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Profit Sharing Agreements in Decentralized Supply Chains:A Distributionally Robust ApproachQi Fu, Chee-Khian Sim, Chung-Piaw Teo

To cite this article:Qi Fu, Chee-Khian Sim, Chung-Piaw Teo (2018) Profit Sharing Agreements in Decentralized Supply Chains: A DistributionallyRobust Approach. Operations Research 66(2):500-513. https://doi.org/10.1287/opre.2017.1677

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OPERATIONS RESEARCHVol. 66, No. 2, March–April 2018, pp. 500–513

http://pubsonline.informs.org/journal/opre/ ISSN 0030-364X (print), ISSN 1526-5463 (online)

Profit Sharing Agreements in Decentralized Supply Chains:A Distributionally Robust ApproachQi Fu,a Chee-Khian Sim,b Chung-Piaw Teoc

aFaculty of Business Administration, University of Macau, Taipa, Macau, China; bDepartment of Mathematics, University of Portsmouth,Portsmouth PO1 3HF, United Kingdom; cNUS Business School and Institute of Operations Research and Analytics, National University ofSingapore, Singapore 119077Contact: [email protected], http://orcid.org/0000-0001-8154-5384 (QF); [email protected],

http://orcid.org/0000-0003-0696-4698 (C-KS); [email protected], http://orcid.org/0000-0002-0534-6858 (C-PT)

Received: January 8, 2015Revised: June 30, 2016; February 22, 2017;June 13, 2017Accepted: August 3, 2017Published Online in Articles in Advance:February 5, 2018

Subject Classifications: inventory/production:policies: ordering/pricing; programming:nonlinear: convexArea of Review: Optimization


Copyright: © 2018 INFORMS

Abstract. How should decentralized supply chains set the profit sharing terms usingminimal information on demand and selling price? We develop a distributionally robustStackelberg game model to address this question. Our framework uses only the firstand second moments of the price and demand attributes, and thus can be implementedusing only a parsimonious set of parameters. More specifically, we derive the relation-ships among the optimal wholesale price set by the supplier, the order decision of theretailer, and the corresponding profit shares of each supply chain partner, based on theinformation available. Interestingly, in the distributionally robust setting, the correlationbetween demand and selling price has no bearing on the order decision of the retailer.This allows us to simplify the solution structure of the profit sharing agreement problemdramatically. Moreover, the result can be used to recover the optimal selling price whenthe mean demand is a linear function of the selling price (cf. Raza 2014) [Raza SA (2014)A distribution free approach to newsvendor problem with pricing. 4OR—A Quart. J. Oper.Res. 12(4):335—358.].

Funding: The first author was supported in part by the University of Macau [Grant MYRG2014-00057-FBA].

Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2017.1677.

Keywords: profit sharing agreements • decentralized supply chains • distributionally robust planning • completely positive program

1. IntroductionProfit sharing agreements are among the most com-mon types of contractual arrangements for firms in asupply chain. Under such agreements, the retailer, as aprovider to a consumer market, engages another firm(the supplier) as a contractor to provide essential sup-plies for the operation. In addition to paying for theessential supplies, the retailer pays the supplier a cer-tain portion of the profit generated from the operation.In this way, profit sharing agreements serve to alignincentives between the retailer and the supplier in asupply chain.A variant of this concept, known as sharecropping,

has been used in agriculture where the landlord allowsthe tenant to use his land in exchange for a specifiedshare of production. However, the terms of paymentcan vary widely. For instance, the landlord can beara portion of the production cost, and thus demand ahigher share of the production, or may bear the entirerisk and employ the farmers as fixed wage workers.Consider, for example, B-BOVID, an agricultural firmoperating in Takoradi, Ghana. In 2014, this firm intro-duced a unique profit sharing model to alleviate theplight of oil palm farmers in the region. The farmers

would sell their palm fruits to B-BOVID for on-the-spotmoney. B-BOVID would then refine the palm fruits forsale and return a portion of the profits generated backto the farmers, based on the quantity supplied by thefarmers to the company. Thus, in addition to reduc-ing poverty, the company incentivises and sustains theinterest of the farmers in oil palm plantations. Stiglitz(1989) presents early economic analysis of these con-tractual forms and confirms the benefits of sharecrop-ping in streamlining the allocation of risk and rewardsamong the different parties involved.

The profit sharing contract is also prevalent in theentertainment industry. See, for instance, Weinstein(1998) and the references therein for a discussion ofthe evolution of profit sharing contracts in Hollywood,and an interesting discussion on the difference between“net profit” sharing and “gross profit” sharing (i.e.,revenue sharing) contract. Filson et al. (2005, p. 367)study the profit sharing contracts written between cer-tain movie distributors and a theatre chain in St. Louis,Missouri, and conclude through empirical analysis thatthe profit sharing contracts “evolved to help distrib-utors and exhibitors share risks and overcome mea-surement problems and not to overcome asymmetricinformation problems.”

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mailto:[email protected]

http://orcid.org/0000-0001-8154-5384


http://orcid.org/0000-0003-0696-4698


http://orcid.org/0000-0002-0534-6858


Fu, Sim, and Teo: Profit Sharing Agreements in Decentralized Supply ChainsOperations Research, 2018, vol. 66, no. 2, pp. 500–513, ©2018 INFORMS 501

In the natural resource industry, the profit sharingagreement (or in the form of production sharing agree-ment, (PSA)) has been practiced for a long period oftime. Under a PSA, the government engages a foreignoil company to operate petroleum exploration. Afterdeducting the royalty and operating costs, the remain-der (called profit oil) will be split between the govern-ment and company.

For many commodity products, such as oil andagricultural commodities, sellers face not only uncer-tain demands but also extreme price fluctuations. Forexample, according to the United Nations Food andAgriculture Organization (FAO), food prices have fluc-tuated wildly over the last few years. The index rosefrom 122 in 2006 to 214 in June 2008 as the 2007–2008food price crisis unfolded. The index then fell rapidlyin the second half of 2008, sinking to 140 inMarch 2009.In the latter half of 2010, it increased considerably, andreaching 215 in December.1 The prices for agriculturalproducts are often largely influenced by supply anddemand factors, such as weather conditions, popula-tion changes, and food stocks, which lead to consider-able risk and uncertainty in the agricultural commod-ity price. Similarly, the price of oil has also fluctuatedsignificantly over the last few years. Therefore, facingsuch high price and demand risks, profit sharing con-tracts allow the supply chain partners to share risks.An additional challenge is to establish a reasonable

method to share profits. One approach is to use theNash bargaining framework, where the total rents ofthe supply chain operation are split between the sup-plier and retailer according to the bargaining strengthof the parties involved, together with the value of out-side options for the partners. The approach has beenapplied in the natural resource industry to provide abenchmark to disentangle the effect of other compet-ing theories on the bargaining outcome (cf. McMillanand Waxman 2007). However, the distribution of prof-its in the Nash bargaining model often depends onsubjective input of the relative bargaining power ofthe parties. More importantly, these models ignorethe fact that contractual elements in the profit shar-ing agreements are often based on full knowledgeof the selling price, demand, and/or cost of opera-tions, which are often not assured because of the dif-ficulty in fully characterizing the distributions of ran-dom elements. Accordingly, in many cases, we mayonly have confidence in estimating some distributionparameters, such as mean and variance. This problemis compounded by the provision of the retailer for theoperational costs in the computation of net profit. Theevaluation of this cost component is often the boneof contention between the two parties involved in theprofit sharing agreements, as often seen in the miningand entertainment industries.

