Procurement Management using Option Contracts ...

Procurement Management using Option Contracts: Random Spot Price and The Portfolio Effect Qi Fu

∗

Chung-Yee Lee

†

Chung-Piaw Teo

‡

Abstract This paper considers the value of portfolio procurement in a supply chain, where a buyer can either procure parts for future demand from sellers using fixed price contracts, option contracts, or tap into the market for spot purchases. We examine a single-period portfolio procurement problem when both the product demand and the spot price are random (and possibly correlated), and construct the optimal portfolio procurement strategy for the buyer. For the general problem, we provide a shortest-monotone path algorithm to obtain the optimal procurement solution and the resulting expected minimum procurement cost. In the event that demand and spot price are independent, the solution algorithm simplifies considerably. More interestingly, the optimal procurement cost function in this case has an intuitive geometrical interpretation which facilitates us to further exploit more managerial insights. We also study the portfolio effect, i.e., the benefit of portfolio contract procurement over a single contract procurement. Finally, we discuss the extension to a two-period problem to examine the impact of inventory on the portfolio procurement strategy. Supplementary materials are available for this article. Go to the publisher’s online edition of IIE Transactions for detailed proofs. Subject Classifications: Stochastic Programming; Procurement Management; Optimization

1

Introduction

Procurement has been identified as a major driver in a company’s bottom line and its ability to compete in the global marketplace. On one hand, despite improvement in forecasting technique, demand for product can be extremely hard to predict in an increasingly competitive environment. On the other hand, there are both volume and price uncertainties at the supply end. One of the greatest challenges confronting many companies these days is the daunting task of coordinating supply of the key inputs to match the demand for the product. For commodities like memory chips, price can double or even triple within a month or two, even though in the long run, a downward trend is fairly predictable. Without binding ∗

Department of Industrial Engineering and Logistics Management, HKUST. Email: [email protected] Department of Industrial Engineering and Logistics Management, HKUST. Email: [email protected] ‡ Department of Decision Sciences, NUS Business School. Email: [email protected] †

1

commitment from the suppliers for quantities or prices, shortfall of supply may occur over time. Most high-tech companies have experienced exposure to such kinds of risks: In September 2000, Sony’s stock price dropped by about 9% when the company announced its inability to meet initial demand for the new PlayStation console, due to parts shortage in capacitors, LCDs, and flash memory chips etc. Similarly, in October 1999, Dell Computer suffered a $470 million earnings shortfall from the effect of DRAM prices on gross margins, which triggered a 13% drop in its stock price (Billington et al. 2002). Many companies are looking for ways to lock in supplies early on to avoid over-reliance on the volatile spot market. Option contract, which can assure the buyer a fixed amount of supply in future at pre-negotiated price, is now being explored by many procurement practitioners in the high-tech industries. Such contracts can postpone procurement until uncertainties are resolved. For example, Hewlett-Packard (HP) has initiated a proactive risk management approach to procurement (PRM), the key to which is segmenting demand based on the risk level, and engaging various supply sources (with a range of price flexibility) to suit each demand segment [18]. For instance, for memory chips, HP designed a customized option contract that pays the suppliers a premium for the option to buy a fixed quantity of memory devices at a capped price. The premium was set at certain percent of the strike price. If spot price increases above forecasted levels, the option is exercised at the fixed strike price for the agreed-upon quantity. Conversely, if spot price falls below the strike price, the procurement teams can let the option lapse and buy in the open market. “There is some cost, but very clearly the benefits far outweigh the cost incurred,” says Venu Nagali, leader of HP’s PRM group (Bartholomew 2005). It has been widely reported that the company has reaped some 425 million dollars in cumulative cost savings from the PRM process (Nagali et al. 2008). The work most relevant to our study is the paper by Mart´ınez-de-Alb´eniz and Simchi-Levi (2005). Motivated by HP PRM process, they examine a multi-period portfolio approach to procurement optimization with option contracts. They assume that both the type and the amount of the portfolio of contracts are fixed in each period and focus on studying the optimal replenishment strategy. In another related work by the same authors (Mart´ınez-de-Alb´eniz and Simchi-Levi (2003)), they study a single-period portfolio procurement problem with the intent to understand the competitive behavior among the suppliers. However, contract selection is studied with the spot market price being constant, and in this way, the spot market acts essentially as a supplier with zero reservation price but a large execution price. The value of option contract is also explored in another related work by Ritchken and Tapiero (1986). Using mean variance approximations to the expected utility function, they addressed the value of option contract in inventory control as a function of the correlation between demand and spot price. They conclude that as the correlation increases, the role of options becomes increasingly more important. However, they restrict their study to the usage of only one option contract, and do not address the associated complexity when the buyer can choose from a portfolio of option contracts. 2

Our work in this paper can be viewed as an integration of this two streams of research: We examine the value of option contracts in the portfolio procurement environment from several different angles: (1) Optimal Contracts - We propose a shortest-monotone path approach to obtain the optimal procurement solution when demand and spot price are random and possibly correlated. For the case of independent demand and spot price, our algorithm reduced to a simple graphical approach. The optimal system cost function under independent case has a nice geometric interpretation. This allows us to obtain more managerial insights and unveil the logic of the optimal portfolio procurement strategy. (2) Value of Option Contracts - We demonstrate that the value of option contracts to the buyer increases with increased volatility of the spot price, as well as increased correlation between spot price and demand (i.e., more option contracts are procured). More surprisingly, the expected procurement cost with option contract actually decreases with the increase in volatility of the spot price. (3) Portfolio Effect - We investigate the extent of the benefits that can be accrued by managing a portfolio of option contracts, instead of sticking to one fixed price contract. In our extensive experiments using more realistic data, by restricting the managers to the usage of up to two option contracts (and the spot market), the procurement cost attained is already very close to the performance under the optimal portfolio of contracts. This shows that there is little need for procurement manager to maintain an extensive list of option contracts with suppliers to hedge against the demand and spot price fluctuation.

2

Literature Review

There is by now a rich body of literature in the supply chain research area, addressing various aspects of the procurement problem. One stream of research focuses on the design of mechanisms to achieve channel coordination or credible information sharing (cf. Eppen (1997) and Barnes et al. (2002)). These papers focus on deriving sufficient conditions on the cost parameters that can achieve the maximum profit of the centralized supply chain system in a decentralized setting. Cheng et al. (2003) develop an option model to quantify and price a flexible supply contract. Cachon and Lariviere (2001) study how to share demand forecasts credibly with a contract consisting of firm commitments and options. The spot market is not modelled in the procurement process. Procurement flexibility is provided only through the use of option contracts. With the addition of the spot market operations, Wu et al. (2002) study the coordination and option pricing issue, where a two-part contract fee structure is assumed: a cost to reserve capacity and a cost to actually supply the commodity. They investigate the bidding and contracting arrangements between one seller and one or more buyers under such cost structure. 3

