (t) = d(t)), discontinuity of x(t DJ ) again implies a continuous adjoint trajectory requiring λ (t DJ ) = p(t DJ ). Using this relation determines the point in time until which the procured batch Q(t JD ) covers demand as the solution of λ (t DJ ) = p(t DJ ), t DJ > t JD , t DJ ∈ [0, T ]. (3.41)

3.3 Procurement Model

47

Figure 3.3: Illustration of the Adjoint Function for the Procurement Model

t DJ is the first intersection point of λ (t) and p(t) right of t JD . λ (t) is continuous throughout, but not necessarily continuously differentiable at t DJ . Due to (3.38) and (3.39), λ (t) is continuously differentiable at t JD . For t JD < t < t DJ , λ (t) < p(t). Using the argument of non-tangentiality, continuity of λ (t) implies a jumpparameter η (θ ) = 0 in the jump-conditions (3.35)-(3.37). This ensures that the relaxed jump-conditions are met by the above solution. As the raw material price is fluctuating and no functional form of p(t) is prespecified, equation (3.39) can have multiple solutions over the entire planning horizon. Thus there can exist multiple time instants where a transition from JITprocurement to inventory holding is advantageous. Let N be the number of candidates identified from (3.39) being denoted by tnJD for n = 1, 2, ..., N. Accordingly, DJ . tnDJ for n = 1, 2, ..., N +1 denotes the points in time solving (3.41), tnDJ ≤ tnJD ≤ tn+1 DJ DJ t1 is defined as the chronologically first, and tN+1 as the last point in time. Note DJ is satisfied as t DJ is defined as first intersection point of λ (t) and that tnJD ≤ tn+1 n+1 p(t) right of tnJD . Knowing both the beginning and the end of destocking, the lot size Q(tnJD ) at time tnJD (where x(t) = ∞) can be expressed as the demand between DJ less the inventory level at time t JD that can result from initial inventotnJD and tn+1 n ries,   Q(tnJD ) = max

 t DJ n+1 tnJD

d(u)du − y(tnJD ); 0 .

(3.42)

The optimal trajectories of control and state variable as well as the adjoint variable and Lagrangian multipliers are summarized in Table 3.1. At t JD , a batch is procured and x∗ (tnJD ) → ∞. Due to (3.40), α1 (t) and α2 (t) are continuous at tnJD .

48

3 Procurement with Deterministic Costs

JIT-procurement x∗ (t)

destocking

d(t)

y∗ (t)

0 DJ

tn+1

0 νt t

t

d(u)du

−ν u du + eν (t−t JD ) p(t JD ) t JD hw (u)e

λ (t)

p(t)

α1 (t)

0

p(t) − λ (t)

α2 (t)

ν p(t) + hw (t) − p(t) ˙

0

e

Table 3.1: Optimal Trajectories of Variables in Model I

Initial and Terminal Inventories DJ ≤ The first and the last phase, i.e., the beginning (0 ≤ t ≤ t1DJ ) and the end (tN+1 t ≤ T ) of the planning horizon, have to be treated separately. If the slope of the raw material price at time t = 0 ( p(0)) ˙ exceeds marginal inventory holding cost rp(0) + hw (0), it is optimal to start with inventory holding and t1DJ = t1JD = 0, 

t2DJ > 0 and a lot size Q(0) = max

t2DJ 0

d(u)du − y0 ; 0

is procured. If p(0) ˙ <

ν p(0) + hw (0), JIT-procurement (t1DJ = 0) in case of y0 = 0 or destocking of initial inventories (t1DJ > 0) in case of y0 > 0 is optimal. If the last lot covers demand DJ = T . Else, the optimal policy terminates with JITuntil the horizon T , then tN+1 JD DJ procurement, tN = tN+1 = T . Solution Algorithm The problem can be decomposed at any point in time during [tnDJ ,tnJD ], i.e., during the phase of JIT-procurement. The remaining subproblems consist of a period of JIT-procurement at the beginning and the end, with a period of destocking in between. This decomposition property is useful to calculate the optimal solution by the following algorithm. 0) Determine all tnJD from (3.39) for n = 1, 2, ..., N and sort them in ascending order. Let n = 1. Repeat 1) Determine tnDJ from (3.41). 2) Determine the corresponding lot-size from (3.42).

3.3 Procurement Model

49

 JD  DJ . 3) n := n + argmin tn+k > tn+1 k=1,2,...

Until n = N. Numerical Examples As a first example, the optimal policy for one subproblem is given in Figure 3.4 showing the functions x(t), y(t), λ (t) with parameter values t ∈ [0, 7], d(t) = 1 + 0.1t, hw (t) = 0.1, ν = 0.05, p(t) = 5 + sin(t − 0.7π ) + 0.1t, y0 = yT = 0. The DJ = 5.25. The NPV of a NPV of total cost is NPVI = 37.71, tnJD = 0.84 and tn+1

7 −ν t pure JIT-strategy is 0 e p(t)d(t)dt = 41.43. 7 p(t) λ(t) x(t) y(t)

6 5 4 3 2 1

tJD n

DJ

tn+1

0 0

1

2

3

4

5

6

t 7

Figure 3.4: Numerical Example for Model I

A second example considers a special case where the adjoint variable λ (t) during destocking beginning at tnJD tangentially reaches the procurement price at the JD ) = ν p(t JD ) + h (t). Either a single lot at time t JD is point in time solving p(t ˙ n+1 w n n+1 DJ or two lots are procured, one covprocured covering demand between tnJD and tn+2 DJ , the other one from t JD = t DJ to t DJ . The two ering the demand from tnJD to tn+1 n+1 n+1 n+2 alternative ordering policies are shown in Figure 3.5 where a modified parameter set is required to construct this special case: t ∈ [0, 7], d(t) = 0.1t, hw (t) = 0.1, ν = 0.0211, p(t) = 5 + sin(t − 0.7π ) + 0.2t. Interpretation of Shadow Prices A first-order Taylor series expansion of eν (t−t ) ≈ 1 + ν (t −t JD ) which disregards compound interest effects is used to provide an economic interpretation of the adjoint variable λ (t). Additionally, hw (t) := hw is assumed to be constant over JD

50

3 Procurement with Deterministic Costs

9 p(t) λ(t) y(t), one lot y(t), two lots

8 7 6 5 4 3 2 1

tJD n

JD tDJ n+1=tn+1

tDJ n+2

0

t 0

2

4

6

8

10

12

Figure 3.5: Numerical Example for a Special Case in Model I 

time. Let t DJ denote the approximate point in time when the inventory depletes to zero. Then equation (3.41) reduces to 

p(t DJ ) − p(t JD ) = ν p(t JD ) + hw .  t DJ − t JD

(3.43)

The left hand side of (3.43) corresponds to the slope of a secant λ (t) between the   points [t JD , p(t JD )] and [t DJ , p(t DJ )], whereas the right hand side turns out to be the slope of p(t) at point t JD from (3.39). That is, the decision maker is indifferent   between buying an extra unit at time t DJ just-in-time at a price of p(t DJ ) or in  advance at a price of p(t JD ) plus extra holding cost hw (t DJ − t JD ) over the period  of length t DJ − t JD . Note that without the approximation λ (t) is convex whereas  the secant λ (t) is linear. Therefore, the exact t DJ is located left of t DJ . Further, an economic interpretation for α1 (t) and α2 (t) is possible. If these Lagrangian multipliers take a positive value, the corresponding variables x(t) or y(t), respectively, take the value zero in order to satisfy the KKT conditions. During destocking, α1 (t) is positive. A linear first-order Taylor series expansion of α1 (t) with hw (t) := hw yields

α1 (t) =

p(t)  current price

− p(t JD ) −[ ν p(t JD ) +       procurement price at t JD

capital cost

hw ] (t − t JD ) .     out of between pocket cost t JD and t

α1 (t) is the fictitious profit of selling raw materials from stock rather than using it to satisfy demand. Figure 3.6 illustrates the argumentation for λ (t) and α1 (t)  DJ = 5.25. with the parameters used in Figure 3.4, t DJ = 5.39 > tn+1

3.3 Procurement Model

51

6.5

6 α1(t)

DJ’

t +

5.5

5

λ (t)’

4.5 tJD + 4 0

1

2

3

4

5

6

t 7

Figure 3.6: Approximate Adjoint Function λ (t) and Lagrangian Multiplier α1 (t)

α2 (t) is positive during JIT-procurement, i.e., it corresponds to the disadvantage of having no short-selling possibility. ν p(t) + hw (t) corresponds to marginal inventory holding cost that could be saved, p(t) ˙ corresponds to marginal procurement cost that would have to be paid extra if the constraint is relaxed.

3.3.2 Model II: Uncapacitated Storage and Backlogging An extension of Model I allows for backlogging of demand, i.e., ymin → −∞. Therefore, constraints and conditions associated with lower bounds on inventories are omitted. This assumption implies that demand satisfaction can be postponed to a future point in time. The economic reason in the raw material procurement application with fluctuating prices is to collect orders and postpone procurement and satisfaction of demand until prices have fallen to a certain level. For ease of presentation, two models are developed in this section. Model II.1 Model II.1 is a special case which does not allow for inventory holding at all (y(t) ≤ 0, y0 = 0 and h(t) = −hb (t)) and uses JIT-procurement and backlogging exclusively. This separate analysis is due to the non-continuous holding cost function h(t) which requires a piecewise analysis of the adjoint function left and right of the point of discontinuity at y(t) = 0. As the adjoint function λ (t) remains continuous, this procedure is feasible (see Feichtinger and Hartl (1986)). Within a second step the analysis is extended to the general case y(t) ∈ ℜ in Model II.2.

52

3 Procurement with Deterministic Costs

The maximization of the Hamilton-function derived in (3.38) still holds. Stating a property similar to Property 3.1 yields that in an optimal solution either intervals of JIT-procurement or backlogging can occur and identifies the transition points between these intervals. Property 3.2 If the marginal raw material price decrease exceeds the benefit of delayed procurement ( p(t) ˙ < ν p(t) − hb (t)), backlogging is preferred over JITprocurement (y(t) < 0). If inventory is negative (y(t) < 0), the procurement rate equals zero (x(t) = 0), i.e., backlogging and JIT-procurement do not occur simultaneously. Proof Assume that y(t) < 0 and x(t) = d(t) hold at the same time. A contradiction similar to the proof of Property 3.1 results. Two major results hold, backlogging (y(t) < 0) and JIT-procurement (x(t) = d(t)) at the same time exclude each other and the condition !

p(t) ˙ = ν p(t) − hb (t) p(t ˙ − ε ) < ν p(t − ε ) + hw (t − ε ), ε → 0+

(3.44)

t ∈ [0, T ] identifies candidates for terminating backorder intervals. In analogy to Model I, this is a point in time where a batch is procured. Let t BJ denote a point in time where the optimal policy switches from backlogging (x(t) = 0) to JIT-procurement (x(t) = d(t)) identified by (3.44). During JIT-procurement, λ (t) = p(t) from (3.38). The optimal control enters the constraint y(t) ≤ 0 at t BJ non-tangentially and therefore the adjoint trajectory is continuous. During backlogging, the adjoint variable λ (t) solves the first-order differential equation λ˙ (t) = νλ (t) − hb (t) similar to Model I with the boundary condition λ (t BJ ) = p(t BJ ) ensuring continuity of λ (t),

λ (t) = eν t

 t BJ t

hb (u)e−ν u du + eν (t−t

BJ )

p(t BJ ).

(3.45)

Let t JB denote a point in time where the optimal policy switches from JIT-procurement (x(t) = d(t)) to backlogging (x(t) = 0). Non-tangentiality of the optimal control at the exit time t JB again requires continuity of the adjoint trajectory. The beginning of the period of backlogging corresponds to the solution of

λ (t JB ) = p(t JB ),

t JB < t BJ ,

t JB ∈ [0, T ]

(3.46)

3.3 Procurement Model

53

where t JB is the first intersection point of λ (t) and p(t) left of t BJ . Again, (3.44) JB and t BJ can have multiple solutions and the following definition is required: tm−1 m JB JB BJ BJ BJ for m = 1, 2, .., M with t0 ≥ 0, tm−1 < tm , tM ≤ T . The lot-size Q(tm ) satisfying JB to t BJ is determined by demand backlogged from tm−1 m Q(tmBJ ) =

 t BJ m JB tm−1

d(u)du.

(3.47)

Numerical Example Figure 3.7 serves as an illustration of Model II.1 with the parameters from Figure 3.4, t ∈ [0, 14], and hb (t) = 0.5. The net present value of total cost is NPVII.1 = JB = 2.61, t BJ = 6.54. 38.83 < NPVJIT = 41.43, tm−1 m p(t) λ(t) x(t) y(t)

6 4 2 tJB m-1

tBJ m

0

t 0

1

2

3

4

5

6

-2 -4 -6

Figure 3.7: Numerical Example for Model II.1

Model II.2 Model II.2 combines the possibilities of inventory holding and backlogging treated separately in Model I and Model II.1. Two different situations characterized by the sequence of policies require a distinction. In the first, simple situation, periods of inventory holding (from Model I) and periods of backlogging (from Model II.1) can be determined separately, that is, an independent construction of intervals does not result in an (infeasible) overlap of intervals. The problem is decomposable during every period of JIT-procurement. The remaining subproblems consist either of JIT-procurement and backlogging or of JIT-procurement and inventory holding and can be solved independently.

54

3 Procurement with Deterministic Costs

Figure 3.8 shows the constellation with independent inventory holding and backlogging intervals with parameters from Figure 3.4 and hb (t) = 1, hw (t) = 0.7. The DJ = 3.24, t JB = resulting optimal values are NPVII.2a = 41.16, tnJD = 1.61, tn+1 m−1 BJ 4.08, tm = 5.9. p(t) λ(t) x(t) y(t)

6

4

2 JD

tDJ n+1

tn

0 0

1

2

3

tJB m-1 4

tBJ m 5

t

6

-2

-4

Figure 3.8: Numerical Example for Model II.2a

Combining the pure backlogging case with the pure inventory holding case may also result in periods of backlogging and inventory holding which would overlap if the interval construction is conducted independently. Two cases are distinguished. In the first one, a phase of backlogging (inventory holding) is completely included JB < t BJ < t DJ (t JB < by a phase of inventory holding (backlogging), i.e., tnJD < tm−1 m n+1 m−1 JD DJ BJ tn < tn+1 < tm ). In an optimal policy the enclosed interval is not considered in order to guarantee a continuous adjoint trajectory which is required for an optimal solution. In the second case, the independently derived inventory holding and JB ≤ t DJ < t BJ . t DB for backlogging intervals overlap partially such that tnJD < tm−1 m u n+1 u = 1, 2, ... is defined as a point in time where a period of backlogging follows a period of inventory holding without JIT-procurement in between. tuDB results from a transition of the intervals of the adjoint variable during destocking and backlogging. The adjoint variable during destocking from (3.40) which equals the marginal cost of holding an additional unit in the warehouse from tnJD to tuDB is defined by λw (t). Analogously, the adjoint variable during backlogging from (3.45) is defined by λb (t) and corresponds to the marginal cost of backlogging an additional unit from tuDB to tmBJ . For an optimal solution equality of these marginal costs is required that simultaneously satisfies the continuity property of the adjoint variable (for a similar approach, see Kiesmüller et al. (2000)). This identifies tuDB as a solution of λw (tuDB ) = λb (tuDB ), tnJD < tuDB < tmBJ . (3.48)

3.3 Procurement Model

55

DJ = t JB := t DB is defined. Note that λ (t) is continuous at t DB In this case, tn+1 u u m−1 anyway as no state constraint exists. Figure 3.9 illustrates an example with parameters from Figure 3.4 and hb (t) = 0.5, hw (t) = 0.1. The net present value is NPVII.2b = 37.1 < NPVI = 37.71 and NPVII.2b < NPVII.1 = 38.83, tnJD = 0.84, tuDB = 4.34, tmBJ = 6.54.

p(t) λ(t) x(t) y(t)

6

4

2 JD

tDB u

tn

0 0

1

2

3

4

tBJ m 5

t

6

-2

-4

Figure 3.9: Numerical Example for Model II.2b

Solution Algorithm The determination of the solution is supported by an algorithm. 0) Tentatively determine tnJD from (3.39) for n = 1, 2, ..., N and tmBJ from (3.44) for m = 1, 2, ..., M and sort them in ascending order. JB from 1) Tentatively determine tnDJ from (3.41) for n = 1, 2, ..., N + 1 and tm−1 (3.46) for m = 1, 2, ..., M and identify tentative intervals of destocking from DJ and of backlogging from t JB to t BJ . tnJD to tn+1 m m−1 JB < t BJ ≤ t DJ or t JB ≤ t JD < 2) Identify overlapping intervals. If tnJD < tm−1 m n n+1 m−1 DJ BJ JB ≤ t DJ < t BJ , determine tn+1 < tm , drop the enclosed interval. If tnJD < tm−1 m n+1 JB := t DB . Else, fix the tentative values. tuDB from (3.48) and set tnDJ = tm−1 u

3) Determine the lot-sizes from (3.42) and (3.47).

3.3.3 Model III: Capacitated Storage, no Backlogging This section considers a limited warehouse capacity ymax , y(t) ≤ ymax < ∞, h(t) = hw (t). Backlogging is not permitted and therefore ymin = 0, y(t) ≥ 0.