Motivated by these concerns, in this paper, wedevelop a distributionally robust Stackelberg gamemo-del to study the profit sharing contract design prob-lem with limited information about demand and pricedistributions. Aghassi and Bertsimas (2006) present adistribution-free, robust optimizationmodel for incom-plete information games, in which the players com-monlyknowanuncertainty set ofpossible gameparam-eter values without knowledge of exact distributionalinformation and each player seeks to maximize theworst-case expected payoff. Their proposed robustgame model presumes that each player uses a robustoptimization, and therefore a worst-case approach tothe uncertainty to enablemutual prediction of the gameoutcomes. We adopt the same assumption that the par-tial distribution information on price and demand iscommon knowledge between the two players in ourbase model analysis (the case of demand informationasymmetry is discussed in the online appendix). Thatis, both the retailer and supplier have access to thesame demand information, such as point-of-sale data,andmarket price information. As the exact distributioninformation on demand and price is not available, bothparties apply a distributionally robust approach tomin-imize their worst-case expected profit.

As a direct application, we show that this approachcan be used to derive testable restrictions on the out-comes in the supply chain setting. More specifically,given the decisions of the supplier and retailer, theunit cost of production (incurred by the supplier) andthe selling price information (mean and variance), butwithout the demand and selling price correlation infor-mation, we develop testable conditions on the out-comes that are consistent with the prediction of thedistributionally robust Stacklberg game based on somerandom demand model. In the traditional Stackelberggame model, this is a difficult problem because find-ing the optimal wholesale price is already a compli-cated problem, and a well-known result by Lariviereand Porteus (2001, p. 295) indicates that the prob-lem is solvable when the demand distribution satis-fies the “increasing-generalized failure-rate” property.A closed-form expression for the optimal wholesaleprice is known only for specific demand distribution(for instance, when demand is uniformwith support ina bounded interval). Otherwise, a closed-form expres-sion is generally not available. Thus, it is difficult to findtestable conditions on the optimal wholesale price andorder quantity. Interestingly, as shown in this paper,this problem becomes tractable in a distributionallyrobust Stackelberg game model. By building a pre-scriptive model on the optimal behavior in the choiceof the contract parameters and operational decisions,we hope to find testable relationships between theseparameters if the parties act in a rational manner asprescribed by the Stackelberg game.

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Fu, Sim, and Teo: Profit Sharing Agreements in Decentralized Supply Chains502 Operations Research, 2018, vol. 66, no. 2, pp. 500–513, ©2018 INFORMS

The remainder of this paper is organized as follows.In Section 2, we review the relevant literature. Sec-tion 3 introduces the distributionally robust model. Wederive the robust ordering decision of the last stageproblem in Section 4, and address the profit sharingcontract design problem in Section 5. We then extendour model to the price dependent demand case in Sec-tion 6. We also discuss other potential applications ofthis model in the online Appendix A. Section 7 reportsnumerical studies. Concluding thoughts are providedin Section 8. All proofs to the technical results are rele-gated to in the online Appendix B.

2. Literature ReviewOur research is related to two streams of literature. Thefirst stream of literature is robust inventory manage-ment, where decisions have to be made with incom-plete distribution information. One common approachto this situation, according to the literature, is the max-min approach, which was pioneered by Scarf (1958)to model the newsvendor problem with only meanand variance of the demand distribution known, aim-ing to find an order quantity to maximize the worst-case expected profit over all possible distributions withthe given mean and variance. Scarf derives a closed-form solution for the robust order quantity. Gallegoand Moon (1993) provide a more concise proof ofScarf’s ordering rule and apply this approach to severalvariants of the newsvendor problem, for example, therecourse case and the fixed ordering cost case. Moonand Choi (1995) apply the max-min approach to thenewsvendor problem with customer balking. A widevariety of extensions to the basic robust newsvenderproblem have been addressed in the literature, includ-ing multiperiod (e.g., Gallego 1992, Moon and Gallego1994), multiechelon (e.g., Moon and Choi 1997), andmulti-item (e.g., Moon and Silver 2000) extensions.Readers may refer to Gallego et al. (2001) for a moredetailed review of the early literature. Mamani et al.(2017) obtain a closed-form solution to a multiperiodinventorymanagement problem, using a class of uncer-tainty sets motivated by the central limit theorem.More recently, Ben-Tal et al. (2005) apply the min-maxapproach to a multiperiod stochastic inventory man-agement problem with a flexible commitment con-tract. Raza (2014) applies the max-min approach tothe newsvendor problem with joint pricing and inven-tory decisions. Han et al. (2014) enrich the analysis bycombining both risk aversion and ambiguity aversionto obtain a closed-form solution. Another approach tomaking decisions with limited distribution informa-tion is the minimax regret principle, which involvesminimizing themaximumopportunity cost of notmak-ing the optimal decision. For example, Yue et al. (2006)apply this regret approach to a newsvendor problem

when only the mean and variance of demand distri-bution are known. Perakis and Roels (2008) furtherinvestigate the minimax regret newsvendor problemby considering more general partial demand informa-tion scenarios. Hanasusanto et al. (2015) use a dis-tributionally robust approach to study the multi-itemnewsvendor problem and obtain a related SDP relax-ation for this NP-hard problem.

Whereas most papers investigate the robust deci-sions from the perspective of a single decision maker,there are several papers examining the robust deci-sions in a game setting. Jiang et al. (2011) study arobust inventory competition problem with stock-outsubstitution and asymmetric information on the sup-port of demand distribution, under the criteria of abso-lute regret minimization. Wagner (2015) employs therobust optimization approach to a supply chain witha price-only contract, where the supplier may not cor-rectly assess the informational state of the retailer.Wagner compares the order quantity and profit ofindividual firms and the supply chain under differentinformational states of the two parties and concludesthat information advantage does not necessarily ben-efit a firm. A common feature in the extant literatureon robust inventory models is that randomness comesonly from demand. This study extends the literatureby investigating the robustmax-min inventory decisionwhen both demand and price are random.

Another related literature stream is the use of profitor revenue sharing contract in supply chains, wherethe retailer pays the supplier a unit price, plus a per-centage of the profit/revenue generated from sales.Pasternack (2002) investigates a newsvendor problemwith both wholesale price and revenue sharing con-tracts. Cachon and Lariviere (2005) provide a compre-hensive study on the use of revenue sharing contractto achieve channel coordination and compare revenuesharing with a number of other supply chain contracts.Tang and Kouvelis (2014) study supply chain coordina-tion in the presence of supply uncertainty, and proposea pay-back-revenue sharing contract to coordinate thesupply chain. These papers focus on the design of thecontract parameters from a central planner perspectiveto coordinate the supply chain. Other papers study rev-enue sharing contracts in decentralized supply chains.For example, Wang et al. (2004) investigate the channelperformance under a consignment contract with rev-enue sharing, where the retailer, acting as the leader,decides the share of sales revenue to be remitted tothe supplier, and the supplier sets the retail price anddelivery quantity. Wang and Gerchak (2003) consideran assembly problem in which the firm assembling thefinal product decides the allocation of sales revenuebetween the firm and several suppliers producing thecomponents.