They model the problem as a von Stackelberg game with the seller as the leader and derive the seller’s optimal bidding and the buyers’ optimal contracting strategies. Kleindorfer and Wu (2003) and Spinler et al. (2003) consider a similar problem as that of Wu et al (2002). Later Wu and Kleindorfer (2005) extend the model and consider multiple suppliers offering option contracts as well as a spot market with random price. However, these models assume a price (and quantity) dependent demand function, which is not randomly or exogenously generated, as assumed in this paper. That is, once its determinants are set, demand is fully determined. With both the demand and the spot price being random, Golovachkina and Bradley (2002) consider a problem with the objective to understand how access to the spot market affects the buyer-seller coordination, but their model consists of only a single supplier and a single manufacturer. Readers may refer to Haks¨oz and Seshadri (2007) for an excellent review and discussions on supply chain operations with spot market consideration. Another stream of research focuses on the procurement side of the supply chain. Ritchken and Tapiero (1986), study a procurement problem with option contract and immediate purchasing subject to price uncertainty which is actually a spot market. Cohen and Agrawal (1999) examine a contract selection problem between a long-term contract with fixed investment and price certainty and a flexible short-term contract with price uncertainty in a multi-period environment. Akella et al. (2001) and Seifert et al. (2004) address this type of scenario in their single-period models and show that a mixed procurement strategy comprising of both longterm contract and spot market procurement is optimal. Mattock (2002) examines a capacity planning problem under uncertain demand and spot price and proposes a graphical technique to solve the problem. Yet, all of the aforementioned papers consider only one supplier with option contract. Schummer and Vohra (2003) consider the mechanism-design problem for a single buyer to procure options from multiple suppliers. They show that the problem can be formulated as a linear program and propose a class of incentive-compatible, efficient auctions for procuring options. Their analysis emphasizes the role played by the distribution of demand. However, they do not consider the spot market. In a single period setting, with random demand and spot price, Haks¨oz and Kadam (2009) study the supply portfolio risk management problem in a rather different framework than the usual supply chain literature. The supply risk is due to contract breach. Using the CreditRisk + framework, they develop the entire loss distribution at the portfolio level, and obtain a quantile supply risk measure, supply-at-risk, for a portfolio of long-term fixed price supply contracts. Table 1 summarizes the related literature and shows the position of our research. We enrich the model by integrating the effect of random demand with random spot price in a portfolio contract procurement environment. The focus in this paper is on the optimal procurement decisions and the managerial insights in a single period setting.

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Table 1: Summary of Related Literature Option Contract

Random Demand

Deterministic Demand Function (one or more suppliers) Single Supplier No Spot Market

Portfolio of Suppliers

Barnes et al. (2002)

Schummer and Vohra (2003)

Cheng et al. (2002)

Cachon and Lariviere (2001) Mart´ınez-de-Alb´ eniz and Simchi-Levi

Fixed Spot Price

Random Spot Price

3 3.1

(2005) Mart´ınez-de-Alb´ eniz and Simchi-Levi (2003)

Wu et al. (2002) Kleindorfer and Wu (2003) Spinler et al. (2003) Wu and Kleindorfer (2005)

Ritchken and Tapiero (1996)

Mart´ınez-de-Alb´ eniz and Simchi-Levi

Cohen and Agrawal (1999) Akella et al. (2001) Mattock (2002) Golovachkina and Bradley (2002) Seifert et al. (2004)

(2005) (General Optimality Conditions) This paper (Correlation between demand and spot price)

The Portfolio Procurement Management Model Model and Assumptions

Consider the following procurement problem: The buyer faces an uncertain demand D. The supplier offers the buyer a selection of n option contracts. The unit reservation cost is ci and the unit execution cost is hi for contract type i. The traditional fixed price (firm commitment) contract corresponds to the special case of execution cost hi being zero. The buyer can also choose to tap into the spot market and purchase the product at the (random) spot price Ps . The demand needs to be filled completely. We assume that the spot market has no limit on capacity, so that the buyer can always fill whatever demand she needs from the spot market if she chooses to. The timing of decision consists of two stages. In stage one, facing uncertain demand D and random spot market price Ps , the buyer would like to purchase options (including firm commitment) from the n contracts to assure certain desired level of supplies in future, say Qi amount of options purchased from contract i. In stage two, the random demand is realized. Then based on the spot market price at that time, the buyer needs to decide how much, up to the quantity specified in the contract, to exercise from each of the contracts signed with the suppliers and how much to procure from the spot market to satisfy the realized demand. Let xi be the amount exercised from contract i and y denote the portion of demand satisfied by spot market purchase. The problem is to design an optimal portfolio of contracts, so that the expected procurement objective is optimized. Note that Qi is determined ahead of demand realization and obviously, the revealed spot market price has a significant influence on the

5

amount exercised from the option contracts. Therefore, the buyer should balance the price he pays for the options, with the uncertain price in an external spot market where there is unlimited supply. Figure 1 depicts graphically the major considerations we need to incorporate in the procurement decision.

Q1

Contract 1

 c1,h1

Contract 2

 c2,h2

Q2

Random D em and D

…

Qn

Contract n

 cn ,hn

SpotM arket

Random SpotPrice

Ps

Figure 1: Procurement Risk Management Model We ordered the contracts such that h1 < h2 < . . . < hn . Note that if the reservation and execution cost of a contract is dominated by another, then the contract will not be engaged in the contract market. Thus we have c1 > c2 > . . . > cn . The first contract with h1 = 0 is a fixed price contract and the remaining are option contracts. Assume that the suppliers can easily scale their capacity according to the amount of options purchased (i.e., no capacity constraints). The portfolio procurement problem can be formulated as follows: ( n  X ) n X min ci Q i + ED,Ps min hi xi + P s y Qi :Qi ≥0,i=1,...,n

xi ,y

i=1

subject to :

xi ≤ Qi ,

i=1

X

xi + y ≥ D, xi , y ≥ 0, ∀ i.

i

It is easy to check that the problem is convex in Qi , i = 1, · · · , n, since Qi ’s appear on the right-hand side of the constraints in a linear programming minimization problem . Remarks: 1. In the setup of our model, we have assumed that the option contract will not be executed if the execution price is larger than the spot price, and that the buyer will purchase the required materials at the spot rate from the spot market. In reality (Nagali 2008), the 6

execution price is implemented as a price cap, i.e., as the maximum price that the buyer would pay to exercise the contract. When the spot price is smaller than the execution price, the buyer will pay the supplier the spot price to exercise those option contracts with larger execution prices. Note that the procurement cost to the buyer remains unchanged in both settings. 2. In the event that the spot price is higher than the execution price, and demand is lower than the total amount of option contracts purchased, technically the buyer can leverage by exercising the contracts and re-sell the parts in the spot market. We do not believe that such speculative motive should drive the procurement decisions, and hence we do not incorporate the added values from such activities into our portfolio procurement objective function.

3.2

Optimality Property

In stage two of the model, D and Ps are realized and can be treated as deterministic. In such a case, we can rank the contract execution costs hi and the spot market price Ps in increasing order. The optimal second stage strategy for the buyer is merely to execute the option amount, or procure from the spot market, in a greedy fashion - starting from the lowest to highest cost among all hi and Ps . Since the spot market is assumed to be a large market with no supply limit, those options with execution cost greater than the realized spot market price will never be executed. The key is to determine the first stage decision variables Qi , i = 1, · · · , n. We next identify a set of optimality conditions on Qi . We denote by Q∗i , i = 1, · · · , n the amount of options purchased from contract i in the optimal solution. Definition 1 Contract i is called active, if Q∗i > 0. Contracts i and j are called consecutive active contracts, if Q∗i > 0, Q∗j > 0, and Q∗r = 0 for i < r < j. Contract i is called the last active contract, if Q∗i > 0, and Q∗r = 0 for r > i. Theorem 1 There exists an optimal solution, such that if contracts i and j are two consecutive active contracts, then we have the following optimality condition : Z ∞h Z ∞ i (ci − cj ) = min(hj , Ps ) − min(hi , Ps ) f (D, Ps ) dPs dD, (1)

P

i r=1

Q∗r

0

and if contract i is the last active contract, we have the optimality condition as follows : Z ∞h Z ∞ i ci = Ps − min(hi , Ps ) f (D, Ps ) dPs dD.