56

3 Procurement with Deterministic Costs

The optimal procurement time in Model III is the same as in Model I determined from (3.39). The main difference to the case with infinite warehouse capacity is that inventory holding y(t) > 0 and procurement x(t) > 0 at the same time do not exclude each other if 1.) the warehouse capacity is binding or 2.) a candidate point for batch procurement appears during positive inventory intervals. Property 3.3 y(t) > 0 and x(t) > 0 can occur simultaneously. Proof Two cases require a distinction. 1.), if y(t) = ymax , α3 (t) > 0 and the contradiction in the proof of Property 3.1 no longer holds. 2.), if x(t) > 0 and 0 < y(t) < ymax , α1 (t) = α2 (t) = α3 (t) = 0, λ (t) = p(t) from (3.38) and therefore λ˙ (t) = p(t). ˙ This yields p(t) ˙ = ν p(t) + hw (t), the condition for candidate points for procuring batches, which is feasible.  The economic rationale is that the binding capacity constraint prohibits the beneficial procurement of additional units. Inventory space released by demands is therefore used to refill the warehouse though inventory is still positive. Let tnlot obtained from (3.39) denote a point in time where inventory is replenished by an infinite rate. According to the notation in Model I, tnJD denotes a transition from DJ a transition from inventory holding to JITJIT-procurement to destocking and tn+1 procurement. Between tnlot and tnJD , the adjoint variable λ (t) takes the value p(t) as demand is still satisfied JIT though inventory is positive. This notation is useful for the distinction of two cases. In the first case, warehouse capacity is not binding, tnlot = tnJD , and the procedure for deriving the optimal inventory interval from tnlot DJ is as outlined in Section 3.3.1. In the second case, warehouse capacity is to tn+1 binding, tnlot < tnJD , α3 (t) ≥ 0 for tnlot ≤ t ≤ tnJD and a batch   (3.49) Q(tnlot ) = max ymax − y(tnlot ); 0 is procured at time tnlot . This lot remains in the warehouse until tnJD and demand beDJ are identified tween tnlot and tnJD is procured JIT. The two points in time tnJD and tn+1 from the solution of the following equations  t DJ n+1 tnJD

d(u)du = ymax ,

DJ DJ λ (tn+1 ) = p(tn+1 ),

λ (t) < p(t),

∀t

DJ tn+1 > tnJD , DJ ∈ (tnJD ,tn+1 ).

(3.50) (3.51) (3.52)

3.3 Procurement Model

57

DJ equals the available ware(3.50) ensures that the demand between tnJD and tn+1 max house capacity y . Further, the adjoint variable equals the raw material price at DJ (3.51) whereas the raw material price exceeds the adjoint variable during time tn+1 JD DJ [tn ,tn+1 ] (3.52). Again, the jump-parameters η (tnlot ) (η (tnJD )) take the value zero as the state constraint k2 (y(t),t) is entered (left) non-tangentially. The argumentation for k1 (y(t),t) equals the argumentation in Section 3.3.2. The construction of the optimal points in time can be performed as follows. DJ from (3.41). If the available First, let tnJD = tnlot as in Model I and determine tn+1 max does not suffice according the first case in Property 3.3, increase capacity y tnJD such that the resulting adjoint variable (3.40) reaches the price curve again (implying cost indifference) but does not intersect (this would imply a cheaper procurement opportunity). In case that the adjoint trajectory reaches the price curve in a candidate point where p(t) ˙ = ν p(t) + hw (t) (the second case in Property 3.3) it is beneficial to refill the inventory with another batch though inventory is positive. The construction of the size of the respective batch is the same as for the initial batch. This construction principle again exploits the continuity of the adjoint variable.

Numerical Example Figure 3.10 shows the optimal policy for the parameter values from Figure 3.4 and ymax = 4. 7 p(t) λ(t) x(t) y(t)

6 5 4 3 2 1 tn

lot

tJD n

1

2

tDJ n+1

0 0

3

4

5

6

t 7

Figure 3.10: Numerical Example for Model III

The optimal time instant tnlot takes the value 0.84 (which equals tnJD in Model I), DJ t JD = 1.86 (which exceeds tnJD in Model I) and tnDJ = 4.85 (which is below tn+1 in Model I). The resulting optimal present value of total cost is NPVIII = 37.99 >

58

3 Procurement with Deterministic Costs

NPVI . If the decision maker neglects optimality of a phase of full inventories and simultaneous JIT-procurement and sets tnlot = tnJD as a simple heuristic, a NPV of 38.65 > NPVIII results. A special case where the adjoint variable tangentially reaches the raw material price can occur as shown in Figure 3.11, the second case in Property 3.3 where a batch is procured though inventory is still positive. The parameter values t ∈ [0, 12], d(t) = 0.5 + 0.01t, hw (t) = 0.05, ν = 0.02, p(t) = 5 + sin(t − 0.7π ) + 0.2t − 1/40(t − 4)2 , ymax = 3.5 are used. 8 p(t) y(t) λ(t)

7 6 5 4 3 2 1 lot

tJD n

tn

0 0

tlot n+1 2

4

6

tDJ n+2 8

t

10

Figure 3.11: Numerical Example for a Special Case in Model III

Solution Algorithm The decomposition approach applies for Model III, too. Separation during the phases of pure JIT-procurement yields subproblems which can be optimized sequentially with the following algorithm. 0) Determine all tnJD from (3.39) for n = 1, 2, ..., N and arrange them in ascending order. Let n = 1. Repeat DJ results from 1) Determine Q(tnJD ) from (3.42). If Q(tnJD ) ≤ ymax , tnJD = tnlot , tn+1 (3.41) and the lot-size from (3.42). Else, calculate tnlot from (3.39), tnJD and DJ from (3.50)-(3.52) and the lot-size from (3.49). tn+1  JD  DJ . 2) n := n + argmin tn+k > tn+1 k=1,2,...

Until n = N.

3.4 Conclusions and Outlook

59

3.3.4 Model IV: Capacitated Storage and Backlogging Finally, the results from the previous sections are combined in Model IV. As the main insights are pointed out in Sections 3.3.1-3.3.3, the analysis is reduced to the major issue, the derivation of an optimal policy. First, a case in which warehousing and backlogging-intervals are non-overlapping and a second, overlapping case are distinguished. As the first mentioned is simple due to the decomposition property, the second is focussed. If the warehouse capacity is not binding, refer to Model II. Else, a period of simultaneous warehousing and JIT-procurement is followed consecutively by destocking, backlogging, and pure JIT-procurement. The candidates for batch-procurement result from (3.39) and (3.44), whereas the points in time tnJD and tuDB are solution of a system of two equations, (3.48) and  t DB u tnJD

d(u)du = ymax .

(3.53)

(3.53) is similar to (3.50) and postulates destocking of inventories before an interval of backlogging starts at tuDB . Continuity of the adjoint trajectory is guaranteed as the state constraint k2 (y(t),t) is entered (left) non-tangentially. At tnlot (tnJD ), the optimal control jumps from d(t) to infinity (from zero to d(t)). Numerical Example Figure 3.12 shows the optimal policy in Model IV with the parameters from Figure 3.9 and ymax = 2. NPVIV = 37.53 > NPVII.2b , tnlot = 0.84, tnDJ = 1.91, tuDB = 3.48, tmBJ = 6.54 result. The benefit of having the option of backlogging compared to a situation with the same parameters excluding backlogging is an NPV of 1.8. Having no warehousing possibility results in a disadvantage of 1.3. If there exists no warehouse capacity constraint, the cost decrease is NPVIV − NPVII.2b = 0.43. Compared to Model II.2b, the point in time where inventory depletes to zero is earlier as the backlogging option benefits from a limited warehouse capacity.

3.4 Conclusions and Outlook This section considered a deterministic, dynamic manufacturing model (Section 3.2) and a procurement model (Section 3.3) in continuous time. In the first model, solution with the theory of optimal control showed that the optimal strategy consists of a combination of the policies just-in-time-manufacturing, inventory holding, and destocking. Optimizing the net present value of total manufacturing and

60

3 Procurement with Deterministic Costs

p(t) λ(t) x(t) y(t)

6 4 2 lot

tJD n

1

2

tn

0 0

tDB u 3

tBJ m 4

5

t

6

-2 -4

Figure 3.12: Numerical Example for Model IV

inventory holding costs resulted in consecutive intervals of these policies where situations with and without zero-inventory-property are possible. It was proven that just-in-time-manufacturing is not optimal during periods with positive inventories. The second model is a special case of the first and combined just-in-time procurement, backlogging, and inventory holding optimally using the theory of optimal control. The minimization of the net present value of total cost of procurement and inventory holding showed that in the case of infinite warehouse capacity, just-in-time procurement and inventory holding are alternative options. Likewise, backlogging and JIT-procurement at the same time exclude each other whereas in the case of limited warehouse capacity inventory holding and JIT- or batchprocurement can occur simultaneously. The continuous time control approach allows for an analytical derivation of these policies as well as an economic interpretation of the optimal solution. These properties are used to construct algorithms to determine the optimal policy numerically for a broad variety of given functional forms of demands and prices which offers a wide field of application. Possible extensions include various aspects which influence the lot-size and thus the advance procurement decision and consist of the aspects fixed ordering/setup cost, non-linear procurement/ manufacturing costs, a relaxation of the assumption of sufficient manufacturing capacity at every time instant, and positive lead-times. Other aspects further limit the generality of the model and address continuity assumptions of the parameters. Setup costs add a trade-off between inventory holding costs, the exploitation of cheap procurement costs, and the number of setups which affects the total amount of setup costs. Non-linear

3.4 Conclusions and Outlook

61

procurement/manufacturing costs allow to include economies of scales, quantity discounts, or a specific costs structure of the manufacturing technology. If the cost function is concave (convex) in the lot-size, the lot-size increases (decreases) in comparison to a linear cost function as advance procurement/manufacturing is more (less) beneficial. The assumption xmax > d(t) implies that instantaneous demand can always be satisfied. If this assumption is relaxed, periods where the manufacturing capacity doesn’t suffice to cover the instantaneous demand are possible. This adds demand satisfaction and the prevention of stock-out periods as motivation for advance procurement. Positive lead-times are relevant if they affect the timing of payments. If, e.g., the quantity ordered in advance has to be paid at the point in time when the order is placed but the order is received later after the lead-time has elapsed, advance procurement becomes less attractive due to discounting and the lot-size decreases in the lead-time. If the cost function p(t) is non-continuous, take step-changes as example, the solution with optimal control theory is no longer possible (see Feichtinger and Hartl (1986)). An approximation function can be adequate to represent the non-continuous cost function.

4 Procurement with Stochastic Costs 4.1 Risk Management for Operational and Financial Commodity Procurement 4.1.1 Introduction This section investigates a commodity procurement problem with operational and financial instruments as two-period, stochastic program in order to determine the influence of risk-aversion when the demand and the procurement price are uncertain. This modelling approach is appropriate in order to draw analytical conclusions. The operational instruments are advance procurement before demand has realized using a warehouse and spot market procurement (JIT-procurement) after the realization of demand. Procurement via financial instruments addresses commodity option contracts. The commodity is traded at a spot market with deterministic current and uncertain future purchasing price. Besides the spot market, there is a derivative market where call option contracts (cp. Section 2.1.3) on the commodity are traded. Both the spot and the derivative market are characterized by arbitrage-free commodity prices (cp. Section 2.1.2). The firm under consideration has access to three different procurement instruments (and mixtures between them) in order to satisfy demand resulting from the available procurement instruments. These strategies imply different levels of mismatch risk (the risk not to meet demand) and price risk (the risk to procure at an uncertain future price) which is relevant for an investigation of risk aspects. As demand always can be satisfied due to the possibility of JIT-procurement, mismatch risk is the risk to have surplus materials. First, the demand can be procured just-in-time. This strategy has no mismatch risk but, on the other hand, price risk as procurement is made at an uncertain future price. Second, units can be procured in advance and stored in the inventory. If the demand exceeds the inventory level, missing units have to be procured JIT. This strategy is connected with a low price risk but, however, a high mismatch risk as excess items in the inventory are possible. Third, the firm can buy option contracts. If the future price is above the exercise price, option contracts are exercised and eventually missing units are procured JIT. Otherwise, all units are procured JIT.

64

4 Procurement with Stochastic Costs

This strategy is affected with (some) mismatch risk and (some) price risk. The influence of mismatch risk depends on the possibility to salvage items exceeding demand. The firm’s problem is illustrated in Figure 4.1.

Figure 4.1: Decision Problem

Three optimization objectives are analyzed in this section. A risk-neutral firm (Section 4.1.3) optimizes the expected net present value of the profit resulting from revenues of demand satisfaction and costs from supply and inventory holding. A firm optimizing a mean-variance criterion is investigated in Section 4.1.4. General analytical results are found for these two optimization objectives. A numerical investigation is provided for the case that the firm’s objective is characterized by a specific (negative exponential) utility function (Section 4.1.5). The key question in this section is how a risk-neutral and a risk-averse firm mix the procurement instruments in order to optimize their purchasing policy. Most of the research done in this context (cp. Section 2.3.1) addresses uncertain future demand but neglects uncertainty of future prices. An investigation of risk issues is of outstanding interest if the decision maker faces different uncertain parameters and has access to different instruments with each having a different ability to reduce mismatch and price risk. This contribution distinguishes from related work by integrating the issues of arbitrage-free procurement prices, procurement with operational and financial instruments, risk-aversion, and the consideration of arbitrarily distributed price and demand.

4.1 Risk Management for Operational and Financial Commodity Procurement

65

4.1.2 Notation and Assumptions A discrete time decision problem with two periods is assumed in order to isolate the effects of advance procurement and JIT-strategies (see Figure 4.2). At time t = 1, the firm sells x units of a product to the sales market at price r. The product’s main ingredient is a commodity which has to be procured at a commodity market. The quotation of the commodity is stochastic and arbitrarily distributed on the interval p1 ≤ p1 ≤ p1 at t = 0. The firm can procure a quantity yw at time t = 0 at price p0 , store this amount at h per unit, and use yw ≤ yw units at t = 1. It is assumed that r > p0 + h (4.1) holds. Otherwise, buying items in advance is ex ante disadvantageous. Likewise, the firm can procure yo European call options with strike price f and exercise yo ≤ yo contracts at t = 1. An option contract provides the right, but not the obligation, to buy one unit of the underlying commodity for the predetermined strike price f . The assumption f < r avoids trivial solutions. One contract is quoted at price c0 at t = 0. The condition p1 < f < p1

(4.2)

is assumed as otherwise no option contract can be constituted (see, e.g., Hull (2006)). JIT-procurement y j is possible at the spot-market at t = 1. Demand d is deterministic at t = 1 and arbitrarily distributed on the interval d ≤ d ≤ d at t = 0. d and p1 are arbitrarily correlated and represented by the joint probability density function φ (d, p1 ). Unsold items at t = 1 are assumed to have no salvage value as no secondary market exists and the firm does not act as bidder on the procurement market or the commodity may be perishable. The influence of this assumption is analyzed in detail. Second period’s payments are discounted with factor 0 < ρ ≤ 1. The assumptions of financial theory hold which applies risk-neutral probabilities and the law of one price for trading strategies (see, e.g., Hull (2006)) such that !

p + h = ρ E(p1 ) = ρ  0   I.)

 p1  d p1

d



p1 φ (d, p1 )dddp1 

II.)

holds, i.e., for a risk-neutral firm results no-arbitrage from buying or (short-) selling the commodity. The two possible strategies to posses one unit of the raw material at t = 1, I.) procuring in advance at t = 0 and storing the unit or II.)

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4 Procurement with Stochastic Costs

Figure 4.2: Time Line of Events (1/2)

procuring JIT at t = 1, must cause the same expected costs in order to provide absence of arbitrage. In this one-period procurement context, the fair price of one option contract on the raw material at t = 0, c0 , is c0 = ρ E(c1 ) = ρ

 p1  d f

d

(p1 − f )φ (d, p1 )dddp1 .

This option price results from the no-arbitrage assumption and is as common for financial models considering a good with holding costs, e.g., a raw material. As a consequence, all procurement instruments are equal in terms of their expected NPV, i.e., conditions (2.1)-(2.4) from Section 2.1.2, which provides an extended exposition of arbitrage-free pricing, hold.

4.1.3 Expected Profit Optimization In the second period of the decision problem, the firm optimizes under certainty. It has the possibility to utilize the entire variety of procurement instruments, from inventories, via option contracts, or JIT, and faces a linear optimization problem at t = 1,

4.1 Risk Management for Operational and Financial Commodity Procurement

max π = rx − p1 y j − f yo subject to x ≤ y j + yw + yo x≤d yo ≤ yo yw ≤ yw y j , yo , yw ≥ 0.

67

(4.3)

The firm maximizes its profit resulting from revenues and the costs from the usage of inventories and option contracts. These may not exceed the quantity available from advance procurement. The optimal decision depends on the realization of price and demand. The solution of the linear program can be summarized to a simple priority-rule which preferentially uses procurement instruments with lower marginal costs.

Figure 4.3: State Space

If p1 ≤ f , option contracts are worthless and the optimal solution is yo∗ = 0, ∗ = min{d, yw } (see 1.) and 2.) in Figure 4.3), and y j = d −yw∗ . The following solution applies if f < p1 ≤ r. For d ≤ yw , demand is completely satisfied from ∗ the inventory, yw∗ = d and yo∗ = y j = 0 (3.)). For yw < d ≤ yw + yo , the entire yw∗

68

4 Procurement with Stochastic Costs

inventory and part of the option contracts is used to satisfy demand, yw∗ = yw , ∗ yo∗ = d − yw∗ , y j = 0 (4.)). For yw + yo > d, the entire inventory and all options ∗ are used, yw∗ = yw and yo∗ = yo , and additional units are procured JIT, y j = d − yw∗ − yo∗ (5.)). If r < p1 , the solution from 3.) and 4.) holds in case d ≤ yw + yo (6.) and 7.)), but additional JIT-procurement is not optimal and yw∗ = yw , ∗ yo∗ = yo , and y j = 0 apply for yw + yo > d (8.)). This is the only case where ∗ x < d. The first period’s objective function in case of risk-neutrality, ΠN1 (yo , yw ), is o w o w ΠN 1 (y , y ) = − co y − (p0 + h)y + ρ

+

+ρ +

 r  yw f

yo  +yw

r

yw +yo

d



 yo  +yw 

 rd − f (d − yw ) φ (d, p1 )dd d    o o w rd − f y − p1 (d − y − y ) φ (d, p1 )dd dp1 rd φ (d, p1 )dd +

 p1  yw 

 d

rd φ (d, p1 )dd

w

 d

+ρ

d

p1

  rd − p1 (d − y ) φ (d, p1 )dd dp1

 d yw

 f  yw 

rd φ (d, p1 )dd +

yw

 yw  +yo  yw

 rd − f (d − yw ) φ (d, p1 )dd 

 r(yo + yw ) − f yo φ (d, p1 )dd dp1

(4.4)

including discounted second-period profit at t = 1 and costs at t = 0 in order to procure and store items in the inventory and to buy option contracts. Property 4.1 A risk-neutral firm has no advantage from procurement via option contracts or inventories. Proof

(4.4) is concavely decreasing in yo and yw (see Appendix A.1.1). 

