European Journal of Operational Research 223 (2012) 680–689

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Production, Manufacturing and Logistics

Aggregate production planning for process industries underoligopolistic competition

Uday S. Karmarkar, Kumar Rajaram ⇑UCLA Anderson School of Management, 110 Westwood Plaza, Los Angeles, CA 90095, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 October 2011Accepted 4 July 2012Available online 16 July 2012

Keywords:Aggregate production planningCompetitionProcess industryNonlinear programming

0377-2217/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.ejor.2012.07.007

⇑ Corresponding author.E-mail addresses: krajaram@anderson.ucla.edu

.ucla.edu (K. Rajaram).

We consider a competitive version of the traditional aggregate production planning model with capacityconstraints. In the general case, multiple products are produced by a few competing producers (oligop-oly) with limited capacities. Production quantities, prices and consequently profits depend on productionand allocation decisions of each producer. In addition, there is competition for the raw material whosesupplies are limited, and where prices reflect these limitations. Such situations have recently occurredin several process industry settings including petro-refining, petrochemicals, basic chemicals, cement,fertilizers, pharmaceuticals, rubber, paper, food processing and metals. We use a successive ‘‘Bertrand–Cournot’’ framework to address this problem and to determine optimal production quantities, pricesand profits at the producers and at the raw material supplier. Our analysis allows a new way to under-stand and evaluate the marginal value of additional capacity when there is competition for the marketand raw materials.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Traditional aggregate production planning models are usuallycast in terms of allocating scarce resources and capacity to compet-ing products (Hax, 1978). They are most commonly formulated aslinear programs that minimize costs for a single producer, withproducts, demand levels and prices given. However, in many pro-cess industries, competition, capacity constraints, and competitionfor raw material supply often cause significant market effects sothat production and allocation decisions themselves affect priceand profit. These situations occur frequently in sectors that pro-duce high volume, commodity products for which capacity isexpensive and not easily expanded. In addition, long planning hori-zons, raw material shortages and modest rates of return on invest-ments leave producers with little incentive to add capacity or tocarry excess capacity. Examples can be found across a variety ofprocess industry sectors such as petro-refining, petrochemicals,basic chemicals, cement, fertilizers, pharmaceuticals, rubber, pa-per, food processing and metals. In some cases, such as the oilrefining and steel industries, there are interactions between capac-ity constraints, shifts in demand curves and raw material availabil-ity (e.g., crude in oil-refining and scrap metal in steel).

Traditional aggregate production planning models are applica-ble in a stable economic environment, with low rates of change.

ll rights reserved.

, kumar.rajaram@anderson

Consequently, factories can operate in a relatively stable manner,with capacity and mix changes being made infrequently. However,in the last decade, we have seen rapid changes in many processindustry sectors usually brought about by competition within thesector and by economic growth in emerging economies, with theconsequent increases in demand. In such situations, raw materialavailability and processing capacity can both be significant con-straints. Here, the traditional aggregate production planning for-mulations are not able to capture the competition in theproduction tier for the raw material, nor the competitive interac-tions between producers on variables such as production quanti-ties, capacity constraints, raw material availability and price.Consequently, it is doubtful whether product mix and market allo-cation decisions made using a single-producer cost minimizationmodel could correctly reflect the interactions between decisionsof competitors and the response of markets and prices to produc-tion quantities. Indeed, we will show that concepts such as La-grange multipliers, reflecting the marginal value of capacity, canbe subject to misinterpretation to the point of being misleading.

In this paper, we formulate a competitive version of the tradi-tional aggregate production planning model with capacity con-straints. We utilize a successive Bertrand–Cournot framework inwhich the raw material supplier sets the price (i.e. the Bertrandmodel) and the producer then sets the production quantity (i.e.the Cournot model). As we will discuss, this framework is very use-ful in incorporating the interactions with the raw material supply-ing sector where supplies are limited, and where prices reflectthese limitations. We first use small examples to provide explicit

U.S. Karmarkar, Kumar Rajaram / European Journal of Operational Research 223 (2012) 680–689 681

solutions that provide insight; larger problems require numericalsolution methods. We show that it is possible to have a situationin which capacity is fully utilized, but for which the multiplier oncapacity is zero and, yet, the value of a marginal expansion ofcapacity can still be positive. Here, Lagrange multipliers can havea very different interpretation then the traditional one of marginalvalue for increases in a constrained resource. In addition, we showthat the multiplier does not capture the additional value of capac-ity for the producer due to competitive interactions between pro-ducers for the raw material used. We next develop acomputational method to solve larger problems and test this meth-od on an illustrative example. This analysis also provides insightinto the impact of production efficiencies (i.e., supplier’s and pro-ducers’ unit production costs) and market parameters (i.e., marketsize and customer price sensitivity) on prices, production quanti-ties and profits for both the supplier and the producers’ tiers.

This paper is organized as follows. In the next section, we re-view the relevant literature. In Section 3, we formulate our model.In Section 4, we examine the special case of a single product, singleraw material with a monopolistic supplier. In this context, we con-sider both homogenous and heterogeneous producers. In Section 5,we develop a computational method to solve general versions ofthis problem with multiple producers, products and raw materials.We test this method on an illustrative example. In the concludingsection, we summarize the major results of the paper and suggestfuture research directions.

2. Literature review

There is substantial literature on aggregate production planningdecisions that address production mix, capacity allocation, sea-sonal inventory planning and distribution planning with annualplanning horizons. Early formulations of employed linear program-ming (Hanssmann and Hess, 1960) and quadratic cost models (Holtet al., 1960) to represent the problem. Extensions of the linear pro-gramming model included its use in hierarchical control schemes,sometimes through column generation techniques (Lasdon andTerjung, 1971). Such models correspond to annual budgeting andsales planning cycles, and usually serve the purpose of inter-func-tional coordination. Recent surveys are presented by Nam and Log-endran (1992), Dauzere-Peres and Lasserre (1994), and Pochet andWolsey (2006). Specific applications to the process industry can befound in Dutta and Fourer (2004) and Dutta et al. (2007). However,none of this work captures the relationship between productionquantities and prices, and the interactions between capacity con-straints, shifts in demand curves and the raw material supply tier.