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Revenue sharing contracts have been shown toallow the supply chain profit to be arbitrarily dividedbetween the supplier and retailer. If the two contractparameters, wholesale price and profit share, are bothset by one party, then this party can extract most of thesupply chain profit, leaving the other party only thereservation profit. Therefore, in this paper, we let eachsupply chain partner determine one of the two contractparameters. As the supplier incurs the costs of makingproducts, he is in a better position to set the whole-sale price. The retailer, as a price taker, determines theprofit share. We model this as a three-stage Stackel-berg game. The retailer, acting as the leader, decides aprofit share that benefits herself, and then the suppliermanipulates his share of the supply chain profit bychoosing thewholesale price. Finally, given the agreed-upon contract, the retailer decides the order quantity.

3. The ModelThere are two common scenarios to model the mar-ket price. One models the case of a monopoly mar-ket, where a dominant seller has the power to endoge-nously set the retail price. Thus, product price is aninternal decision, and demand is typically modelledas a price dependent function. Another considers anoligopoly market with a few key competing sellers ora market with a number of small sellers, whose actionsjointly determine a competitive market price. In thiscase, product price is typically exogenous and ran-dom in advance, subject to some factors that cannotbe observed at the time strategic decisions are made,such as competitors’ prices, total market supply anddemand. We consider the latter case with both productdemand and price being random in our base model,and analyze the monopolistic case in Section 6.

We modify the setup in McMillan and Waxman(2007) to find the optimal split of profit between theretailer and supplier. In this setting, with only the firstand second moments information on the demand andselling price of the product, the retailer would like todevise a profit sharing formula with a key supplier tomaximize her own profit. Through sharing, the retailerhopes to procure from the supplier at a lower whole-sale price.

To maximize supply chain profit, the retailer wouldalso need to determine the capacity/quantity invest-ment decision, Q, in advance. Let D(>0) denote thehitherto unknown demand and Ps(>0) denote theuncertain selling price. Both Ps and D are unknownwhen the contractual decisions are made, and theretailer cannot sell more than the installed capacity Q.The capacity is sourced from an external supplier witha unit cost of f . The supplier charges a wholesaleprice c(Q) � wQ, for Q units of capacity. The whole-sale price w is a decision by the supplier that the

retailer would like to influence through the profit shar-ing arrangement.

Let ΠC denote the expected profit made by the sup-ply chain in a centralized setting:

ΠC � EPs ,D

[Ps︸︷︷︸Price

min(Q ,D)︸︷︷︸Quantity Sold

]− f ×Q︸︷︷︸

Cost

. (1)

In the traditional Nash bargaining approach, theprofits allocated to the retailer (ΠR) and supplier (ΠS)are obtained by solving

maxα

{(ΠR − g)δ(ΠS − b)1−δ |ΠR �αΠC ,ΠS � (1−α)ΠC

},

assuming all parameters are known, where g and b arethe outside option value for the retailer and supplier,respectively. The parameter δ is the relative bargainingpower of the retailer. The profit ΠR of the retailer isthus obtained by solving the Nash bargaining solution.

This analysis however requires qualitative judgmentof the relative bargaining power of the retailer and thesupplier and ignores the other economic and strate-gic considerations of the parties involved (e.g., thesetting of the wholesale price w by the supplier). Itassumes that the bargaining outcome has no effect onthe profitΠ in the decentralized setting, which is oftenviolated in practice, as the profit share decision affectsthe capacity investment decision Q made by the firm,and thus affects the profit Π. Furthermore, the sup-plier can also extract rents through judicious choice ofwholesale price decision w.

In the decentralized setting, consider the net profitof the retailer, denoted by

Π� EPs ,D[Psmin(Q ,D)] −w ×Q. (2)

The retailer allocates a share γ of the total profit Πto the supplier. In this paper, we model directly theimpact of profit share 1 − γ accrued by the retailer,on the investment decision Q made by the retailer inresponse to thewholesale price decision w made by thesupplier. A very common form of the wholesale pricein practice is a linear wholesale price. The popularityof linear contract has been addressed by many papersin the economics literature. For example, Holmströmand Milgrom (1987) state that the popularity of linearcontracts probably lies in its great robustness. A recentpaper, Carroll (2015), using a principal-agent frame-work with worst-case performance objective, showsthat out of all possible contracts, a linear contract offersthe principal the best guarantee. Therefore, we restrictour analysis to a linear wholesale price. The problem issolved in three stages.

• In stage 1, the retailer determines the profit share1− γ she would retain on the profit made. The goal isto maximize her share of the profit pie ΠR :� (1− γ)Π,allocating γΠ to the supplier.

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• In stage 2, the supplier determines the wholesaleprice w(>0) per unit, for the items supplied to theretailer. Note that the actual cost per unit is f (>0) forthe supplier. The firm is thus interested in maximizingthe total profit

ΠS :� γΠ+ (w − f )Q.

• In stage 3, the retailer chooses the optimal capacitydecision Q, to maximize the profit Π, which is givenby Equation (2). Any unused capacity will be lost withzero salvage value.In this way, the protracted bargaining process

between the retailer and the supplier on the profit shar-ing agreement is modelled using a multistage Stackel-berg game. Our goal in this paper is to develop verifi-able relationships between the model parameters andthe optimal decisions obtained under the multistageStackelberg game, in a distributionally robust setting.

We are interested in the situation when demand andprice distributions are not known precisely at the timethe contract is being negotiated. Instead, we only haveinformation on the means, variances, and covarianceof price Ps and demand D. These parameters are givenby a matrix Σ, called the moments matrix:

Σ�©«

E(P2s ) E(Ps D) E(Ps)

E(Ps D) E(D2) E(D)E(Ps) E(D) 1

ª®¬ .We assume that both parties in the Stackelberg gameare ambiguity averse and plan for the worst case.There is some distinction between risk aversion andambiguity aversion. Risk refers to the situation thatthe probabilities of possible outcomes are known, i.e.,known risks, while ambiguity refers to the situationthat the probability distribution of possible outcomesis unknown, i.e., unknown risks. An ambiguity averseindividual prefers known risk over unknown risk. Inour problem setting, without knowing the price anddemand distributions, the retailer and supplier face anambiguity situation and both plan against the worst-case distribution, which could be characterized.It is easy to see that Σ is positive semidefinite

with nonnegative entries for any random Ps ,D > 0.Throughout this paper, capital letters in bold fontdenote matrices. Let

Φ(Σ0)�{(Ps ,D); Ps ,D > 0,Ps ,D have moments matrix Σ�Σ0

}denote the space of distributions with the prescribedmoments conditions, where Σ0 is given. All decisionsare evaluated against the worst possible distributionin Φ(Σ0). That is, the first and second moments of thedemand and price, E(Ps),E(D),E(P2

s ),E(D2),E(Ps D),should take values from the entries of Σ0.

The distributionally robust Stackelberg game modelreduces to the following problem: Let

Q(w)

:�argmaxQ>0

{−wQ+ inf

(Ps ,D)∈Φ(Σ0)EPs ,D(Ps min(Q ,D))

}, (3)

Π(w) :�maxQ>0

{−wQ+ inf


}.

(4)

Our goal is to choose the profit sharing formula γbased on solving the following model:

(Model M): maxγ>0(1− γ)Π(w(γ)), (5)

where w(γ), for given γ, is chosen to maximize thesupplier’s profit function:

w(γ)

:�

Supplier’s Problem︷︸︸︷argmax

w>0

[(w− f )Q(w)

+γ{maxQ>0

{−wQ + inf


}}︸︷︷︸

Retailer’s Stage 3 Problem

].

