Competitive and Cooperative Inventory Poicies in a Two-Stage Suppy Chain Gérard P Cachon Pau H Zipkin The Fuqua Schoo of Business Duke University Durham North Caroina 7708 gpc@maidukeedu zipkin@maidukeedu We investigate a two-stage seria suppy chain with stationary stochastic demand and fixed transportation times Inventory hoding costs are charged at each stage and each stage may incur a consumer backorder penaty cost eg the upper stage (the suppier) may disike backorders at the ower stage (the retaier) We consider two games In both the stages independenty choose base stock poicies to minimize their costs The games differ in how the firms track their inventory eves (in one the firms are committed to tracking echeon inventory; in the other they track oca inventory) We compare the poicies chosen under this competitive regime to those seected to minimize tota suppy chain costs ie the optima soution We show that the games (neary aways) have a unique Nash equiibrium and it differs from the optima soution Hence competition reduces efficiency Furthermore the two games equiibria are different so the tracking method infuences strategic behavior We show that the system optima soution can be achieved as a Nash equiibrium using simpe inear transfer payments The vaue of cooperation is context specific: In some settings competition increases tota cost by ony a fraction of a percent whereas in other settings the cost increase is enormous We aso discuss Stackeberg equiibria (Suppy Chain; Game Theory; Mutiecheon Inventory; Incentive Contracts) Introduction How shoud a suppy chain manage inventory? If the members care ony about overa system performance they shoud choose poicies to minimize tota costs ie the optima soution Whie this approach is appeaing it harbors an important weakness Each member may incur ony a portion of the suppy chain s costs so the optima soution may not minimize each member s own costs For exampe a suppier may care more than a retaier about consumer backorders for the suppier s product or the retaier s cost to hod inventory may be higher than the suppier s Whie the firms may agree in principa to cooperate each may face a temptation to deviate from any agreement to reduce its own costs Supposing each firm can anticipate these temptations how wi the firms behave? 936 Management Science/Vo 45 No 7 Juy 999 Furthermore to what extent wi these temptations ead to suppy chain inefficiency? This paper studies the difference between goba/ cooperative and independent/competitive optimization in a seria suppy chain with one suppier and one retaier (We assume there are two independent firms but the mode aso appies to independent agents within the same firm) Consumer demand is stochastic but independent and stationary across periods There are inventory hoding costs and consumer backorder penaty costs but no ordering costs There is a constant transportation time between stages and the suppier s source has infinite capacity Inventory is tracked using either echeon inventory or oca inventory (A firm s oca inventory is its on-hand inventory and its echeon inventory is its oca inventory pus a 005-909/99/4507/0936$0500 Copyright 999 Institute for Operations Research and the Management Sciences
Competitive and Cooperative Inventory Poicies inventory hed ower in the suppy chain) In the optima soution the firms choose base stock poicies described in 3 These poicies can be impemented by tracking either echeon inventory or oca inventory To mode independent decision making we consider two games the Echeon Inventory (EI) game and the Loca Inventory (LI) game In both games the firms simutaneousy choose their base stock eves This is their ony strategic decision and it cannot be modified once it is announced The suppier pays hoding costs for inventory in its possession or in-transit to the retaier and the retaier pays hoding costs on units it possesses Both firms are concerned about consumer backorders; the suppier pays a consumer backorder penaty as does the retaier This is an important assumption because it aows us to study how the firms reative preferences infuence their strategic behavior and in turn the performance of the system (Section 3 discusses this modeing issue) The EI and LI games differ in ony one way: In the EI game both firms are committed to tracking echeon inventory whereas in the LI game both firms track oca inventory The firms in each game pay a Nash equiibrium (A pair of strategies is a Nash equiibrium if each firm minimizes its own cost assuming the other payer chooses its equiibrium strategy) Thus each firm makes an optima decision given the behavior of the other firm and therefore neither firm has an incentive to deviate uniateray from the equiibrium We find that in each game there is (usuay) a unique Nash equiibrium We compare the games equiibria to each other and to the optima soution The optima soution is typicay not a Nash equiibrium so competitive decision making degrades suppy chain efficiency We evauate the magnitude of this effect with an extensive numerica study Impementation of the cooperative soution requires that the firms eiminate the incentives to deviate ie they shoud modify their costs so that the optima soution becomes a Nash equiibrium This goa can be achieved by a contract that specifies inear transfer payments based on easiy verifiabe performance measures ike inventory and backorders We deveop a set of inear contracts that meet this objective and briefy discuss other techniques for aigning incentives In these games neither payer dominates the other and the firms simutaneousy choose their strategies We aso study Stackeberg versions of the games in which one dominant payer chooses its strategy before the other The next section reviews the reated iterature and 3 formuates the mode Section 4 describes the system optima soution and 5 anayzes the two games Section 6 compares the games equiibria with the optima soution Section 7 describes contracts that make the system optima soution a Nash equiibrium Section 8 discusses the numerica study Section 9 anayzes the Stackeberg games and 0 concudes Literature Review The iterature on suppy chain inventory management mosty assumes poicies are set by a centra decision maker to optimize tota suppy chain performance Three exceptions are Lee and Whang (996) Chen (997) and