The instruments inventory holding, option contracts, and JIT-procurement would all be affected with an equal expected NPV in case that the usage of the item in the second period was certain. In presence of cases where the procurement price p1 realizes above the selling price r, this does not hold. JIT-procurement thus is the most flexible instrument and dominates inventories and option contracts. This result is illustrated in Figure 4.4 with the parameter values f = 3, h = 0.2, r = 5, ρ = 0.9, d and p1 distributed normally and uncorrelated with means μ (d) = 10 and

4.1 Risk Management for Operational and Financial Commodity Procurement

69

μ (p1 ) = 3 and standard deviations σ (d) = 3 and σ (p1 ) = 1. For these parameters, the values p0 = 2.5 and c0 ≈ 0.359 result from (2.1) and (2.2).

Figure 4.4: Numerical Example for a Risk-Neutral Firm

Analysis of the Salvage Value The influence of the salvage value for items left over at t = 1 is crucial. Let z denote the fraction of the salvage value in comparison to p1 with 0 ≤ z ≤ 1, i.e., 0 ≤ zp1 ≤ p1 is the salvage value per item. This approach applies if items are not fully perishable or can be sold back to the procurement market at a given discount. (4.4) corresponds to the extreme case z = 0. For z > 0, the objective function

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4 Procurement with Stochastic Costs

o w w o ΠN 1 (y , y , z) = − (p0 + h)y − c0 y   w f y   +ρ rd + zp1 (yw − d) φ (d, p1 )dd p1

+ + + + + + + +

d

 d

rd + p1 (y − d) φ (d, p1 )dd dp1

yw 





f z

f

yw  

d

 yo  +yw yw 

 d

r z f z





 rd + zp1 (yw − d) φ (d, p1 )dd

(rd − f (d − yw )φ (d, p1 )dd



yo  +yw



o



yo  +yw 

d

r



yo  +yw

r z

 

yo  +yw

d

o





φ (d, p1 )dd dp1

  rd + zp1 (yo + yw − d) − f yo φ (d, p1 )dd dp1

  rd − f yo − p1 (d − yo − yw ) φ (d, p1 )dd dp1



d

 p1  d  r z

w

rd − p1 (d − y − y ) − f y

 r  d f z





w

o

w

o

r(y + y ) − f y o

w

o

zp1 (y + y ) − f y







φ (d, p1 )dd dp1 



φ (d, p1 )dd dp1 (4.5)

applies. In another extreme case, z = 1, the expected profit is independent of the procurement decision as the model is characterized by a decomposition property. Setting z = 1 in (4.5), the problem splits up into two parts, a maximization of revenues and a procurement problem. Due to the no-arbitrage assumption (2.1), the procurement problem is associated with an expected profit of zero. The remaining problem for revenue maximization therefore is independent from the procurement decisions and a scalar in yo and yw with profit    ΠN1 (yo , yw , z = 1) = ρ

r

p1 d

d

d(r − p1 )dddp1 .

(4.6)

4.1 Risk Management for Operational and Financial Commodity Procurement

71

The influence of z can be analyzed generally by the first derivative of the ex∂ ΠN (yo ,yw ,z)

1 pected profit function with respect to z. ≥ 0 holds (see Appendix ∂z A.1.2) and thus an increasing fraction of the salvage value has a positive influence on the expected profit. This is illustrated in Figure 4.5 for yo := 0 and in Figure 4.6 for yw := 0 with the parameter values from Figure 4.4.

Figure 4.5: Influence of the Salvage Value on the Expected Profit in Case yo = 0

Figure 4.6: Influence of the Salvage Value on the Expected Profit in Case yw = 0

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4 Procurement with Stochastic Costs

4.1.4 Mean-Variance Optimization A risk-averse firm optimizing a mean-variance criterion incorporates, in addition to the expected profit, the risk exposure of its operational and financial positions in terms of the profit’s variance, Var(ΠN1 (yo , yw )) := Var(ΠN1 ), and has the objective MV (yo , yw ) = ΠN1 (yo , yw ) − ω Var(ΠN1 )

(4.7)

where increasing (decreasing) ω (ω ≥ 0) increases (decreases) the influence of the variance on the objective function (see Markowitz et al. (2000)). Such a riskaverse firm prefers higher expected profit and lower variance. The latter denotes Var(ΠN1 ) =E[Π(yo , yw  )2 ] − E[Π(yo , yw  )]2   f  yw  2 = ρ dr − c0 yo  − (p0 + h)yw φ (d, p1 )dd p1

+ +



d

yw 



ρ (d(r − p1 ) + p1 yw  ) − c0 yo  − (p0 + h)yw



r f

+ + +

d





yo +yw

+



yw





d

yo  +yw



+



yw

d



p1

r



yw 





d

yo  +yw

  − ρ 

+ρ





r

yw



p1

d

φ (d, p1 )dd

2

drφ (d, p1 )dddp1 +

drφ (d, p1 )dd +





d

yw

yo  +yw 

yw

2



φ (d, p1 )dd dp1

φ (d, p1 )dd 2

(4.8)

φ (d, p1 )dd

2



φ (d, p1 )dd dp1 

(d(r − p1 ) + p1 yw )φ (d, p1 )dd dp1 (dr − f (d − yw ))φ (d, p1 )dd 

(dr − f yo  − p1 (d − yo − yw ))φ (d, p1 )dd dp1



yw

r

d

d



yo  +yw

ρ dr − c0 yo  − (p0 + h)yw

yw

d



yo  +yw

2

ρ ((yo  + yw )r − f yo  ) − c0 yo  − (p0 + h)yw

p1

d

φ (d, p1 )dd

ρ (dr − f (d − yw )) − c0 yo  − (p0 + h)yw



f



+ρ +

f



φ (d, p1 )dd dp1

ρ (dr − p1 (d − yo − yw ) − f yo ) − c0 yo  − (p0 + h)yw

yw

yo +yw

2

ρ (dr − f (d − yw )) − c0 yo  (p0 + h)yw

d



+

ρ dr − c0 yo  − (p0 + h)yw

2

drφ (d, p1 )dd +



yo  +yw

yw



 dr − f (d − yw ) φ (d, p1 )dd

 2  (yo  + yw )r − f yo  φ (d, p1 )dd dp1 − c0 yo − (p0 + h)yw .

4.1 Risk Management for Operational and Financial Commodity Procurement

73

Property 4.2 A mean-variance optimizer can benefit from advance procurement. Proof As (4.8) has a minimum for yo ≥ 0, yw ≥ 0 and (4.4) a maximum for yw = yo = 0, the optimal decision of a mean-variance optimizer is located between yo = yw = 0 and the minimum of the variance which is characterized by 0 ≤ yw ≤ d, 0 ≤ yo ≤ d (see Appendix A.2.1). For ω → ∞, the optimal yo and yw tend to the variance-minimizing values which are strictly positive.  Variance can be minimized by a mixture of inventories and option contracts as illustrated in Figure 4.8. This result is crucial for the behaviour of a risk-averse firm as the decision minimizing the variance is at least yo = yw = 0 and at most yo = yw = d and thus in a region where the expected NPV is decreasing in both variables. In the consequence, every firm optimizing a mean-variance criterion trades-off decreasing expected profit and decreasing risk exposure (e.g., using a pre-specified mean-variance function) such that it finds an optimal solution in this interval as illustrated in Figure 4.7 for the case of pure inventory holding. This

Figure 4.7: Location of the Optimal Solution Under Mean-Variance Optimization

holds likewise in case of option contracts. If ω = 0, this optimum equals the optimal decision of a risk-neutral firm and thus is yo = yw = 0. For increasing ω , the optimal solution shifts to the variance-minimizing decision and, therefore, increasing risk-aversion increases the level of advance procurement. The variance is exemplified in Figure 4.8 with the parameter values used in Figure 4.4.

Isolated Analysis of Mismatch Risk and Price Risk A distinction of the variance function in two cases where only the price or only the demand is uncertain sheds light on the influence of price risk and mismatch risk. Assume that only the procurement price at t = 1, p1 , is uncertain and thus risk is only composed of price risk. For ease of presentation, let yo = 0. (4.8) simplifies

74

4 Procurement with Stochastic Costs

Figure 4.8: Numerical Example for the Variance

to Var(ΠN1 ) := Var p (ΠN1 ) with Var p (ΠN1 ) =

 r 

2  ρ dr − p1 (d − yw ) − (p0 + h)yw φ (p1 )dp1

p1

+ − +

 p1

r

2 ρ yw r − (p0 + h)yw φ (p1 )dp1

r p1

 p1 r

(4.9)

ρ (dr − p1 (d − yw ))φ (p1 )dp1 2

ρ y rφ (p1 )dp1 − (p0 + h)y w

w

for yw ≤ d and Var p (ΠN1 ) =

 p1 p1

−



2 ρ dr − (p0 + h)yw φ (p1 )dp1 p1

p1

2

(4.10)

ρ drφ (p1 )dp1 − (p0 + h)y

w

for yw > d. It is decreasing in yw (see Appendix A.2.2) as illustrated in Figure 4.9 for the parameters from Figure 4.4 and d = 10.

4.1 Risk Management for Operational and Financial Commodity Procurement

75

Figure 4.9: Variance in Case of Certain Demand and Uncertain Procurement Price

Consequently, price risk can be reduced by advance procurement. If the entire (certain) demand is satisfied from inventories, the firm procures all items at the certain price p0 at t = 0 and thus carries no price risk at all. A similar rationale holds for option contracts. If the entire demand is covered with option contracts, the firm exercises these contracts only in case of p1 ≤ f and therefore carries some price risk when demands are satisfied JIT in case of p1 > f . The reduction of risk is thus higher when inventories are utilized instead of option contracts. Assume vice versa that p1 is certain and the demand d at t = 1 is the only uncertain parameter in order to isolate the effect of mismatch risk. In this case, no option contract can be constituted and (4.8) simplifies to Var(ΠN1 ) := Vard (ΠN1 ) with Vard (ΠN1 ) =

 yw d

+

2 ρ dr − (p0 + h)yw φ (d)dd

 d yw

2 ρ (dr − p1 (d − yw )) − (p0 + h)yw φ (d)dd

 

− ρ

d

yw

rd φ (d)dd +

−(p0 + h)yw

2

 d yw

 (rd − p1 (d − yw ))φ (d)dd

(4.11)

76

4 Procurement with Stochastic Costs

if p1 ≤ r is assumed in order to avoid trivial solutions. This function is illustrated in Figure 4.10 with parameter values as in Figure 4.4 and p1 = 3. Its analysis (see Appendix A.2.2) shows that Vard (ΠN1 ) is an increasing function in yw , and therefore, mismatch risk is increasing in an increasing usage of inventories.

Figure 4.10: Variance in Case of Uncertain Demand and Certain Procurement Price

Procuring more inventories than d results in a positive probability of surplus materials at t = 1 which is responsible for mismatch risk. This risk further increases when more items are procured in advance. Influence of Correlation Between Price and Demand Correlation between price and demand has an impact on the variance of the total profit. A high negative or high positive coefficient of correlation τ between those parameters makes the probability of extreme cases, i.e., the simultaneous realization of high or low values of d and p1 , more likely to happen. If correlation is high, a high (low) procurement price and high (low) demand are likely to realize simultaneously. This has a balancing effect on the profit’s variance as a high procurement price is disadvantageous whereas a high demand is advantageous. If, on the other hand, correlation is low, high (low) procurement price and low (high) demand are likely to occur simultaneously. The two extreme cases amplify the

4.1 Risk Management for Operational and Financial Commodity Procurement

77

profit’s variance as the concurrent realization of a high procurement price and low demand is a strong disadvantage whereas a low price and high demand is a strong advantage for the firm’s profit. This phenomenon is exemplified in Figure 4.11 for a bivariate normal distribution (and parameter values from Figure 4.4) and different coefficients of correlation showing that the variance is increasing in decreasing correlation.

Figure 4.11: Variance for Different Levels of Correlation

Comparison to Newsvendor Models The standard Newsvendor model assumes deterministic prices and advance procurement via inventories as only procurement possibility. Extensions incorporate a second ordering occasion, i.e., the possibility of JIT-procurement at the point in time when demand realizes. Usually, it is assumed that the costs of the second ordering occasion are larger than the costs of advance procurement (see Cachon and Terwiesch (2007)). The expected profit and its standard deviation σ (ΠN1 ) are compared for different Newsvendor models and the model analyzed in this section in Figure 4.12 with the following parameters: h = 0.2, r = 5, ρ = 0.9, d distributed normally with μ (d) = 10, σ (d) = 3. p1 = 4 in Model B and C and p1 distributed normally with μ (p1 ) = 4, σ (p1 ) = 1 in Model D, p0 = 3.4 in Model A, B, and D, p0 = 3.2 in Model C.

78

4 Procurement with Stochastic Costs

Figure 4.12: Expected Profit and Standard Deviation in Newsvendor Models

• Model A exemplifies a situation where a Newsvendor can only procure in advance and has no possibility of JIT-procurement. As, in case of an order quantity of zero, the profit is zero with certainty, this decision results in zero variance. An increasing order quantity involves uncertain profit and thus an increasing variance. Under mean-variance optimization, the optimal order quantity decreases in increasing risk-aversion. • Model B illustrates a Newsvendor with second ordering occasion for a (discounted) deterministic price which is above the price of advance procurement. Advance procurement thus is cheaper than JIT-procurement such that the optimal advance ordering level is positive. Procuring zero items in advance is connected with positive variance as in this case pure JIT-

4.1 Risk Management for Operational and Financial Commodity Procurement

79

procurement results which is connected with full demand variability. Increasing advance procurement further increases the variance as it involves the possibility of items left over. Optimizing a mean-variance criterion results in a lower advance ordering level than under risk-neutrality and increasing risk-aversion further decreases this level. • Model C is a Newsvendor with second ordering occasion for a deterministic price. The discounted costs of JIT-procurement equal those of advance procurement, p0 + h = ρ p1 . Therefore, JIT-procurement dominates advance procurement as it enables the Newsvendor to satisfy demand exactly whereas advance procurement involves the possibility of items left over. Procuring zero items in advance is connected with positive variance as pure JIT-procurement is connected with full demand variability. Increasing advance procurement further increases the variance as it involves the risk of items left over. Both optimization objectives risk-neutrality and meanvariance result in an ordering level of zero. • Model D is similar to the model analyzed in this section and presents a Newsvendor with second ordering occasion for a stochastic price. The discounted expected costs of this procurement possibility equal the costs of advance procurement. Consequently, JIT-procurement dominates advance procurement in terms of the expected profit as the latter involves the possibility of items left over. The following trade-off holds for the variance. Inventories reduce price risk as advance procurement is possible for a deterministic price in contrast to JIT-procurement for a stochastic price. On the other hand, inventories increase mismatch risk as they involve the risk of items left over. Consequently, the variance is decreasing for low inventories and increasing for high inventories. Therefore, a risk-neutral Newsvendor orders zero units in advance whereas a mean-variance optimizer orders at least zero units in advance and increases his order level in increasing riskaversion. For infinite risk-aversion, the order quantity minimizing the variance is optimal. In comparison to Model D, the model analyzed in this section adds option contracts as second advance procurement instrument. As the maximum profit results for zero advance procurement and the minimum of the variance is located above this amount, the ordering level thus increases due to risk-aversion. This effect results from the additional price uncertainty which can be reduced by advance procurement.

80

4 Procurement with Stochastic Costs

Analysis of the Salvage Value Considering a salvage value z as in Section 4.1.3 reduces the minimum of the variance. If the only advance procurement instrument is inventory holding, the minimum of the variance decreases in an increasing salvage value as illustrated in Figure 4.13. The same observation holds for an increasing salvage value if op-

Figure 4.13: Influence of the Salvage Value on the Variance in Case yo = 0

tion contracts are the only advance procurement instrument as illustrated in Figure 4.14. This effect is due to a reduction of the impact of mismatch risk. Though mismatch risk itself cannot be reduced, the consequences of excess items at t = 1 are less severe when they can be sold at the primary market at (a fraction of) the current price. In the consequence, an increasing salvage value increases the level of advance procurement as it increases the expected profit, decreases the minimum of the variance, and shifts it to a higher amount of advance procurement.

4.1.5 Expected Utility Optimization Due to theoretical deficiencies of the mean-variance approach as realistic concept for real-world applications (see, e.g., Lau (1980)), this section presents the results if the objective is to maximize the expected utility in terms of a negative exponential utility function which is among the class of the so-called CARA utility

4.1 Risk Management for Operational and Financial Commodity Procurement

81

Figure 4.14: Influence of the Salvage Value on the Variance in Case yw = 0

functions (constant absolute risk-aversion) implying a linear risk tolerance,   1 − e−aΠ1 (y U ΠN1 (yo , yw ) = a N

o ,yw )

,

(4.12)

where a > 0 is the Arrow-Pratt measure of absolute risk-aversion. The utility is zerofor ΠN1 (yo , yw ) = 0 and tends to ΠN1 (yo , yw ) for a → 0. The expected utility EU ΠN1 (yo , yw ) can be expressed by a Taylor-series expansion o

1 e−aE(Π1 (y − a a N

o w EU(ΠN 1 (y , y )) =

,yw ))



N−1 n a gn (−1)n 1 1 + a2 Var(ΠN 1 )+ ∑ 2 n! n=3

 + RN (4.13)

(See Burghardt (2008)). gn is defined as n−th moment such that  n  gn := E ΠN1 (yo , yw ) − E(ΠN1 (yo , yw )) holds and RN as residual of the Taylor-series. The expected profit (the variance) has a positive (negative) influence on the expected utility. The key implication is that the analytical results gained from Section 4.1.4 are approximately valid if higher moments like skewness or kurtosis which are neglected by the truncated Taylor-series expansion have a low influence.