(6)

This allows us to study the profit sharingmodel using avery parsimonious setup that uses onlymoments infor-mation on demand and selling price. Surprisingly, thismodel permits a very simple solution.

4. The Robust Newsvendor Problem:Closed-Form Solution

In the rest of the paper, we formally derive the mainresults obtained in this paper. Given the wholesaleprice w per unit of capacity from the supplier, theretailer’s problem in the last stage is to maximize theworst-case profit of the supply chain over the space ofdemand and price distributions with the moment con-straint, by choosing the optimal capacity decision Q.The robust newsvendor problem (PR) can be formu-lated as

(PR): maxQ>0

{Π�−wQ+ inf


}.

In the literature (Moon andGallego 1994, Scarf 1958),a closed-form solution is derived for the robust news-boy problem, when only the demand is random andwe have partial information regarding its mean andvariance. However, the problem here is more challeng-ing as there are two random variables with only infor-mation about their means, variances, and covariance.To tackle the problem, we use an indirect approach

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by first transforming it into a dual completely positiveprogram.Let I(1) denote the event {D 6 Q} and I(2) denote

the event {D >Q}. Define the variables

©«

y(i)P(i)D(i)

PP(i)DD(i)PD(i)

ª®®®®®®¬�

©«

Prob(I(i))E(Ps | I(i))Prob(I(i))E(D | I(i))Prob(I(i))E(P2

s | I(i))Prob(I(i))E(D2 | I(i))Prob(I(i))

E(Ps D | I(i))Prob(I(i))

ª®®®®®®¬,

for i � 1, 2. (7)

Using conditional expectation, we can decomposethe second stage profit function into

EPs ,D(min{Ps D ,PsQ})� EPs ,D(Ps D | D 6 Q)Prob(D 6 Q)+EPs ,D(PsQ | D >Q)Prob(D >Q)

� PD(1)+QP(2). (8)

Proposition 1. The inner problem in the robust newsven-dor problem (PR) can be formulated as the following com-pletely positive program2 (CPP):

minMi , i�1, 2

{PD(1)+QP(2)}

subject to

(A): Mi �©«

PP(i) PD(i) P(i)PD(i) DD(i) D(i)P(i) D(i) y(i)

ª®¬ � 0, i � 1, 2,

(B): Mi > 0, i � 1, 2,(C): M1 +M2 �Σ0.

Remark 1. A symmetric matrix A of size n is com-pletely positive (denoted A �CP 0) if it can be writtenas BBT , where B ∈ <n×m has nonnegative entries. It isknown that if n 6 4, then A is completely positive if andonly if it is positive semidefinite and has all entries non-negative (Berman and Shaked-Monderer 2003). Hence,the minimization problem in the above proposition isa completely positive program.

Proposition 1 implies that the inner problem in therobust problem (PR) can be rewritten as

minMi , i�1,2

Tr©«

0 1/2 01/2 0 00 0 0

ª®¬M1+©«

0 0 Q/20 0 0

Q/2 0 0

ª®¬M2

(9)

subject to constraints (A), (B), and (C).We can find the dual of this completely positive pro-

gram. Let Y1 ,Y2 denote the dual variables associated

with constraint (B) andY denote the dual variable asso-ciated with constraint (C). By taking the dual of Prob-lem (9), we have

maxY1 ,Y2 ,Y

−Tr(Σ0Y) (10)

s.t. Y � −©«0 0 Q/20 0 0

Q/2 0 0

ª®¬+Y2 ,

Y � −©«0 1/2 0

1/2 0 00 0 0

ª®¬+Y1 , Y1 > 0, Y2 > 0.

Note that since Problem (10) is convex and strictlyfeasible, strong duality holds (Boyd and Vandenberghe2004), with its optimal value equal to that ofProblem (9).

Adding the outer problem, we can obtain the dualCPP formulation of the robust problem (PR) as shownin the following theorem.

Theorem 1. The robust problem (PR) can be reformulatedas a dual CPP as follows:

maxQ>0,Y1 ,Y2 ,Y

{−wQ −Tr(Σ0Y)

}(11)

s.t. Y � −©«0 0 Q/20 0 0

Q/2 0 0

ª®¬+Y2 ,

Y � −©«0 1/2 0

1/2 0 00 0 0

ª®¬+Y1 , Y1 > 0,Y2 > 0.

To simplify the formulation, we perform a lineartransformation on Y in Problem (11), by adding

©«0 1/2 0

1/2 0 00 0 0

ª®¬to it, to obtain the following equivalent problem, whichwe will consider from this point onward:

maxQ>0,Y1 ,Y2 ,Y

{−wQ −Tr(Σ0Y)+E(Ps D)

}(12)

s.t. Y � 12©«

0 1 −Q1 0 0−Q 0 0

ª®¬+Y2 ,

Y � Y1 , Y1 > 0,Y2 > 0.

Let (Q∗ ,Y∗1 ,Y∗2 ,Y∗) be an optimal solution to Prob-lem (12). Note that Problem (12) is a dual CPP withmatrices of size three. Let Π(w) denote the optimalobjective value to Problem (12).

In the following proposition, we observe an impor-tant property of Problem (12).

Proposition 2. Π(w) is convex decreasing in w.

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To solve Problem (12), and thus the robust prob-lem (PR), we first solve a simpler version of the prob-lem, and then use the optimal solution constructedto recover the solution to the original robust prob-lem (PR). By setting Y1 ,Y2 in Problem (12) to zero, thesimpler problem is obtained as follows:

maxQ>0,Y

{−wQ −Tr(Σ0Y)+E(Ps D)

}(13)

s.t. Y � 12©«

0 1 −Q1 0 0−Q 0 0

ª®¬ , Y � 0.

Let (Q∗r ,Y∗r) be an optimal solution to Problem (13).For ease of exposition, we define the following terms:

Definition 1. Define α(w)� E(Ps)/2−w, β � E(P2s )/4.

Theorem 2. For Problem (13), the optimal solution Q∗r isgreater than zero if and only if

w <12

[√E(P2

s )E(D2)E(D)+E(Ps)

], (14)

in which case,

Q∗r � E(D)+ α(w)√β− α(w)2

σ(D), (15)

and the worst-case profit is

Πwst�−wQ∗r + inf(Ps ,D)∈Φ(Σ0)

E(Ps min(D ,Q∗r))

�α(w)E(D)− σ(D)√(β−α(w)2)+ 1

2 E(Ps D), (16)

where σ(D) is the standard deviation of demand D.

Remark 2. Theorem 2 in fact presents a closed-form expression for the robust newsvendor problemwhen Ps and D are not restricted to be nonnegative. Inthis case, when the condition (14) is not met, namely,w is larger or equal to the given threshold, no invest-ment will be made. This places a bound on the whole-sale price set by the supplier. Otherwise, we can obtainthe optimal capacity, Q∗r , and the resulting worst-caseprofit, Πwst, in closed form.

In the following theorem, we recover the solution toProblem (12) (and the original robust capacity invest-ment problem (PR)), using the dual constructed fromthis simpler problem, Problem (13).