Porteus (997) In Lee and Whang (996) the firms use echeon stock poicies and a backorder penaties are charged to the owest stage The upper stage incurs hoding costs ony Therefore with competitive seection of poicies the upper stage carries no inventory thereby minimizing its own cost They deveop a noninear transfer payment contract that induces each firm to choose the system optima base stock poicies Our mode differs from theirs on severa dimensions We assume the upper stage (the suppier) may care about consumer backorders so it may carry inventory even when inventory poicies are chosen competitivey Hence the competitive decisions are nontrivia We distinguish between echeon inventory and oca inventory and investigate how these different methods for tracking inventory infuence strategic behavior Finay we deveop inear transfer payment contracts They consider a setup cost at the upper echeon whie we do not Porteus (997) studies a mode simiar to Lee and Whang s mode but he proposes a different coordination scheme caed responsibiity tokens Chen (997) studies a game simiar to the popuar Beer Game (Sterman 989) except that the demands in Management Science/Vo 45 No 7 Juy 999 937
Competitive and Cooperative Inventory Poicies different periods are independent random variabes with a common distribution that is known to a payers Unike our payers his share the objective of minimizing tota system costs; they have no competing interests He outines an accounting scheme that aows each payer to optimize its own costs and yet choose the system optima soution This scheme is more compex than ours in some ways though simper in others as expained in 7 He aso studies the behavior of boundedy rationa payers whereas we ony assume rationa payers Severa other papers address reated issues yet their modes are significanty different Lippman and Mc- Carde (997) study competition between two or more firms in a one-period setting where a consumer may switch among firms to find avaiabe inventory Parar (988) and Li (99) aso study the roe of inventory in the competition among retaiers In a mutiecheon mode with mutipe retaiers Muckstadt and Thomas (980) Hausman and Erkip (994) and Axsäter (996) investigate a centraized contro system that aows each firm to optimize its own costs and sti choose an outcome desirabe to the centra panner The behavior of a centra panner has aso been investigated in settings with mora hazard (eg Porteus and Whang 99 Kouveis and Lariviere 996) Many papers investigate how a suppier can induce a retaier to behave in a manner that is more favorabe to the suppier (eg Donohue 996 Tsay 996 Ha 996 La and Staein 984 Moses and Seshadri 996 Narayanan and Raman 996 Pasternack 985) Chen et a (997) study competitive seection of inventory poicies in a mutiecheon mode with deterministic demand 3 Mode Description Consider a one-product inventory system with one suppier and one retaier The suppier is Stage and the retaier is Stage Time is divided into an infinite number of discrete periods Consumer demand at the retaier is stochastic independent across periods and stationary The foowing is the sequence of events during a period: () shipments arrive at each stage; () orders are submitted and shipments are reeased; (3) consumer demand occurs; (4) hoding and backorder penaty costs are charged There is a ead time for shipments from the source to the suppier L and from the suppier to the retaier L Each firm may order any nonnegative amount in each period There is no fixed cost for pacing or processing an order Each firm pays a constant price per unit ordered so there are no quantity discounts The suppier is charged hoding cost h per period for each unit in its stock or enroute to the retaier The retaier s hoding cost is h h per period for each unit in its stock Assume h 0 and h 0 Unmet demands are backogged and a backorders are utimatey fied Both the retaier and the suppier may incur costs when demand is backordered The retaier is charged p for each backorder and the suppier ( ) p 0 The parameter p is the tota system backorder cost and specifies how this cost is divided among the firms The parameter is exogenous These backorder costs have severa possibe interpretations a standard They may represent the costs of financing receivabes if customers pay ony upon the fufiment of demands (This requires a discounted-cost mode to represent exacty but the approximation here is standard in the average-cost context anaogous to the treatment of inventory financing costs) Aternativey they may be proxies for osses in customer good-wi which in turn ead to ong-run decines in demand Such costs need not affect the firms equay which is why we aow the fexibiity to choose [0 ] Finay they provide a crude approximation to ost saes (It woud be better of course to mode ost saes directy but that introduces considerabe anaytica difficuties Even the optima poicy is unknown) In period t before demand define the foowing for stage i: in-transit inventory IT it ; echeon inventory eve IL it is a inventory at stage i or ower in the system minus consumer backorders; oca inventory eve IL it is inventory at stage i minus backorders at stage i (the suppier s backorders are unfied retaier orders); echeon inventory position IP it IP it IL it IT it ; and oca inventory position IP it IP it IL it IT it Each firm uses a base stock poicy Using an echeon base stock eve each period the firm orders a sufficient amount to raise its echeon inventory position pus outstanding orders to that eve A firm s oca 938 Management Science/Vo 45 No 7 Juy 999
