# zbMATH — the first resource for mathematics

Pricing and inventory management in a system with multiple competing retailers under ($$r, Q$$) policies. (English) Zbl 1208.90006
Summary: We consider a multi-retailer system operated on an infinite horizon, in which each retailer faces stochastic demand following a Poisson process and adopts a continuous-review $$(r, Q)$$ policy for replenishing inventory to satisfy customer demand. The system involves decisions of pricing and inventory management with the goal of maximizing profit, which equals the sales revenue minus the purchase and inventory costs. Taking Cournot competition into account, models are formulated to optimize simultaneously the expected sales volumes and $$(r, Q)$$ policies of all retailers. An efficient approach is proposed to calculate the approximate inventory cost. Based on this approach, solution methods for centralized and decentralized scenarios are developed. A great number of numerical computations are provided to evaluate the efficiency of the solution methods, and their performance in the two scenarios. Moreover, system performance under sequential decisions (first pricing and then inventory management) is also investigated.

##### MSC:
 90B05 Inventory, storage, reservoirs 91B26 Auctions, bargaining, bidding and selling, and other market models 60K10 Applications of renewal theory (reliability, demand theory, etc.)
Full Text:
##### References:
 [1] Amir, R.; Jin, J.Y., Cournot and bertrand equilibria compared: substitutability, complementarity and concavity, International journal of industrial organization, 19, 3, 303-317, (2001) [2] Araman, V.F.; Caldentey, R., Dynamic pricing for nonperishable products with demand learning, Operations research, 57, 5, 1169-1188, (2009) · Zbl 1233.91099 [3] Aydin, G.; Porteus, E.L., Joint inventory and pricing decisions for an assortment, Operations research, 56, 5, 1247-1255, (2008) · Zbl 1167.90478 [4] Bernstein, F.; Federgruen, A., Pricing and replenishment strategies in a distribution system with competing retailers, Operations research, 51, 409-426, (2003) · Zbl 1163.90358 [5] Bernstein, F.; Federgruen, A., Dynamic inventory and pricing models for competing retailers, Naval research logistics, 51, 2, 258-274, (2004) · Zbl 1055.90005 [6] Bernstein, F.; Federgruen, A., A general equilibrium model for industries with price and service competition, Operations research, 52, 6, 868-886, (2004) · Zbl 1165.91432 [7] Bernstein, F.; Federgruen, A., Decentralized supply chains with competing retailers under demand uncertainty, Management science, 51, 1, 18-29, (2005) · Zbl 1232.90171 [8] Bish, E.K.; Wang, Q., Optimal investment strategies for flexible resources, considering pricing and correlated demands, Operations research, 52, 6, 954-964, (2004) · Zbl 1165.91428 [9] Cachon, G.P., Stock wars: inventory competition in a two-echelon supply chain with multiple retailers, Operations research, 49, 5, 658-674, (2001) · Zbl 1163.90319 [10] Carr, S.M.; Karmarkar, U.S., Competition in multiechelon assembly supply chains, Management science, 51, 1, 45-59, (2005) · Zbl 1232.90029 [11] Chan, L.M.A.; Shen Max, Z.J.; Simchi-Levi, D.; Swann, J., Coordination of pricing and inventory decisions: a survey and classification, () · Zbl 1125.91322 [12] Chao, X.; Zhou, S.X., Joint inventory-and-pricing strategy for a stochastic continuous-review system, IIE transactions, 38, 401-408, (2006) [13] Chen, F.; Federgruen, A.; Zheng, Y.S., Near-optimal pricing and replenishment strategies for a retail/distribution system, Operations research, 49, 6, 839-853, (2001) [14] Chen, F.; Federgruen, A.; Zheng, Y.S., Coordination mechanisms for a distribution system with one supplier and multiple retailers, Management science, 47, 5, 693-708, (2001) · Zbl 1232.90109 [15] Chen, X.; Simchi-Levi, D., Coordinating inventory control and pricing strategies with random demand and fixed ordering cost: the finite horizon case, Operations research, 52, 6, 887-896, (2004) · Zbl 1165.90308 [16] Corbett, C.J.; Karmarkar, U.S., Competition and structure in serial supply chains with deterministic demand, Management science, 47, 7, 966-978, (2001) · Zbl 1232.90175 [17] Dastidar, K.G., Comparing Cournot and bertrand in a homogenous product market, Journal of economic theory, 75, 1, 205-212, (1997) · Zbl 0888.90027 [18] Federgruen, A.; Heching, A., Combined pricing and inventory control under uncertainty, Operations research, 47, 3, 454-475, (1997) · Zbl 0979.90004 [19] Federgruen, A.; Zheng, Y.S., An efficient algorithm for computing an optimal (r, Q) policy in continuous review stochastic inventory systems, Operations research, 40, 4, 808-813, (1992) · Zbl 0758.90021 [20] Gallego, G.; Ryzin, G., Optimal dynamic pricing of inventories with stochastic demand over finite horizons, Management science, 40, 8, 999-1020, (1994) · Zbl 0816.90054 [21] Gallego, G.; Huh, W.T.; Kang, W.; Phillips, R., Price competition with the attraction demand model: existence of unique equilibrium and its stability, Manufacturing & service operations management, 8, 4, 359-375, (2006) [22] Guo, L.; Villas-Boas, J.M., Consumer stockpiling and price competition in differentiated markets, Journal of economics & management strategy, 16, 4, 827-858, (2007) [23] Hadley, G.; Whitin, T.M., Analysis of inventory systems, (1963), Prentice-Hall Englewood Cliffs, NJ · Zbl 0133.42901 [24] Hobbs, B.F.; Pang, J.S., Nash – cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints, Operations research, 55, 1, 113-127, (2007) · Zbl 1167.91356 [25] Hopp, W.J.; Xu, S., A static approximation for dynamic demand substitution with applications in a competitive market, Operations research, 56, 3, 630-645, (2008) · Zbl 1167.90330 [26] Lin, K.Y.; Sibdari, S.Y., Dynamic price competition with discrete customer choices, European journal of operational research, 197, 969-980, (2009) · Zbl 1176.90341 [27] Liu, L.; Parlar, M.; Zhu, S.X., Pricing and lead time decisions in decentralized supply chains, Management science, 53, 5, 713-725, (2007) · Zbl 1232.91252 [28] Majumder, P.; Srinivasan, A., Leadership and competition in network supply chains, Management science, 54, 6, 1189-1204, (2008) · Zbl 1232.90118 [29] Ohkawa, T.; Okamura, M., On the uniqueness of the welfare-maximizing number of firms under Cournot oligopoly, Bulletin of economic research, 55, 2, 209-219, (2003) [30] Okuguchi, K.; Szidarovszky, F., Changes in demand function in Cournot oligopoly, Pacific economic review, 10, 3, 371-378, (2005) [31] Parlar, M.; Weng, Z.K., Coordinating pricing and production decisions in the presence of price competition, European journal of operational research, 170, 211-227, (2006) · Zbl 1079.90041 [32] Roberts, J.; Sonnenschein, H., On the existence of Cournot equilibrium without concave profit functions, Journal of economic theory, 13, 1, 112-117, (1976) · Zbl 0341.90011 [33] Rosen, J.B., Existence and uniqueness of equilibrium points for concave N-person games, Econometrica, 33, 3, 520-534, (1965) · Zbl 0142.17603 [34] Singh, N.; Vives, X., Price and quantity competition in a differentiated duopoly, RAND journal of economics, 15, 4, 546-554, (1984) [35] Tang, C.S.; Yin, R., Joint ordering and pricing strategies for managing substitutable products, Production and operations management, 16, 1, 138-153, (2007) [36] Tyagi, R.K., On the effects of downstream entry, Management science, 45, 1, 59-73, (1999) · Zbl 1231.91131 [37] Vives, X., Oligopoly pricing, (2000), The MIT Press Cambridge, Massachusetts London, England [38] Wang, Q.; Wang, R., Quantity discount pricing policies for heterogeneous retailers with price sensitive demand, Naval research logistics, 52, 7, 645-658, (2005) · Zbl 1154.91404 [39] Wang, Y., Joint pricing-production decisions in supply chains of complementary products with uncertain demand, Operations research, 54, 6, 1110-1127, (2006) · Zbl 1167.90360 [40] Weng, Z.K., Channel coordination and quantity discounts, Management science, 41, 9, 1509-1522, (1995) · Zbl 0861.90067 [41] Wu, J.; Zhai, X.; Huang, Z., Incentives for information sharing in duopoly with capacity constraints, Omega, 36, 6, 963-975, (2008) [42] Xiao, T.; Qi, X., Price competition, cost and demand disruptions and coordination of a supply chain with one manufacturer and two competing retailers, Omega, 36, 5, 741-753, (2008) [43] Zhao, X., Coordinating a supply chain system with retailers under both price and inventory competition, Production and operations management, 17, 5, 532-542, (2008) [44] Zhao, X.; Atkins, D.R., Newsvendors under simultaneous price and inventory competition, Manufacturing & service operations management, 10, 3, 539-546, (2008) [45] Zhao, W.; Zheng, Y., Optimal dynamic pricing for perishable assets with nonhomogeneous demand, Management science, 46, 3, 375-388, (2000) · Zbl 1231.91106 [46] Zhu, K.; Thonemann, U.W., Coordination of pricing and inventory control across products, Naval research logistics, 56, 2, 175-190, (2009) · Zbl 1158.90310 [47] Zipkin, P.H., Foundations of inventory management, (2000), McGraw-Hill New York · Zbl 1370.90005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.