Models of Cournot competition have a long history going back tothe original paper by Cournot (1838). There are several streams of re-search that address multi-tier competition. Machlup and Taber(1960) provide an early overview of successive monopoly in two tiers.Much of the literature since has addressed vertical integration andcompetitiveness. Greenhut and Ohta (1979) show that vertical inte-gration of successive oligopolists leads to higher output and lowerprices. Abiru (1988) extends this to a wider range of cases. Ziss(1995) studies horizontal mergers in two-tier supply chains withtwo entrants at each tier, two part pricing and the lower tier faces lin-ear demand. They then show that upstream merger leads to higher fi-nal prices due to higher supply prices resulting from the monopoly atthe upper tier. On the other hand, if the monopoly is formed by mergerat the lower tier, the two entrants at the upper tier compete and re-duce supply prices to such an extent final process are reduced. Thesuccessive Cournot model is extended to an arbitrary number of serialtiers, by Corbett and Karmarkar (2001) who also consider entry andvertical integration. There are other substantial streams of literatureaddressing topics like vertical restraints, vertical relationships and

channel coordination. Bernstein and Federgruen (2005) study distri-bution systems with competing retailers, considering discounts andmore general contracting schemes. Carr and Karmarkar (2005) con-sider large assembly networks along with the vertical and lateral ef-fects of entry. Bernstein and DeCroix (2004) study the effect ofoutsourced sub-assembly in assembly systems. None of these abovepapers consider multi-product production planning with capacityconstraints.

The literature on capacity constrained competition is quite lim-ited. Moreover, managerial issues are rarely addressed in this liter-ature. Haskel and Martin (1994) examine the behavior of capacityconstrained producers, and conclude that they exhibit Cournot-likebehavior. Herk (1993) formulates a two-stage duopoly model withcapacity choice followed by price competition and shows thatcapacity choices exhibit Cournot behavior. Of the papers that aresimilar to the setting of our work are Karmarkar and Pitbladdo(1993, 1994) who study a two-stage, single-tier formulation withentry at the first stage and multi-product Cournot competitionwith capacity constraints at the second stage. Our work differsfrom theirs in that we focus on the tactical (second stage) problemwith fixed (given) capacity, permit heterogeneity of producers andconsider multiple products, multiple capacity constraints andinteractions with the raw material supply tier.

Previous work that is closest to our model is that of Zappe andHorowitz (1993) who examine a multi-product, multi-marketmodel and study the effect of capacity on competitive response un-der quasi-Cournot conjectures. In addition, they embed a capacitydecision into the Cournot quantity decision. As a consequence, theylimit their analysis to a problem with two identical producers, oneproduct, two capacity levels and two markets, and restrict theirinvestigation to the symmetric case. Further, they do not considerthe two-tier case that includes the raw material supplier. In con-trast, our emphasis is on developing methods to analyze planningdecisions in a multi-tier setting consisting of producers and theraw material supplier.

This paper extends the traditional aggregate planning model toinclude the case when there are a few competing producers (oli-gopoly) who produce multiple products under capacity con-straints. The production quantity of each product across allproducers determines the price and eventually profits. Further,there is competition in the production tier for the raw materials.Such situations have most recently been observed in the petro-refining (Coy, 2004) and copper processing sectors (Morrison,2005). To the best of our knowledge, this is the first paper in the lit-erature to consider this problem. In this context, this paper makesthe following contributions:

(1) We develop methods to address large-sized real problemsefficiently. Our methods explicitly calculate prices and pro-duction quantities at the producers and raw material sup-plier level and show how they can be different from thetraditional aggregate production planning model withoutcompetition. This is important to researchers and practitio-ners as if we ignore competitive effects the results are mis-leading and could lead to the wrong production planningquantities. Our results are relevant across several processindustry sectors such as petro-refining and metalsprocessing.

(2) Our methods develop managerial insight by developing anew way to understand and evaluate the marginal value ofadditional capacity when there is competition for the marketand raw materials. This new understanding of the marginalvalue of capacity expansion is very important to make thecorrect production planning and capacity expansion deci-sions to succeed in a competitive environment with restric-tions on raw material availability.

682 U.S. Karmarkar & K. Rajan / European Journal of Operational Research 223 (2012) 680–689

(3) We conduct extensive numerical analysis based on a realis-tic example and provide managerial insight on how marketparameters and production efficiencies affect price, produc-tion quantity and profitability at the supplier and the pro-ducer’s tier.

3. Model formulation

Consider m producers indexed by i 2 (1,2, . . . ,m) producing ncommodity products indexed by j 2 (1,2, . . . ,n). Let variable qij rep-resent the production quantity for product j at producer i andQj ¼

Pmi¼1qij denote the total amount of product j available in the

market. For example, in the context of the petrochemical industry,these producers can be considered as the refiners of crude oil (e.g.Shell, Exxon Mobil, Chevron, etc.), while products could be the dif-ferent grades of gasoline (e.g. Regular, Plus and Supreme). We as-sume that demand for product j is characterized by an affineinverse demand function pj = aj � bjQj, where pj is the price forproduct j, while aj and bj are parameters. Here, aj/bj can be consid-ered as the market size and 1/bj can be regarded as the customerprice sensitivity. This can also be viewed as an affine approxima-tion of the actual demand function and has been commonly usedin the literature (Karmarkar and Pitbladdo, 1993; Corbett and Kar-markar, 2001). We are given:

vij: production cost per unit of product j at producer i ($/unit),cij: capacity required per unit of product j at producer i (capacity

units/unit), anddi: total capacity available at producer i.We assume quantity competition at the producers to model

production planning decisions under competition, as price compe-tition with more than one producer will lead to marginal cost pric-ing. If raw material prices are assumed to be exogenous, we canplausibly utilize a Cournot framework in which producers makedecisions on production quantities by allocating capacity acrossproducts and by choosing how to allocate production across mar-kets. Given a demand curve, this establishes product prices. How-ever, in process industry setting, it is unrealistic to assume rawmaterial prices are exogenous. This is because there are interac-tions with the raw material supplying sector, where supplies arelimited, and where raw material prices reflect these limitations.This situation occurs in oil production, where the supply of crudeis often controlled by a group of countries such as the OPEC (Grif-fin, 1985). It also occurs in a number of other industries, for whichthe item in limited supply could be a raw material commodity likecopper, a rare material like uncut diamonds or a manufacturedproduct like DRAM chips.