Competitive and Cooperative Inventory Poicies base stock eve is simiar except the oca inventory position repaces the echeon inventory position Define s i as stage i s echeon base stock eve and s i as its oca base stock eve Let D denote random tota demand over periods and denote mean tota demand over periods Let and be the density and distribution functions of demand over periods respectivey We assume ( x) is continuous increasing and differentiabe for x 0 so the same is true of 0 Furthermore (0) 0 so positive demand occurs in each period Math notation foows: [ x] max{0 x}; [x] max{0 x}; [a b] is the cosed interva from a to b; and E[ x] is the expected vaue of x A prime denotes the derivative of a function of one variabe 4 System Optima Soution The system optima soution minimizes the tota average cost per period Cark and Scarf (960) Federgruen and Zipkin (984) and Chen and Zheng (994) demonstrate that an echeon base stock poicy is optima in this setting The optima soution is found by aocating costs to the firms in a particuar way Then each firm chooses a poicy that minimizes its cost function This section briefy outines this method Let Ĝ o (IL t D ) equa the retaier s charge in period t where Ĝ o x h x h px Aso in period t define G o (IP t ) as the retaier s expected charge in period t L where G o y EĜ o y D L Define s o as the vaue of y that minimizes G o ( y): L s o h p h h p () This is the retaier s optima base stock eve Define the induced penaty function and define G o y G o mins o y G o s o Ĝ o y h y G o y In period t charge the suppier G o (IP t ) where G o y EĜ o y D L The suppier s optima echeon base stock eve s o minimizes G o 5 Echeon and Loca Inventory Games In the Echeon Inventory (EI) game the two stages are independent firms or payers In the game s ony move the payers simutaneousy choose their strategies s i [0 S] where s i equas payer i s echeon base stock eve is payer i s strategy space and S is a very arge constant (S is sufficienty arge that it never constrains the payers) A joint strategy s is a pair s ) After their choices the payers impement their poicies over an infinite horizon In addition a mode parameters are common knowedge In the Loca Inventory (LI) game the suppier and the retaier choose oca base stock eves s s Again strategies are chosen simutaneousy the payers are committed to their strategies over an infinite horizon and a parameters are common knowedge The payers know which game they are paying; the choice between the EI and LI games is not one of their decisions Define H i s ) as payer i s expected per-period cost when payers use echeon base stock eves s ) When s s s and s s the oca base stock pair (s s ) is equivaent to s ) in the sense that H i s ) H i (s s s ) Since any echeon base stock pair can be converted into an equivaent oca pair there is no need to define distinct cost functions with oca arguments We wi frequenty switch a pair of base stock eves from one tracking method to another to faciitate comparisons Athough there is itte operationa distinction between echeon and oca base stock poicies we ater show that they differ strategicay (However the operationa equivaence of echeon and oca base stocks does depend on the assumption of stationary demand In a nonstationary demand environment it may not be possibe to run the system optimay with oca base stock poicies) Management Science/Vo 45 No 7 Juy 999 939
Competitive and Cooperative Inventory Poicies For the EI game the best repy mapping for firm i is a set-vaued reationship associating each strategy s j j i with a subset of according to the foowing rues: r s s H s s min x r s s H s s min x H x s H s x Likewise for the LI game the best repy mappings are r s s H s s s min x r s s H s s s min x H x s x H s x s A pure strategy Nash equiibrium is a pair of echeon base stock eves e s e ) in the EI game or oca base stock eves (s s ) in the LI game such that each payer chooses a best repy to the other payer s equiibrium base stock eve: s e r s e s s e r s e r s s r s (We do not consider mixed strategies We generay find a unique pure strategy equiibrium) 5 Actua Cost Functions In each period the retaier is charged h h per unit hed in inventory and p per unit backordered Define Ĝ (IL t D ) as the sum of these costs in period t Ĝ y h h y py Define G (IP t ) as the retaier s expected cost in period t L G y EĜ y D L h h y L h h p y x y L xdx a Define s as the vaue that minimizes this function that is the base stock eve that minimizes the retaier s costs assuming retaier orders are shipped immediatey s a arg min y G y Differentiation verifies that G is stricty convex so s a is determined by G a ) 0 L s a p h h p The retaier s true expected cost depends on both its own base stock as we as the suppier s base stock We use a standard derivation After the firms pace their orders in period t L the suppier s echeon inventory position equas s After inventory arrives in period t but before period t demand the suppier s echeon inventory eve equas s D L Hence E[s D L ] is the suppy chain s expected inventory eve (average suppy chain inventory minus average backorders) When s D L s the suppier can competey fi the retaier s period t order so IP t s When s D L s the suppier cannot fi a of the retaier s order and IP t s D L Hence H s s EG mins D L s L s s G s ss L xg s xdx Define Ĝ (IL t D ) as the suppier s actua period t backorder cost Ĝ y py and G (IP t ) as the suppier s expected period t L backorder cost Define so G y EĜ y D L Ĥ s x h L h x G s minx 0 H s s EĤ s s s D L 940 Management Science/Vo 45 No 7 Juy 999
Competitive and Cooperative Inventory Poicies h L h 0s s s s x L xdx L s s G s ss L xg s xdx The first term above is the expected hoding cost for the units in-transit to the retaier (from Litte s Law) the second term is the expected cost for inventory hed at the suppier and the fina two terms are the expected backorder cost charged to the suppier We mentioned above the operationa equivaence of oca and echeon base stock poicies when s s and s s s However the change in payer i s cost due to a shift in payer j s strategy depends on the inventory tracking method For exampe hoding s constant the suppier s expected on-hand inventory is independent of s but when s stays constant the suppier s inventory decines as s increases Furthermore the tota system inventory depends on s ony So hoding s fixed the retaier s s ony infuences the aocation of inventory between the suppier and the retaier However hoding s fixed the retaier can increase tota system inventory by raising s 5 Echeon Inventory Game Equiibria with Shared Backorder Costs In this section we assume that each firm incurs some backorder cost ie 0 (We subsequenty consider the extreme cases 0 and ) We begin with some preiminary resuts on the payers cost functions and best repy mappings Lemma Assuming H s ) is stricty convex in s s 0 and H s ) is quasiconvex in s Proof Fix D L function of s : and s Consider the foowing h s s D L G mins D L s Both terms are convex whie the second term is stricty convex in the interva s [D L D L s ] Now take the expectation