82

4 Procurement with Stochastic Costs

The optimal ordering policy under a negative exponential utility function is investigated numerically. Exemplarily, the influence of risk-aversion, price volatility, and demand volatility is analyzed in Figure 4.15 for the parameter values from Figure 4.4 and a risk-aversion parameter a = 0.02. The key results are summarized as follows • Increasing risk-aversion increases the optimal level of advance procurement. • Increasing price uncertainty increases the proportion of inventories and decreases the proportion of option contracts. • Increasing demand risk decreases the optimal level of advance procurement, decreases the proportion of inventories, and increases the proportion of option contracts. • Increasing salvage value increases the optimal level of advance procurement, increases the proportion of inventories, and decreases the proportion of option contracts. An increasing risk-aversion parameter a increases the optimal level of advance procurement. This effect depends on the arbitrage-free and uncertain prices. As prices are arbitrage-free, JIT-procurement is beneficial in terms of the expected costs. On the other hand, advance procurement is affected with certain procurement costs and thus advance procurement allows to reduce risk. Increasing riskaversion shifts the proportion to advance procurement in order to exploit the reduction of price risk. The influence of uncertain prices on the optimal advance procurement decisions (p1 ) is plotted against the coefficient of variation of p1 , σμ (p in Figure 4.15. If prices 1) are more risky, the optimal policy shifts the proportion between the advance procurement instruments in order to exploit the chance to reduce price risk. The proportion of inventories (option contracts) increases (decreases) in increasing price uncertainty. The economic rationale is a trade-off between price risk and mismatch risk. If price risk is low, it is preferentially covered with option contracts as the influence of mismatch risk prevails. For higher price risk, mismatch risk is outweighed and the proportion of advance procurement is shifted to the instrument with the strongest reduction of price risk, i.e., inventory holding. While the optimal total amount of advance procurement is increasing in the (d) , the optimal proportion of inventories is coefficient of variation of demand, σμ (d) decreasing and the optimal proportion of option contracts is increasing. The economic rationale is that increasing demand risk increases the mismatch risk. In the consequence, the optimal procurement policy shifts to procurement instruments

4.1 Risk Management for Operational and Financial Commodity Procurement

83

Figure 4.15: Influence of Risk-Aversion, Uncertain Prices, and Uncertain Demands on the Advance Procurement Decisions

with lower mismatch risk, i.e., JIT-procurement is favored over advance procurement and option contracts are favored over inventory holding. Analysis of the Salvage Value The salvage value is responsible for the level of advance procurement and its proportioning as illustrated in Figure 4.16. This result is due to an ambiguous influence of the salvage value. First, a higher salvage value increases the expected profit and decreases the variance and thus is an argument for a more intensive engagement in advance procurement. The total

84

4 Procurement with Stochastic Costs

Figure 4.16: Influence of the Salvage Value on the Advance Procurement Decision

amount of advance procurement thus is increasing in the salvage value. On the other hand, salvage value connects inventory holding and option contracts with higher price risk as unsold items have an uncertain value (whereas a salvage value of zero implies a certain revenue of zero). This effect is responsible for the proportioning of the procurement instruments as increasing salvage value yields a larger proportion of inventories which allow for the best reduction of price risk.

4.1.6 Conclusions and Outlook In this section, a two-period commodity procurement problem with uncertain demand and uncertain procurement prices has been formulated. Stochastic future demand can be satisfied by inventory holding, via option contracts, or JIT when the procurement market is characterized by arbitrage-free prices. Both a risk-neutral and a risk-averse firm have been studied and compared Buying zero units in advance is optimal for a risk-neutral firm as JIT-procurement is the most flexible instrument. A risk-averse firm optimizing a mean-variance criterion procures at least zero units in advance. Increasing advance procurement decreases price risk but increases mismatch risk. If risk-aversion is expressed by a negative exponential utility function, a numerical investigation showed that the optimal order quantity is increasing in risk-aversion as price risk can be reduced. Guidelines for future research address the aspects of the two-stage modeling approach and the no-arbitrage assumption. A multi-period model formulation allows to analyze the aspect of intertemporal risk allocation and to determine the salvage value endogenously. If the available procurement instruments are not arbitragefree, this can result from different holding cost parameters of the procurement market and the firm under consideration, i.e., inventory holding can be cheaper (or more expensive) for the firm such that inventories provide an advantage (disad-

4.2 Operational and Financial Commodity Procurement in Competition

85

vantage) in expected costs compared to option contracts. If inventory holding is cheaper for the firm, a speculation profit is possible. Else, the alternative of option contracts is becoming more attractive as these contracts allow to avoid inventories and are priced using the lower holding cost parameter of the procurement market.

4.2 Operational and Financial Commodity Procurement in Competition 4.2.1 Introduction Due to soaring volatile global commodity markets, raw material procurement problems have received increasing attention. The impact of raw material prices on global trade, the wealth of a nation, and the profitability of enterprises nowadays is highly visible. The purpose of this chapter is to investigate the usage of procurement instruments, i.e., just-in-time (JIT) procurement, procurement in advance and subsequent inventory holding, and procurement via call options, to answer the questions of profitability and substitutability of these instruments and to show under which circumstances and market conditions the participation in advance procurement markets and derivative markets is beneficial. The reasons of holding inventories are multi-layered (see Silver et al. (1998)), e.g., to protect against changing prices. A firm expecting an increasing future price for a commodity can speculate on this price movement anticipating profits. Inventories further serve to protect against demand uncertainty. To ensure a temporary shortage doesn’t affect the capability for production, firms can keep safety stocks. In competition, inventories may serve as quantity commitment and provide the owner market power. Call options cannot serve equally concerning certain goals of storing inventories. Nevertheless, financial theory claims their equality regarding expected procurement costs (see, e.g., Geman (2005)) due to freedom of arbitrage. The benefit of the instruments is analyzed in different settings of competition, a monopoly, a Cournot duopoly, and a Bertrand duopoly. In game theoretic models, neither firm can choose a strategy without taking into account the resulting reaction of the other firm. Therefore, the aim is also to determine whether competition has an impact on the profitability of procurement using operational or financial instruments. The question to answer is whether the benefit of either instrument depends on the presence and mode of competition. The duopoly model analyzed in this chapter assumes two asymmetric firms with exclusive technology competing with a homogeneous product on a common sales market characterized by a linear price-response function. This product has negli-

86

4 Procurement with Stochastic Costs

gible production costs and is composed of a commodity as main ingredient. The firms have to procure this commodity at a procurement market underlying the assumptions of a perfect financial market, i.e., homogeneous expectations of market participants, equality of debit and credit interest, and absence of transaction costs (see Section 2.1.1). In a two-period model, firm 1 can build up inventories and buy call options on the commodity in the first period one period before the competition in the second period. Both firms can procure JIT at a spot market in the competition period. This asymmetry can result if the first firm is an incumbent on its market and the second firm enters the market in the second period. From the view of the first period, procurement price and demand in the second period are stochastic. Both firms’ objective is to maximize expected profit consisting of revenues minus procurement costs. Only the two firms have access to both the procurement and the sales market. The model studied fits to practical applications where purchasing agents in trading companies or industrial firms procure commodities such as agricultural products like wheat, sugar, or coffee and have access to a spot and a derivatives market for their procurement. The key results are applicable whether the company acts as monopolist or faces quantity or price competition on its sales market. As an example, take a flour producer. During the wheat harvest, he can procure wheat in advance or he can procure derivatives on the future wheat price. This investigation distinguishes from related research by integrating the issues of procurement with operational and financial instruments, arbitrage-free prices on the procurement market, the assumption of risk-neutral firms, and a price-sensitive and competitive sales market. The chapter is structured as follows. Section 4.2.2 provides the assumptions and recapitulates the required concepts from financial theory. The procurement model is analyzed for a monopoly in Section 4.2.3, a Cournot duopoly in Section 4.2.4, and a Bertrand duopoly in Section 4.2.5.

4.2.2 Notation and Assumptions A discrete time decision problem with two periods is assumed in order to isolate the effect of commitment and just-in-time strategies (see Figure 4.17). The two duopolists s = 1, 2 sell a homogeneous product, characterized by negligible production costs, with quantities xs at time t = 1 to the sales market. The main ingredient of this product has to be procured at a commodity market. The first firm can procure a quantity yw at time t = 0 at price p0 , store this amount at h per unit, and use yw ≤ yw units at t = 1. Likewise it can procure yo European call options with strike price f and exercise yo ≤ yo contracts at t = 1. JIT-procurement y j is pos-

4.2 Operational and Financial Commodity Procurement in Competition

87

sible for firm 1 at the spot-market at t = 1. The second firm can only procure JIT. From Section 4.1.2, the assumptions of arbitrarily jointly distributed procurement price and market demand, salvage value, discounting, and conditions (2.1)-(2.4) hold likewise in this section. The price-response-function is linear with slope −b

Figure 4.17: Time Line of Events (2/2)

and has - depending on the mode of competition - two different functional forms, r(x1 , x2 ) = d − b(x1 + x2 )

(4.14)

applies in case of Cournot quantity competition and xs =

d − r(x1 , x2 ) , b

s = 1, 2

(4.15)

in case of Bertrand price competition. As it is common in the Industrial Organization literature (see, e.g., Fudenberg and Tirole(1991)), we assume that d ≥ 2p1

(4.16)

holds in case of Cournot competition and d ≥ 2bp1

(4.17)

88

4 Procurement with Stochastic Costs

holds in case of Bertrand competition which is equivalent to a participation condition of both firms at the sales market for every realization of price and demand. If (4.16) or (4.17) is violated, a complete displacement of the second firm is possible which requires a piecewise statement of its objective function. All parameters are observable for both adversaries.

4.2.3 Procurement Strategy in a Monopoly At t = 1 the monopolist fixes his sales quantity given the price-response function (4.14) (setting x2 := 0), decides whether to strike yo option contracts, sell yw inventories to the market, and procure y j further units JIT. The available number of option contracts and items in the inventory is exogenous at t = 1 as it results from o w decisions made at t = 0, yo and yw . The total profit function ΠM 1 (y , y ) is the o sum of procurement and inventory holding costs, −c0 y − (p0 + h)yw , and the discounted expected profit E(π1M ),  M o w o w (4.18) ΠM 1 (y , y ) = −c0 y − (p0 + h)y + ρ E π1 . The second-period objective function of the monopolist for a given price p1 and market potential d is

π1M = (d − bx1 )x1 − p1 y j − f yo

(4.19)

which is subject to capacity constraints x1 ≤ y j + yo + yw

(4.20)

yo ≤ yo

(4.21)

y ≤y

(4.22)

w

w

and non-negativities yw ≥ 0, yo ≥ 0, y j ≥ 0, x1 ≥ 0. The optimization problem for the second period is solved using a Lagrange-approach (cp. Appendix B.1). Formally, the optimal second-period sourcing decision for a given sales quantity x1∗ is yw∗ = min{x1∗ , yw }, yo∗ = min{yo , x1∗ − yw∗ } for p1 > f and yo∗ = 0 else, ∗ y j = x1∗ − yw∗ − yo∗ . If the optimal sales quantity is above the amount available via option contracts and inventories, the difference is procured JIT (cp. region 1a.) and 1b.) in Figure 4.18). In case of p1 ≤ f , all the items in the inventory are sold d which results from marginal procurement costs (region 4a.)) up to the quantity 2b of zero and revenue maximization. The excess is wasted (region 5a.)). In case of p1 > f , all items available via option contracts and inventories are sold up to the

4.2 Operational and Financial Commodity Procurement in Competition

89

f quantity d− 2b (region 2.)) which results from marginal procurement costs of f due f to the exertion of option contracts. If larger quantities than d− 2b are available, part f of the option contracts is not exercised (region 3.)). If larger quantities than d− 2b are available only via inventories, the entire amount is sold (region 4b.)) up to the d quantity 2b and units exceeding this quantity are wasted (region 5b.)). The optimal

Figure 4.18: State Space in the Monopoly

sales quantity x1∗ (yo , yw ) in the second period can be expressed as a function of the first-period decisions yo and yw and realizations of price and demand. The solution is ⎧ d−p 1 1 ⎪ for yo + yw ≤ d−p 1b.) ⎪ 2b ⎪ o2b w d− f ⎪ d−p o w 1 ⎪ for 2.) ⎨ y +y 2b < y + y ≤ 2b d− f d− f o + yw and yw ≤ d− f x1∗ (yo , yw ) = for < y 3.) 2b 2b 2b ⎪ d− f ⎪ w w ≤ d o ⎪ y for < y ∀y 4b.) ⎪ 2b 2b ⎪ ⎩ d d w o for ∀y 5b.) 2b 2b < y (4.23)

90

4 Procurement with Stochastic Costs

for p1 > f and x1∗ (yo , yw ) =

⎧ ⎪ ⎨ ⎪ ⎩

d−p1 2b yw

for for for

d 2b

yw ≤ < yw ≤ < yw

d−p1 2b d 2b

d−p1 2b d 2b

1a.) 4a.) 5a.)

(4.24)

for p1 ≤ f regardless of yo . Insertion of x1∗ (yo , yw ) from (4.23) and (4.24) into π1M provides the optimal profit at t = 1 depending on the first period’s decisions yo and yw such that the objective function (4.18) becomes o w o w ΠM 1 (y , y ) = − c0 y − (p0 + h)y + ρ

+ + +

 2byw  +p1 2byw 

2byw  +p1



p1

2byw 

d

d2 φ (d, p1 )dd 4b

  (d − p1 )2 w + p1 y φ (d, p1 )dd dp1 4b

 p1  2byw  2 d

φ (d, p1 )dd +

4b  2b(yo  +yw )+ f  f

f

(d − byw )yw φ (d, p1 )dd



 d



d

 2byw  + f 2byw 

(d − byw )yw φ (d, p1 )dd

 (d − f )2 + f yw φ (d, p1 )dd 4b 2byw  + f  2b(yo  +yw )+p1   d(yo + yw ) − b(yo + yw )2 − f yo φ (d, p1 )dd + 2b(yo  +yw )+ f      d (d − p1 )2 + + p1 yw + (p1 − f )yo φ (d, p1 )dd dp1 . 4b 2b(yo  +yw )+p1 +

(4.25)

Property 4.3

Proof

There is no benefit from advance procurement in a monopoly.

See Appendix B.2. 

In a monopolistic market, the firm will set the optimal monopoly sales quantity at t = 1. The state yielding the lowest profit is characterized by lowest possible market potential and largest possible procurement price, (d, p1 ). The amount sold in the worst-case represents the lowest and thus secure sales quantity and the lowest profit. Intuitively, the monopolist is indifferent how to procure this amount as

4.2 Operational and Financial Commodity Procurement in Competition

91

he faces an arbitrage-free market and maximizes his expected profit. If he procures additional quantities in advance, their usage at t = 1 is uncertain, i.e., there is a positive probability of items left over. The implication is that all units in the inventory and all option contracts above the secure quantity resulting from the worst-case state are detrimental.

4.2.4 Procurement Strategy in a Cournot Duopoly

This section considers a sales market with Cournot competition in the second period. The firm’s objective functions in the second period are

π2C = [d − b(x1 + x2 )] x2 − p1 x2 for the second firm and

π1C = [d − b(x1 + x2 )] x1 − p1 y j − f yo for the first firm which optimizes subject to constraints (4.20)-(4.22) and nonnegativities. As concept of solution, the Nash-equilibrium is determined. For firm ∗ 1, yw∗ = min{x1∗ , yw }, yo∗ = min{yo , x1∗ − yw∗ }, y j = x1∗ − yw∗ − yo∗ results for o∗ p1 > f . y = 0 holds in case of p1 ≤ f (cp. Appendix C.1). If the optimal sales quantity is above the available amount of advance procurement, the shortage is procured JIT and the marginal procurement cost is p1 (cp. region 1a.) and 1b.) 1 in Figure 4.19). If in case of p1 ≤ f larger quantities than d−p 3b are available via inventories, the entire stock is sold (region 4a.)) up to the demand which 1 corresponds to marginal costs of zero, d+p 3b . Available inventories exceeding this level are rather wasted than sold to the market (region 5a.)). In case of p1 > f , 1 quantities of yo + yw above d−p 3b are sold up to the demand which corresponds to d+p1 −2 f (region 2.)). If more items are available, part of the marginal costs of f , 3b 1 −2 f option contracts is rather wasted than exercised (region 3.)). If more than d+p3b items are available from inventories, the entire amount is sold up to the demand 1 which corresponds to marginal costs of zero, d+p 3b (region 4b.)), and inventories exceeding this quantity are rather wasted (region 5b.)). The optimal sales quantity

92

4 Procurement with Stochastic Costs

Figure 4.19: State Space in Cournot Competition

thus is

x1∗ (yo , yw ) =

⎧ d−p1 ⎪ 3b ⎪ ⎪ ⎪ o w ⎪ ⎨ y +y d+p1 −2 f

3b ⎪ ⎪ ⎪ yw ⎪ ⎪ ⎩ d+p1 3b

for for for for for

d−p1 3b d+p1 −2 f 3b d+p1 −2 f 3b d+p1 3b

< < < <

1 yo + yw ≤ d−p 3b 1 −2 f yo + yw ≤ d+p3b 1 −2 f yo + yw and yw ≤ d+p3b 1 yw ≤ d+p ∀yo 3b w o y ∀y

1b.) 2.) 3.) 4b.) 5b.) (4.26)

for p1 > f and x1∗ (yo , yw ) =

⎧ ⎪ ⎨ ⎪ ⎩

d−p1 3b yw d+p1 3b

for for for

d−p1 3b d+p1 3b

yw ≤ < yw ≤ < yw

d−p1 3b d+p1 3b

1a.) 4a.) 5a.)