P

i r=1

Q∗r

0

7

(2)

We note that the above result is implicit in the work by Mart´ınez-de-Alb´eniz and SimchiLevi (2005), although they do not formally state the result. Their paper of 2003 contains a similar optimality condition, for the case when the spot price is deterministic. Proof : We use standard perturbation argument to show that the above properties hold for any optimal solution. For details, we refer the readers to the online supplementary materials.  P The cumulative amount of option contracts ordered ( ir=1 Q∗r in (1)) can be obtained using P a simple line search algorithm, since the right hand side is monotone in ir=1 Q∗r . Furthermore, the value depends on the cost parameter of the two consecutive active contracts or the last active contract. Hence provided we know how to identify the set of active contracts, we can P determine exactly the value of the terms ir=1 Q∗r , for i = 1, . . . , n (and hence Q∗i , . . . , Q∗n ), by applying these equations recursively. Note also that the cumulative option reservation amounts determined by the system of optimality equations must satisfy a monotonicity property, as Q∗r is non-negative.

4

Solution Procedure

We discuss in this section how to construct the optimal procurement solution to our model.

4.1

Demand and Spot Price are independent

For a buyer of standard products, the effect of its demand on the spot market price is usually negligible due to its limited size. In this case, we can treat the demand and the spot price mechanism as independent. We let f (D) and g(Ps ) denote the demand and spot price distribution functions, respectively. Definition 2 We call h0i the effective execution cost which is defined as h0i = E[min(hi , Ps )], for i = 1, ..., n,

and

h0n+1 = E[Ps ].

By Definition 2, we map the original execution cost into the effective execution cost defined above, which incorporates the effect of random spot price on the contract market. Note that if the spot price turns out to be cheaper than the execution cost hi , then the buyer will purchase at price Ps from the spot market, so h0i can also be interpreted as the expected execution cost. Corollary 1 There exists an optimal solution, such that if contracts i and j are two consecutive active contracts, then we have the following optimality condition : 

(ci − cj ) = P D >

i X r=1

8

Q∗r



 h0j − h0i ,

(3)

and if contract i is the last active contract, we have the optimality condition as follows : 

ci = P D >

i X

Q∗r



 h0n+1 − h0i .

(4)

r=1

Proof : The corollary can be proved by treating D and Ps as independent in Equation (1) and (2) and by Definition 2. 

Lemma 1 If we represent the set of contracts by points (h0i , ci ), i = 1, · · · , n and the spot market by point (h0n+1 , 0) on a two dimensional plane, then the set of active contracts can be viewed as points on the lower convex hull. For ease of exposition, the “lower convex hull” mentioned in this paper refers to the portion with downward slope between 0 and 1. Furthermore, we ignore the sign of “slope” (cf. Figure 2).

c i § · cicj P¨ D ! ¦Qr* ¸ ' ' r1 © ¹ hj hi

ci cj cl

l c § · P¨ D ! ¦Qr* ¸ ' l ' r1 © ¹ hn1  hl

h'

45q

hi' h'j

hl'

hn'1

A ctive C ontracts N on A ctive C ontracts

  P Figure 2: Active suppliers lie on the lower convex hull. P D > ir=1 Q∗r corresponds to the slope of the solid line. The active contracts can thus be determined geometrically by finding the lower convex hull of all the corresponding points, and the slope of the lines joining consecutive active contracts determines the cumulative optimal procurement solution. This optimality condition is similar to the Critical Ratio solution for the classic newsvendor problem, but for the portfolio procurement problem, we have a series of monotone Critical Ratios. For any two consecutive active contracts

9

i and j, i < j, we must have c i − cj < 1 ⇒ ci + h0i < cj + h0j . h0j − h0i This implies that flexibility is not free: The first active contract (fixed price contract) has the cheapest unit procurement cost. In the sequence of the contract index, the flexibility of the contract increases, while the expected unit procurement cost also increases. Starting from the first active contract, the next one can be identified by searching the points with steepest slope. Continuing the search, we can screen out all the active contracts and determine the procurement quantities using the slopes. The solution procedure is formally presented in Algorithm 1. Algorithm 1 Input:

(cj , hj ), j = 1, · · · , n, F (D) and G(Ps ).

Output: active contracts and their order quantities. Step 1: Compute h0i , i = 1, · · · , n + 1, using Definition 2. Let cn+1 = 0 Step 2: Find m = argmini≤n+1 (ci + h0i ). If m = n + 1, quit and output ”No active contract”. Otherwise, m is the first active contract. Set Q∗i = 0, i = 1, · · · , m − 1. Step 3: Let l = m. Find the next active contract s which is determined by the following equation s = argmaxl 0

r=1

i X

Q∗r

r=1

C(Q ) = E(D)E(Ps ) + +

N −1 X

P

Z

i r=1

Q∗r

E[min(hi , Ps )] − E[min(hi+1 , Ps )]

i=1



Q∗r ,

r=1

the optimal objective value can be simplified as ∗

i X

Z

P

0

i=2



0

N i=1

E[min(hN , Ps )] − E[Ps ] 0 ( Z Z Q∗  X N 1 = h01 h0i Df (D)dD +

Df (D)dD

P P

Q∗i

Df (D)dD i r=1

Q∗r

Df (D)dD

i−1 r=1

Q∗r

)

+ h0N +1

Z

∞

P

Df (D)dD. N i=1

Q∗i

 Figure 3 illustrates the geometric interpretation of Theorem 2 and its relationship with the optimality conditions. The figure on the left hand side illustrates the optimality conditions, using which we can identify the N active contracts together with the N critical ratios: F

X i r=1

Q∗r





=1−P D >

i X

Q∗r



=1−

r=1

11

ci − ci+1 , h0i+1 − h0i

for i = 1, · · · , N.

F D

1

c 1

ci  cj h'j  hi'

F

¦

i r1

Q r*



# Probability Area ofthis region

¦ r 1Q r*

Slope

³¦

ci cj

i

i1

Q* r1 r

D f D dD



F Q 1*

45q

hi'

h'j

hi'

hn'1

h'

"

# "

h1' Q

* 1

D

i

¦Q

* r

i1

A ctive contracts

Figure 3: Geometric View of the Moments Decomposition

The figure on the right hand side shows the cdf for the demand, F (D). For points F



Pi

∗ r=1 Qr

Pi



,

i = 1, . . . , N, on the vertical axis, the corresponding points on the horizontal axis are r=1 Q∗r , i = 1, . . . , N . This is the geometric way to obtain Q∗i , i = 1, . . . , N . More importantly, the area of the whole region above the curve F (D) is the mean demand E(D), which is partitioned into N + 1 subregions by the horizontal lines at the critical ratio points, where the areas of these subregions are just the conditional moments of demand shown in Equation (5). The cost h0i , i = 1, · · · , N + 1, shown in the center of each subregion is the weight associated with that region in the optimal cost function. Then, by the above characterization, the optimal objective value is simply the sum of each subregion’s area multiplied by its corresponding effective execution cost. Another feature for the independent problem is that the effect of the random demand can be decoupled from the effect of the random spot price. The optimal procurement strategy is to segment demand based on the corresponding risk levels and meet each segmentation by an appropriate contract type. For example, for the portion of demands that are almost certain, we use less flexible contracts with high commitments (e.g. fixed price or option contracts with large reservation costs), so that the buyer can benefit from the lower unit procurement cost. Those somewhat uncertain demands are met with more flexible option contracts, and in this way, risk can be shared by both parties. The most unlikely demand is managed via the spot market, which has the most flexibility but with the highest risk. This is the gist behind the portfolio approach: managing a diversified contract portfolio to hedge against risk. Though the buyer pays a higher price for a more flexible contract, the benefit of risk reduction may 12

far exceed the cost incurred. The managerial insights obtained from our analytical results also mirror HP’s Procurement Risk Management practices. Our analysis validates the logic behind the PRM approach and provides a rigorous scheme to screen out the active contracts and to segment the demand.