over D L The first term h E[(s s D L ) ] is convex and stricty convex for s s The second term E[G (min{s D L s })] is convex and stricty convex when s 0 and s 0 Hence H s ) is stricty convex in s 0 Consider H When s s H is constant with respect to s Assume s s and differentiate H H L s s s G s When s s a H is decreasing for s s and constant for s s When s s a H is decreasing for s s a increasing for s a s s and constant for s s Hence H is quasiconvex in s The foowing emma characterizes the suppier s best repy mapping Lemma Assuming r ) is a function r ) s and 0 r ) Proof From Lemma H is stricty convex in s so r ) is a function (ie H has a unique minimum) and is determined by the first-order condition H h s L s s s L xg s xdx 0 s This condition cannot hod at s s because then L (s s ) 0 and G ( y) 0 Therefore s r ) s Given s s from the impicit function theorem r s H s s H s where H s s H s s ss L xg s xdx H s h G s L s s ss L xg s xdx H s s L s s h G s () Management Science/Vo 45 No 7 Juy 999 94
Competitive and Cooperative Inventory Poicies The cross partia of H is negative because G 0 h and L (s s ) 0 for s s Since G 0 0 r ) Athough in the EI game the suppier does not aways fi the retaier s orders immediatey the suppier s echeon base stock eve has itte infuence over the retaier s strategy Figure Reaction Functions 030 p 5 h h 05 L L Lemma 3 For the EI game the retaier s best repy mapping is r s s a a s s a s S s s Proof Reca that G ( y) is stricty convex and minimized by y s a Let x D L When s x s a s s minimizes G (min{s x s }) When s x s a a ony s s minimizes G (min{s x s }) When s s a s D L s a sor (s ) [s S] When s s a ony s s a minimizes G (min{s x s }) for a x sor (s ) s a The retaier s best repy is not necessariy unique but there is ony one Nash equiibrium Figure Reaction Functions 090 p 5 h h 05 L L Theorem 4 Assuming 0 in the EI game e s a s e r a )) is the unique Nash equiibrium Proof From Theorem in Fudenberg and Tiroe (99) a pure strategy Nash equiibrium exists if () each payer s strategy space is a nonempty compact convex subset of a Eucidean space and () payer i s cost function is continuous in s and quasiconvex in s i By the assumptions and Lemma these conditions are met so there is at east one equiibrium From Lemma in any equiibrium (s e s e ) s e r (s e ) s e If s e s a Lemma 3 impies s e s e a contradiction Hence s e s a but from Lemma 3 this impies s e s a e Since r is a function there is ony one s r (s a ) Therefore the equiibrium is unique Figures and pot the firms reaction functions and the resuting equiibrium for two exampes 53 Loca Inventory Game Equiibria with Shared Backorder Costs The anaysis of the LI game aso begins by characterizing the cost functions and the best repy mappings Lemma 5 H (s s s ) is stricty convex in s and H (s s s ) is stricty convex in s Proof Set s s s and s s Differentiation of H (s s s ) reveas that H s s s s H s s s s H s s s ; H s s s (3) From Lemma H s ) is stricty convex in s so H (s s s ) is stricty convex in s Differentiate H (s s s ) 94 Management Science/Vo 45 No 7 Juy 999
Competitive and Cooperative Inventory Poicies H s s s L s s G s s H s s s L s s G s s L xg s s xdx; L xg s s xdx Since G is stricty convex H (s s s )/s 0 which means that H is stricty convex in s The next two emmas characterize the best repy mappings Lemma 6 Assuming r ) s r ); r ) 0; and r ) 0 Proof For the suppier r ) s r ) because H (s s s ) H s ) whenever s s s and s s From Lemma r ) s which impies that r ) 0 From the same emma 0 r ) Aso r ) r ) so r ) 0 Lemma 7 Assuming 0 r (s ) s a When s 0 r (s ) 0; and when s 0 r (s ) Proof The retaier s best repy is determined by the first order condition H (s s s )/s 0 see (4) When s s a G (s ) 0 and therefore H (s s s )/s 0 Hence r (s ) s a From the impicit function theorem r s H s s s s s H s s s s s L xg s s xdx L s G s s L xg s s xdx Assume s 0 Since L (s )G (s ) 0 the numerator above is positive and the numerator equas the second term in the denominator r (s ) 0 (4) When s 0 L (s )G (s ) 0 and therefore r (s ) When 0 there is a unique Nash equiibrium in the LI game Theorem 8 Assuming 0 (s s ) is the unique Nash equiibrium Proof Lemma 5 confirms the required conditions for the existence of an equiibrium (in the proof of Theorem 4) First from Lemma 6 r ) r ) s 0 so s 0 Now suppose there are two equiibria (s s ) and (s* s* ) Without oss of generaity assume s s* From Lemma 7 this impies that s* s From the same emma r (s ) so s* s* s* s s s But from Lemma r is increasing so s* s impies that s* s a contradiction Hence there is a unique equiibrium Figures and aso dispay the reaction functions in the LI game as we as the Nash equiibrium 54 Equiibria Under Extreme Backorder Cost Aocations Suppose the retaier is charged a of the backorder costs ie In this situation the Nash equiibrium in the EI game is no onger unique Theorem 9 For in the EI game the Nash equiibria are e [s e S] s e [0 s a ]) Proof The existence proof in Theorem 4 appies even when so a pure strategy equiibrium exists When the suppier incurs no backorder costs ony hoding costs Hence the suppier picks s s ie r ) [0 s ] Suppose (s* s* ) is an equiibrium where s* s a From Lemma 3 r (s* ) s a but an equiibrium ony occurs when s* s* so s* s a cannot be an equiibrium Suppose s* s a From Lemma 3 r (s* ) [s* S] so for any s* s a (s* [s* S] s* ) is an equiibrium In the LI game there is a unique equiibrium even when the retaier incurs a of the backorder cost Theorem 0 Assuming in the LI game (s r (0) s 0) is the unique Nash equiibrium Proof When the suppier chooses s 0 Since r (s ) is a function r (0) is unique When the suppier incurs a backorder costs there Management Science/Vo 45 No 7 Juy 999 943