Theorem 3. For the robust problem (PR), if

w 612

[E(Ps)

+E(Ps D)E(D) − σ(D)

√E(P2

s )E(D2) −E(Ps D)2E(D2)

],

(17)

the optimal solution Q∗ is given as

Q∗ � E(D)+ α(w)√β− α(w)2

σ(D), (18)

with the worst-case profitΠ(w)�Πwst being the same as thatpresented in Theorem 2 by replacing Q∗r with Q∗. OtherwiseQ∗ � 0.

Definition 2. Define wUB �12 [E(Ps) + (E(Ps D)E(D) −

σ(D)√

E(P2s )E(D2) −E(Ps D)2)/E(D2)].

We assume the parameters are chosen such thatwUB > f . From Theorem 3, the optimal solution Q∗ forthe original problem is the same as that of the simplerproblem, Problem (13), but with a different thresholdcondition on w. We can check that the upper boundwUB here is lower than the upper bound given in The-orem 2, so the nonnegative constraints on Ps ,D reducethe feasible region of w. The reason is that for the sim-pler problem, Y1 ,Y2 are set to zero, which reduces thefeasible region. An interesting observation is that theoptimal robust solution Q∗ only depends on the firstand second moments of the demand and selling pricedistributions. The correlation term E(Ps D) does not affectthe optimal choice of Q in the distributionally robust prob-lem. If the threshold condition (17) is not satisfied, theretailer is better off not ordering anything.

Scarf (1958) obtains the min-max solution for anewsvendor problem with unit cost c, and unit sell-ing price p. Note that the newsvendor problem is aspecial case of our problem, with Ps being constant.In this case, let Ps � p and w � c. It can be verifiedthat our result in Theorem 3 reduces to Scarf’s formula,with exactly the same Q∗ and the corresponding con-dition. Thus our result generalizes Scarf’s formula to amore general setting when both demand and price arerandom.

The following useful inequality (Scarf’s bound) iswell known:

E[(D −Q)+]6 1

2

[E(D) −Q +

√(E(D) −Q)2 + (E(D2) −E(D)2)

].

This bound has found applications in a variety ofdomains fromfinance to supply chainmanagement. Asa by-product, our analysis actually yields an extensionof this classical inequality to the case when the price Psis also random.

Corollary 1. If price Ps and demand D are both random,the following bound holds:

E[Ps(D −Q)+]6 1

2

[E(Ps D) −QE(Ps)+

√E(P2

s )√(E(D) −Q)2 + (E(D2) −E(D)2)

].

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The above bound follows from Proposition 1 and theproof of Theorem 2.

It is well known in the literature that for the min-max newsvendor problem, the worst possible distribu-tion of demand only has positive mass at two points.We find that the worst-case distribution for the robustproblem (PR) is a three-point distribution as character-ized in Proposition 6 in the online appendix. We verifythat under the special case when Ps becomes a con-stant, our three-point worst-case distribution reducesto the two-point distribution of Gallego and Moon(1993) (Lemma 2, Remark 2). Hence, our three-pointdistribution generalizes the literature on the min-max newsvendor problem to a more challenging set-ting when the random elements are two dimensional(demand D and price Ps).

Themin-max order quantity obtained by Scarf (1958)has been pointed out to be too conservative, becausethe order quantity would be zero under certain con-dition. However, as shown in Perakis and Roels (2008)and Wagner (2015), when we focus on the range ofcontract prices that induce a positive order quantity,the min-max order quantity is not much different thanthe risk-neutral order quantities underwell-knowndis-tributions. For our robust newsvendor problem withboth demand and price being random, we also makethe comparison. Consider the following problem set-ting: E(D) � 100, σ(D) � 30, E(Ps) � 40, σ(Ps) � 15,ρ � 0.5, f � 5. Figure 1 compares the robust order quan-tity given by Equation (18) with the risk-neutral orderquantity obtained assuming bivariate normal distribu-tion with the above given moments, by changing thewholesale price w. In this setting, the wholesale priceshould be in the range of production cost f and theexpected selling price E(Ps). However, as the upperbound on the wholesale price to guarantee a positiverobust order quantity is wUB � 36.79, we will restrict w

Figure 1. (Color online) Comparison Between the Robustand the Risk-Neutral Order Quantities

5 10 15 20 25 30 35 4050

60

70

80

90

100

110

120

130

140

150

w

Ord

er q

uant

ity

Robust quantity

Risk-neutal quantity

in this range. In fact, for many problem cases, wUB

is found to be quite close to E[Ps], so only when thewholesale price is close to the selling price, it will resultin not ordering in the robust setting. We can observefrom the comparison that the two order quantities arequite close, with the robust quantity slightly lower. Inthe following sections, we will consider the case thatthe supplier chooses a wholesale price that induces apositive order quantity from the retailer.

5. The Profit Sharing ContractDesign Problem

For the supplier, in anticipation of the retailer’s ro-bust decision Q(w) for given wholesale price w, hisambiguity-averse solution is to choose an optimalwholesale price w > f , for the given profit share γ, suchthat his worst-case profit ΠS is maximized, where

ΠS � (w − f )Q(w)+ γΠ(w)

�

(w − f )Q(w)+ γ[−wQ(w)

+ inf(Ps ,D)∈Φ(Σ0)

EPs ,D(Ps min{D ,Q(w)})]

if w 6 wUB ,

0, if w > wUB.

Note that

Π(w)�−wQ(w)+ inf(Ps ,D)∈Φ(Σ0)

EPs ,D(Ps min{D ,Q(w)})

� α(w)E(D) − σ(D)√β− α(w)2 + 1

2 E(Ps D),

for w 6 wUB, by Theorem 3.Let Π1(w) � (w − f )Q(w). The supplier’s worst-case

profit, under the profit sharing agreement, is thereforegiven by

ΠS(w , γ)�Π1(w)+ γΠ(w), for w 6 wUB.

To determine the optimal wholesale price w, the sup-plier needs to solve

w(γ) :� argmaxf 6w6wUB

{Π1(w)+ γΠ(w)}. (19)

Figure 2 presents the general shape of the supplier’srevenue (when f � 0) as a function of the wholesaleprice w and profit share parameter γ. For each γ, find-ing the corresponding optimal wholesale price for thesupplier is a complex nonlinear problem.

5.1. Relationship Between w and Q, γWe now examine how the optimal wholesale price wchanges with γ. Note that when γ� 1, the supply chainis vertically integrated and the supplier chooses w � fto maximize the supply chain profit.3 On the otherhand, when γ � 0, the problem reduces to the classical

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Figure 2. (Color online) The Worst-Case Revenue of theSupplier as a Function of (γ,w)

1.0 0.8 0.6 0.4 0.2 0 10 20 30 40

w�

Firm

’s r

even

ue

4,000

3,500

3,000

2,000

1,500

1,000

500

0

2,500

wholesale price optimization problem. For intermedi-ate value of γ, we can project out the inner retailer’soptimization problem in the supplier’s problem, tosolve a single variable optimization problem as shownin Equation (19) to obtain w(γ) as a function of theprofit sharing formula γ.

We present the following propositions.

Proposition 3. Q(w) is strictly decreasing in w ∈ [ f ,wUB].

This proposition is in line with our intuition that theretailer orders less, if the wholesale price increases.

Proposition 4. (a) w(γ) > f for all γ ∈ [0, 1).(b) The supplier’s optimal wholesale price w(γ) is decreas-

ing in γ ∈ [0, 1).