4.2

Demand and Spot Price are correlated

In some high-tech industries, the spot price is significantly influenced by a few top buyers, so correlation between demand and spot price must be factored in when making the procurement decision. We next exploit how to identify the set of active contracts. Definition 3 We call a subset of contracts S = {i1 < i2 < . . . < i|S| } feasible if there are Q∗ik ≥ 0 satisfying the optimality conditions: Z ∞h Z ∞ i (cik − cik+1 ) = min(hik+1 , Ps ) − min(hik , Ps ) f (D, Ps ) dPs dD,

P

k r=1

Q∗ir

0

for each k ∈ {1, . . . , |S| − 1}, and Z Z ∞ ci|S| =

P

|S| r=1

Q∗ir

∞h

i Ps − min(hi|S| , Ps ) f (D, Ps ) dPs dD.

0

Note that the set of active contracts in the optimal solution corresponds to a feasible set, but not all feasible sets are optimum. Our challenge is to find a feasible set with the smallest procurement cost. For ease of exposition, suppose there are N = |S| contracts in a feasible set S re-indexed in increasing order of execution cost hi . Theorem 3 Let {Q∗i } denote the procurement quantity corresponding to feasible set S. The procurement objective function in this case is Z h1 Z ∞ CS (Q∗ ) = Ps D f (D, Ps ) dD dPs 0 0 N −1 X

+

hi

Z

∞Z

+

Z

hi+1

P hi

+hN

Z

∞Z

hN

i r=1

Q∗r

Df (D, Ps ) dD dPs − hi+1

0

hi

i=1

P

N r=1

Z

∞

P

i r=1

Ps D f (D, Ps ) dD dPs

Z

!

∞ hi+1

P

Z

i r=1

Q∗r

Df (D, Ps ) dD dPs 0

Q∗r

Q∗r

Df (D, Ps ) dD dPs +

0

Z

∞Z ∞ hN

P

N r=1

Ps D f (D, Ps ) dD dPs .

(6)

Q∗r

The procurement cost function using feasible set S (with N = |S| active contracts) decomposes into N + 1 groups. The first term is independent of Q∗i , i = 1, · · · , N , depending only on the execution cost of the first active contract. Each term in the next N groups is related only 13

P to ir=1 Q∗r for i = 1, · · · , N . Note that for consecutive active contracts i and i + 1, we can P identify ir=1 Q∗r by invoking the optimality condition. Consider a directed graph G with arcs (i, j) joining two nodes i and j, for all 1 ≤ i < j. For each arc (i, j), it is associated with a pair of parameters (ki,j , di,j ), where ki,j is a scalar P and di,j is the distance of the arc. Intuitively, we would like ki,j to keep track of ir=1 Q∗r , the aggregate quantity purchased up until contract i, if i, j are two consecutive active contracts, i.e., setting ki,j to be the solution to Z ∞Z ∞h i (ci − cj ) = min(hj , Ps ) − min(hi , Ps ) f (D, Ps ) dPs dD, (7) ki,j

0

and di,j keeps track of a component in the procurement cost function, i.e., di,j = hi

Z

∞ Z ki,j hi

Df (D, Ps ) dD dPs − hj

0

Z

∞ Z ki,j

hj

Df (D, Ps ) dD dPs +

0

Z

hj hi

Z

∞

Ps D f (D, Ps ) dD dPs , (8)

ki,j

if i, j are consecutive active contracts. We add an origin and a destination nodes O and D to G respectively, with an arc from O to each node in G, and an arc from each node in G to D. For arc (O, j), we set kO,j = 0, and Rh R∞ dO,j = 0 j 0 Ps D f (D, Ps ) dD dPs . For arc (j, D), set kj,D to be the solution to Z ∞ Z ∞h i cj = Ps − min(hj , Ps ) f (D, Ps ) dPs dD, kj,D

and dj,D = hj

Z

∞ Z kj,D

hj

0

Df (D, Ps ) dD dPs +

0

Z

∞Z ∞ hj

Ps D f (D, Ps ) dD dPs .

kj,D

A path is called a monotone path, if it starts from node O and ends in node D, with ki,j monotonously increasing along the path. Furthermore, the shortest-monotone path is a monotone path having the shortest path distance with respect to distance measure {di,j }. Theorem 4 The general portfolio procurement problem can be solved by finding the shortestmonotone path from the origin O to the destination D using nodes in G. The nodes in G lying on the shortest-monotone path are the active contracts and the corresponding distance equals the optimal objective value. The shortest-monotone path in the graph constructed in Theorem 4 satisfies the optimality conditions with the lowest cost, and therefore it is optimal. A dynamic programming type algorithm to solve the shortest-monotone path problem is provided in Algorithm 2. The algorithm first computes (ki,j , di,j ) for each arc. ki,j is computed using bi-section search, so the complexity is T log2 Dmax , assuming that T is a bound on the time required for computing the integral on the right hand side in (7) with fixed ki,j , and Dmax is the maximum value of demand. Once ki,j is found, di,j can be computed easily in time O(T ). Thus the overall complexity of computing 14

(ki,j , di,j ) is O(n2 T log2 Dmax ). Then the algorithm finds the shortest-monotone path based on (ki,j , di,j ). The main idea is that we need to keep track of the shortest-monotone distance to each node with all possible predecessors, so that we can compare the ki,j ’s along the path to enforce the monotonicity property. For example, when we find the shortest monotone distance to node j with predecessor i, denoted by Di,j , we need to compare all the shortest monotone distances to node i, i.e., Ds,i for all possible predecessor s of i, and set Di,j = min{s:ks,i cx X) if and only if E[v(Y )] > E[v(X)] for all convex functions v(·). If Y >cx X and E(Y ) = E(X), it follows that V ar(Y ) > V ar(X). The convex order therefore orders random variables with the same mean by variability. More precisely, if F˜ and F denote the distribution function for P˜s and Ps respectively, then Z h Z h ˜ F (t)dt ≥ F (t)dt for all h ≥ 0; 0

0

Z

∞

F˜ (t)dt =

0

Z

∞

F (t)dt.

0

Intuitively, P˜s is a mean preserving spread of Ps if P˜s = Ps + Z, where Z denotes a noise term (with zero mean) added to Ps . Theorem 5 Suppose demand is independent of the spot price. If P˜s >cx Ps and E(P˜s ) = E(Ps ), then the optimal system cost decreases under P˜s , whereas the total quantity contracted P (i.e., i Q∗i ) increases. This strange phenomenon exists because of the presence of the contract markets. The proper use of the option contracts allows the buyer to cushion the impact of high spot price scenarios, whereas the flexibility of the contract market allows the buyer to exploit the situation when the spot price is low. Proof : Note that h0i = E[min(hi , Ps )] = hi −

Z

hi

F (P )dP ≥ hi − 0

Z

hi 0

16

˜0 . F˜ (P )dP = E[min(hi , P˜s )] = h i

Let Q∗ denote the optimal ordering quantities when the contract parameters are (ci , hi ), and spot price distribution is modelled by Ps . The procurement cost can be expressed as ∗

C(Q ) = (c1 −

c2 )Q∗1

+ (c2 −



c3 )(Q∗1 h

+



Q∗2 )

+E[Ps ] E[D] − E min D,

+ · · · + (cN −1 − cN )