Competitive and Cooperative Inventory Poicies is a unique equiibrium in both games and they are identica Theorem Assuming 0 (s e 0 s e r (0)) is the unique Nash equiibrium in the EI game and (s s e s s e s ) is the unique Nash equiibrium of the LI game Proof Since the retaier incurs no backorder cost e s s 0 The suppier s best repy mapping is a function in either game so s e r (0) Furthermore s e s s e 6 Comparing Equiibria This section compares the equiibria in the LI and EI games to each other as we as to the optima soution To faciitate these comparisons convert the LI game equiibrium (s s ) into the equivaent pair of echeon base stock eves (s s ) where s s and s s s 6 Competitive Equiibria The firms choose higher base stock eves in the LI game than in the EI game Theorem Assuming 0 the base stock eves for both firms are higher in the LI game equiibrium than in the EI game equiibrium ie s s e and s s e Proof The equiibrium in the EI game is (s e s a s e r (s e )) From Lemma 7 r (s ) s a which impies that s s e s a From Lemma r ) is increasing in s sos r (s ) r (s a ) s e The retaier s cost in the LI game equiibrium can be more or ess than in the EI game equiibrium (The numerica study confirms this) However the suppier hasadefinite preference for the LI game Theorem 3 Assuming 0 the suppier s cost in the LI game equiibrium is ower than its cost in the EI game equiibrium Proof In the EI game the suppier chooses r ) In the LI game the suppier chooses r ) s as its oca base stock eve and the equivaent echeon base stock eve is r ) Differentiate the suppier s cost function with respect to the retaier s base stock eve assuming the suppier chooses s r ): dh H H r s H ds s s s s L s s h G s since H r ))/s 0 From Lemma r ) s so L (s s ) 0 and G 0 so dh /ds 0 Thus the suppier s cost decines as s increases Since s s e H is ower at s Why does the suppier prefer the LI game equiibrium? The suppier aways prefers the retaier to increase its base stock thereby increasing the retaier s inventory and decreasing the suppier s backorder costs The retaier aways chooses a ower base stock in the EI game than it does in the LI game hence the suppier is aways better off in the LI game 6 Competitive Equiibria and the Optima Soution In the EI game the retaier s base stock eve is ower than in the optima soution Theorem 4 In an EI game equiibrium the retaier s base stock eve is ower than in the optima soution Proof Note that Ĝ o ( x) Ĝ ( x) for a x Hence G o ( y) G ( y) for a y Since both G o ( y) and G ( y) are increasing in y s o s a s e In the LI game either s s o or s s o is possibe (In Figure the retaier chooses s s o but in Figure s s o ) However when backorder costs are charged to the suppier the suppier s base stock eve is ower than in the system optima soution in both games Theorem 5 Assuming the suppier s base stock eve in both the LI and the EI equiibria is ower than in the system optima soution Proof In any equiibrium s s e soitissufficient to show that s o s For x s and Ĥ x G s x p Ĝ o x h G o x h G o x p 944 Management Science/Vo 45 No 7 Juy 999
Competitive and Cooperative Inventory Poicies For s x 0 Ĥ x G s x p 0 and again Ĝ o ( x) p For x 0 Ĥ / x h and Ĝ o x h G o x h with strict inequaity for x s o So in a cases Ĝ o ( x) Ĥ / x with strict inequaity for s x s o Therefore G o ( x) H / x with strict inequaity for s s So s o s Reca that the suppier s echeon base stock determines the suppy chain s average inventory eve From Theorem 5 it foows that in either game s equiibrium the suppy chain s average inventory eve wi be ower than in the optima soution suggesting that competition wi aso tend to ower the suppy chain s average inventory The numerica study confirms this observation When the suppier incurs no backorder costs the suppier s base stock eve is no greater than in the system optima soution Theorem 6 Assuming s e s o and s s o e e a o Proof In the EI game s s s s s o hence s e s o When in the LI game s s so the proof of Theorem 5 demonstrates s s o Theorem 7 Assuming the system optima soution is not a Nash equiibrium e Proof From Theorem 5 s s s o so the optima soution is not a Nash equiibrium in either game When the suppier incurs no backorder costs the system optima soution can be a Nash equiibrium under a very specia condition Theorem 8 Assuming the system optima soution is a Nash equiibrium in the LI game ony when LL s o p h h p Proof When the LI game Nash equiibrium is s s ) Soving for s LL s p h h p It is possibe that s s o because L L stochasticay dominates L and p/(h h p) (h p)/(h h p) For the suppier s o s o when G o (s o s o ) 0 This occurs precisey when LL s o In that case s o s o s p h h p s 7 Cooperative Inventory Poicies According to Theorem 7 the optima soution is virtuay never a Nash equiibrium Hence the firms can ower tota costs by acting cooperativey There are severa methods that enabe the firms to minimize tota costs and sti remain confident that the other firm wi not deviate from this agreement For instance the firms coud contract to choose o s o )as their base stock eves But since each firm has an incentive to deviate from this contract (because it is not a Nash equiibrium) the contract must aso specify a penaty for deviations Such stipuations are hard to enforce Aternativey the firms coud write a contract that specifies transfer payments which eiminate incentives to deviate from the optima soution There are severa schemes to achieve this goa: a per period fee for the suppier s backorder; a per unit fee for each unit the suppier does not ship immediatey; a per unit fee per consumer backorder; or a subsidy for each unit of inventory in the system (Noninear payment schedues as in Lee and Whang (996) coud aso be considered but these are necessariy more cumbersome For instance the induced penaty function G o is noninear) Transfer payments can aso be imposed on the retaier eg a subsidy on retaier inventories or a subsidy on backorders Payments coud aso be based on the inventory and backorder eves that woud have occurred had the suppier performed certain actions Chen (997) uses this approach Define accounting inventory and accounting backorders as the inventory and backorder eves assuming the suppier fis retaier orders im- Management Science/Vo 45 No 7 Juy 999 945