To include interactions with the raw material supply tier, weconsider a monopolist price setting supplier and consider r rawmaterials indexed by ke(1,2, . . . ,r). Further, let pk represent theprice of these raw materials, vk represent production costs per unitof the raw material and rijk be a consumption factor representingthe number of units of raw material consumed for the productionof a unit of product j at producer i. We model the interaction be-tween the raw material supplier and the producers using a succes-sive ‘‘Bertrand–Cournot’’ framework. In this framework, thesupplier sets raw material prices to maximize profits subject tocapacity constraints on raw material production. At this price,the raw material production quantity equals the total require-ments across all producers. Given these raw material prices, theproducers solve the Cournot game to determine the optimal pro-duction quantities by allocating capacity across a set of products.Given a demand curve for the product, this establishes prices forthe end customer. The successive ‘‘Bertrand–Cournot’’ frameworkis similar in concept to the successive Cournot framework com-monly used in the literature (Machlup and Taber, 1960; Greenhutand Ohta, 1979; Abiru, 1988; Corbett and Karmarkar, 2001).

However the successive Cournot framework cannot be always em-ployed in our context, as in general, the raw material demand func-tion may not be invertible.

Using the successive ‘‘Bertrand–Cournot’’ framework, producer iconsiders the following problem:

ðP1Þ Pi ¼ MaxXn

j¼1

pj � v ij þXr

k¼1

pkrijk

! !qij

( )ð1Þ

S:t:Xn

j¼1

cijqij 6 di ð2Þ

qij P 0; 8j ð3Þ

Objective function (1) maximizes profits at producer i by choos-ing the appropriate production quantity, qij, given the productionquantities at the other producers (i.e., qpj, "p – qi). Constraint (2)ensures capacity limits at producer i, while non-negativity con-straints are enforced by (3).

When raw material prices pk, "k is set by the supplier, then theequilibrium production quantities and prices at the producers canbe easily found. Here we substitute pj ¼ aj � bj

Pmi¼1qij

� �in (1) and

note that (P1) is a concave optimization problem. We then relaxconstraint (2) by introducing Lagrange multiplier ki to get the fol-lowing dual problem for the ith producer:

PDi ¼MaxqijP0;kiP0

Xn

j¼1

aj�bj

Xm

i¼1

qij� v ijþXr

k¼1

pkrijk

!�kicij

!qij

( )þkidi

ð4Þ

From (4), the sufficient first order conditions with respect to qij canbe found by setting @PD

i@qij¼ 0, so that:

2qij þXt–i

qtj ¼aj � v ij þ

Prk¼1pkrijk

� �� kicij

bj; 8i; j ð5Þ

Proposition 1. There exists a unique vector of equilibrium orderquantities q�ij; 8i; j.

Proof All omitted Proofs are provided in the supplemental on-line appendix. h

In light of Proposition 1, we can find the equilibrium productionquantities q�ij; 8i; j, by solving the system of equations defined in(5) to get:

q�ij¼aj�m v ijþ

Prk¼1pkrijkþkicij

� �þXt–i

v tjþPr

k¼1pkrtjkþktctj� �( )

ðmþ1Þbj8i;j ð6Þ

The total quantity q�j of product j in the market is given by:

q�j ¼X

i

q�ij

¼ maj

ðmþ1Þbj

þP

i

Pt–i v tjþþ

Prk¼1pkrtjkþktctj

� ��m v ijþ

Prk¼1pkrijkþkicij

� �� �ðmþ1Þbj

ð7Þ

As expected, it can be observed from (6) and (7) that for a given j; q�ijand q�j increase when market size increases, and when productioncosts, capacity requirements and costs of capacity expansion in-crease for competing producers (i.e. "t, t – i). On the other hand,q�ij and q�j decrease when the number of competing producers

U.S. Karmarkar, Kumar Rajaram / European Journal of Operational Research 223 (2012) 680–689 683

increase, and when capacity requirements, costs of capacity expan-sion and production costs increase at producer i. The equilibriumprice p�j for product j is given by:

p�j ¼aj�bj

Xi

q�ij

¼aj�

Pi

Pt–i v tjþ

Prk¼1pkrtjkþktctj

� ��m v ijþ

Prk¼1pkrijkþkicij

� �� �ðmþ1Þ 8j

ð8Þ

It can be seen from (8) that the equilibrium price increases as mar-ket size increases, but decreases as the number of producers in-creases. To find the marginal value of capacity expansion underthe equilibrium production quantities, we can use (5) to find ki,the Lagrange multiplier corresponding to producer i as:

ki ¼aj � v ij þ

Prk¼1pkrijk

� �� bj 2q�ij þ

Pt–iq

�tj

� �cij

: ð9Þ

This shows that for a given producer, the Lagrange multiplier repre-senting the marginal value of capacity expansion is increasing withmarket size for the product, but decreasing with production costsand production quantities at this producer. In addition, observe thatthe marginal value of capacity expansion at any producer is reducedby the total equilibrium production quantities produced by thecompetitors. This effect is not captured by the traditional, non-com-petitive Lagrange multiplier.

Next, we compare the marginal value of capacity expansion andthe equilibrium production quantities and prices with the NonCompetitive (NC) or monopolistic case. To perform this analysis,we set i = (NC), m = 1 in (6) and (8) to get the production quantitiesand prices in this case as:

q�ðNCÞj ¼aj � v ðNCÞj þ

Prk¼1pkrðNCÞjk � kðNCÞcðNCÞj

2bjð10Þ

p�ðNCÞj ¼aj þ v ðNCÞj þ

Prk¼1pkrðNCÞjk þ kðNCÞcðNCÞj

2ð11Þ

Proposition 2. When the capacity constraint is not binding and whenproduction costs are equal across all producers, the monopolistproduces less of the product at a higher price.