N X

Q∗i

i

N −1 X

Q∗i

+ cN

i=1

N X

Q∗i

i=1

i=1

+

N X

 h i i−1 i h  X i  X ∗ ∗ E[ min(hi , Ps )] E min D, Qr − E min D, Qr

i=2

h

+E[ min(h1 , Ps )]E min



D, Q∗1

r=1

i

.

r=1

(9)

˜ 0 , the same order quantities Q∗ will incur a lower Since the effective execution cost h0i ≥ h i expected procurement cost in the environment when the spot price is changed to P˜s . Thus the optimal procurement cost for the latter environment will be lower. Note that the last active contract N is determined by   ci N = argmini E[Ps ] − h0i when the spot price is modelled by Ps . Similarly,   ci ˜ = argmini N ˜0 E[P˜s ] − h i is the last active contract when the spot price is modelled by P˜s . The total quantity contracted increases under P˜s , since cN˜ cN cN ≥ ≥ . 0 0 ˜ ˜0 ˜ ˜ E[Ps ] − hN E[Ps ] − hN E[Ps ] − h ˜ N  The above results show that the portfolio procurement approach using option contracts is more valuable to the buyer when the spot price becomes more volatile, for the independent problem. However, when the spot price remains unchanged, but the demand becomes more volatile, the portfolio procurement approach is unfortunately less valuable. ˜ >cx D and E(D) ˜ = E(D), Theorem 6 Suppose demand is independent of the spot price. If D ˜ and the total quantity contracted (i.e., P Q∗ ) then the optimal system cost increases under D, i i will increase or decrease. ˜ denote the cdf for demand D and D ˜ respectively. Note that for all Q, Proof : Let G and G Z Q Z Q ˜ ˜ E[min(Q, D)] = Q − G(x)dx = E[min(Q, D)]. G(x)dx ≥ Q − 0

0

17

˜ ∗ denote the optimal ordering quantities when the contract parameters are (ci , hi ), and Let Q ˜ The procurement cost can be expressed as the demand is modelled by D. ˜ ∗ ) = (c1 − c2 )Q ˜ ∗1 + (c2 − c3 )(Q ˜ ∗1 + Q ˜ ∗2 ) + · · · + (cN −1 − cN ) C(Q

N −1 X i=1

˜ +E(D)E[P s] +

˜ ∗i + cN Q

N X

˜ ∗i Q

i=1

N −1  X

i i  h  X ˜ ˜∗ E[min(hi , Ps )] − E[min(hi+1 , Ps )] E min D, Q r r=1

i=1



N i  h  X ˜ ∗i . ˜ + E[min(hN , Ps )] − E[Ps ] E min D, Q i=1

˜ ∗ will incur a lower exSince E[min(hi , Ps )] < E[min(hi+1 , Ps )], the same order quantities Q pected procurement cost in the environment when the demand is changed to D. Note that the last active contract N is determined by   ci N = argmini , E[Ps ] − h0i and remains unchanged in both environments. However, the total number of contracts ordered cN ˜ −1 (1 − under both environments depends on G−1 (1 − E[PcsN]−h0 ) and G E[Ps ]−h0N ) respectively. N The value E[PcsN]−h0 thus determines whether the total quantity contracted has increased or N decreased in a more volatile environment. 

1 hN' hN'

F D! 1

F D

!

A

h1' D, D

Figure 4: Geometric View of Theorem 6 Theorem 6 can also be explained graphically as shown in Figure 4. For the independent problem with the same spot price distribution, the series of critical ratios are the same for whatever demand distributions, so the horizontal partition of the subregions above the demand cdf curves and the weighs associated with the subregions are the same (c.f. Figure 3). Recall 18

that the optimal cost function is a weighted sum of the subregions’ area, and the weight increases ˜ the cdf curve is more stretched out bottom-up. In a more volatile demand environment D, than the cdf curve for D, but the area above both curves is the same, due to the same mean. Therefore, the two curves must crossover each other at some point, say point A. Then curve ˜ is above curve F (D) to the left of point A and is below curve D to the right of point A. F (D) ˜ some area shifts from the lower Then we can observe that in the volatile demand environment D, weight region to the higher weight region compared with D, and therefore the optimal cost is higher. The total quantity contracted depends on the highest critical ratio, which corresponds to the horizontal line that has an intercept closest to 1. If this intercept is below point A, then ˜ is the total quantity under D is larger; if it is above point A, then the total quantity under D larger.

5.2

Impact of Correlation between Demand and Spot Price

In what follows we conduct a numerical study to investigate the effect of correlation on the optimal solution and the objective value. The correlated demand and spot price are modelled by a truncated Bivariate Normal distribution with correlation ρ, demand D ∼ N (100, 50) and spot price Ps ∼ N (20, 7). There are 3 different contracts with c1 = 10, h1 = 0, c2 = 5.3272, h2 = 6.7810 and c3 = 1.1580, h3 = 16.7230. The first contract is a fixed price contract, while the other two are option contracts. The optimal solution and expected cost function under a range of values of correlation between demand and spot price are shown in Table 2. Correlation ρ 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

Fixed contract Q1 77.7344 77.3438 77.3438 76.9531 76.5625 76.5625 76.1719 76.1719 76.1719

Q2 33.2031 32.2266 30.4688 28.5156 26.5625 24.2188 22.2656 20.3125 16.9922

Optimal solution Option contract Q3 Total options 58.2031 91.4062 52.5391 84.7657 46.0938 76.5626 39.8438 68.3594 32.4219 58.9844 24.2188 48.4376 15.6250 37.8906 5.8594 26.1719 0 16.9922

Optimal expected cost 1432.1 1424.9 1410.8 1393.8 1373.8 1348.5 1320.4 1285.8 1248.7

Percent increase in optimal expected cost without option contract 7.00 5.56 4.47 3.45 2.56 1.86 1.11 0.75 0.38

Table 2: Impact of Correlation

As can be observed from the table, the amount purchased using option contracts increases with the correlation ρ. This phenomenon is not hard to imagine. A larger correlation implies that when demand is high, it is more likely that the spot price will also be high, so more options are required to avoid buying at a higher price from the spot market. Moreover, the optimal expected system cost also increases with the degree of correlation, because more premium needs

19

to be set aside to purchase the options. This suggests that option contracts are more valuable when demand and spot price are positively correlated. The last column in the table shows the percent cost increase in the optimal cost function if only the fixed price contract and the spot market are used. Comparing the last two columns, we can also observe that the performance gap between with and without options increases with the correlation. These phenomena are similar to those of Ritchken and Tapiero (1986), where they consider a single option contract together with a fixed price contract using mean-variance objective. In a risk-averse setting, their numerical study shows that the benefit of utilizing option contracts increases in the price and demand correlation. Our result demonstrates that their observation can be carried over to the risk-neutral setting. In addition, as we consider multiple option contracts, the magnitude of the correlation effect increases with the contract index: contract 3 exhibits the most change, ranging from 58.2031 to 0, while contract 1 is affected only slightly. This demonstrates that the more flexible the option contract, the more significant the impact of correlation, because highly flexible contracts are only valuable for hedging the high demand and high spot price scenarios.