Part (a) of Proposition 4 indicates that in the distribu-tionally robust setting under a profit sharing contract,the wholesale price charged by the supplier is strictlylarger than his production cost. Part (b) states that thetwo contract parameters, w and γ, move in oppositedirections. From Proposition 2, we know that the totalprofit, Π(w), decreases in the wholesale price, so theintuition here is that if the supplier gets a larger profitshare γ, he will charge a lower wholesale price w toinduce the retailer to order more to increase the overallprofit.

5.2. Relationship Between γ and QIn anticipation of the supplier’s optimal choice of thewholesale price w∗(γ), the retailer’s problem is to deter-mine the optimal profit share parameter γ∗ to maxi-mize her worst-case profit given by

ΠR � (1− γ)Π(w(γ)).

By Proposition 2,Π(w) decreases with w. On the otherhand, w(γ) is decreasing in γ by Proposition 4. HenceΠ(w(γ)) is increasing in γ. More interestingly, theprofit (1 − γ)Π(w(γ)) is the product of two functions,one decreasing in γ, and another increasing in γ. Intu-itively, the optimal profit share parameter γ∗ for theretailer should be an interior solution. However, asthere is no closed-form expression for w∗(γ), we cannotwrite the retailer’s profit ΠR as an explicit function ofthe profit share γ, so finding the optimal γ∗ turns outto be a challenging problem and can only be obtainedthrough numerical methods.

We solve the problem indirectly by addressing thefollowing question next: How should the retailer choosethe right profit share parameter γ to ensure that the supplierwill respond with a wholesale price w(γ) such that Q �

Q(w(γ))?Let α(w) � E(Ps)/2 − w. For given wholesale price

w 6 wUB, the optimal order quantity is

Q(w)� E(D)+ σ(D) α(w)√β− α(w)2

.

We can invert the above expression to obtain

α(Q)�

√β

/√1+

(σ(D)

Q −E(D)

)2

, if Q > E(D),

−√β

/√1+

(σ(D)

Q −E(D)

)2

, if Q < E(D).

We know that the retailer will choose capacity invest-ment level of Q if the supplier sets the wholesaleprice at

w(Q)�E(Ps)/2−α(Q)

�

E(Ps)

2 −√β

/√1+

(σ(D)

Q−E(D)

)2

, if Q >E(D),

E(Ps)2 +

√β

/√1+

(σ(D)

Q−E(D)

)2

, if Q <E(D).

On the other hand, the profit seeking supplier willset the wholesale price to be w∗ if it optimizes itsprofit ΠS. Note that the first order condition of opti-mization problem (19) is necessary (butmay not be suf-ficient) for optimality, and it has to hold at w∗. We haveγ(Q)� 0 if and only if w∗ � w0, where w0 ( f <w0 6 wUB)is defined in Definition 3, while γ(Q) � 1 if and only ifw∗ � f . For γ(Q) ∈ (0, 1), by Proposition 4, w∗ is in theinterior of f and wUB. In this case, we have w∗ satisfiesthe first order optimality condition:

∂ΠS

∂w(w∗ , γ)� dΠ1

dw(w∗)+ γ dΠ

dw(w∗)� 0.

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Hence, to induce the profit seeking supplier to set awholesale price of w∗ � w(Q), the retailer only needs tochoose

γ(Q)�−(dΠ1/dw)(w(Q))(dΠ/dw)(w(Q)) . (20)

This set of parameters will ensure that the retailer’sorder quantity in the Stackelberg game is precisely Q.An explicit expression for γ(Q) is given in Theorem 4.Note that in the above arguments, w(Q) must lie in[ f ,w0], while Q must lie in [Ql ,Qu], where w0, Ql,and Qu are defined as follows:Definition 3. Define w0 � w(γ � 0) and Qu � Q( f ),Ql � Q(w0).Putting everything together, we have the following

main result:Theorem 4. The outcome of the Stackelberg game is Q ∈[Ql ,Qu]4 when the retailer chooses

γ � 1−(E(Ps)/2− f − α(Q))σ(D)β

(β− α(Q)2)3/2Q, for Q ∈ (Ql ,Qu),

where

α(Q)�

√β

/√1+

(σ(D)

Q −E(D)

)2

, if Q > E(D),

−√β

/√1+

(σ(D)

Q −E(D)

)2

, if Q < E(D).

For Q � Ql , then choose γ � 0 and for Q � Qu , then chooseγ � 1.This result is interesting as it posits a nonlinear con-

nection between 1 − γ, the profit share accrued bythe retailer, and the level of capacity investment Q bythe retailer and the wholesale price w by the supplierinvolved in the supply chain.We can now answer the main question posed in this

paper: How should the retailer determine the profit shareparameter γ to maximize the profit the retailer can obtain?Instead of optimizing over γ, which is difficult, we opti-mize over α. For a given α, where α � E(Ps)/2−w, theprofit that the retailer obtains from the supply chaincan be rewritten as a function of α by substitutingΠ�Πwst:

(1−γ)Π�(E(Ps)/2− f −α)σ(D)β

(E(D)+ (α/√β−α2)σ(D))(β−α2)3/2

×(αE(D)− σ(D)√β−α2 +E(Ps D)/2). (21)

By optimizing the above function over α ∈ [E(Ps)/2−w0,E(Ps)/2− f ] to obtain α∗, we can set the optimal profitshare parameter γ∗ as

γ∗ � 1−(E(Ps)/2− f − α∗)σ(D)β

(E(D)+ (α∗/√β− (α∗)2)σ(D))(β− (α∗)2)3/2

.

(22)

Figure 3. (Color online) The Retailer’s Profit as a Function ofα and f

Ret

aile

r’s p

rofit

300

250

200

150

100

50

0

0

5

20

10 20

f�

1510

–10

The retailer’s profit in Equation (21) is a complicatedfunction in α. The following example shows the generalshape of the retailer’s profit function with the changeof α and f .

Example 1. Consider the case when demand and priceare independent, with E(D)�100, σ(D)�50, E(Ps)�40,σ(Ps)� 15, γ � 0.2. We can plot the function (1− γ)Π interms of the parameter f and α in Figure 3.

6. Price Dependent StochasticDemand Model

In the previous analysis, we focus on the model inwhich both the demand and selling price are exoge-nous and random and can be arbitrarily correlated.The general moments matrix captures the price anddemand correlation with the cross moment termE[Ps D]. In this section we investigate the case thatthe selling price is an internal decision of firms as inmonopoly markets. In the literature, price dependentstochastic demand models have been widely used toaddress the joint pricing and inventory decisions (read-ers may refer to Petruzzi and Dada 1999 for an excel-lent review of earlier works along this line of research).Raza (2014) investigates the distribution-free approachto the newsvendor problem with pricing decision.With the linear price dependent demand model, theauthor shows that the profit function is quasi-concavein price and demand, and proposes a sequential searchprocess to solve the optimal price and robust inven-tory decisions. To internalize the pricing decision, wemodel explicitly the dependency between the sellingprice and demand by a linear price dependent stochas-tic demand function, and show that this problem is aspecial case of our general model.

To see the connection, consider the case when bothdemand and selling price are functions of a commonfactor τ, say Ps(τ) � τ, and D(τ) � a − bτ + ε, where ε

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is a random component with zero mean and standarddeviation σ. The objective is to jointly determine theselling price Ps , i.e., τ, and the order quantity Q tomaximize the resulting worst-case expected profit inthe distributionally robust setting. This setup impliesthe following conditions on the first twomoments of Psand D:

E(Ps)� τE(P2

s )� τ2

E(D)� a − bτE(D2)� (a − bτ)2 + σ2

E(Ps D)� τ(a − bτ).