(4.27)

1 else, regardless of yo . Note, that x1 = d+p 3b equals the quantity a Stackelberg leader would offer. The optimal decision of the second firm in combination with the first firm is

4.2 Operational and Financial Commodity Procurement in Competition

x2∗ =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

d−p1 3b d−p1 −b(yo +yw ) 2b d−2p1 + f 3b d−p1 −byw 2b d−2p1 3b

1 x1∗ = d−p 3b ∗ o x1 = y + yw 1 −2 f x1∗ = d+p3b x1∗ = yw 1 x1∗ = d+p 3b

for for for for for

93

(4.28)

in case of p1 > f . An increasing sales quantity x1 is related to decreasing sales of the second firm. I.e., the first firm displaces its opponent increasingly by offering a higher quantity and obtains a higher market share. On the other hand, the total sale 1 1 ≤ x1 + x2 ≤ 2d−p of both firms is an increasing function in x1 with range 2d−2p 3b 3b yielding a decreasing sales price. The firms have to trade-off these two effects (a similar result is found in case of p1 ≤ f ). Note, that (4.16) is relevant for ∗ 1 x2∗ = d−2p 3b . If d > 2p1 holds, x2 is strictly positive in each state and the second firm is not entirely displaced from the market.

The total profit function of firm 1, ΠC1 (yo , yw ), is determined by costs for procurement in the first period and discounted expected profit from sales in the second period, ΠC1 (yo , yw ) = − c0 yo − (p0 + h)yw + ρ



f p1



3byw  −p1

d

(d + p1 )2 φ (d, p1 )dd 9b

 (d + p1 − byw )yw φ (d, p1 )dd 2 3byw  −p1     d (d − p1 )2 + + p1 yw φ (d, p1 )dd dp1 9b 3byw  +p1   p1  3byw −p1 (d + p1 )2 + φ (d, p1 )dd 9b f d +

+ +

 3byw  +p  1 1

 3byw  −p1 +2 f 1 3byw −p1

2

(d + p1 − byw )yw φ (d, p1 )dd

 3b(yo  +yw )−p +2 f  1 (d + p1 − 2 f )2 3byw  −p1 +2 f

 + f yw φ (d, p1 )dd

9b    1 + d + p1 − b(yo + yw ) (yo + yw ) − f yo φ (d, p1 )dd 3b(yo  +yw )−p1 +2 f 2      d (d − p1 )2 + + p1 yw + (p1 − f )yo φ (d, p1 )dd dp1 . 9b 3b(yo  +yw )+p1  3b(yo  +yw )+p1



(4.29)

94

4 Procurement with Stochastic Costs

Property 4.4 Advance procurement is beneficial in Cournot competition. The optimal solution always consists of a positive amount of inventories above the worst-case sales quantity and zero option contracts.

Proof

See Appendix C.2. 

A sufficient amount of inventories or option contracts can always generate a better solution than a pure JIT-strategy and advance procurement is advantageous. Inventories, however, dominate option contracts as procurement instrument in Cournot competition. The economic rationale is due to the reliability of commitment. Inventories provide a stronger quantity commitment than option contracts as the optimal solution includes zero option contracts. Procuring sufficient inventories results in zero marginal procurement costs in the second period while procuring a sufficient amount of option contracts still requires a payment of f in the second period in order to exercise them. The commitment of inventories thus is more reliable. In comparison to the monopoly, a commitment advantage can be exploited from a sufficient usage of advance procurement instruments. The analytical results are exemplified numerically in Figure 4.20 with the parameter values b = 0.5, d = 8, d = 14, f = 3, h = 0.2, ρ = 0.9, p1 = 2, p1 = 4, d and p1 distributed uniformly and independently. c0 = 0.225 and p0 = 2.5 result for these values. The Influence of Salvage Value The influence of the salvage value on the optimal level of advance procurement is analyzed through the parameter z representing a fraction of the procurement price at t = 1 with 0 ≤ z ≤ 1 such that each item not sold to the market attains a salvage value of zp1 . This is the case if the firm has transaction costs for selling items left over. The first firm trades off whether to sell items to the sales market or to salvage these items. The salvage value thus corresponds to marginal procurement costs as items have an alternative usage. Exemplarily, z = 0 implies marginal costs of zero whereas z = 1 implies marginal costs of p1 as JIT-procurement does. In case that only inventory holding is possible, the first firm’s objective function in the second period is π1C (z) = [d − b(x1 + x2 )] x1 − p1 y j + zp1 (y j + yw − x1 ) with optimal decisions as follows (see Appendix C.3). If the optimal sales quantity is above the amount available from inventories (region 1.) in Figure 4.21), the 1 shortage is procured JIT. Quantities of yw above d−p 3b are sold up to the amount d+p1 (1−2z) 3b

(region 2.)). Additional inventories are rather salvaged than sold to the

4.2 Operational and Financial Commodity Procurement in Competition

95

Figure 4.20: Numerical Example for the Cournot Duopoly

sales market (region 3.)). The first-period objective function is ΠC1 (yw , z) = − (p0 + h)yw   p1  3byw  −p1 (1−2z)  (d + p1 − 2p1 z)2 +ρ + zp1 yw φ (d, p1 )dd 9b p1 d  3byw  +p1

 1 d + p1 − byw yw φ (d, p1 )dd 2     d (d − p1 )2 w + + p1 y φ (d, p1 )dd dp1 . 9b 3byw  +p1 +

3byw  −p1 (1−2z)

(4.30)

96

4 Procurement with Stochastic Costs

Figure 4.21: State Space in Cournot Competition for yo = 0 and Salvage Value z = 0.5

Note, that in case of z = 0 (4.30) equals (4.29) with yo = 0. Figure 4.22 plots (4.30) against yw for different levels of z and shows that the maximum profit of firm 1 is decreasing in an increasing salvage value fraction and that (4.30) is constant in yw for z = 1. A similar effect is observed when analyzing the case of advance procurement only with option contracts which is not presented here. Salvage value corresponds to the value of the outside option to sell items back to the procurement market rather than to sell them to the sales market. An increasing salvage value increases the value of this outside option, but, on the other hand, makes commitment unreliable. In an extreme case with z = 1, the model has a decomposition property as a separation into a pure procurement model and a pure

4.2 Operational and Financial Commodity Procurement in Competition

97

Figure 4.22: Influence of the Salvage Value on the Expected Profit in Case yo = 0

Cournot model is possible. The optimization problem then is ΠC1 (yw , z = 1) = − (p0 + h)yw + ρ =ρ 

 p1  d  (d − p1 )2 p1

 p1  d (d − p1 )2 p1

d

9b 

9b

d

 + p1 yw φ (d, p1 )dddp1

φ (d, p1 )dddp1 

Cournot model

−(p0 + h)yw + ρ 

 p1  d p1

d

p1 yw φ (d, p1 )dddp1 .





procurement model

Due to the no-arbitrage assumption (2.1), the expected profit from the procurement model is zero and the Cournot duopolist has no reliable commitment possibility 1 such that he always sells x1∗ = d−p 3b units to the market.

4.2.5 Procurement Strategy in a Bertrand Duopoly This section considers a duopolistic sales market characterized by price competition, i.e., a Bertrand game. As analyzed in the previous sections, risk aversion and

98

4 Procurement with Stochastic Costs

Cournot competition are drivers for the usage of advance procurement instruments. A risk neutral monopolist cannot benefit from inventory holding or procurement via option contracts, neither on a price insensitive (Section 4.1.3), nor a price sensitive sales market (Section 4.2.3). On the other hand, a risk-averse monopolist and a Cournot duopolist benefit from these instruments (Sections 4.1.4-4.1.5 and 4.2.4). The question to answer in the following is whether a Bertrand duopolist can profit from the possibility of advance procurement. Can he exploit a commitment strategy? Can he profit to the same extend as a Cournot duopolist? Is the benefit from advance procurement in the consequence contingent on the mode of duopolistic competition? In case that both firms charge the same price, an assignment rule is required. In this case, it is assumed that firm 1 is assigned demands with priority which provides an advantage for this firm. (This is a standard assumption as made in Boccard and Whauty (2000)). The price-response function is as given in (4.15). In a standard Bertrand duopoly, firms compete by prices offered to the market using the price-response function (4.15). The price a firm charges results from its marginal procurement costs (see, e.g., Gibbons (1999)). Firm 2 only has the possibility to procure JIT at price p1 which characterizes the price it will charge. Thus r2 = p1 holds, firm 2 ends in a trivial solution and realizes a profit of zero. Firm 1, on the other hand, eventually has procured in advance at less than p1 at t = 0 and thus can charge a price below p1 and gain a higher market share. Its non-linear optimization problem is to maximize revenues

π1B = r1 (y j + yo + yw ) − p1 y j − f yo

(4.31)

subject to y j + yo + yw = d − br1 yo ≤ yo yw ≤ yw r1 ≤ p1 y j , yo , yw , r ≥ 0. It is solved by a Lagrange approach and Karush-Kuhn-Tucker conditions provided in Appendix D.1. The solution is as follows. In general, firm 1 can identify three different strategies concerning the price offered to the market, r1 . First, it can charge a price above the prevailing procurement price p1 . In this case, the competitor will charge a lower price such that it is assigned the entire demand. This strategy is not op-

4.2 Operational and Financial Commodity Procurement in Competition

99

timal as it is affected with zero revenues. Second, firm 1 can charge the current price (as it is assumed that it can sell all items available via advance procurement instruments up to the demand at this price). If it procures additional items at the spot market at t = 1 in order to sell them, such an activity is associated with zero revenues and neglected in the further considerations. Third, firm 1 can set a price below p1 in order to sell additional items. This strategy is promising if sufficient items are available via advance procurement instruments. The usage of the available procurement instruments is prioritized by increasing marginal costs. Using items from the inventory results in marginal costs of zero, exercising option contracts with costs of f , and procuring JIT with costs of p1 . For different realizations of price and demand, different optimal policies result as summarized in the following (numbering corresponds to the state space as provided in Figure 4.23). 1. If firm 1 holds less items in the inventory than requested by the market at the current price p1 , yw ≤ d − bp1 , it sells all these items to the market, charges price p1 , i.e., yw∗ = yw , and makes a profit p1 yw . 2. If more items are available in the inventory than requested by the market, the optimal second-stage strategy additionally is dependent on the (fictitious) d as realizations of p1 both below and optimal monopoly price which is 2b above this price are possible in general. The economic rationale is as follows. Assume the current market price p1 (which equals the second firm’s d and the first firm has a sufmarginal costs) is above the monopoly price 2b ficient amount of inventories (exceeding the monopoly quantity d2 ). Then it prefers to sell the monopoly quantity at the monopoly price with correspond2 ing profit d4b rather than to sell all available inventories at a lower price. 3. If yw < d2 , price r1 = yw∗

= yw ,

d−yw b

is charged and all items in the inventory are sold,

with corresponding profit

d−yw w b y .

d , firm 1 prefers to waste additional units rather than to accept a 4. If p1 < 2b lower sales price and makes a profit of p1 (d − bp1 ).

5. A similar rationale holds if firm 1 owns inventories and option contracts. If the sum of both does not exceed the demand at price p1 , all items are sold to the market at price p1 and a profit of p1 (yo + yw ) − f yo results. f 6. Else, it sells at the monopoly price d−b 2b if this value is below the current f market price and an amount above the monopoly quantity d−b 2 is available

100

4 Procurement with Stochastic Costs

via advance procurement instruments with corresponding profit f yw .

d−b f d−yo −yw is charged, all items are sold 2 , price r1 = b d−yo −yw o o∗ o w∗ w y = y , y = y , and a profit of (y + yw ) results. b

7. If yo + yw < that

(d−b f )2 4b

+

such

f 8. If p1 < d−b 2b , firm 1 prefers to waste additional units rather than to accept a lower sales price and makes a profit of (p1 − f )(d − bp1 ) + f yw .

Figure 4.23: State Space in Bertrand Competition

From the view of t = 0, the first firm optimizes yo and yw associated with costs c0 and p0 + h per item and the discounted expected second stage revenues E(π1B ) contingent on yo and yw . The first-stage objective function thus is

4.2 Operational and Financial Commodity Procurement in Competition

ΠB1 (yo , yw ) = − c0 yo − (p0 + h)yw + ρ E(π1B )   f

= − c0 yo − (p0 + h)yw + ρ + + + + +

+ + +

d

yw  +bp1

p1 yw φ (d, p1 )dddp1

 p1  yw  +bp1  p1  2yw  2 d − yw w d y φ (d, p1 )dddp1 + yw φ (d, p1 )dddp1 yw 

b

2yw 

yw b

 2bp1 2 d

p1

b

4b

d

φ (d, p1 )dddp1 +

 p  d 1 f





yo  +yw +bp1

2yo  +yw  +f b yo  +yw +f b

 p1

b



yw b

 yw +bp1

p1

2bp1

d

4b

p1 (d − bp1 )φ (d, p1 )dddp1

p1 yw + (p1 − f )yo φ (d, p1 )dddp1

 yo  +yw +bp  1 d − yo − yw 2(yo  +yw )+b f

b

 yo  +yw +bp1  d − yo − yw

2yo +yw + f yw +bp1 b  yo  +yw  + f  2bp1 b yw + f yw  +d p1 b  2yo  +yw  + f  2(yo  +yw )+b f b yo  +yw yw +bp1 +f b  yw + f  yo  +yw +bp1 b



+

p1

101

b

(d − b f )2 4b 

 (yo + yw ) − f yo φ (d, p1 )dddp1

 (yo + yw ) − f yo φ (d, p1 )dddp1

 + f yw φ (d, p1 )dddp1

 (d − b f )2 + f yw φ (d, p1 )dddp1 4b



f



yo  +yw  b yw + f b

yw +bp1

(p1 − f )(d − bp1 ) + f yw φ (d, p1 )dddp1

+ f  yo  +yw +bp1 2bp1 −b f

⎤

(p1 − f )(d − bp1 ) + f yw φ (d, p1 )dddp1 ⎦ . (4.32)

This function is illustrated by a numerical example in Figure 4.24 using the parameter values from Figure 4.20. Property 4.5 Advance procurement is not beneficial in a Bertrand competition. Proof

ΠB1 (yo , yw ) is decreasing in yo and yw , see Appendix D.2. 

The economic intuition is as follows. If the only supply option is JIT, the Bertrand duopolist makes zero profit as his price charged to the market equals the procurement price. If he has access to advance procurement, e.g., via inventories, he has a chance to procure at a lower price at t = 0 than charged to the market at t = 1 but, on the other hand, a risk to procure at a higher price. In case of arbitrage-free prices, the chance to gain and the risk to loose from this specula-

102

4 Procurement with Stochastic Costs

Figure 4.24: Numerical Example for the Bertrand Duopoly

tion is balanced and, therefore, an expected profit of zero results. In both modes of competition, Bertrand and Cournot, the procurement of a sufficient amount of advance instruments allows to partially displace the competitor from the market, i.e., to gain market share. As these amounts are affected with positive profits in a Cournot game, quantity commitment is advantageous. In a Bertrand game, however, there is no benefit from gaining the competitor’s market share as every item sold results in an expected profit of zero. Concluding, a firm facing price competition cannot benefit from advance procurement instruments, neither via operational nor financial instruments, when it faces an arbitrage-free procurement market. The implication is that managers in this situation can abstain from an engagement in advance procurement and thus concentrate on a pure JIT strategy.

4.2 Operational and Financial Commodity Procurement in Competition

103

4.2.6 Conclusions and Outlook In this section, a commodity procurement model under competition on the sales market with three procurement instruments, in advance via inventories and call options, and just-in-time, was developed. Both procurement price and demand are stochastic and jointly distributed at the point in time when the amount of call options and the inventory level have to be decided. The firms are asymmetric as the first firm can use all of these instruments whereas the second firm only has the possibility to procure just-in-time. In a monopoly, neither advance procurement instrument is beneficial. In Cournot quantity competition, both instruments inventory holding and call options generate a positive contribution to the optimal procurement plan of the first firm. If both are available, pure inventory holding combined with JIT-procurement is a dominant strategy as it provides a stronger quantity commitment although procurement takes place at an arbitrage-free market and all procurement instruments are equal in terms of their expected NPV. The first firm can obtain a better position by building up sufficient stocks and, therefore, market-power. The commitment advantage is dependent on the salvage value as an increasing salvage value decreases the reliability of quantity commitment. In Bertrand price competition, neither advance procurement instrument is beneficial. Though advance procurement allows to increase the market share, all sales are connected with zero expected profits and thus no advantage can be taken from this market share. This result doesn’t imply expendability of option markets in general as other participants at the commodity market may have different objectives such as riskaversion and face a different sales market or the inventory might be subject to a limited capacity or require an investment in order to install the warehouse. If managers in the quantity competitive setting have a warehouse, then they should not participate in the financial market but should use the warehouse to build up market power. Managers without a warehouse should participate in the financial market by procuring option contracts in advance. Managers in a monopolistic or price competitive situation should concentrate on a pure JIT-strategy. Guidelines for future research address the aspects of symmetric competition and the no-arbitrage assumption. In a symmetric competition both firms have access to advance procurement instruments and thus the possibility to make a strategic choice before entering competition. Simple analytical solutions for a general distribution of price and demand a questionable as the Nash equilibrium on the second stage can be ambiguous and thus requires a distinction of cases (see Pal (1991) for a deterministic investigation of this problem). If the assumption of an arbitrage-free procurement market does not hold, i.e., due to a different holding cost parameter

104

4 Procurement with Stochastic Costs

for the firm and the procurement market, the expected NPV of inventories can be above the expected NPV of option contracts in case that the firm can store items for higher costs than assumed by the market. In this case, the usage of option contracts becomes more attractive which can compensate the commitment advantage of inventories such that a mixture of inventories and option contracts optimizes the procurement policy.