6

Portfolio Effect

We study next the value of managing a portfolio of contracts, versus a single contract, in this procurement environment. We first consider the case when demand and spot price are ∗ and C ∗ denote the optimal expected system cost of using the portfolio independent. Let Cm s approach and a single contract respectively, along with the opportunities for spot purchases. We are interested in determining the Portfolio Effect defined as follows: PE =

∗ Cs∗ − Cm , ∗ Cm

which is the relative error of an optimal single contract procurement cost compared to the optimal portfolio procurement cost. Let R s −1 F (1 − t)dt G(s) := 0 . s Note that G(s) is decreasing in s. Furthermore, it can be shown that G(s) is convex whenever the CDF function F (·) is concave. It turns out that the performance of the portfolio procurement problem is intimately related to this parameter. Lemma 2 Let s1 ≥ s2 ≥ . . . ≥ sm denote the slopes (ignoring sign) determined by the active contracts with parameters (c1 , h1 ), . . . , (cm , hm ). For ease of exposition, we set cm+1 = 0 and i+1 h0m+1 = E[Ps ]. Then it follows that si = hci0 −c−h 0 , i = 1, 2, · · · , m. i+1

Cm ≥

h01 E[D]

i

+ c1

R s1

20

0

F −1 (1 − t)dt . s1

If G(s) is convex, Cm ≥ h01 E[D] + c1

R

P

m ci −ci+1 si i=1 c1

0

Pm

i=1

F −1 (1 − t)dt

ci −ci+1 si c1

.

Proof : We use the characterization of the optimal system procurement cost, and re-write Z 1 Z s1 Z sm Cm = h01 F −1 (1 − t)dt + (h02 − h01 ) F −1 (1 − t)dt + . . . + (h0m+1 − h0m ) F −1 (1 − t)dt 0 R s01 −1 R sm −1 0 Z 1 F (1 − t)dt F (1 − t)dt = h01 F −1 (1 − t)dt + (c1 − c2 ) 0 + . . . + (cm ) 0 . s1 sm 0 The bound follows from the fact that G(s) is non-increasing in s.  The above lemma points out the special role of the first active contract in the determination of the savings from portfolio procurement. Theorem 7 Let C1 denote the procurement cost using only the first contract and the spot market. Then h01 E[D] + c1 G( E[Pcs1]−h0 ) G( E[Pcs1]−h0 ) C1 1 1 ≤ ≤ . Cm h01 E[D] + c1 G(s1 ) G(s1 ) In particular, if G(s) is convex, h01 E[D] + c1 G( E[Pcs1]−h0 ) G( E[Pcs1]−h0 ) C1 1 1 ≤ 0 ≤ . P Pm ci −ci+1 ci −ci+1 Cm h1 E[D] + c1 G( m s ) G( si ) i i=1 i=1 c1 c1 Proof : This follows from the previous lemma and C1 = h01 E[D] + (E[Ps ] − h01 )

Z

c1 /(E[Ps ]−h01 )

F −1 (t)dt = h01 E[D] + c1 G(

0

c1 ). E[Ps ] − h01 

Note that from Cauchy-Schwarz inequality, X X m m (ci − ci+1 )2 i=1

Thus

h0i+1 − h0i

(h0i+1



− h0i )

i=1

≥

X m

2

(ci − ci+1 )

.

i=1 m

X ci − ci+1 c1 ≤ si . 0 E[Ps ] − h1 c1 i=1

We scrutinize the above bound for some special cases of demand distributions. Lemma 3 For all symmetric and bounded demand distributions,

21

C1 Cm

≤ 2.

That is, using the first active contract alone guarantees that we will not deviate more than 100% away from the optimal procurement cost using all the contracts available. Proof : Suppose demand is bounded in the range [µ − kσ, µ + kσ], with mean µ and µ ≥ kσ, then we have G( E[Pcs1]−h0 ) C1 G(0) µ + kσ 1 ≤ ≤ ≤ ≤ 2. Cm G(s1 ) G(1) µ  Remark: In practice, it is rare to find c1 + h01 much smaller than p. When demand is exponentially distributed with rate λ, G(s) = (1 − ln(s))/λ and hence when c1 /(E[Ps ] − h01 ) ≥ α > 0, G( E[Pcs1]−h0 ) C1 1 ≤ ≤ 1 − ln(α). Cm G(1) If the fixed price contract has c1 at least within 90% of E[Ps ], then CCm1 ≤ 1 − ln(0.9) ≈ 1.105. Hence the buyer cannot hope to generate more than 9.5% more savings even with a portfolio procurement approach in this case. In general, there are however environments where the benefit of portfolio approach can be enormous. We exhibit next an environment where all the contracts are needed in order to maintain a low procurement cost, i.e., elimination of any contract from the optimal portfolio may lead to a huge increase in the procurement cost. Theorem 8 The worst case error of deleting any active contract from the optimal portfolio is ∞. Proof : This can be proved through the following example. Suppose there are n contracts and P P k let m be a large number. The problem parameters are as follows: (ci , hi ) = ( nk=i m−k , i−1 k=0 m ), Pn P for i = 1, · · · , n, Ps = k=0 mk , D = {0, di , i = 1, · · · , n}, where di = ik=1 mi , P (D = di ) = P m−(2i−1) and P (D = 0) = 1 − ni=1 P (D = di ). For ease of exposition, in the following we illustrate the case of n = 3, i.e., a problem with three contracts. Similar analysis can be applied to the n contracts case. (c1 , h1 ) = (c2 , h2 ) = (c3 , h3 ) = Ps =

1 1 1 ( + 2 + 3 , 1), m m m 1 1 ( 2 + 3 , 1 + m), m m 1 ( 3 , 1 + m + m2 ), m 1 + m + m 2 + m3 .

P (D = 0) = 1 −

1

m

+

1  1 , + m3 m5

1 P (D = m) = , m P (D = m + m2 ) =

1 , m3

P (D = m + m2 + m3 ) =

22

1 . m5

It can be shown that all contracts are active and the optimal solution is as follows: 1 Q∗1 = F¯ −1 ( 2 ) = m, m

1 Q∗2 = F¯ −1 ( 4 ) − m = m2 , m

1 Q∗3 = F¯ −1 ( 6 ) − m − m2 = m3 . m

The resulting reservation cost Cr and expected execution cost Ce are 1 1 1 1 1 1 + 2 + 3 ) + m2 ( 2 + 3 ) + m3 ( 3 ) = O(1), m m m m m m 1 1 1 1 1 1 2 = ( + 3 + 5 ) · m · 1 + ( 3 + 5 ) · m · (1 + m) + 5 · m3 · (1 + m + m2 ) = O(1). m m m m m m

Cr = m( Ce

Hence, the optimal system cost is O(1). Suppose that contract 1 is missing, then the optimal solution will be Q∗1 = 0,

1 Q∗2 = F¯ −1 ( 4 ) − m = m + m2 , m

1 Q∗3 = F¯ −1 ( 6 ) − m − m2 = m3 . m

The reservation cost Cr and expected execution cost Ce will be 1 1 1 + 3 ) + m3 ( 3 ) = O(1), 2 m m m 1 1 1 1 1 1 = ( + 3 + 5 ) · m · (1 + m) + ( 3 + 5 ) · m2 · (1 + m) + 5 · m3 · (1 + m + m2 ) = O(m). m m m m m m

Cr = (m + m2 )( Ce

Therefore the resulting expected total cost will be O(m). The case of missing contract 2 (contract 3) can be analyzed similarly to show that the expected total cost will be O(m). Thus the worst case error of eliminating any contract in the optimal portfolio is limm→∞ O(m) O(1) = ∞.  Note that the above worst case example arises when the contract parameters and the demand distribution are carefully chosen to complement each other. For more natural classes of demand distributions, and more reasonable set of contract parameters, the portfolio effect P E is more manageable. For this, we present some numerical examples in Table 3. The contract parameters are randomly generated. There are 6 problem cases in the table, tested under both truncated Bivariate Normal distribution and correlated Lognormal distribution. For each setting, we record the number of contracts in the optimal portfolio, the optimal cost function, and the relative error of using the best single contract (best-1) and the best two contracts (best-2). The optimal cost function under correlated Lognormal distribution changes much more significantly with the correlation than that under truncated Bivariate Normal distribution. Though the performance error is case dependent, the data exhibit some patterns. The error of best-1 contract is rather sensitive to the change in problem settings and exhibits a larger variance, ranging from 0.94% to 11.62%. In most cases, the performance of best-1 contract deteriorates with the increase of correlation, because a larger correlation requires more diversified contracts to meet different demand segments. Exceptions occur for some cases under Lognormal distribution. 23