Our analysis thus far considers any raw material prices pk, "kset by the supplier. However, finding the optimal pk is intricateand requires constructing the demand function Qk(pk) for eachraw material k, and then solving the supplier’s problem:

ðP2Þ Po ¼ MaxXr

k¼1

fðpk � vkÞQ kðpkÞg ð12Þ

S:t:Xr

k¼1

ckQ kðpkÞ 6 do ð13Þ

pk P 0

Here, ck is the capacity required to produce one unit of raw materialk and d0 is the total units of capacity available at the supplier. Tosolve (P2), we first need to construct Qk(pk), "k. To develop this

function, let Qk ¼Pm

i¼1

Pnj¼1rijkqij represent the total amount of

raw material k required across all producers. We can use (6) in

Qk ¼Pm

i¼1

Pnj¼1rijkqij to construct Qk(pk). We can then substitute

function Qk(pk) into (12) and (13) and solve (P2) to find the optimalraw material prices. The equilibrium producer’s quantities andprices can be calculated by using these optimal raw material pricesand employing (6) through (8) along with the conditionPn

j¼1cijq�ij ¼ di; 8i. In Section 5, we develop an efficient method to

execute this procedure for large real problems. We next considersmaller versions of this problem to develop insight into how prices,production quantities and capacity constraints interact across theproduction and supply tiers.

4. Special cases

There are some very significant strategic interactions that occurbetween the raw material supplier tier and the product processingtier. Some of these interactions are best revealed by examiningsimplified versions of the problem. As a start, we consider theproblem in which there are multiple producers making a singleproduct with a capacity constraint, and in which there is a singleraw material supplied by a monopolist supplier. This setting allowsus to derive insights into the relationship between capacity con-straints and supply decisions and into the interpretation of La-grange multipliers in a multi-tier competitive setting.

In the context of this simplified problem, we are able to demon-strate that, when all producers are homogenous with respect tocapacity limits, there are systematic situations for which the La-grange multipliers for the capacity constraints are zero, but a mar-ginal increase in capacity can result in positive benefits for theproducers. In short, the usual interpretation of Lagrange multipli-ers is not valid. We further show that when producers are hetero-geneous and have varying capacity constraints, the Lagrangemultipliers are generally not zero, but, again, their interpretationrequires care. We show that in the latter case there is a criticalmarginal producer, whose capacity constraint has a significant ef-fect on the entire production tier.

4.1. Homogenous capacity across producers

Consider the scenario in which there are m producers in theproduction tier, a single product is manufactured and all producershave equal variable manufacturing costs and equal capacity so thatvi = v and di = d, "i = 1,2, . . . ,m. In addition, there is a single rawmaterial required (i.e., k = 1), and we can assume without loss ofgenerality that one unit of the raw material is required for one unitof the product (i.e., r = 1). Further, let p0 represent the price of thisraw material. The problem for producer i is:

ðP3Þ ePi ¼Max a� bX

i

qi � ðv þ p0Þ !

qi

( )S:t: qi 6 d and qi P 0

The supplier considers the following problem:

ðP4Þ ePo ¼Max ðp0 � voÞeQ ðp0Þn o

S:t: eQ ðp0Þ 6 d0; and p0 P 0

Proposition 3. The only possible solutions for (P3) are "i, orq�i ¼ q� < d 8i.

Proposition 3 implies that if the supplier has capacity d0 > md,then they would sell at price p0 = p⁄, so that at this price, each pro-ducer would order d and the total supplier production quantitycorresponds to md. When d0

6md, then the supplier would sellat price po, so that p⁄ 6 po

6 a � v. At this price, each producer or-ders q⁄, where q⁄ 6 d and the total raw material production quan-tity is mq⁄. Fig. 1 shows the demand function faced by themonopolist supplier.

To determine q⁄ and p⁄, we consider the problem for producer iand note that due to Proposition 3, this now reduces toPi = (a � bmq � v � po)q. The optimal production quantity q⁄ can

be obtained by setting @Pi@q ¼ 0, so that q� ¼ ða�v�p0Þ

bðmþ1Þ . The total

684 U.S. Karmarkar & K. Rajan / European Journal of Operational Research 223 (2012) 680–689

quantity of product in the market is Q ¼ mq� ¼ mðmþ1Þ

ða�v�p0Þb . From

Fig. 1, note that p0 = p⁄ is chosen so that Q ¼ mðmþ1Þ

ða�v�p�Þb ¼ md. This

implies that p⁄ = a � v � b(m + 1)d. We next analyze the marginalvalue of capacity expansion in this context by examining the La-grange multipliers in (P3).

Proposition 4. At the equilibrium price, the Lagrange multipliers onthe capacity constraints in the producers’ decision problems (P3) areidentically zero, i.e., ki = 0, "i.

The usual interpretation of a Lagrange multiplier is that it rep-resents the marginal increase in value resulting from a marginalchange in a constrained resource. If that interpretation held here,Proposition 4 would imply that there would be no increase in valuefrom increased production capacity, under any circumstance. Weshow that this is not the case under the conditions defined in Prop-osition 5.

Proposition 5. When the supplier’s capacity constraint is not bindingand when d < d� ¼ ða�v�voÞ

2bðmþ1Þ , there is a positive change in the ith

producer’s profit when its capacity is increased by o, o ? 0.

If a producer solves (P3) in a myopic single-tier manner, as ex-pected by Proposition 4, the Lagrange multiplier would be zero andthey could conclude that the marginal value of capacity expansionis zero. However, as shown by Proposition 5, this conclusion maynot be true if we consider multi-tier effects. Propositions 4 and 5together imply that the usual interpretation of Lagrange multipli-ers is not valid in this competitive, multi-tier context. In this set-ting, as described in the proof of Proposition 4, the best responseto supplier price p⁄(d) is qi = d. This implies that although the pro-ducers are at capacity, they would not gain by producing more.However, if p⁄ were anything else, say p⁄(d + d), then the Lagrangemultipliers will not be zero. This is shown in Proposition 5 whichgives the comparative statics in equilibrium.