The robust order quantity Q∗(τ) can be solved by Equa-tion (15) for moments given in the above equations:

Q(τ,w)� a − bτ+τ/2−w√

τ2/4− (τ/2−w)2σ,

and the worst-case expected profit is given by Equa-tion (16) as follows:

Π(τ,w)� (τ/2−w)(a − bτ)− σ

√(τ2/4− (τ/2−w)2)+ 1

2τ(a − bτ).

The optimal price can be solved by the first order con-dition of function Π(τ,w) for each w. This recovers thesolution obtained by Raza (2014) for the pricing prob-lem with limited distribution information, when priceand demand are linearly related.Our approach actually recovers more. By Equa-

tions (21) and (22), the retailer can actually jointly opti-mize theprice and theprofit sharing formula as follows:

maxα, τ

{σ(τ2/4)(τ/2− f − α)

(a − bτ+ σ(α/√τ2/4− α2))

√(τ2/4− α2)3

×((a − bτ)α− σ

√τ2

4 − α2 + τ(a − bτ)/2

)}(23)

to obtain the optimal α∗ and τ∗, and the optimal profitsharing formula γ∗ is then obtained by

γ∗ � 1−σ((τ∗)2/4)(τ∗/2− f − α∗)

Q(τ∗ , τ∗/2− α∗)√((τ∗)2/4− (α∗)2)3

. (24)

7. Numerical ExperimentsWe use a set of numerical experiments to under-stand how the proposed technique works. The follow-ing common parameters are used in the example: themean and standard deviation of the demand and pricedistributions are as follows, E(D) � 100, σ(D)� 50,E(Ps)� 40, σ(Ps)� 15. Since the optimal capacity invest-ment decision is independent of the demand-price cor-relation, without loss of generality, we set the corre-lation coefficient ρ � 0.5, and the partial demand and

Figure 4. (Color online) The Supplier’s Optimal WholesalePrice w∗ as Functions of γ

0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

�

w*

f = 0f = 5f = 10f = 15f = 20

price information is given by the following momentsmatrix:

Σ�©«

E(P2s ) E(Ps D) E(Ps)

E(Ps D) E(D2) E(D)E(Ps) E(D) 1

ª®¬�©«1,825 4,375 404,375 12,500 100

40 100 1

ª®¬ .Figure 4 depicts the supplier’s optimal choice of

wholesale price w∗ � w(γ) as functions of the profitsharing parameter γ. As expected, w∗ decreases with γ,reflecting the trade-off between the supplier’s twosources of profit. The supplier charges a higher whole-sale price w when it takes a smaller share of the profitmade by the retailer. In addition, the optimal whole-sale price increases with the supplier’s internal capac-ity cost f , for fixed γ. This makes sense, as the supplierneeds to charge a higher wholesale price to recover thehigher capacity cost.

Figure 5 shows the retailer’s optimal capacity invest-ment decision Q∗ � Q(w(γ)) as functions of the profitshare parameter γ with the corresponding wholesaleprice chosen optimally by the supplier. We can observethat Q∗ increases with γ, since the wholesale pricedecreases with γ, and Q∗ decreases with the wholesaleprice. Furthermore, Q∗ decreases with the supplier’scapacity cost f for fixed γ, because of the higherwhole-sale price charged by the supplier for higher capac-ity cost.

Figures 6 and 7 present how the worst-case profitsof the supplier and the retailer change with γ. Interest-ingly, the supplier’s worst-case profit increases with γ,so the supplier prefers the profit share γ to be as largeas possible. However, the retailer’s profit is unimodal,attaining the maximum for some γ strictly between 0and 1. Thus, the retailer would select the profit sharingparameter γ corresponding to this maximum point onits profit performance curve. This point would predict

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Figure 5. (Color online) The Retailer’s Optimal CapacityInvestment Quantity Q∗ as Functions of γ

60

80

100

120

140

160

180

200

220

240

Q*

0 0.2 0.4 0.6 0.8 1.0�

f = 0f = 5f = 10f = 15f = 20

Figure 6. (Color online) The Supplier’s Worst-Case Profit asFunctions of γ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

�

Sup

plie

r’s p

rofit

f = 0f = 5f = 10f = 15f = 20

the equilibrium of the three-stage Stackelberg game. Itis not surprising that the retailer’s profit decreaseswiththe unit cost f .

8. ConclusionIn this paper, we solve the profit sharing agree-ment problem in an ambiguity averse supply chainwith price and demand uncertainty. We formulate theproblem as a Stackelberg game, and show that com-pared with the case without profit sharing (γ � 0,i.e., the wholesale price case), the Stackelberg equilib-rium chooses a profit share parameter γ∗ in (0, 1), suchthat (i) the retailer has a higher worst-case profit and(ii) the supplier has a higher worst-case profit, and

Figure 7. (Color online) The Retailer’s Worst-Case Profit asFunctions of γ

0

100

200

300

400

500

600

700

Ret

aile

r’s p

rofit

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0�

f = 0f = 5f = 10f = 15f = 20

hence the supply chain is more efficient. If the profitshare of the supplier is smaller than this threshold γ∗,then both the retailer and the supplier will be worseoff in terms of the profit accrued to each of them. Onthe other hand, a share higher than γ∗ accrued to thesupplier will only benefit the supplier, but hurt theretailer. Thus a careful calibration of the contractualelements in the profit sharing agreements is crucial forthe retailer. Furthermore, the worst-case profit for boththe retailer and the supplier dominates the purewhole-sale price model (with no profit sharing). These resultsindicate that in an ambiguity averse supply chain, theprofit sharing agreement approach is generally morebeneficial to both parties involved, compared with thetraditional wholesale price contract.

Our paper also develops new analytical insights onthe profit share parameter and the optimal capacityinvested in the supply chain. By projecting out the roleof wholesale price w, we show that the profit shareparameter γ and the corresponding optimal order deci-sion Q should satisfy the relationship

1− γ �(E(Ps)/2− f − α(Q))σD(E(P2

s )/4)(E(P2

s )/4− α(Q)2)3/2Q,

where

α(Q)�

√

E(P2s )/4

/√1+

(σD

Q −E(D)

)2

, if Q > E(D)

−√

E(P2s )/4

/√1+

(σD

Q −E(D)

)2

, if Q < E(D).

The results obtained in this paper generalize a longseries of work in the area of distribution-free newsven-dor problems (initiated by Scarf 1958), including the

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price optimization variant studied in Raza (2014).These results can also be used to check whether theoutcomes (ordering decision, wholesale price) are con-sistent with a distributionally robust Stackelberg gamemodel, and can be extended to handle price optimiza-tionwhen themean demand is related to price througha factor model, and to deal with information asymme-try in case the supplier and retailer possess differentbeliefs on the moment conditions on demand and sell-ing price.Themodel proposed in this paper can be extended in

different directions. In our approach, we have assumedthat the supplier uses a linear wholesale price w tosell to the retailer. In practice, the supplier can alsooffer a quantity discount or other kind of contracts toextract the appropriate profits from the retailer. Thiscan be handled by splitting the order size into distinctsegments, with a different wholesale price for each seg-ment. Optimizing thewholesale prices over the distinctsegments turns out to be technically challenging, andwe are not aware of a tractable solution to this variantof the problem considered in this paper.