5 Conclusions and Outlook 5.1 Conclusions This thesis considered deterministic commodity procurement and production problems and stochastic commodity procurement problems in presence of a financial market in order to isolate dynamic and stochastic effects. The deterministic problems investigated dynamic manufacturing and procurement models in continuous time (Sections 3.2 and 3.3) and were solved using an optimal control approach. The key result is a crucial trade-off for the decisions of timing the production/procurement and determining the production/procurement quantity which holds for general price, cost, and demand functions and is as follows. • In the dynamic manufacturing problem, the production decision results from the trade-off between marginal production costs, inventory holding costs, and costs on capital tied in the inventory. Production though inventories are available can be optimal if the manufacturing costs are strongly increasing in time as the production rate is subject to limited capacity. • In the dynamic procurement problem, the trade-off between marginal procurement costs, inventory holding costs, and costs on capital tied in the inventory is responsible for the procurement decision. Additional procurement in presence of positive inventories can be optimal if the warehouse capacity is limited. These optimality properties have been derived analytically and are exploited in solution algorithms to solve the trade-off. The stochastic models shed light on the influence of risk and competition aspects. The interface of operations and finance comes up when the restrictive assumptions of a perfect capital market are relaxed. The aspects of risk-aversion and a competitive sales markets in presence of a perfect capital market have been investigated. Section 4.1 considered the risk management aspect when a firm has access to both operational and financial procurement instruments under arbitrage-free

106

5 Conclusions and Outlook

conditions on the procurement market for various optimization objectives (riskneutrality and risk-aversion). The following results are analytically proven for an arbitrary and joint distribution of price and demand. • A risk-neutral firm does not benefit from advance procurement. • For a risk-averse firm optimizing a mean-variance criterion, a mixture of inventories, option contracts, and JIT-procurement can be optimal. For an exponential utility function which is among the class of CARA utility functions, the following numerical results were found. • Increasing risk-aversion increases the optimal level of advance procurement. • Increasing price uncertainty increases the optimal level of inventory holding and decreases the optimal amount of option contracts. • Increasing demand uncertainty decreases the optimal level of advance procurement. • An increasing salvage value implying a lower degree of perishability of the commodity increases the optimal level of advance procurement. Chapter 4.2 addressed procurement with financial and operational instruments on an arbitrage-free procurement market when the firm’s sales market is pricesensitive and competitive and the firms are asymmetric. The following key results were derived analytically for an arbitrarily joint distribution of procurement price and market demand. • A monopolist cannot benefit from advance procurement, neither with operational nor financial instruments. • In Cournot quantity competition, the firm benefits from the usage of advance procurement instruments. Inventories dominate option contracts as procurement instruments as they allow for a stronger quantity commitment and thus market power. The competitor can be partially displaced from the market by a sufficient amount of advance procurement. This quantity commitment becomes more reliable and thus beneficial when the salvage value of excess items is decreasing. • In Bertrand price competition, the firm cannot benefit from advance procurement. Though a partial displacement of the competitor is possible by a sufficient amount of advance procurement, the market share gained is connected with zero marginal revenues due to the characteristics of Bertrand competition and arbitrage-free procurement prices.

5.2 Outlook

107

5.2 Outlook Promising extensions of deterministic, continuous time control problems in Chapter 3 are fixed ordering/setup cost, non-linear procurement/manufactur-ing costs, the possibility of capacity shortages, and positive lead-times. The presence of (time-varying) fixed ordering/setup costs adds an additional trade-off between inventory holding costs, the exploitation of cheap procurement costs, the exploitation of cheap setup costs, and the number of setups which affects the total amount of setup costs. If procurement/manufacturing costs are non-linear, the cost function can be concave or convex in the order quantity and thus tend to increase the order quantity in the first and decrease it in the latter case. If periods with capacity shortages are considered, i.e., instantaneous demand may be above the instantaneous manufacturing capacity, avoiding stockout periods is an additional motivation for inventory holding. Positive lead-times affecting the timing of payments, i.e., the payment of an order is before its delivery, reduce the order quantity in case of discounting with a positive interest rate. Promising fields for research on the stochastic procurement problems in Chapter 4 are a multi-period model formulation, the investigation of different holding costs for the market and the individual firm, and firms’ symmetry in competition. A multi-period model framework allows to isolate the influence of the salvage value as excess items can be used for supply in future periods and thus salvage value can be determined endogenously. If the firm’s inventory holding decision is subject to a different holding cost parameter than valid for the market and thus the commodity’s price fixing, the law of one price does not hold for all trading strategies and the no-arbitrage property is lost. If, e.g., the firm’s holding costs are above the market’s holding costs, inventory holding is less attractive in comparison to spot market procurement and the procurement via option contracts. A modelling approach which provides both market participants in Cournot and Bertrand competition with a strategic decision before entering the competition, i.e., both firms can procure inventories and option contracts, allows for a more detailed analysis of commitment, the multitude of equilibria, and the interaction of financial and operational procurement instruments.

A Risk Management A.1 Expected Profit Optimization A.1.1 Analysis of the Expected NPV of the Profit To prove Property 4.1, the slope and concavity of (4.4) is analyzed. o w o • First partial derivative of ΠN 1 (y , y ) with respect to y :   o w r d ∂ ΠN 1 (y , y ) = − c0 + ρ (p1 − f )φ (d, p1 )dddp1 o ∂y f yo  +yw    p1

+

r

d

yo  +yw

(r − f )dddp1 .

o w w • First partial derivative of ΠN 1 (y , y ) with respect to y :   o w f d ∂ ΠN 1 (y , y ) = − (p0 + h) + ρ p1 φ (d, p1 )dddp1 ∂ yw p1 yw   o  w  

+ +

y +y

r

f



yw

f φ (d, p1 )dd +

 p1  yo  +yw r

yw 

f φ (d, p1 )dd +

d

yo +yw



p1 φ (d, p1 )dd dp1

 d yo +yw





rφ (d, p1 )dd dp1 .

110

A Risk Management

o w o ΠN 1 (y , y ) is decreasing in y (this derivation exploits that φ (d, p1 ) = 0 for d ≤ d and for d ≤ d) as it is...

... decreasing for 0 ≤ yo + yw < d: #   o w # r d ∂ ΠN 1 (y , y ) # = − c0 + ρ (p1 − f )φ (d, p1 )dddp1 # o # o w ∂y f d 0≤y +y
+

d

r

d

≤ − c0 + ρ

(r − f )dddp1



p1  d f

d

 (p1 − f )φ (d, p1 )dddp1

=0 due to (2.4). ... decreasing for d ≤ yo + yw < d: #   o w # r d ∂ ΠN 1 (y , y ) # = − c0 + ρ (p1 − f )φ (d, p1 )dddp1 # # o w ∂ yo f yo  +yw d≤y +y
p1

d

yo  +yw

r

≤ − c0 + ρ ≤ − c0 + ρ

(r − f )dddp1

 p1  d f

yo  +yw

f

d

 p1  d

(p1 − f )φ (d, p1 )dddp1

(p1 − f )φ (d, p1 )dddp1

=0 due to (2.4). ... decreasing for d ≤ yo + yw : # o w # ∂ ΠN 1 (y , y ) # # # o ∂ yo

= − c0 + ρ

d≤y +yw

+

  r

 p1  d r

d

f

d

d

(p1 − f )φ (d, p1 )dddp1 

(r − f )dddp1

= − c0 < 0 and constant.

A.1 Expected Profit Optimization

111

o w w ΠN 1 (y , y ) is decreasing in y d ≤ d) as it is...

... decreasing for 0 ≤ yo + yw < d which implies 0 ≤ yw < d:

  # f d ∂ ΠN1 (yo , yw ) ## = − (p + h) + ρ p1 φ (d, p1 )dddp1 0 # o w ∂ yw p1 d 0≤y +y
+

+

f

d



d

 p1  d r

d

+

 r d f

d



f d p1

p1 φ (d, p1 )dd dp1

 d d





rφ (d, p1 )dd dp1

p1 φ (d, p1 )dddp1

d

p1 φ (d, p1 )dddp1

 p1  d d

r

d

f φ (d, p1 )dd +

≤ − (p0 + h) + ρ +

d

f φ (d, p1 )dd +

 p1 φ (d, p1 )dddp1

=0 due to (2.3).

... decreasing for d ≤ yo + yw and yw < d:

  # f d ∂ ΠN1 (yo , yw ) ## #d≤yo  +yw  = − (p0 + h) + ρ p yw p1 φ (d, p1 )dddp1 ∂ yw 1 yw
+

y +y

r

f

 p1

yw



yo +yw

yw 

r

≤ − (p0 + h) + ρ + +

  r  f

 p1

d

d

 d

r

+

f

d



f φ (d, p1 )dd + f d p1

d

d

f d p1

d

 d yo +yw





rφ (d, p1 )dd dp1

p1 φ (d, p1 )dd dp1

d

 d d





rφ (d, p1 )dd dp1

p1 φ (d, p1 )dddp1

p1 φ (d, p1 )dddp1 +

=0 due to (2.3).

p1 φ (d, p1 )dd dp1



 d

f φ (d, p1 )dd + 

yo +yw



p1 φ (d, p1 )dddp1

f φ (d, p1 )dd +

≤ − (p0 + h) + ρ  r d

f φ (d, p1 )dd +

d

 p1  d r

d

 p1 φ (d, p1 )dddp1

112

A Risk Management

... decreasing for d ≤ yw :

  # f d ∂ ΠN1 (yo , yw ) ## p1 φ (d, p1 )dddp1 = − (p + h) + ρ 0 # w ∂ yw p1 d d≤y     + +

r

f

d



f φ (d, p1 )dd +

d

 p1  d r

d

f φ (d, p1 )dd +

d

d

p1 φ (d, p1 )dd dp1

 d d





rφ (d, p1 )dd dp1

= − (p0 + h) < 0 and constant. o w ΠN 1 (y , y ) is concave as it is ...

... concave in yo : o w o The second partial derivative of ΠN 1 (y , y ) with respect to y is o w ∂ 2 ΠN 1 (y , y ) =ρ o2 ∂y



+

r

( f − p1 )φ (yo + yw , p1 )dp1  o w ( f − r)φ (y + y , p1 )dp1

f  p1 r

<0. ... concave in yw : o w w The second partial derivative of ΠN 1 (y , y ) with respect to y is  o w f ∂ 2 ΠN 1 (y , y ) =ρ −p1 φ (yw , p1 )dp1 ∂ yw2 p1  r  + ( f − p1 )φ (yo + yw , p1 ) − f φ (yw , p1 ) dp1 f   p1   + ( f − r)φ (yo + yw , p1 ) − f φ (yw , p1 ) dp1 r

<0. ... jointly concave in yo and yw : The joint second derivative is o w o w ∂ 2 ΠN ∂ 2 ΠN 1 (y , y ) 1 (y , y ) = =ρ w o o ∂y ∂y ∂ y ∂ yw



+ <0.

r

( f − p1 )φ (yo + yw , p1 )dp1  ( f − r)φ (yo + yw , p1 )dp1

f  p1 r

(A.1)

A.1 Expected Profit Optimization

113

o w The corresponding Hessian Matrix is negative definite and thus ΠN 1 (y , y ) is concave throughout. Determinants of the Hessian Matrix: # # # ∂ 2 ΠN (yo , yw ) # # # 1 # #<0 # # ∂ yw2

and

# # # # # #

=ρ

∂ 2 ΠN1 (yo  ,yw  ) ∂ yw 2 ∂ 2 ΠN1 (yo  ,yw  ) ∂ yo  ∂ yw



f p1

# # # # # #

∂ 2 ΠN1 (yo  ,yw ) ∂ yw  ∂ yo  ∂ 2 ΠN1 (yo  ,yw ) ∂ yo 2

−p1 φ (yw , p1 )dp1 +

 r f

( f − p1 )φ (yo + yw , p1 )

  p1    ( f − r)φ (yo + yw , p1 ) − f φ (yw , p1 ) dp1 − f φ (yw , p1 ) dp1 + r   r  p1 ( f − r)φ (yo + yw , p1 )dp1 ·ρ ( f − p1 )φ (yo + yw , p1 )dp1 + r

f

  r  ( f − p1 )φ (yo + yw , p1 )dp1 + − ρ

r

f

>0 ⇔ρ



+

f p1

−p1 φ (yw , p1 )dp1 +

 p1  r

 r f

p1

2 ( f − r)φ (y + y , p1 )dp1 o

w

 ( f − p1 )φ (yo + yw , p1 ) − f φ (yw , p1 ) dp1

 ( f − r)φ (yo + yw , p1 ) − f φ (yw , p1 ) dp1



 r   p1 ( f − r)φ (yo + yw , p1 )dp1 < 0 ( f − p1 )φ (yo + yw , p1 )dp1 + −ρ r f   

⇔ρ

f

p1

p1 φ (yw , p1 )dp1 +

p1

f

f φ (yw , p1 )dp1 > 0.

114

A Risk Management

A.1.2 The Influence of the Salvage Value The first derivative of (4.5) with respect to z is  f  w  o w y ∂ ΠN z 1 (y , y , z) =ρ p1 (yw − d)φ (d, p1 )dddp1 ∂z p1 d + +



r z f z

 yo  +yw d

 p1  yo  +yw r z

d

p1 (yo + yw − d)φ (d, p1 )dddp1 

(A.2)

p1 (yo + yw )φ (d, p1 )dddp1

≥0 as all elements are non-negative when substituting d with the respective upper integration limit.

A.2 Mean-Variance Optimization

115

A.2 Mean-Variance Optimization A.2.1 Analysis of the Variance of the Profit To prove Property 4.2, the first partial derivatives of (4.8) are analyzed. First partial derivative with respect to yo :

∂ Var(ΠN 1) o ∂y

=

 p1  yw  p1

+ + +

d

  −2 ρ dr − (p0 + h)yw − c0 yo c0 φ (d, p1 )dddp1

 f d p1

yw 

  −2 ρ (dr − p1 (d − yw )) − (p0 + h)yw − c0 yo c0 φ (d, p1 )dddp1

 p1  yo  +yw f

 r d f

  −2 ρ (dr − f (d − yw )) − (p0 + h)yw − c0 yo c0 φ (d, p1 )dddp1

yw 

yo  +yw

  2 ρ (dr − f yo − p1 (d − yo − yw )) − (p0 + h)yw − c0 yo

· (ρ (p1 − f ) − c0 ) φ (d, p1 )dddp1 +

 p1  d yo  +yw

r

  2 ρ (r(yo + yw ) − f yo ) − (p0 + h)yw − c0 yo

· (ρ (r − f ) − c0 ) φ (d, p1 )dddp1    r

−2 ρ + + + +

 p1  yo  +yw yw 

f

 r d f

r

yw 

drφ (d, p1 )dddp1

 − f yo − p1 (d − yo − yw ) φ (d, p1 )dddp1  r(yo + yw ) − f yo φ (d, p1 )dddp1 



−p1 (d − y )φ (d, p1 )dddp1 − (p0 + h)y − c0 y w

w

d

 p1  d r

d

− f (d − yw )φ (d, p1 )dddp1

yo  +yw

f

r



yo  +yw

 f d

 p1  yo  +yw



yo  +yw

 p1  d

p1

drφ (d, p1 )dddp1 +

p1 d

   r · ρ +

d

yo  +yw

(p1 − f )φ (d, p1 )dddp1 



(r − f )φ (d, p1 )dddp1 − c0 .

o

116

A Risk Management ∂ Var(ΠN1 ) ∂ yo

There is no closed-form solution for a first-order condition that in direction yo , Var(ΠN 1 ) is ...

= 0. It can be stated

... decreasing for yo = yw = 0: #   # r d ∂ Var(ΠN 2 1 )# =2ρ d(r − p1 )(p1 − f )φ (d, p1 )dddp1 # ∂ yo # o  w f d y =y =0

− ·

 r d

+

d(r − p1 )φ (d, p1 )

p1 d

  r f

d

d

 p1  d r

(A.3)

(p1 − f )φ (d, p1 )dddp1

d





(r − f )φ (d, p1 )dddp1 dddp1

<0 for p1 < p1 and

... zero for yw ≥ d:

# # ∂ Var(ΠN 1 )# # o # ∂y

= yw  ≥d

 p1  d p1

d

  −2 ρ dr − (p0 + h)yw − c0 yo

  · c0 φ (d, p1 )dddp1 − 2 ρ

p1  d p1

−(p0 + h)yw − c0 yo (−c0 ) =0.

d

drφ (d, p1 )dddp1

A.2 Mean-Variance Optimization

117

First partial derivative with respect to yw :

∂ Var(ΠN 1) w ∂y

=

 p1  yw p1

+

2 ρ dr − (p0 + h)yw − co yo (−p0 − h)φ (d, p1 )dddp1

d

 f d p1

yw

2 ρ (dr − (d − yw )p1 ) − (p0 + h)yw − c0 yo

· (ρ p1 − p0 − h)φ (d, p1 )dddp1 +

 p1  yo  +yw

2 ρ (dr − f (d − yw )) − (p0 + h)yw − co yo

yw

f

· (ρ f − p0 − h)φ (d, p1 )dddp1 +

 r d f

yo  +yw

2 ρ (dr − f yo − p1 (d − yo − yw )) − (p0 + h)yw − c0 yo

· (ρ p1 − p0 − h)φ (d, p1 )dddp1 +

 p1  d

2 ρ ((yo + yw )r − f yo ) − (p0 + h)yw − c0 yo

yo  +yw

r

· (ρ r − p0 − h)φ (d, p1 )dddp1    r

−2 ρ + + +

p1 d

 p1  yo  +yw yw

f

 r d f

yo  +yw

 p1  d r

 

· ρ +

d

yw

 r d f

yo  +yw

r

d

− f (d − yw )φ (d, p1 )dddp1 +

drφ (d, p1 )dddp1

 f  p1 p1

yw 

−p1 (d − yw )φ (d, p1 )dddp1



(r(y + y ) − f y )φ (d, p1 )dddp1 − (p0 + h)y − c0 y o

f d

 p1  yo  +yw

(− f yo − p1 (d − yo − yw ))φ (d, p1 )dddp1 

yo  +yw

p1

drφ (d, p1 )dddp1 +

w

o

p1 φ (d, p1 )dddp1 + p1 φ (d, p1 )dddp1 +

w

 p1  yo  +yw f

yw

 p1  d f

yo  +yw

o

f φ (d, p1 )dddp1 



rφ (d, p1 )dddp1 − (p0 + h) .