We now consider the supplier under the conditions outlined inProposition 5. The supplier’s initial profit is Po(md) = (p⁄ � vo)md =(a � v � b(m + 1)d � vo)md. When the supplier changes capacity tomeet the increased supply requirement of d by each of the m pro-ducers, their new capacity is m(d + @). The resulting profit is nowPo(m(d + @)) = (a � v � b(m + 1)(d + @) � vo)m(d + @). The change

in profit due to this capacity change is lim@!0Poðmðdþ@ÞÞ�PoðmdÞ

@

¼ p�m� bmðmþ 1Þd, where p⁄ = a � v � b(m + 1)d. Therefore,

lim@!0Poðmðdþ@ÞÞ�PoðmdÞ

@> 0, if p⁄ > db(m + 1). This will hold when

the optimal supplier’s price is sufficiently high, which would enticethe supplier to marginally expand capacity and sell more productor when the producer’s capacity d is sufficiently low providingthe basis for capacity expansion. This analysis shows that under

Supp

lier’s

Pro

duct

ion

Qua

ntity

Supplier’s Pricep0

Q

)( va −*p

md

Fig. 1. Demand function for the supplier with homogenous producers.

these circumstances, it could be profitable for both producersand the supplier to expand capacity by a small amount. However,were the producers to only consider the myopic problem (P3), byProposition 3, they might erroneously conclude that the marginalvalue of capacity expansion is zero. This could lead to the scenariowhen both the supplier and producers may not even marginally in-crease their capacity, when this could be profitable for both ofthem.

4.2. Heterogeneous capacity across producers

Now consider a scenario in which capacity differs across theproducers, but without loss of generality the producers still haveidentical variable manufacturing costs. The problem for produceri is now given by

ðP5Þ bPi ¼Max ða� bX

i

qi � ðv þ pÞÞqi

( )S:t: qi 6 di and qi P 0

The supplier considers the following problem:

ðP6Þ bPo ¼ Max ðpo � voÞbQ ðp0Þn o

S:t: bQ ðp0Þ 6 d0 and p0 P 0

Note that (P6) is similar in structure to (P4), but the imputed

demand function bQ ðp0Þ is different. Fig. 2 shows the demand func-tion for this case. We explain how Fig. 2 is constructed in the Sup-plemental Online Appendix and show that the demand functionbQ ðp0Þ ¼minu¼1 to mfQuðp0Þg. Here each line segment Qu(p0) =

Cu � Dup0, with Cu ¼ m�uþ1m�uþ2

� �a�v�b

Pu�1

i¼1di

b

� and Du ¼ m�uþ1

m�uþ2

� �1b

� �. In

addition, Qu(p0) is defined in the range Qu < Qu(p0) < Q(u�1), "u = 1

to m, where the break points Q u ¼Pu�1

i¼1 di þ ðm� uþ 1Þdu foru = 1 to m and Q0 = 0.

The corresponding inverse demand function for the raw mate-

rial is pu(Q) = Au � BuQ, where Au ¼ a� v � bPu�1

i¼1 di

� �and Bu ¼

bðm�uþ2Þðm�uþ1Þ . Here, each line segment pu (Q) is defined in the range

pou < puðQÞ < po

ðu�1Þ, where break points p0u ¼ a� v � b

Pu�1i¼1 di � b

ðmþ 2� uÞdu for u = 1 to m and p00 ¼ a� v .

Proposition 6. The supplier’s problem with heterogeneous capacityacross producers is a concave optimization problem.

The proof of Proposition 6 shows that the equilibrium solutioncan either lie at a break point on the demand curve for the supplierat which some producer w just reaches capacity, or between breakpoints (w � 1) and w for some w at which producers 1 to (w � 1)are at capacity, while producers w to m are below capacity. We de-note the wth producer as the marginal producer and let cw be themarginal value of capacity expansion at the two cases of the equi-librium solution. Further, let kw denote the Lagrange multiplier gotby solving (P5) for firm w.

Proposition 7. When the equilibrium solution lies between the (w �1)th and wth break point, ki = ci = 0,"i = w to m and ki < ci, "i = 1 to

w � 1, where ci ¼ ki þ b ðm�wþ2Þðm�wþ1Þ.

Observe from Proposition 7 that ci ¼ ki þ b ðm�wþ2Þðm�wþ1Þ, which repre-

sents the additional value of capacity expansion to this producerdue to the competitive interactions between producers for theraw material. Note that this additional value increases as the num-ber of producers who are at capacity increase (i.e., as (w � 1) in-creases), so that this producer uses this additional capacity to

Supp

lier’s

Pro

duct

ion

Qua

ntit y

Supplier’s Price p0)( va −0

mp

∑=

=m

iim dQ

1

uQ

2

12)1( dm

dQ−

+=

11 mdQ =

0up

02p

01p

Qm1

Q

Fig. 2. Demand function for the supplier with heterogeneous producers.

U.S. Karmarkar, Kumar Rajaram / European Journal of Operational Research 223 (2012) 680–689 685

strengthen their market position. The multiplier ki does not cap-ture this effect and thus caution must be used when we interpretthis parameter in this multi-tier, competitive context.