AcknowledgmentsThe authors thank the area editor, the associate editor, andthe referees for their valuable comments and suggestions onimproving this manuscript.

Endnotes1http://carnegieendowment.org/2011/01/13/how-to-stop-rise-in-food-price-volatility.2The study of completely positive programs has gained considerableinterests in recent years. Relevant literature includes Burer (2012)and Dü (2010).3 In this case, note that the retailer will be indifferent to the choiceof w and Q, since it gets no share of the profit. We assume thesolution concept adopted is the Strong Stackelberg Equilibrium, i.e.,the retailer will pick Q to maximize the payoff of the supplier, if thereare multiple optimal solutions available for the retailer.4The outcome of the Stackelberg game Q cannot lie outside [Ql ,Qu]when each party seeks to maximize its worst-case profit.

ReferencesAghassi M, Bertsimas D (2006) Robust game theory. Math. Program-

ming 107(1–2):231–273.Ben-Tal A, Golany B, Nemirovski A, Vial J-P (2005) Retailer-supplier

flexible commitments contracts:Arobustoptimizationapproach.Manufacturing Service Oper. Management 7(3):248–271.

Berman A, Shaked-Monderer N (2003) Completely Positive Matrices(World Scientific Publishing, Singapore).

Boyd S, Vandenberghe L (2004)Convex Optimization (Cambridge Uni-versity Press, Cambridge, UK).

Burer S (2012) Copositive programming. Anjos MF, Lasserre JB, eds.Handbook on Semidefinite, Conic and Polynomial Optimization(Springer, New York), 201–218.

Cachon GP, Lariviere MA (2005) Supply chain coordination withrevenue-sharing contracts: Strengths and limitations. Manage-ment Sci. 51(1):30–44.

Carroll G (2015) Robustness and linear contracts. Amer. Econom. Rev.105(2):536–563.

Dü M (2010) Copositive programming—A survey. Diehl M,Glineur F, Jarlebring E, Michiels W, eds. Recent Advances in Opti-mization and Its Applications in Engineering (Springer, Berlin),3–20.

Filson D, Switzer D, Besock P (2005) At the movies: The economicsof exhibition contracts. Econom. Inquiry 43(2):354–369.

Gallego G (1992) A minimax distribution free procedure for the(Q ,R) inventory model. Oper. Res. Lett. 11(1):55–60.

Gallego G, Moon I (1993) The distribution free newsboy problem:Review and extentions. J. Oper. Res. Soc. 44(8):825–834.

GallegoG, Ryan JK, Simchi-Levi D (2001)Minimax analysis for finite-horizon inventory models. IIE Trans. 33(10):861–874.

Han Q, Du D, Zuluaga LF (2014) Technical note—A risk- andambiguity-averse extension of the max-min newsvendor orderformula. Oper. Res. 62(3):535–542.

Hanasusanto GA, KuhnD,Wallace SW, Zymler S (2015) Distribution-ally robust multi-item newsvendor problems with multimodaldemand distributions.Math. Programming 152(1):1–32.

Holmström B, Milgrom P (1987) Aggregation and linearity inthe provision of intertemporal incentives. Econometrica 55(2):303–328.

Jiang H, Netessine S, Savin S (2011) Technical note: Robust newsven-dor competition under asymmetric information. Oper. Res.59(1):254–261.

Lariviere EL, Porteus M (2001) Selling to the newsvendor: An anal-ysis of price-only contracts. Manufacturing Service Oper. Manage-ment 3(4):293–305.

Mamani H, Nassiri S, Wagner M (2017) Closed-form solutions forrobust inventory management.Management Sci.63(5):1625–1643.

McMillan MS, Waxman A (2007) Profit sharing between govern-ments and multinationals in natural resource extraction: Evi-dence from a firm-level panel. NBER Working Paper 13332.

Moon I, Choi S (1995) The distribution free newsboy problem withbalking. J. Oper. Res. Soc. 46(4):537–542.

Moon I, Choi S (1997) Distribution free procedures formake-to-order,make-in-advance, and composite policies. Internat. J. ProductionEconom. 48(1):21–28.

Moon I, Gallego G (1994) Distribution free procedures for someinventory models. J. Oper. Res. Soc. 45(6):651–658.

Moon I, Silver EA (2000) The multi-item newsvendor problem witha budget constraint and fixed ordering costs. J. Oper. Res. Soc.51(5):602–608.

Pasternack BA (2002) Using revenue sharing to achieve channelcoordination for a newsboy type inventory model. Geunes J,Pardalos PM, Romeĳin HE, eds. Supply Chain Management: Mod-els, Applications, and Research Directions (Kluwer Academic Pub-lishers, London), 117–136.

Perakis G, Roels G (2008) Regret in the newsvendor model with par-tial information. Oper. Res. 56(1):188–203.

Petruzzi NC, Dada M (1999) Pricing and the newsvendor problem:A review with extensions. Oper. Res. 47(2):183–194.

Raza SA (2014) A distribution free approach to newsvendor problemwith pricing. 4OR—A Quart. J. Oper. Res. 12(4):335–358.

Scarf H (1958) A min-max solution to an inventory problem. Studiesin the Mathematical Theory of Inventory and Production (StanfordUniversity Press, Stanford, CA), 201–209.

Stiglitz JE (1989) Sharecropping. Eatwell J, Milgate M, Newman P,eds. Economic Development (Macmillan, London).

Tang SY, Kouvelis P (2014) Pay-back-revenue-sharing contract incoordinating supply chains with random yield. Production Oper.Management 23(12):2089–2102.

WagnerM (2015) Robust policies and information asymmetry in sup-ply chains with a price-only contract. IIE Trans. 47(8):819–840.

Wang Y, Gerchak Y (2003) Capacity games in assembly systemswith uncertain demand.Manufacturing Service Oper. Management5(3):252–267.

Wang Y, Jiang L, Shen ZJ (2004) Channel performance under consign-ment contractwith revenue sharing.Management Sci. 50(1):34–47.

WeinsteinM (1998) Profit-sharing contracts in Hollywood: Evolutionand analysis. J. Legal Stud. 27(1):67–112.

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137.

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http://carnegieendowment.org/2011/01/13/how-to-stop-rise-in-food-price-volatility

http://carnegieendowment.org/2011/01/13/how-to-stop-rise-in-food-price-volatility


Yue JF, Chen BT, Wong MC (2006) Expected value of distribu-tion information for the newsvendor problem. Oper. Res. 54(6):1128–1136.

Qi Fu is an assistant professor in the Faculty of Busi-ness Administration at the University ofMacau. Her researchinterests include supply chain management and applicationsof optimization theory in operations management.

Chee Khian Sim is currently a senior lecturer with theDepartment of Mathematics, and a member of the Centre forOperational Research and Logistics (CORL), at the University

of Portsmouth. His research interests include semidefiniteprogramming: theory and applications, supply chain opti-mization, and subgradient methods.

Chung Piaw Teo is Provost’s Chair Professor at NUS Busi-ness School. He is spearheading an effort to develop an Insti-tute on Operations Research and Analytics at NUS as partof the University’s strategic initiatives in the Smart NationResearch Program. His research interests lie in service andmanufacturing flexibility, discrete optimization, ports con-tainer operations, matching and exchange, and healthcare.

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