118

A Risk Management

There is no closed-form solution for a first-order condition that in direction yw , Var(ΠN 1 ) is... ... decreasing for yo = yw = 0: # # ∂ Var(ΠN 1 )# # ∂ yw # o  w

=2ρ 2

y =y =0

 r d p1 d



· p1 − −

r

p1 d

d

= 0. It can be stated

d(r − p1 )φ (d, p1 )

 r d

 p1  d

∂ Var(ΠN1 ) ∂ yw

p1 φ (d, p1 )dddp1

(A.4)



rφ (d, p1 )dddp1 dddp1

<0 for p1 < p1 . ... and zero for yw ≥ d: #  p1  d  #  ∂ Var(ΠN 1 )# = 2 ρ dr − (p0 + h)yw − c0 yo # w # w ∂y p1 d y ≥d

· (−p0 − h))φ (d, p1 )dddp1   

−2 ρ +ρ

r

p1 d

 p1  d r

d

d

drφ (d, p1 )dddp1 

drφ (d, p1 )dddp1 − (p0 + h)y − c0 y w

o

· (−p0 − h) =0. o w These statements are useful to define an extremum property. ΠN 1 (y , y ) is a continuous function without extreme values in this range and the variance is a quadratic transformation of this function. Therefore Var(ΠN 1 ) is continuous either and characterized by the following property: o • Var(ΠN 1 ) has a solution satisfying the first-order condition in the range 0 ≤ y , 0 ≤ w w o y , y ≤ d, y ≤ d or

• is minimal for yo ≥ d, yw ≥ d.

A.2 Mean-Variance Optimization

119

A.2.2 Isolated Analysis of Mismatch and Price Risk o For an isolation of mismatch and price risk, the function Var(ΠN 1 ) for y := 0 can be analyzed when... p N ... only the price is uncertain such that Var(ΠN 1 ) := Var (Π1 ). In this case, the first derivative of the variance as given in (4.9) and (4.10) is

∂ Var p (ΠN 1) = ∂ yw

 r

2 ρ (dr − p1 (d − yw )) − (p0 + h)yw (ρ p1 − p0 − h)φ (p1 )dp1

p1

+

 p1 r

−2



r p1

 p1

+ r 

·

2 ρ ryw − (p0 + h)yw (ρ r − p0 − h)φ (p1 )dp1

r

p1

  ρ dr − p1 (d − yw ) φ (p1 )dp1 

ρ yw rφ (p1 )dp1 − (p0 + h)yw

ρ p1 φ (p1 )dp1 +

 p1 r



ρ rφ (p1 )dp1 − (p0 + h)

for yw ≤ d and

∂ Var p (ΠN 1) = ∂ yw

 p1 p1

−2

2 ρ dr − (p0 + h)yw (−p0 − h)φ (p1 )dp1



p1 p1



ρ drφ (p1 )dp1 − (p0 + h)y

w

(−p0 − h)

=0 for yw > d. The first derivative is zero for yw = d: #  r # ∂ Var p (ΠN 1 )# =2 (ρ dr − (p0 + h)d) ρ p1 φ (d, p1 )dp1 # w # w ∂y p1 y =d

+ 2 (ρ dr − (p0 + h)d) − 2 (ρ dr − (p0 + h)d) + =0.

 p1 r

 p1

r



ρ rφ (d, p1 )dp1

r p1

ρ rφ (d, p1 )dp1 ρ p1 φ (d, p1 )dp1

120

A Risk Management

... only the demand is uncertain. The first derivative of (4.11) is

∂ Vard (ΠN 1) = ∂ yw

 yw  d

  −2(p0 + h) ρ dr − (p0 + h)yw φ (d)dd

 d

 −2(p0 + h − ρ p1 ) ρ (dr − p1 (d − yw ))  −(p0 + h)yw φ (d)dd $ yw   rd φ (d)dd − 2 −(p0 + h)yw + ρ d   +

yw 

+ 

d

yw 

(rd − p1 (d − yw ))φ (d)dd

· −(p0 + h) + ρ =2ρ p1 2

−



yw   d

d

 d  yw  yw d

yw

 d yw

 p1 φ (d)dd



 dr − p1 (d − yw ) φ (d)dd φ (d)dd 

drφ (d)dd φ (d)dd

≥0. The variance thus is zero for yw = 0 and increasing in yw .

(A.5)

A.3 Expected Utility Optimization

121

A.3 Expected Utility Optimization o w For sparse notation, let ΠN 1 (y , y ) := Π. Then

U(Π) =

1 − e−aΠ a

is the utility function as given in (4.12). Substituting Π with its expectation E(Π) and the deviation from the expectation h such that Π := E(Π) + h and applying a Taylor-series results in U(Π) =U(E(Π) + h) U  (E(Π)) U  (E(Π)) (Π − E(Π)) + (Π − E(Π))2 1! 2! U  (E(Π)) (Π − E(Π))3 + . . . + 3! ∞ U (n) E(Π)(Π − E(Π)n ) . =∑ n! n=0

=U(E(Π)) +

U (n) (•) is defined as n−th derivative of U(•) with respect to (•). The expected utility thus is U  (E(Π)) U  (E(Π)) E(Π − E(Π)) + E(Π − E(Π))2 1! 2! U  (E(Π)) + E(Π − E(Π))3 + . . . 3! ∞ U (n) E(Π)E(Π − E(Π)n ) . =∑ n! n=0

E(U(Π)) =U(E(Π)) +

As E(Π − E(Π))n is the n−th moment defined as gn , ∞

U (n) E(Π)gn n! n=0

E(U(Π)) = ∑

∞ U (n) E(Π)gn 1 =U(E(Π)) + U  (E(Π))Var(Π) + ∑ 2 n! n=3

results. A second-order Taylor-series expansion yields E(U(Π)) ≈ (see Burghardt (2008)).

1 e−aE(Π) − a a

$

1 1 + Var(Π)a2 2

% (A.6)

B Monopoly

B.1 Analysis of the Second Period

Lagrange-function: L(y j , yo , yw , x1 ) = (d − bx1 ) x1 − p1 y j − f yo + α1 (y j + yo + yw − x1 ) + α2 (yo − yo ) + α3 (yw − yw ). First-order Karush-Kuhn-Tucker conditions:

∂L ≤ 0 ⇔ d − 2bx1 − α1 ≤ 0, ∂ x1

∂L x1 = 0 ⇔ (d − 2bx1 − α1 )x1 = 0 ∂ x1

∂L ≤ 0 ⇔ −p1 + α1 ≤ 0, ∂yj

∂L j y = 0 ⇔ (−p1 + α1 )y j = 0 ∂yj ∂L ∂L o ≤ 0 ⇔ − f + α1 − α2 ≤ 0, y = 0 ⇔ (− f + α1 − α2 )yo = 0 ∂ yo ∂ yo ∂L ∂L w ≤ 0 ⇔ α1 − α3 ≤ 0, y = 0 ⇔ (α1 − α3 )yw = 0 ∂ yw ∂ yw ∂L ∂L ≥ 0 ⇔ y j + yo + yw − x1 ≥ 0, α1 = 0 ⇔ (y j + yo + yw − x1 )α1 = 0 ∂ α1 ∂ α1 ∂L ∂L ≥ 0 ⇔ yo − yo ≥ 0, α2 = 0 ⇔ (yo − yo )α2 = 0 ∂ α2 ∂ α2 ∂L ∂L ≥ 0 ⇔ yw − yw ≥ 0, α3 = 0 ⇔ (yw − yw )α3 = 0 ∂ α3 ∂ α3 α1 ≥ 0, α2 ≥ 0, α3 ≥ 0, x1 ≥ 0, y j ≥ 0, yo ≥ 0, yw ≥ 0.

(B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (B.7)

The resulting optimal decisions in the monopoly’s second period are as summarized in Table B.1 (the numbering of regions corresponds to Figure 4.18).

124

B Monopoly

yj

∗

yo∗

yw∗

x1∗

r∗ d+p1 2 d+p1 2

d−p1 w 2b − y d−p1 o w 2b − y − y

0

yw

yo

yw

d−p1 2b d−p1 2b

2.)

0

yo

yw

yo + yw

d − b(yo + yw )

3.)

0

− yw

yw

d− f 2b

d+ f 2

4a.), 4b.)

0

0

yw

yw

d − byw

5a.), 5b.)

0

0

d 2b

d 2b

d 2

1a.) 1b.)

d− f 2b

Table B.1: Optimal Second-period Decisions in the Monopoly

B.2 Analysis of the First Period

125

B.2 Analysis of the First Period o w To prove Property 4.3, the profit ΠM 1 (y , y ) as given in (4.25) is shown to deo w crease in y and y . The first partial derivatives are   p1  2b(yo +yw )+p1 o w

∂ ΠM 1 (y , y ) d − f − 2b(yo + yw ) φ (d, p1 )dd = − c0 + ρ o o w ∂y f 2b(y +y )+ f   d

+



2b(yo +yw )+p1

(p1 − f )φ (d, p1 )dd dp1 



I.)

(B.8) and o w ∂ ΠM 1 (y , y ) = − (p0 + h) + ρ ∂ yw

+



f p1

 

2byw +p1

2byw

(d − 2byw )φ (d, p1 )dd 



 d 2byw +p1



II.)

p1 φ (d, p1 )dd dp1



+

+

 p1  2byw + f f



2byw

 2b(yo +yw )+ f 2byw + f

(d − 2byw )φ (d, p1 )dd 



III.)

(B.9)

f φ (d, p1 )dd

 2b(yo +yw )+p1

+



2b(yo +yw )+ f

d − 2b(yo + yw ) φ (d, p1 )dd   IV.)

+

 d 2b(yo +yw )+p1





p1 φ (d, p1 )dd dp1 .

o w In direction yo , ΠM 1 (y , y ) is... d−p1 2b : d−p1 o w 2b yields 2b(y +y )+ p1

... constant for yo + yw ≤ yo +yw ≤

≤ d and thus 2b(yo +yw )+ f ≤ d.

126

B Monopoly

As φ (d, p1 ) = 0 for d ≤ d, #  p1  d o w # ∂ ΠM 1 (y , y ) # = − c + ρ (p1 − f )φ (d, p1 )dddp1 0 # o w d−p1 ∂ yo f d y +y ≤ 2b

= − c0 + ρ E(c1 ) = 0 due to (2.2) (B.10) results. ... decreasing for yo + yw >

d−p1 2b :

o w 1 yo + yw > d−p 2b yields 2b(y + y ) + p1 > d. Comparing the elements in (B.8) and (B.10), the following upper bound holds by substitution of d with the largest possible term resulting from the upper integration limit:

I.) = ≤

 p1  2b(yo +yw )+p1 f

2b(yo +yw )+ f

f

2b(yo +yw )+ f

o w

d − f − 2b(yo + yw ) φ (d, p1 )dddp1

 p1  2b(yo +yw )+p1

2b(yo + yw ) + p1 − f

−2b(y + y ) φ (d, p1 )dddp1

=

 p1  2b(yo +yw )+p1 f

2b(yo +yw )+ f

(p1 − f )φ (d, p1 )dddp1 .

Therefore, all elements are at most p1 − f and # o w # ∂ ΠM 1 (y , y ) # # o w d−p1 ≤ (B.10) = 0 ∂ yo y +y > 2b

holds. ... linearly decreasing for yo + yw ≥

d− f 2b

d− f 2b

:

yields 2b(yo + yw ) + f ≥ d and thus 2b(yo + yw ) + p1 ≥ d yo + yw ≥ for p1 ≥ f . As φ (d, p1 ) = 0 for d ≥ d, # o w # ∂ ΠM 1 (y , y ) # # o w d− f = −c0 < 0 and constant. ∂ yo y +y ≥ 2b

B.2 Analysis of the First Period

127

In direction yw , the function is...

... constant for yo + yw ≤

d−p1 2b :

o w w 1 yo + yw ≤ d−p 2b yields 2b(y + y ) + p1 ≤ d and thus 2by + p1 ≤ d, w o w w 2by − p1 ≤ d, 2b(y + y ) + f ≤ d, and 2by + f ≤ d. As φ (d, p1 ) = 0 for d ≤ d,

#  p1  d o w # ∂ ΠM 1 (y , y ) # = − (p + h) + ρ p1 φ (d, p1 )dddp1 0 # o w d−p1 ∂ yw p1 d y +y ≤ 2b

= − (p0 + h) + ρ E(p1 ) = 0 due to (2.1). (B.11)

... decreasing for yo + yw >

d−p1 2b :

o w 1 yo + yw > d−p 2b yields 2b(y + y ) + p1 > d. Comparing the elements in (B.9) and (B.11), the following upper bounds hold by substitution of d with the largest possible term, i.e., the upper integration limit:

I.) = ≤ =

 f  2byw +p1 p1

2byw

p1

2byw

p1

2byw

 f  2byw +p1  f  2byw +p1

II.) = ≤ ≤

(d − 2byw )φ (d, p1 )dddp1 (2byw + p1 − 2byw )φ (d, p1 )dddp1 p1 φ (d, p1 )dddp1

 p1  2byw + f f

2byw

f

2byw

f

2byw

 p1  2byw + f  p1  2byw + f

(d − 2byw )φ (d, p1 )dddp1 (2byw + f − 2byw )φ (d, p1 )dddp1 p1 φ (d, p1 )dddp1

128

B Monopoly

III.) = ≤

 p1  2b(yo +yw )+p1 f

2b(yo +yw )+ f

f

2b(yo +yw )+ f

d − 2b(yo + yw ) φ (d, p1 )dddp1

 p1  2b(yo +yw )+p1

2b(yo + yw ) + p1 − 2b(yo + yw )

· φ (d, p1 )dddp1

=

 p1  2b(yo +yw )+p1 f

2b(yo +yw )+ f

p1 φ (d, p1 )dddp1 .

Therefore, all elements are at most p1 and # o w # ∂ ΠM 1 (y , y ) # # o w d−p1 < (B.11) = 0 ∂ yw y +y > 2b

holds. ... linearly decreasing for yw ≥

d 2b :

d yw ≥ 2b yields 2byw ≥ d and thus all integration limits in (B.8) for integration of d are at least d and can be substituted as φ (d, p1 ) = 0 for d ≥ d. # o w # ∂ ΠM 1 (y , y ) # # w d = −(p0 + h) < 0 and constant. ∂ yw y ≥ 2b

C Cournot Duopoly

C.1 Analysis of the Second Period

Lagrange-function of the first firm: L(y j , yo , yw , x1 , x2 ) = [d − b(x1 + x2 )] x1 − p1 y j − f yo + λ1 (y j + yo + yw − x1 ) + λ2 (yo − yo ) + λ3 (yw − yw ). First-order Karush-Kuhn-Tucker conditions:

∂L ∂L ≤ 0 ⇔ d − 2bx1 − bx2 − λ1 ≤ 0, x1 = 0 ⇔ (d − 2bx1 − bx2 − λ1 )x1 = 0 ∂ x1 ∂ x1 and (B.1) - (B.7). Objective-function and first-order condition of the second firm:

π2C = [d − b(x1 + x2 )] x2 − p1 x2 ,

∂ π2C ! = d − b(x1 + 2x2 ) − p1 = 0. ∂ x2

The resulting simultaneous conditions are x1∗ =

d + p1 − 2λ1 , 3b

x2∗ =

d − 2p1 + λ1 . 3b

The optimal second-period decisions in the Cournot duopoly case are summarized in Table C.1 (the numbering of regions corresponds to Figure 4.19).

130

C Cournot Duopoly

yj 1a.) 1b.)

d−p1 3b d−p1 3b

∗

− yw

−y −y o

w

yo ∗

yw∗

x1∗

x2∗

r∗

0

yw 

d−p1 3b

d−p1 3b

d+2p1 3

o

y

yw 

d−p1 3b

d−p1 3b

d+2p1 3

yw 

yo  + yw

d−p1 −b(yo +yw  ) 2b

d+p1 −b(yo +yw ) 2

yw 

d+p1 −2 f 3b

d−2p1 + f 3b

d+p1 + f 3 d+p1 −byw 2 d+p1 3

2.)

0

yo 

3.)

0

d+p1 −2 f 3b

− yw

4a.), 4b.)

0

0

yw 

yw

d−p1 −byw 2b

5a.), 5b.)

0

0

d+p1 3b

d+p1 3b

d−2p1 3b

Table C.1: Optimal Second-period Decisions in the Cournot Duopoly

C.2 Analysis of the First Period

131

C.2 Analysis of the First Period To prove Property 4.4, the profit ΠC1 (yo , yw ) as given in (4.29) is shown to be constant in a first, increasing in a second, and decreasing in a third interval in yo and yw , and the directional derivative in direction (yw , −yo ) is shown to be non-negative. The first partial derivatives of (4.29) are   p1  3b(yo +yw )+p1 $ ∂ ΠC1 (yo , yw ) d + p1 − 2 f = − c0 + ρ ∂ yo 2 f 3b(yo +yw )−p1 +2 f % (C.1) −b(yo + yw ) φ (d, p1 )dd   +

d

3b(yo +yw )+p1

(p1 − f )φ (d, p1 )dd dp1

and

∂ ΠC1 (yo , yw ) = − (p0 + h) + ρ ∂ yw +

 d 3byw +p1



 f  3byw +p1 $ d + p1



p1 φ (d, p1 )dd dp1



+ρ + + +

 p1  3byw −p1 +2 f $ d + p1 f

2

3byw −p1

p1

% − byw φ (d, p1 )dd

2

3byw −p1

 3b(yo +yw )−p1 +2 f 3byw −p1 +2 f

 3b(yo +yw )+p1 3b(yo +yw )−p1 +2 f

 d

3b(yo +yw )+p1

% − byw φ (d, p1 )dd

f φ (d, p1 )dd

$

% d + p1 o w − b(y + y ) φ (d, p1 )dd 2 

p1 φ (d, p1 )dd dp1 . (C.2)

132

C Cournot Duopoly

The shape of ΠC1 (yo , yw ) is as follows. In direction yo , the function is... ... constant for yo + yw ≤

d−p1 3b :

o w o w 1 yo + yw ≤ d−p 3b yields 3b(y + y ) + p1 ≤ d and thus 3b(y + y ) − p1 + 2 f ≤ d for p1 ≥ f . As φ (d, p1 ) = 0 for d ≤ d,

#  p1  d ∂ ΠC1 (yo , yw ) ## = − c + ρ (p1 − f )φ (d, p1 )dddp1 0 # o w d−p1 ∂ yo f d y +y ≤ 3b

= − c0 + ρ E(c1 ) = 0 due to (2.2). ... increasing for yo + yw = yo + yw =

d−p1 +ε 3b

d−p1 +ε : 3b

yields 3b(yo + yw ) + p1 = d + ε and thus,

# ∂ ΠC1 (yo , yw ) ## lim # d−p1 +ε ∂ yo ε →0+ 3b   p1  d+ε $ d + p1 − 2 f = lim −c0 + ρ + 2 ε →0 f d % −b(yo + yw ) φ (d, p1 )dd   +

d

d+ε

= lim ρ ε →0+

(C.3)

(p1 − f )φ (d, p1 )dd dp1

 p1  d+ε $ d − p1 f

d

2

% − b(yo + yw ) φ (d, p1 )dddp1

due to (2.2). For lowest possible d (d), largest possible p1 (p1 ), and substitution of yo + yw = d−p3b1 +ε results # ∂ ΠC1 (yo , yw ) ## lim # d−p1 +ε ∂ yo ε →0+ 3b $ %%  p1  d+ε $ d − p1 + ε d − p1 −b ≥ lim ρ φ (d, p1 )dddp1 2 3b ε →0+ f d $ %  p1  d+ε d − p1 − ε φ (d, p1 )dddp1 > 0 = lim ρ 6 ε →0+ f d due to (4.16) and thus (C.3) is positive.