Now consider the supplier. Initially, the optimal profit for the

supplier is bPo ¼ ðp̂0 � voÞbQ . If the supplier increases the supplyof raw material to accommodate producer i, then the supplier’s

profit is bPonew ¼ ðp̂0 � vo � ðbþ BwÞ@ÞðbQ þ @Þ. Therefore,

lim@!0

bPo�bPonew

@¼ ðp̂0 � voÞ � ðAw þ voÞ ðm�wþ3=2Þ

ðm�wþ2Þ . This implies that

lim@!0bPo�bPo

new@

> 0, if ðp̂0 � voÞ > ðAw þ voÞ ðm�wþ3=2Þðm�wþ2Þ . This inequality

shows that it is profitable for the supplier to increase the rawmaterial supply to accommodate producer i when the existingprofit margin is greater than a threshold. Note that as w or thenumber of producers who are at capacity increases, this thresholddecreases. This is because the price reduction due to increase insupply affects only producers who are not at capacity and now thiswould impact fewer high volume producers who are not yet atcapacity. This analysis shows that while it is possible that boththe producer and the supplier can make higher profits, solvingthe myopic, single-tier problem underestimates the marginal valueof capacity expansion and the producers, if myopic, may notchoose to increase capacity.

Proposition 8. When the equilibrium solution lies at the wth breakpoint, ki = ci = 0, "i = w + 1 to m and ki < ci, "i = 1 to w, whereci = ki + bdi.

Observe from Proposition 8 that ci = ki + bdi, which representsthe additional value of capacity expansion to this producer dueto the competitive interactions between producers for the rawmaterial. The multiplier ki will not capture this effect and thus cau-tion must be used when we interpret this parameter in this multi-tier, competitive context.

Now consider the supplier. Initially, the optimal profit for the

supplier is bPo ¼ p0w � vo

� �Q w. If the supplier increases the supply

of raw material to accommodate producer w, then the supplier’s

profit is now bPonew ¼ p0

w � vo � b@ðm�wþ 2Þ� �

ðQ w þ ðm�wþ 1Þ

@Þ. Therefore, lim@!0bPo�bPo

new@

¼ p0w � vo

� �� b ðm�wþ2Þ

ðm�wþ1ÞQ w. This implies

that lim@!0bPo�bPo

new@

> 0, if p0w � vo

� �> b ðm�wþ2Þ

ðm�wþ1ÞQw. This inequality

shows that it is profitable for the supplier to accommodate pro-ducer w only when the current profit margin is greater than athreshold. Note that, unlike the case when the solution is in be-tween break points, as w or the number of producers who are atcapacity increases, this threshold also increases. This is becausethere would be fewer producers to sell the additional supply, whileprice reduction due to this additional supply is felt across all pro-

ducers. This analysis again shows that while it is possible that boththe producer and supplier can increase their profits, the producermay not increase capacity even by a small amount, perhaps be-cause solving the myopic, single-tier problem underestimates themarginal value of capacity expansion and because they are also un-sure if the supplier would provide additional raw material.

In both the scenarios discussed in Propositions 7 and 8, pre-dicting whether the supplier would provide additional raw mate-rial is complicated. This is because it requires knowledge of theraw material supplier’s margin and demand function, and under-standing whether the raw material price is located at a breakpoint or at a line segment and knowing how many producersare below and above this region. This, coupled with the underes-timation of the marginal value of expansion at the producerstier, may explain in part why both producers and the suppliercould be reluctant to increase capacity even by small amounts.Such reluctance is particularly evident in the US petro-refiningindustry where the last refinery was built in 1976 (Hargreaves,2007).

5. Computational method

We next develop a method to solve the general versions of theproducer’s problem (i.e. (P1)) and the supplier’s problem (i.e. (P2)).For raw material prices pk, k = 1 to r, the equilibrium productionquantities are given by (6). Let si be the slack variable associatedwith capacity constraint (2). The complimentary slackness condi-tions associated with these constraints at the equilibrium produc-tion quantities are:

Xn

j¼1

cijq�ij þ si ¼ di; 8i ¼ 1 to m; and ð14Þ

kisi ¼ 0 8i ¼ 1 to m: ð15Þ

Rather than explicitly construct the inverse demand function,Qk(pk), required to solve (P2) and determine the optimal pk, "k,we implicitly represent this function by defining Qk ¼

Pmi¼1

Pnj¼1

rijkq�ij and introduce this as constraints along with (6), (14) and(15) in (P2) to get:

ðP7Þ Po ¼MaxXr

k¼1

fðpk � vkÞQ kÞg

S:t: ð6Þ; ð14Þ; ð15Þ; andXr

k¼1

ckQ k6 do

; ð16Þ

Q k ¼Xm

i¼1

Xn

j¼1

rijkq�ij; 8k ð17Þ

pk; ki; si P 0; 8i; k:

Proposition 9. Solving (P7) is equivalent to solving (P1) and (P2).

Proof of Proposition 9 observe that (P7) consists of (P2) and def-initional constraints (6), (14) and (15) that represent the optimalsolution of (P1) that is obtained for any value of pk, including itsoptimal value. Therefore solving (P7) is equivalent to solving (P1)and (P2). h

We tried to solve realistic sized instances of (P7) using leadingcommercial software programs such as GAMS (Brooke et al., 1992)and CPLEX (1995) loaded on a Dell Optiplex PC. However, we were

686 U.S. Karmarkar & K. Rajan / European Journal of Operational Research 223 (2012) 680–689

unable to generate feasible solutions due to complementarity con-straints (15). In general, solving non linear programs with comple-mentarity constraints are complicated and require specializedtechniques (Harker and Pang, 1990). Consequently, we developeda procedure to solve (P7). The following steps outline this procedure.

Step 1: Initialization: Set a ¼ 0; k0i ¼ 0 and ki ¼ kðaÞi ¼ 0.

Step 2: Solve:

ðP8Þ Po ¼MaxXr

k¼1

fðpk � vkÞQkÞg

S:t: ð6Þ; ð16Þ; ð17Þ and pk P 0; 8k

Note that in (P8), Qk can be substituted in the objective function andin (16) using (6) and (17). This reduces (P8) to a standard concavequadratic programming problem in pk, which can be solved usingcommercially available software such as Matlab (MathWorks Inc.,1998). Let p

_k8k represent the optimal solution to this problem.Step 3: Use p

_k in (6) to find q_�

ij; 8i; j: and use q_�

ij in (17) to findbQ k.Step 4: Use q

_�ij in (16) to compute s

_

i8i. Let P ¼ fp j p_

p < 0g. If P = {},set ki = 0, "i and stop. Otherwise, set ki = 0, "i R P andkðaþ1Þ

i ¼ kðaÞi þ @; @ 2 Rþ, set ki ¼ kðaþ1Þi ;a! aþ 1; 8i 2 P. Go to Step 2.