Competitive and Cooperative Inventory Poicies mediatey Suppose the suppier pays a of the retaier s actua costs and the suppier charges the retaier h per unit of accounting inventory and h p per accounting backorder Then the retaier wi choose s o Since the suppier incurs a actua costs and the retaier chooses s o the suppier chooses s o With this scheme the retaier s decision is independent of the suppier s; there is no strategic interaction between the firms However this approach creates a chaenging accounting probem We study inear transfer payments based on actua inventory and backorder eves Whie our approach avoids the probem of tracking accounting inventory and backorders Chen s method uses ony cost parameters Ours aso requires a demand parameter Thus his technique may be easier to impement in some situations ours in others 7 Linear Contracts Suppose the firms track oca inventory and they adopt a transfer payment contract with constant parameters ( ) This contract specifies that the period t transfer payment from the suppier to the retaier is I t B t B t where I t is the retaier s on-hand inventory and B it is stage i s backorders a measured at the end of the period There are no a priori sign restrictions on these parameters eg 0 represents a hoding cost subsidy to the retaier and 0 represents a hoding fee (We ater impose some restrictions on the parameters) We aso assume the optima soution is common knowedge Define T (IP t ) as the expected transfer payment in period t L due to retaier inventory and backorders where T y E y D L y D L y L y x y L xdx Define T(s s ) as the expected per period transfer payment from the suppier to the retaier Ts s E s D L T s min0 s D L s s L xx s dx L s T s L xt s s xdx Note that s infuences the retaier inventory and backorders but not the suppier s backorders Let H c i (s s s ) be payer i s costs after accounting for the transfer payment H c s s s H s s s Ts s H c s s s H s s s Ts s We wish to determine the set of contracts ( ) such that (s c s c ) is a Nash equiibrium for the cost functions H i c (s s s ) where s c s o and s c s c s o With these contracts the firms can choose (s c s c ) thereby minimizing tota costs and aso be assured that no payer has an incentive to deviate To find the desired set of contracts first assume that H i c is stricty convex in s i given that payer j chooses s j c j i Then determine the contracts in which s i c satisfies payer i s first order condition thereby minimizing payer i s cost Finay determine the subset of these contracts that aso satisfy the origina strict convexity assumption The foowing are the first order conditions: H c s H c s 0 L s G s T s s L xg s s x 0 h L s s T s s xdx; (5) L xg s s x 946 Management Science/Vo 45 No 7 Juy 999
Competitive and Cooperative Inventory Poicies T s s xdx (6) Define L o (s s o ) (This is the suppier s in-stock probabiity essentiay its fi rate) Furthermore the suppier s first order condition in the optima soution is 0 p p h L s o s o h h p T s s xdx 0; c H h s L s s L xg s s x or s o o s L x L s o xdx T s s xdx 0 The first inequaity reduces to h h p 0 s o o s L x L s o xdx p p h h h p (7) Using (7) (5) and (6) yied the foowing two equations in three unknowns p p h h (8) h h h h (9) It remains to ensure that the costs functions are indeed stricty convex Theorem 9 When the firms choose ( ) to satisfy (8) and (9) and the foowing additiona restrictions appy (i) h h 0 (ii) 0 (iii) p p then the optima poicy (s c s c ) is a Nash equiibrium Proof When the foowing second order conditions are satisfied H i c is stricty convex in s i assuming s j s j c j i: c H L s s G s T s s L xg s s x Substituting (8) yieds h h and p For the suppier sufficient conditions are p 0; h p p L s o 0 Combining the first inequaity with (8) yieds 0 and ( ) p The second inequaity aong with (8) and (9) yieds 0 These are quite reasonabe conditions: The first requires that the retaier s inventory subsidy not eiminate retaier hoding costs; the second stipuates that the suppier be penaized for its backorders; and the third states that the suppier shoud not fuy reimburse the retaier s backorder costs and the retaier shoud not overcompensate the suppier s backorder costs To hep interpret these resuts consider the three extreme contracts where one of the parameters is set to zero: i) 0 h p ii) h h 0 p iii) h h h 0 (Of these three contracts the second does not meet the conditions in Theorem 9 because the suppier fuy compensates the retaier for a of its costs The retai- Management Science/Vo 45 No 7 Juy 999 947
Competitive and Cooperative Inventory Poicies er s incentive to choose the optima poicy is weak: s o is a Nash equiibrium strategy but any s is too) With the first contract the retaier fuy reimburses the suppier for the suppier s consumer backorder penaty However the suppier sti carries inventory because it pays a penaty for its oca backorders With the third contract the suppier subsidizes the retaier s hoding costs but not fuy (provided 0) In addition the suppier is penaized for its backorders but ess than in the first contract When the retaier incurs a backorder costs (ie ) ony a suppier backorder penaty is required h /( ) Incidentay (8) and (9) can be written p h h h h p h h h p h h p Consider the first identity The quantity p is the retaier s backorder cost and its hoding cost incuding transfer payments is h h h h which is written more simpy as h h So the eft-hand side is the retaier s critica ratio The righthand side is the critica ratio for the tota system costs controed by the retaier These ratios must be identica to induce the retaier to minimize tota system costs The second identity specifies a critica ratio for the suppier is its (oca) backorder cost The hoding cost h is mutipied by the factor ( /(h h )) which is the fraction of actua hoding costs paid by the retaier taking the transfer payment into account Thus this rue effectivey reduces both stages hoding costs by the same fraction 7 Additiona Contracting Issues Theorem 9 detais the contracts that make the optima soution a Nash equiibrium but this does not impy a unique equiibrium Nevertheess even if there were additiona Nash equiibria the one corresponding to the optima soution Pareto dominates any other Hence the payers can coordinate on this equiibrium (There is experimenta evidence that payers coordinate on a Pareto dominant equiibrium when they are abe to converse before paying the game eg Cooper et a 989 Cachon and Camerer 996) Athough tota costs decine when the firms coordinate one firm s cost may increase This firm wi be unwiing to participate in the contract uness it receives an additiona transfer payment To maintain the strategic baance of the contract this payment shoud be independent of a other costs and actions For exampe the firms coud transfer a fixed fee each period Aternativey one coud seek a contract ( ) such that each firm s cost is no greater than in the origina Nash equiibrium Finay this anaysis assumes the firms use