C.2 Analysis of the First Period

133

... linearly decreasing for yo + yw ≥

d−2 f +p1 : 3b

yo + yw ≥ d−23bf +p1 yields 3b(yo + yw ) − p1 + 2 f ≥ d and 3b(yo + yw ) + p1 ≥ d for p1 ≥ f . Due to φ (d, p1 ) = 0 for d ≤ d, # ∂ ΠC1 (yo , yw ) ## # o w d−2 f +p1 = −c0 < 0 and constant ∂ yo y +y ≥ 3b

results. In direction yw , the function is... ... constant for yo + yw ≤

d−p1 3b :

o w 1 yo + yw ≤ d−p 3b yields 3b(y + y ) + p1 ≤ d and, in combination with φ (d, p1 ) = 0 for d ≤ d,

#  p1  d ∂ ΠC1 (yo , yw ) ## = − (p + h) + ρ p1 φ (d, p1 )dddp1 0 # o w d−p1 ∂ yw p1 d y +y ≤ 3b

= − (p0 + h) + ρ E(p1 ) = 0 due to (2.1) results. ... increasing for yo + yw =

d−p1 +ε : 3b

# ∂ ΠC1 (yo , yw ) ## # d−p1 +ε ∂ yw ε →0+ 3b  f  = lim −(p0 + h) + ρ lim

ε →0+

+

d+ε

+ρ +

p1



 d

 d d+ε

$

d

% d + p1 − byw φ (d, p1 )dd 2

p1 φ (d, p1 )dd dp1

 p1  d+ε $ d + p1 f

d+ε

d

2 

(C.4) %

− b(y + y ) φ (d, p1 )dd

p1 φ (d, p1 )dd dp1

o

w

134

C Cournot Duopoly

= lim ρ ε →0+

+ρ 



 f  d+ε $ d − p1

2

d

p1

 p1  d+ε $ d − p1 f

d

2

% − byw φ (d, p1 )dddp1 



I.)

% − b(yo + yw ) φ (d, p1 )dddp1 .   II.)

due to (2.1). For lowest possible d (d), largest possible p1 ( f in I.) and p1 in II.)), and substitution of yo + yw = d−p3b1 +ε results  f  d+ε $ d− f

%% d − p1 + ε o lim I.) ≥ lim ρ φ (d, p1 )dddp1 −b −y 2 3b ε →0+ ε →0+ p1 d %  f  d+ε $ d − 3 f + 2p1 + byo − ε φ (d, p1 )dddp1 > 0 = lim ρ 6 ε →0+ p1 d $

due to (4.2) and (4.16) and % d − p1 + ε φ (d, p1 )dddp1 2 3b ε →0+ f d %  p1  d+ε $ d − p1 − ε φ (d, p1 )dddp1 > 0 = lim ρ 6 ε →0+ f d

lim II.) ≥ lim ρ

ε →0+

 p1  d+ε d − p1

$

−b

due to (4.16) which are in sum a lower bound for (C.4) which thus is positive. ... linearly decreasing for yw ≥

d+p1 3b :

w 1 yw ≥ d+p 3b yields 3by − p1 ≥ d and, combined with φ (d, p1 ) = 0 for d ≤ d,

# ∂ ΠC1 (yo , yw ) ## # w d+p1 = −(p0 + h) < 0 and constant ∂ yw y ≥ 3b

results. The optimal solution is as follows. Decreasing the number of option contracts by ε and increasing the number of items in inventory by ε doesn’t decrease the profit (ε > 0). This procedure is permissible due to the 1 : 1 substitution of yo and yw in (C.1) (yo and yw appear in the form yo + yw ) which every candidate solution must satisfy and verified if the directional derivative satisfies ∇ := ∂ ΠC1 (yo ,yw ) ∂ yo

∂ ΠC1 (yo ,yw ) ∂ yw

−

≥ 0. From (C.1), (C.2), and substitution of c0 and p0 from (2.1) and

C.2 Analysis of the First Period

135

(2.2) results

∂ ΠC1 (yo , yw ) ∂ ΠC1 (yo , yw ) − ≥0 ∂ yw ∂ yo   w $ % f 3by +p1 d − p 1 − byw φ (d, p1 )dd ⇔ρ 2 3byw −p1 p1   w   w +

+

3by −p1

d

 3byw −p1 +2 f $ d + p1 − 2 f

2

3byw −p1

3by −p1

p1

−p1 φ (d, p1 )dd dp1 +

f

d

% − by

w



(C.5)

− f φ (d, p1 )dd

(C.6)



φ (d, p1 )dd dp1 ≥ 0.

∇ is independent of yo . Condition (C.6) holds if 3byw − p1 ≤ d

and

d − p1 − byw ≥ 0 for all (d, p1 ). 2

(C.7)

(C.2) with yo = 0 and substitution of c0 and p0 with (2.1) and (2.2) yields  # %  p1  3byw +p1 $ ∂ ΠC1 (yo , yw ) ## d − p1 w − by φ (d, p1 )dddp1 # o =ρ p ∂ yw 2 3byw −p1 1 y =0    w +

3by −p1

p1

p1

d

−p1 φ (d, p1 )dddp1 .

This expression can only be zero (representing a first-order condition) in the candidate region (C.7) and thus ∇ ≥ 0 holds in this candidate region either.

136

C Cournot Duopoly

C.3 Salvage Value Lagrange-function: L(y j , yw , x1 ) = (d − b(x1 + x2 )) x1 − p1 y j + zp1 (y j + yw − x1 ) + λ1 (y j + yw − x1 ) + λ3 (yw − yw ). First-order Karush-Kuhn-Tucker conditions: ∂L ∂L ≤ 0 ⇔ d − 2bx1 − bx2 − zp1 − λ1 ≤ 0, x1 = 0 ⇔ (d − 2bx1 − bx2 − zp1 − α1 )x1 = 0 ∂ x1 ∂ x1

∂L ≤ 0 ⇔ −p1 + zp1 + α1 ≤ 0, ∂yj

∂L j y = 0 ⇔ (−p1 + zp1 + α1 )y j = 0 ∂yj

∂L ≤ 0 ⇔ zp1 + α1 − α3 ≤ 0, ∂ yw

∂L w y = 0 ⇔ (zp1 + α1 − α3 )yw = 0 ∂ yw

∂L ≥ 0 ⇔ y j + yw − x1 ≥ 0, ∂ α1 ∂L ≥ 0 ⇔ yw − yw ≥ 0, ∂ α3

∂L α1 = 0 ⇔ (y j + yw − x1 )α1 = 0 ∂ α1 ∂L α3 = 0 ⇔ (yw − yw )α3 = 0 ∂ α3

α1 ≥ 0, α3 ≥ 0, x1 ≥ 0, y j ≥ 0, yw ≥ 0. The resulting simultaneous conditions are d + p1 − 2α1 − 2zp1 , 3b

x1∗ =

x2∗ =

d − 2p1 + α1 + zp1 3b

which yield optimal decisions in the second period as summarized in Table C.2 (the numbering of regions corresponds to Figure 4.21). ∗

yw∗

x1∗

x2∗

r∗

− yw

yw

d−p1 3b

d−p1 3b

d+2p1 3

2.)

0

yw

yw

d−p1 −byw 2b

3.)

0

d+p1 −2zp1 3b

d+p1 −2zp1 3b

d−2p1 +zp1 3b

d+p1 −byw 2 d+p1 (1+z) 3

yj 1.)

d−p1 3b

Table C.2: Optimal Second-period Decisions in the Cournot Duopoly with Salvage Value

D Bertrand Duopoly D.1 Analysis of the Second Period This section derives the optimal decisions in the Bertrand duopoly model at t = 1, the second stage, as solution of (4.31). The Lagrange function includes the objective function π1B = r1 (y j + yo + yw ) − p1 y j − f yo , constraints y j + yo + yw ≤ d − br1 , yo ≤ yo , yw ≤ yw , and r1 ≤ p1 such that L(r1 , y j , yo , yw ) =r1 (y j + yo + yw ) − p1 y j − f yo + α1 (d − br1 − y j − yo − yw ) + α2 (yo − yo ) + α3 (yw − yw ) + α4 (p1 − r1 ). First-order Karush-Kuhn-Tucker conditions: ∂L ∂L ≤ 0 ⇔ y j + yo + yw − bα1 − α4 ≤ 0, r1 = 0 ⇔ (y j + yo + yw − bα1 − α4 )r1 = 0 ∂ r1 ∂ r1

(D.1)

∂L ∂L j ≤ 0 ⇔ r1 − p1 − α1 ≤ 0, y = 0 ⇔ (r1 − p1 − α1 )y j = 0 ∂yj ∂yj

(D.2)

∂L ∂L ≤ 0 ⇔ r1 − f − α1 − α2 ≤ 0, o yo = 0 ⇔ (r1 − f − α1 − α2 )yo = 0 ∂ yo ∂y

(D.3)

∂L ∂L ≤ 0 ⇔ r1 − α1 − α3 ≤ 0, w yw = 0 ⇔ (r1 − α1 − α3 )yw = 0 ∂ yw ∂y

(D.4)

∂L ∂L ≥ 0 ⇔ d − br1 − y j − yo − yw ≥ 0, α1 = 0 ⇔ (d − br1 − y j − yo − yw )α1 = 0 ∂ α1 ∂ α1

(D.5)

∂L ≥ 0 ⇔ yo − yo ≥ 0, ∂ α2 ∂L ≥ 0 ⇔ yw − yw ≥ 0, ∂ α3 ∂L ≥ 0 ⇔ p1 − r1 ≥ 0, ∂ α4

∂L α2 = 0 ⇔ (yo − yo )α2 = 0 ∂ α2 ∂L α3 = 0 ⇔ (yw − yw )α3 = 0 ∂ α3 ∂L α3 = 0 ⇔ (p1 − r1 )α4 = 0 ∂ α4

α1 ≥ 0, α2 ≥ 0, α3 ≥ 0, α4 ≥ 0, r1 ≥ 0, y j ≥ 0, yo ≥ 0, yw ≥ 0.

(D.6) (D.7) (D.8) (D.9)

138

D Bertrand Duopoly

The resulting optimal second-period decisions for the Bertrand duopoly case are summarized in Table D.1. r1∗

yw∗

yo∗

1.)

p1

yw

0

2.)

d 2b

d 2

0

d−yw b

yw

0

p1

d − bp1

0

5.)

p1

yw

yo

6.)

d−b f 2b

yw

7.)

d−(yo +yw ) b

yw

yo

8.)

p1

yw

d − bp1 − yw

3.) 4.)

d−b f 2

− yw

Table D.1: Optimal Second-period Decisions in the Bertrand Duopoly

For each of these eight strategies, specific conditions for the first-period decision variables yo and yw and the parameters d and p1 have to be satisfied as provided in Table D.2. 1.)

yw < d − bp1 ∩ p1 < f

2.)

yw ≥ d − bp1 ∩ yw <

3.) 4.) 5.) 6.)

yw ≥

d 2

∩ p1 ≥

d 2

d 2b

yw ≥ d − bp1 ∩ p1 <

d 2b

yo + yw < d − bp1 ∩ p1 ≥ f yo + yw ≥ d − bp1 ∩ yw < d − bp1 ∩ yo + yw <

− b2f ∩ p1 ≥ f

− b2f ∩ p1 ≥

7.)

yw < d − bp1 ∩ yo + yw ≥

8.)

yo + yw ≥ d − bp1 ∩ yw < d − bp1 ∩

d 2

d 2

f d 2b + 2 ∩ p1 ≥ f d p1 < 2b + 2f ∩ p1 ≥ f

Table D.2: Conditions for the Optimal Second-period Decisions in the Bertrand Duopoly

D.2 Analysis of the First Period

139

D.2 Analysis of the First Period To prove property 4.5, (4.32) is shown to be decreasing in yo and yw . The first partial derivatives of ΠB1 (yo , yw ) are ∂ ΠB1 (yo , yw ) = − c0 + ρ ∂ yo



p1  d yo  +yw +bp1

f

(p1 − f )φ (d, p1 )dddp1

% d − 2yo − 2yw − f φ (d, p1 )dddp1 yo +yw b +f 2(yo  +yw )+b f b %   p1  yo  +yw +bp1 $ d − 2yo − 2yw − f φ (d, p1 )dddp1 + 2yo +yw b + f yw  +bp1 b (D.10) +



2yo  +yw b

+ f  yo  +yw +bp1

and ∂ ΠB1 (yo , yw ) = − (p0 + h) + ρ ∂ yw + + + + + + + +



f d p1

 p1  d f

yo  +yw +bp1

yw  b

2yw 

yw  +bp1

2yo +yw b yo  +yw b

 p1 



yw +bp1

yw  b +f yo  +yw  b yw  b



2(yo  +yw ) +b f

2bp1 −b f + f  2bp1 yw  +bp1

2yo +yw b yo  +yw b

b

yw +bp1

(D.11)

f φ (d, p1 )dddp1

f φ (d, p1 )dddp1

+ f  2yo  +2yw +b f

+f

φ (d, p1 )dddp1

f φ (d, p1 )dddp1

+ f  yo  +yw +bp1

+f

d − 2yo − 2yw φ (d, p1 )dddp1 b

 yo  +yw +bp1 d − 2yo − 2yw



yo  +yw  b

φ (d, p1 )dddp1

+ f  yo  +yw +bp1

2yo +yw + f yw  +bp1 b yw yo  +yw +bp1 b +f

f



b

+f

p1 φ (d, p1 )dddp1

p1 φ (d, p1 )dddp1

 p1  yw  +bp1 d − 2yw 

$

 f φ (d, p1 )dddp1 .

The shape of ΠB1 (yo , yw ) is as follows. For yo = 0 and yw = 0, an expected profit of ΠB1 (0, 0) = 0 results.

140

D Bertrand Duopoly

In direction yo , the function is... ... constant for yo + yw ≤ d − bp1 . This implies yo + yw + bp1 ≤ d and, in connection with φ (d, p1 ) = 0 for d ≤ d, (D.10) simplifies to 



p1 d ∂ ΠB1 (yo , yw ) = − c + ρ (p1 − f )φ (d, p1 )dddp1 0 ∂ yo d f = − c0 + ρ Ec1 = 0 due to (2.2).

(D.12)

... decreasing for d − bp1 < yo + yw ≤ d − bp1 which yields to

∂ ΠB1 (yo , yw ) ≤ρ ∂ yo



2yo +yw +f b yo +yw +f b

 yo +yw +bp1  d − 2yo − 2yw

b

2(yo +yw )+b f

 −(p1 − f ) φ (d, p1 )dddp1

+

 p1 2yo +yw +f b

 yo +yw +bp1  d − 2yo − 2yw yw +bp1

  −(p1 − f ) φ (d, p1 )dddp1 .

(D.13)

b

The right-hand side is non-positive for d − 2yo − 2yw − (p1 − f ) ≤ 0 b

(D.14)

which is satisfied as d ≥ b(p1 + f ) due to (4.17). ... linear and decreasing for d − bp1 ≤ yo + yw which is equivalent to d ≤ yo + yw + bp1 and, combined with Φ(d, p1 ) = 0 for d ≥ d, (D.10) simplifies to ∂ ΠB1 (yo , yw ) = −c0 < 0 and constant. (D.15) ∂ yo

D.2 Analysis of the First Period

141

In direction yw , the function is... ... constant for yo + yw ≤ d − bp1 . This yields yo + yw + bp1 ≤ d and, as φ (d, p1 ) = 0 for d ≤ d, (D.11) simplifies to

∂ ΠB1 (yo , yw ) = − (p0 + h) + ρ ∂ yw

 p1  d p1

d

p1 φ (d, p1 )dddp1

= − (p0 + h) + ρ Ep1 = 0

(D.16)

due to (2.1).

... decreasing for d − bp1 < yo + yw ≤ d − bp1 . In this case, 



 d − 2yw − (p − f ) φ (d, p1 )dddp1 1 yw b yo +yw +bp1 b  2yo +yw + f  yo +yw +bp1  b d − 2yo − 2yw yo +yw b +f 2(yo +yw )+b f b  −p1 φ (d, p1 )dddp1

∂ ΠB1 (yo , yw ) ≤ρ ∂ yw

+

p1  d

 yo +yw +bp1  d − 2yo − 2yw

 p1 2yo +yw +f b

b

yw +bp1

 −p1 φ (d, p1 )dddp1 +

 p1 f

 ( f − p1 )φ (d, p1 )dddp1 (D.17)

holds. The right-hand side is non-positive for • •

d−2yo −2yw − p1 ≤ 0 which holds due to the above argumentation. b w d−2y − (p1 + f ) ≤ 0 which holds as d ≥ b(p1 + f ) due to (4.17). b

• f − p1 ≤ 0 for p1 ≥ f which is satisfied. ... linear and decreasing for d − bp1 ≤ yo + yw . This condition yields d ≤ yo + yw + bp1 and, due to φ (d, p1 ) = 0 for d ≥ d, (D.11) simplifies to

∂ ΠB1 (yo , yw ) = −(p0 + h) < 0 and constant. ∂ yw

(D.18)

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