Proposition 10. Steps 1 through 4 provide an optimal solution to(P7).

Proof of Proposition 10 observe that for any value of raw mate-rial prices pk, "k, the producer’s problem is a concave optimizationproblem. Therefore, the Kuhn–Tucker conditions are necessary andsufficient for this problem. Since this procedure enforces theKuhn–Tucker conditions for any value of raw material prices,including the optimal raw material prices p

_k; 8k, this procedureprovides an optimal solution to (P7). h

In light of Propositions 9 and 10, we can use this procedure tofind the optimal solution to problems (P1) and (P2). We use thisprocedure in the following illustrative example.

5.1. An illustrative example

To better understand how production efficiencies (i.e., sup-plier’s and producers’ unit production costs) and market parame-ters (i.e., market size and customer price sensitivity) affectprice, production quantities and profits at both the supplier and

Table 1Summary of parameters used in illustrative example.

ProducersProductsRaw materialsRaw material cost per unitCapacity required to produce unit of raw materialTotal capacity available at supplier

Producer i = 1 i = 2 i = 3 i = 4

Production cost product 1 v11 = 2 v21 = 0.5 v31 = 0.5 v41 = 1Production cost product 2 v12 = 1.5 v22 = 1 v32 = 1 v42 = 0.5Production cost product 3 v13 = 1 v23 = 2 v33 = 2.5 v43 = 1.5Raw material consumption factor

– product 1r111 = 0.03 r211 = 0.03 r311 = 0.02 r411 = 0.0

Raw material consumption factor– product 2

r121 = 0.01 r221 = 0.01 r321 = 0.01 r421 = 0.0

Raw material consumption factor– product 3

r131 = 0.01 r231 = 0.03 r331 = 0.01 r431 = 0.0

Capacity required product 1 c11 = 1.5 c21 = 2.5 c31 = 0.5 c41 = 2.5Capacity required product 2 c12 = 0.5 c22 = 1.5 c32 = 2.5 c42 = 2.5Capacity required product 3 c13 = 2 c23 = 1 c33 = 1 c43 = 1

Total capacity 1000 1500 1000 1500

producers tiers, we considered a 10-producer, three-product,one-raw material problem. This example was chosen to be realisticand correspond to the 10 major petrochemical refining companiesin the United States (Platts, 2006), each of whom produce threegrades of gasoline (i.e., regular, plus and supreme) refined fromcrude oil (i.e., raw material). The parameters used in this exampleare summarized in Table 1.

For this example, we first solved (P7) using Steps 1–4 of this pro-cedure, which was programmed in Matlab. We then changed thesupplier’s unit production cost from the base level in incrementsof 10% from�50% to 50%. Fig. 3 illustrates the corresponding changein price and production quantity for the raw material and products.As expected, the price of the raw material increases with an increasein raw material unit production costs. But this, in turn, leads to lessdemand for the raw material and, consequently, lower profits for thesupplier. For the producer, an increase in raw material price in-creases the price offered to end customers. This lowers end customerdemand and hence their production quantities for the products.However, the impact on producer’s profit depends on the consump-tion factor (i.e., rijk) representing the rate at which the raw material isconsumed by the producer to provide the end product. In particular,profits go up for producers with lower consumption factors, whilethey go down for producers with higher consumption factors. Theimplications in the petro-refining industry are that producers couldbenefit from technologies that decrease consumption factors (i.e.,improve yields) particularly when raw material prices (i.e., crudeprices) increase.

To understand the impact of a producer’s unit production costson price, quantity and profit at the supplier and producers, we var-ied the value of this parameter across the three products and 10producers from the base level in increments of 10% from �50% to50%. Fig. 4 illustrates the corresponding change in price and pro-duction quantity for the raw material and products, and the result-ing profits for the supplier and producers. With an increase in unitproduction costs at the producer, these costs are passed onto cus-tomers and end product prices increase. This reduces end productdemand and consequently the producer’s production quantity. Thisin turn leads to a decline in producer’s profits. As the total produc-tion quantities decrease, demand for raw material decrease, whichcauses raw material production quantities and prices to drop. Thiscontributes to a decline in supplier profits. Thus, a decrease in pro-duction efficiencies at the producers causes profits for both thesupplier and producers to decline.

m = 10; i = 1–10n = 3; j = 1–3r = 1; k = 1v1 = 1c1 = 1d0 = 11,000

i = 5 i = 6 i = 7 i = 8 i = 9 i = 10

v51 = 0.5 v61 = 0.5 v71 = 2.5 v81 = 0.5 v91 = 0.5 v(10)1 = 2v52 = 2.5 v62 = 2.5 v72 = 1 v82 = 1.5 v92 = 0.5 v(10)2 = 1.5v53 = 2.5 v63 = 2 v73 = 0.5 v83 = 2.5 v93 = 0.5 v(10)3 = 2

2 r511 = 0.02 r611 = 0.02 r711 = 0.01 r811 = 0.01 r911 = 0.03 r(10)11 = 0.02

2 r521 = 0.02 r621 = 0.01 r721 = 0.03 r821 = 0.01 r921 = 0.02 r(10)21 = 0.01

3 r531 = 0.02 r631 = 0.02 r731 = 0.02 r831 = 0.01 r931 = 0.02 r(10)31 = 0.02

c51 = 0.5 c61 = 1.5 c71 = 0.5 c81 = 2 c91 = 1 c(10)1 = 2.5c52 = 1 c62 = 1.5 c72 = 1.5 c82 = 0.5 c92 = 0.5 c(10)2 = 1.5c53 = 2.5 c63 = 2 c73 = 1 c83 = 2.5 c93 = 2 c(10)3 = 1

1000 1500 1000 1500 1000 1500

-5%

-4%

-3%

-2%

-1%

0%

1%

2%

3%

4%

5%

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50%

Change in Unit Raw Material Production Costs (%)

Chan

ge in

Pric

e (%

)

Raw MaterialProduct 1Product 2Product 3

-5%

-4%

-3%

-2%

-1%

0%

1%

2%

3%

4%

5%

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50%

Change in Unit Raw Material Production Costs (%)

Chan

ge in

Pro

duct

ion

Qua

ntity

(%)

Raw Material Product 1 Product 2 Product 3

Fig. 3. Change in prices and production quantities with changes in unit costs of raw material.