1

2

3

4

5

6

0.8 0.2 -0.5 0.8 0.2 -0.5 0.8 0.2 -0.5 0.8 0.2 -0.5 0.8 0.2 -0.5 0.8 0.2 -0.5

Correlated Lognormal Log(D) ∼ N (4, 1), Log(Ps ) ∼ N (3.2, 0.5) ρ N Optimal Error of Error of cost best-1 best-2 0.8 5 2064.2 4.69% 0.25% 0.2 4 1720.3 5.38% 0.27% -0.5 4 1345.0 5.86% 0.30% 0.8 4 2036.1 6.72% 0.26% 0.2 4 1724.9 7.43% 0.23% -0.5 4 1360.0 5.32% 0.24% 0.8 5 1991.9 10.9% 0.99% 0.2 5 1723.5 11.32% 0.76% -0.5 4 1370.4 4.68% 0.52% 0.8 5 2004.2 11.33% 1.92% 0.2 5 1725.3 10.12% 1.65% -0.5 5 1366.1 4.00% 0.65% 0.8 6 2008.4 11.62% 1.52% 0.2 7 1714.9 10.32% 0.79% -0.5 5 1349.2 3.16% 0.55% 0.8 7 2038.5 11.18% 1.95% 0.2 8 1747.3 9.96% 1.54% -0.5 7 1378.5 4.41% 0.25%

Table 3: Portfolio Effect

11 10

1 2

9

Contract 2 : low scenario

3

8 reservation cost c

ρ

Truncated Bivariate Normal D ∼ N (100, 50), Ps ∼ N (20, 7) N Optimal Error of Error of cost best-1 best-2 6 1396.3 8.34% 0.58% 6 1359.1 5.30% 0.42% 6 1280.3 2.50% 0.43% 4 1413.1 7.81% 0.50% 4 1375.9 4.64% 0.39% 4 1294.9 1.61% 0.29% 5 1423.4 7.36% 1.31% 5 1387.4 3.93% 0.81% 4 1303.9 1.01% 0.36% 3 1420.0 6.75% 1.06% 5 1380.9 4.30% 1.04% 3 1295.5 1.58% 0.37% 6 1392.7 4.98% 0.89% 5 1356.0 2.60% 0.60% 5 1274.3 0.57% 0.27% 8 1430.2 7.14% 0.88% 5 1394.1 3.43% 0.47% 3 1303.8 0.94% 0.15%

Spot market : high scenario

7 6

4

5

Contract 6 : base scenario

4

5

3

6

2 1

0

2

4

6 8 execution cost h

10

12

5 14

Figure 5: Contract parameters for problem case 1

24

The errors of best-2 contracts are rather stable, and are never more than 2% in all cases. In Figure 5, we provide an example in the case of Bivariate Normal distribution with correlation of 0.8. The six contracts as shown in the figure provide a range of cost and flexibility to the buyer. 1 2 3 4 5 6

1 9.74%

2 9.28% 9.30%

3 7.79% 7.79% 8.34%

4 4.83% 4.71% 5.09% 10.98%

5 2.84% 2.70% 2.98% 8.82% 13.59%

6 0.64% 0.58% 0.77% 6.55% 11.22% 16.25%

Table 4: Relative error of using arbitrary one or two contracts (case 1: ρ = 0.8)

Table 4 shows the relative error of employing arbitrary one or two contracts among the 6 active contracts (together with the spot market), where the diagonal terms correspond to the single-contract settings and the remaining terms are under dual-contract settings. From the table, any single contract still results in certain error, but two contracts are generally good enough (0.58% error) if the two contracts are carefully selected (contracts 2 and 6 in this example).

7

Inventory in Portfolio Procurement Problem

Procurement is often managed on an ongoing basis, and buyers can use inventory and demandbacklogging as additional levers to match supply with demand. For instance, if the current spot price is low, buyers can procure and store the products for future demand; on the contrary, if both current demand and spot price are high, buyers can backlog some demand to wait for a favorable procurement price in the future. In this case, buyers should be more forward looking, in anticipation of market dynamics in future periods. Extending the portfolio procurement problem into a multi-period setting adds new complexities to the problem, e.g., dynamics of option and spot prices, demand and price correlation across period, etc., and requires different modelling and analysis approaches. Mart´ınez-de-Alb´ eniz and Simchi-Levi (2005) consider a multi-period portfolio procurement model where the optimal replenishment strategy for a given set of contracts as well as the structure of the optimal contract portfolio from a pool of suppliers are presented. However, in their model both the type and quantities of the portfolio of contacts are predetermined at the beginning of the planning horizon and fixed over the entire horizon. Fu et al. (2008) extended the model to allow for dynamic adjustment of contract quantities in each period, and obtain general properties of the optimal contract replenishment policies. In the rest of this paper, we discuss a two-period portfolio procurement problem as an illustrative example to highlight the critical role of contract replenishment across periods, and how the

25

approach described in the earlier section can be used in the computational routine to solve this two period problem. Our model differs from Fu et al. (2008), as we focus on procurement cost minimization, whereas their model looks at profit maximization with prices as decision variables. Nevertheless the proof techniques are similar, and we refer the readers to Fu et al. (2008) for the technical details. Let D1 and D2 be the random demand in period 1 and 2 respectively. The buyer reserves options at the beginning of each period before observing demand, and exercises options or procures from the spot market after demand realization. The spot market price Pst is random in each period and may be correlated across periods. Unmet demand in the first period can be fully backlogged and excess inventory can be carried over to the second period with some penalty and holding costs, respectively. We denote by b and k the unit backlogging and holding cost per unit inventory, and I(·) the associated one period inventory cost function. Note that ¯ ·,t = (Q1,t , · · · , QN,t ) I(·) is convex. Let Qi,t be the quantity reserved from contract i, and Q be the reservation vector in period t, t = 1, 2. Let z be the ending inventory level of period 1 (or the starting inventory level of period 2). If z > 0, then there is leftover inventory; if z < 0, demand is backlogged. We denote by V (z|D1 , Ps1 ) the optimal expected second period cost function, with starting inventory level z. t (q, Q ¯ ·,t , Pst ) denote the replenishment cost function in period t, t = 1, 2, given the Let CR ¯ ·,t , and any realization of spot total replenishment quantity q, reservation quantity vector Q t (q, Q ¯ ·,t , Pst ) can be expressed as follows: price Pst . CR ( P Pj + t + h1 q + · · · + (hj − hj−1 )(q − j−1 t t i=1 Qi,t ) + (Ps − hj )(q − i=1 Qi,t ) , if j > 0, ¯ CR (q, Q·,t , Ps ) = Pst q if j = 0, where j = argmax0≤i≤N {hi < Pst }, h0 = 0 (artificial for ease of exposition) and x+ = min{x, 0}. Those options with execution cost higher than the spot market price will not be exercised. It t is a piecewise linear convex function. is easy to see that CR We formulate the problem as follows: min

¯ ·,1 ≥0 Q

X N

ci Qi,1 + ED1 ,Ps1

i=1



h i 1 1 1 ¯ min CR (D1 + z, Q·,1 , Ps ) + I(z) + V (z|D1 , Ps ) z

where V

(z|D1 , Ps1 )

= min

X N

¯ ·,2 ≥0 Q

ci Qi,2 + ED2 ,Ps2

i=1

h

2 CR (D2

i 2 1 ¯ − z, Q·,2 , Ps ) D1 , Ps .