oca base stock eves In this context oca poicies have severa advantages over echeon poicies Reca that s has no infuence on B t the suppier s backorder penaty when firms use oca base stock eves However with echeon stock base stock eves s does infuence B t hoding s constant This can create a perverse incentive Suppose is arge By increasing s s the retaier can make B t arbitrariy arge (assuming S the imit on s is arge too) The B t transfer payment coud easiy dominate the additiona retaier inventory cost (Furthermore once s s the retaier can increase s without increasing its inventory) There is a soution to this probem The transfer payment coud assume that the retaier chooses s s o Hence the retaier woud receive no additiona benefit by raising s above s o Ceary this increases the compexity of the contract Loca measurements avoid this probem atogether 8 Numerica Study The system optima soution is virtuay never a Nash equiibrium but how arge is the difference between their costs? To answer this question we conducted a numerica study One period demand is normay distributed with mean and standard deviation /4 (There is ony a tiny probabiity of negative demand) The remaining 948 Management Science/Vo 45 No 7 Juy 999
Competitive and Cooperative Inventory Poicies parameters are chosen from the 65 possibe combinations of the foowing: 0 0 03 05 07 09 L 4 8 6 p 5 5 h 0 03 05 07 09 L 4 8 6 h h Note that h h in a of the probems; therefore the retaier hoding cost is constant For each probem three soutions are evauated: () the system optima soution; () the Nash equiibrium of the EI game; and (3) the Nash equiibrium of the LI game (These data can be obtained from http:// wwwdukeedu/ gpc/) (When there are mutipe Nash equiibria in the EI game; we choose the one with the argest s ) Tabe summarizes the percentage increase in cost of the Nash equiibrium over the system optima soution We ca this percentage the competition penaty Severa resuts are evident from the tabe First when the payers care about backorder costs equay (ie 05) the Nash equiibrium is cose to the system optima soution: In the EI and LI games the median competition penaty is 6% and 3% respectivey and the maximum is 3% and 8% respectivey The competition penaty increases as the backorder cost aocation becomes more asymmetric In the EI game the median percentage is 84% when the retaier incurs no backorder penaty (ie 0) and 483% when the retaier incurs a of the backorder penaty (ie ) The competitive outcome is poor when 0 because the retaier chooses s s 0 so a consumer demands are backordered When the competition penaty is substantia because the suppier refuses to carry inventory thereby hampering the retaier s effort to mitigate consumer backorders The LI game s response to changes in is sighty different The performance of the competitive soution deteriorates rapidy as the retaier incurs ower backorder cost (ie decines) However as the retaier incurs higher backorder cost (ie increases) the maximum competition penaty increases rapidy whie the minimum and median penaties do not Consider the extreme case where a backorder costs are aocated to the retaier In the competitive soution the suppier carries no inventory Nevertheess this behavior is not aways harmfu Look at Figure 3 which dispays the percentage increase in tota system costs as a function of Tabe The Competition Penaty Under Different Aocations of Backorder Costs Competition Penaty: Percentage Increase in Cost of the Nash Equiibrium Over the System Optima Soution Minimum 5th Percentie Median 95th Percentie Maximum 0 07% 7% 804% 5930% 0939% 0 5% 9% 4% 97% 0% Echeon 03 % 4% % % 7% Inventory 05 % % 6% % 3% Game 07 % % 4% % 9% 09 % % 8% 40% 66% % 5% 483% 97% 34493% 0 07% 7% 804% 5930% 0939% 0 5% 8% 37% 96% 9% Loca 03 % 3% 9% 9% 6% Inventory 05 % % 3% 6% 8% Game 07 0% 0% % 4% 9% 09 0% 0% % 7% 45% 0% 0% % 34% 6% Management Science/Vo 45 No 7 Juy 999 949
Competitive and Cooperative Inventory Poicies Figure 3 Competition Penaty When the Retaier Incurs A Backorder Costs ( ) 9 One Dominant Payer In the LI and EI games the payers choose their poicies simutaneousy In the Stackeberg version of either game one of the payers chooses its base stock eve first announces its choice to the other payer and then the other payer chooses its base stock eve As in the EI and LI games a payer is committed to its choice ie the first payer cannot change its decision after observing the second payer s We seek sub-game perfect equiibria ie the second payer chooses an optima response to the first payer s strategy and the first payer (correcty) anticipates this behavior The Stackeberg version represents a situation where one payer is the dominant member of the suppy chain (eg WaMart Inte) L L L h h h (0) What does (0) measure? When L /(L L ) is arge the suppier s ead time is a sma fraction of the tota system ead time so the suppier s decisions have a sma impact When h /(h h ) is arge there is itte benefit to hoding inventory at the suppier rather than the retaier As (0) increases the suppier s sef serving behavior does itte damage therefore the competitive soution s performance is neary as good as the optima soution s Overa the competition penaty is high when one of the firms has a substantia infuence over a major portion of tota system costs but itte incentive to hep manage that cost Tabe presents data on the percentage change in average suppy chain inventory in the two equiibria reative to the optima soution Average inventories in the competitive soutions are generay ower than in the optima soution except in some cases when the retaier cares itte about backorders ( is sma) Nevertheess competition raises suppy chain inventory by at most 4% 9 Suppier Stackeberg Games The Stackeberg game with the suppier eading is caed either the Echeon Inventory Suppier game (EIS) or the Loca Inventory Suppier game (LIS) depending on the inventory tracking method In the EIS (LIS) game the suppier chooses s (s ) to minimize its cost given that it anticipates the retaier wi choose r (s )(r (s )) According to the next theorem there is itte difference between the EIS and EI games Theorem 0 When in the EIS game {s a r a )} is the unique Stackeberg equiibrium When {s [s S] s [0 s a ]} are the Stackeberg equiibria Proof The proof of Theorem 3 shows that if the suppier coud choose s it woud choose s as arge as possibe When r (s ) s a so the suppier shoud anticipate s s a Therefore the suppier chooses s r (s a ) and the retaier chooses s s a When the suppier wishes to carry no inventory so it chooses s [0 s a ] since then the retaier wi choose s s The suppier cannot choose s s a because then the retaier chooses s s a eaving the suppier with some expected inventory In the LIS game the suppier anticipates that the retaier wi choose r (s ) Hence the suppier s cost is H s def H r s r s s EĤ r s s D L Since this is continuous in s there exists s* such that 950 Management Science/Vo 45 No 7 Juy 999