U.S. Karmarkar, Kumar Rajaram / European Journal of Operational Research 223 (2012) 680–689 687

Finally, we wanted to analyze how market size and customerprice sensitivity affect price, production quantity and profit atthe supplier and producers. Recall that the market size and cus-tomer price sensitivity for product j are aj/bj, and 1/bj, respectively.To vary market size for a product, we varied aj across the threeproducts from the base level in increments of 10% from �50% to50%. This analysis showed that as market size increased, as ex-pected, prices, production quantities and profits increased at boththe supplier and the producers, and the converse also holds. Thisseems consistent with trends in the petrochemical industry. Here,an increase in market size due to emerging markets like China anddue to gas guzzling vehicles such as SUV’s (U.S. Census BureauNews, 2004), has been observed to increase prices, productionquantities and profits for refiners and the crude oil supplier (En-ergy Information Administration, 2006). We also analyzed the im-pact of customer price sensitivity by varying bj across the threeproducts from the base level in increments of 10% from �50% to50%. Here, we also changed aj as required to ensure that marketsize remained unchanged. This analysis showed that as customerprice sensitivity decreased, prices, production quantities and prof-its increased at both the supplier and the producers, and the con-verse also holds. Again, this seems consistent with trends in thepetrochemical industry, in which customers seem more insensitiveto gas prices due to life style choices (Victorian Transport PolicyInstitute, 2006) and this has led to an increase in prices, productionquantities and profits for refiners and the crude oil supplier (En-ergy Information Administration, 2006).

6. Conclusions

We have considered a competitive version of the traditionalaggregate production planning with capacity constraints. We de-velop a model to include interactions with the raw material sup-plying sector for which supplies are limited and prices reflectthese limitations. Here, we use a successive ‘‘Bertrand–Cournot’’framework in which a monopolist supplier sets raw material pricesto maximize profits so that the raw material production quantityequals the total requirements across all producers. Given theseraw material prices, the producers solve the Cournot game todetermine the optimal production quantities by allocating capacityacross a set of products.

To understand the strategic interactions between the raw mate-rial supplier and the product processing tier, we examine simpli-fied versions of this problem. In particular, we consider thesingle-raw-material, single-product problem with homogenousand heterogeneous capacity across producers. When producersare homogenous with respect to capacity limits, there are system-atic situations for which Lagrange multipliers for capacity con-straints are zero, but a marginal increase in capacity can result inpositive benefits for the producers. Therefore, the usual interpreta-tion of Lagrange multipliers is not valid. We also show that it isprofitable for the supplier to increase capacity marginally and sup-ply to the producers when the optimal price set by the supplier ishigh or the producer’s capacity is sufficiently low. While such mar-ginal increases in capacity could be profitable to both the supplier

-60%

-40%

-20%

0%

20%

40%

60%

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50%

Change in Average Unit Production Costs of Products (%)

Cha

nge

in P

rice

(%)

Raw MaterialProduct 1Product 2 Product 3

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

25%

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50%

Change in Average Unit Production Costs of Products (%)

ytitnauQ

noit cudorPn i

eg nah C(%

)

Raw MaterialProduct 1 Product 2 Product 3

-40%

-30%

-20%

-10%

0%

10%

20%

30%

40%

-50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50%

Change in Average Unit Production Costs of Products(%)

Cha

nge

in P

rofit

s (%

)

Supplier

Producer's Average

Fig. 4. Change in prices, production quantities and profits with changes in unit costs of products.

688 U.S. Karmarkar & K. Rajan / European Journal of Operational Research 223 (2012) 680–689

and producers, this is not apparent if the producer does not con-sider multi-tier effects in their decision making process.

We further show that when producers are heterogeneous withvarying capacity constraints, Lagrange multipliers are not generallyzero, but again, their interpretation requires care. In particular, weshow that the multiplier does not capture the additional value ofcapacity for the producer due to competitive interactions betweenproducers for the raw material used. We also show that it is prof-itable for the supplier to marginally increase capacity when theprofit margin on the raw material is greater than a threshold;

and this threshold changes with the number of producers at capac-ity. Here again, we find that while capacity expansion could beprofitable for both the supplier and producers, this is not evidentif the producer does not consider multi-tier effects in their decisionmaking process.

We also present a computational method to solve the generalproblem. We use this method on an illustrative example to betterunderstand how production efficiencies at the supplier and pro-ducers affect production quantities and prices of the raw materialand the product, and profits at the supplier and producer’s tiers.

U.S. Karmarkar, Kumar Rajaram / European Journal of Operational Research 223 (2012) 680–689 689

We also consider the impact of market size and customer pricesensitivity on these aspects.

This paper presents several avenues for future research. First,this problem could be extended to incorporate multi-period effectsusing inventory constraints. Second, we could consider the impactof yield uncertainty at both the supplier and producer tiers. Boththese extensions would require significant modifications to thecomputational method to solve the general problem. Finally, an-other direction could be to extend this problem to multiple suppli-ers and producers. In this case, the entire structure of the supplierproblem has to be changed, as price competition with more thanone supplier would lead to marginal cost pricing. One plausible ap-proach could be to change the supplier problem to quantity ratherthan price competition. However, the optimal solution for such aproblem may not be always computable, as in general, the rawmaterial demand function may not be invertible. We hope this pa-per provides the stimulus and building blocks to examine thesenew, exciting and challenging avenues for future research.

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