We start with solving the second period problem. For convenience, let ¯ ·,1 , z|D1 , P 1 ) = C 1 (D1 + z, Q ¯ ·,1 , P 1 ) + I(z) + V (z|D1 , P 1 ) J(Q s R s s N h i X 2 2 1 ¯ ·,2 |D1 , P 1 ) = ¯ U (z, Q C c Q + E (D − z, Q , P ) D , P 2 i i,2 2 ·,2 1 D2 ,Ps s R s s . i=1

26

Theorem 9 For the second period problem, we have the following properties: ¯ ·,2 |D1 , Ps1 ) (a) The optimal option reservation quantity Q∗i,2 , i = 1, · · · , N , minimizing U (z, Q can be solved using the method described in Section 4.2 by replacing D with D2 − z. ¯ ·,2 |D1 , Ps1 ) is convex in (z, Q ¯ ·,2 ). (b) U (z, Q (c) If D2 and Ps2 are independent and conditional on D1 and Ps1 , then the optimal Q∗i,2 can be determined as follows (Q∗0,2 = 0): Q∗i,2

=

Q∗N,2 =



F



−1



ci − ci+1  1− 0 −z hi+1 − h0i

 F −1 1 −

+

 cN −z h0N +1 − h0N

−

i−1 X

Q∗j,2 , i = 1, · · · , N − 1

j=0

+

−

N −1 X

Q∗j,2

j=0

where F −1 (·) is the inverse cdf of the second period demand and h0i can be computed based on the second period spot price distribution. Theorem 10 For the first period problem, we have the following properties: (a) V (z|D1 , Ps1 ) is convex and decreasing in z. ¯ ·,1 , z|D1 , Ps1 ) is convex in (Q ¯ ·,1 , z). (b) J(Q ¯ ·,1 , z)]) is convex in Q ¯ ·,1 . (c) ED1 ,Ps1 (minz [J(Q Theorem 9 and 10 characterize the structural properties for the two-period problem. Convexity of the problem ensures that we can solve for the optimal procurement decisions using the first order conditions. Since the overall demands have to be met at the end of the second period, the problem boils down essentially to an allocation problem. The second period problem is easy and reduces to a single period problem with initial inventory or demand backlogged. When D2 and Ps2 are independent and conditional on D1 and Ps1 , the strategic role of the various types of option contracts is particularly noteworthy. Note that when the decision is to backlog first period demand (i.e. z < 0), the amount −z will be met solely using the cheapest but least flexible contract in the second period, i.e. mini {ci + E(min(hi , Ps2 )|D1 , Ps1 )}. In a way, the cheapest contract in period 2 can be viewed as an alternative spot market for period 1 demand. If Ps1 is high, the buyer has the option to defer the fulfilment to the next period, using the cheapest contract available. On the other hand, if there are unused option contracts already reserved in the first period, or if spot price in the first period is low, then the buyer may use these left over option contracts as sources of new fixed price contracts in the second period, at a penalty of incurring holding cost. Effectively, this creates a buffer of “future” demand that the first period procurement strategy can exploit to hedge against the situation when demand or spot price in the first period are lower than expected. To gain some insights into this problem further, we analyze the simplest case when the spot price Pst is constant in both periods, and demands are independent. A fixed price contract (unit

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price cf ) and the spot market are used to meet the uncertain demands in the two periods. We further assume cf < Ps , I(z) = kz + + bz − , and k = 0. The problem can be formulated as follows:   h i + − minQ1 ≥0 cf Q1 + ED1 minz Ps (D1 + z − Q1 ) + bz + V (z)  h i + V (z) = minQ2 ≥0 cf Q2 + ED2 Ps (D2 − z − Q2 ) The optimal second stage decision is given by Theorem 9(c): cf Q∗2 = (F −1 (1 − ) − z)+ . Ps In fact, from Theorem 2, it is easy to see that    V (0) + cf × (−z) if z < 0 c V (z) = V (0) − cf × (z) if 0 ≤ z < F −1 (1 − Pfs )   c Ps E(D2 − z)+ if z ≥ F −1 (1 − Pfs ). For the first period, we consider next the problem h i min Ps (D1 + z − Q1 )+ + bz − + V (z) . z

(10)

(a) If the spot price is less than the unit price plus backlogging cost, i.e., Ps < b + cf , it is never optimal to backlog demand. In the event that D1 > Q1 , z = 0 attains the minimum in (10) since it is never optimal to hold inventory by purchasing from the spot market. Otherwise, since k = 0, and V (z) < V (0) for all z > 0, it is optimal for z = (Q1 − D1 ) whenever Q1 > D1 . Hence h i min Ps (D1 + z − Q1 )+ + bz − + V (z) = Ps (D1 − Q1 )+ + V ((Q1 − D1 )+ ). z

The optimality condition for the first period problem reduces to Z Q∗ 1 ∗ cf − Ps P (D1 > Q1 ) + V 0 (Q∗1 − t)fD1 (t)dt = 0. 0

(b) If Ps > b + cf , then it is optimal to backlog all excess demand, so if Q1 > D1 we will hold the extra inventory, and if Q1 < D1 , we will backlog the excess demand. Therefore, we have z = Q1 − D1 for whatever demand realization and h i min Ps (D1 + z − Q1 )+ + bz − + V (z) = b(D1 − Q1 )+ + V ((Q1 − D1 )). z

The optimality condition for the first period problem reduces to Z ∞ cf − bP (D1 > Q∗1 ) + V 0 (Q∗1 − t)fD1 (t)dt = 0. 0

Q∗1

The optimal solution can be obtained via a line search procedure, to look for a solution to the optimality condition. Convexity of the objective function ensures tha this is the optimal solution. Note that the solution approach requires the derivative of V (t) = E(D2 − t)+ , a function which has no nice analytical form for most distribution of the r.v. D2 . 28

8

Conclusions

The optimal portfolio procurement decisions when both the demand and the spot price are random (either correlated or independent) are investigated in a single period setting. We provide an approach helping companies to properly segment demand facing a portfolio of contracts, evaluate the resulting cost performance, and visualize the optimal decisions and the associated cost components through an intuitive graphical approach. A two-period model is also presented to gain some insight into the multi-period problem. We show that the correlation between spot price and demand can affect the segmentation significantly. With higher demand-spot price correlation, option contracts play a more important role and can take care of a larger segment of demand. Moreover, option contracts can serve as a risk buffer and their value increases with the volatility of spot price. In reality, it may not be practical for companies to maintain a full portfolio of suppliers, due to the time and effort to maintain the relationships with suppliers. In that case, the buyer should identify a few key suppliers. Our analysis of the portfolio effect shows that two carefully selected suppliers are already enough to produce near-optimal performance in most cases.

Acknowledgments The authors thank the anonymous associate editor and two referees for their constructive comments which greatly improved this paper. We are also grateful to V. Nagali, D. Sanghera, J. Hwang, M. Gaskins, T. Thurston, etc., from HP’s PRM group, for their helpful discussions and suggestions. The HP group also shared with us some of the issues they faced while implementing the PRM approach in the industry. This research is supported in part by Hong Kong RGC Earmark Grant 615607.

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