Competitive and Cooperative Inventory Poicies Tabe Change in Suppy Chain Inventory Percentage Change in Average Suppy Chain Inventory in the Nash Equiibrium Reative to the Optima Soution Minimum 5th Percentie Median 95th Percentie Maximum 0 8% % % % 4% 0 0% 7% 3% % 3% Echeon 03 8% 5% % 0% 0% Inventory 05 3% 9% 3% % 0% Game 07 % 5% 4% % 0% 09 37% 7% 7% % % 73% 56% 7% 3% % 0 8% % % % 4% 0 0% 7% % % 3% Loca 03 7% 5% % 0% % Inventory 05 0% 7% % % 0% Game 07 8% % % % 0% 09 34% 5% % 0% 0% 38% 6% % 0% 0% H (s* ) inf s [0S] H (s ) Hence there exists a Stackeberg equiibrium Let {s s s s } be an equiibrium and et {s s s s } be the equivaent pair of echeon base stock eves ie s s s s and s s s s s s Theorem When in the LIS game the suppier chooses a base stock eve ower than in the LI game ie s s s and s s s Proof Assuming s r (s ) differentiate the suppier s cost function with respect to s dh H r ds s s H s L s G s s L xg s s xdx r s H s From Lemma 6 r (s ) 0 When G ( y) 0 So (H /s )r (s ) 0 When s r (s ) H /s 0 Hence for s r (s ) dh /ds 0 Therefore s s r (s ) s Since r (s ) s is decreasing in s s s r (s s s ) s s s r (s ) 9 Retaier Stackeberg Games In the Echeon Inventory Retaier (EIR) and Loca Inventory Retaier (LIR) games the retaier anticipates the suppier wi choose r ) and r (s ) respectivey Since r ) s r (s ) when s s the retaier s cost for any base stock eve is the same in the two games Hence when the retaier is dominant it is immateria whether the firms use echeon or oca inventory measurements Existence of an equiibrium is straightforward Theorem In the EIR and LIR games the retaier chooses a base stock eve that is higher than in the EI game e ) but ower than in the LI game ) Proof The retaier anticipates that the suppier wi choose r ) Differentiate the retaier s cost function: dh s r s ds H s r s s H s r s r s s L s s G s r s ss L xg s x Management Science/Vo 45 No 7 Juy 999 95
Competitive and Cooperative Inventory Poicies Since r ) 0 the above is negative for a s s a s e hence the optima s is arger than s e Since r ) and in the LI game equiibrium it hods that H s s s L s s G s s L xg s s x 0 dh s r s ds 0 Hence the retaier chooses a base stock eve ower than s 0 Concusion When both payers care about consumer backorders the suppy chain optima soution is never a Nash equiibrium so competitive seection of inventory poicies decreases efficiency Athough the payers may agree to cooperate and choose suppy chain optima poicies at east one of them has a private incentive to deviate from the agreement Furthermore there is a unique Nash equiibrium in either the EI game or the LI game and these equiibria differ Hence whie there is itte operationa distinction between tracking echeon inventory or oca inventory (since we assume stationary demand) there is a significant strategic difference The suppier prefers oca inventory but the retaier s preference depends on the parameters of the game In the games we study competition generay owers suppy chain inventory reative to the optima soution In other words if firms cooperate and choose the optima soution they wi tend to increase inventory This is a surprising resut since many authors suggest the opposite (eg Buzze and Ortmeyer 995 Kumar 996) The rationae is that inventory is a pubic good: Each firm benefits from more inventory but each wants the other to invest in it It is we known that participants tend to underinvest in the provision of pubic goods (see Kreps 990) In other settings cooperation may ead to ower inventory For instance cooperative firms coud share saes information and this might enabe better poicies than those avaiabe to competitive firms Nevertheess inventory remains a pubic good even here; we suspect there is aways a strong tendency for competitive firms to choose ower inventory than in the optima soution Shoud the payers wish to choose the optima soution cooperativey we characterize a set of simpe inear contracts which eiminate each payer s incentive to deviate These contracts are based on actua inventories and backorders Impementation of these contracts wi not provide dramatic improvements when the payers have simiar preferences for reducing consumer backorders We draw this concusion from a sampe of 65 probems For each probem we measured the competition penaty the percentage increase in tota cost of the Nash equiibrium over the optima soution When the payers view consumer backorders as equay costy (ie 05) the median competition penaty in the EI game is ony 6% and in the LI game it is 3% However when the payers have divergent backorder costs the competition penaty can be huge For instance when the suppier is indifferent to consumer backorders the median competition penaty in the EI game is 483% These resuts highight an important esson for managers: Whie the ack of cooperation/coordination impies the system wi not perform at its best efficiency the magnitude of the efficiency oss is context specific The authors woud ike to thank the seminar participants at Rochester University the University of Michigan the University of Chicago and the 997 Muti-Echeon Inventory Conference at New York University The hepfu comments of Eric Anderson Marty Lariviere the referees and the associate editor are aso graciousy acknowedged References Axsäter S 996 A framework for decentraized muti-echeon inventory contro Working Paper Lund University Lund Sweden Buzze R G Ortmeyer 995 Chane partnerships streamine distribution Soan Management Rev 36 Cachon G C Camerer 996 Loss-avoidance and forward induction in experimenta coordination games Quart J Econom 65 94 Chen F 997 Decentraized suppy chains subject to information deays To appear in Management Sci 95 Management Science/Vo 45 No 7 Juy 999