Research status and challenges of inventory control problems with nonlinear ordering cost

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  • 1. CEMS, MADIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;
    2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received date: 2021-02-22

  Online published: 2021-09-26

Abstract

Inventory management is one subject based on Operations Research, and has been one of the most popular topics in the area of Operations Research and Management Science in the last few decades. Ordering cost is one class of essential costs in inventory systems, and it includes product cost, shipping cost, loading/unloading cost, etc. The ordering cost is often a nonlinear function of order quantity. This paper will introduce several well-known nonlinear ordering cost functions, such as quantitydependent fixed/setup cost, incremental quantity discount, all-unit quantity discount, truckload discount, convex ordering cost, etc. We review the literature of inventory models with nonlinear ordering cost based on periodic-review and continuous-review models, respectively. Although the inventory models with nonlinear ordering cost functions have been studied in recent decades, the optimal policies of many models haven't been fully characterized due to their complexity. This paper tries to discuss the challenges and chances in this topic, by reviewing related works.

Cite this article

YAO Dacheng . Research status and challenges of inventory control problems with nonlinear ordering cost[J]. Operations Research Transactions, 2021 , 25(3) : 105 -118 . DOI: 10.15960/j.cnki.issn.1007-6093.2021.03.006

References

[1] Zipkin P H. Foundations of Inventory Management[M]. Boston:McGraw-Hill, 2000.
[2] Nahmias S. Production and Operations Analysis[M]. Boston:McGraw-Hill. 2009.
[3] Harris F. How many parts to make at once[J]. Factory, The Magazine of Management, 1913, 10:135-136, 152.
[4] Arrow K J, Harris T, Marschak J. Optimal inventory policy[J]. Econometrica, 1951, 19:250-272.
[5] Arrow K J, Karlin S, Scarf H, et al. Studies in Mathematical Theory of Inventory and Production[M]. Stanford:Stanford University Process, 1958.
[6] Scarf. H. The optimality of (S, s) policies in the dynamic inventory problems[M]//Mathematical Methods in the Social Sciences. Stanford:Stanford University Press, 1960:196-202.
[7] Iglehart D L. Optimality of (s, S) policies in the infinite horizon dynamic inventory problem[J]. Management Science, 1963, 9:259-267.
[8] Veinott A F Jr. On the optimality of (s, S) inventory policies:New conditions and a new proof[J]. SIAM Journal on Applied Mathematics, 1966, 14:1067-1083.
[9] Porteus E L. Foundation of Stochastic Inventory Theory[M]. Stanford:Stanford University Press, 2002.
[10] Lippman S A. Optimal inventory policy with subadditive ordering costs and stochastic demands[J]. SIAM Journal on Applied Mathematics, 1969, 17:543-559.
[11] Lippman S A. Economic order quantities and multiple set-up costs[J]. Management Science, 1971, 18:39-47.
[12] Iwaniec K. Inventory model with full load ordering[J]. Management Science, 1979, 25:374-384.
[13] Lee C-Y. A solution to the multiple set-up problem with dynamic demand[J]. IIE Transactions, 1989, 21:266-270.
[14] Alp O, Tim Huh W, Tan T. Inventory control with multiple setup cost[J]. Manufacturing & Service Operations Management, 2014, 16:89-103.
[15] Chao X, Zipkin P H. Optimal policy for a periodic-review inventory system under a supply capacity contract[J]. Operations Research, 2008, 56:59-68.
[16] Zhou B, Katehakis M N, Zhao Y. Managing stochastic inventory systems with free shipping option[J]. European Journal of Operational Research, 2009, 196:186-197.
[17] Caliskan-Demirag O, Chen Y, Yang Y. Ordering policies for periodic-review inventory systems with quantity-dependent fixed costs[J]. Operations Research, 2012, 60:785-796.
[18] He S, Yao D, Zhang H. Optimal ordering policy for inventory systems with quantity-dependent setup costs[J]. Mathematics of Operations Research, 2017, 42:979-1006.
[19] Porteus E L. On the optimality of generalized (s, S) policies[J]. Management Sciences, 1971, 17:411-426.
[20] Porteus E L. The optimality of generalized (s, S) policies under uniform demand densities[J]. Management Sciences, 1972, 18:644-646.
[21] Benjaafar S, Chen D, Yu Y. Optimal policies for inventory systems with concave ordering costs[J]. Naval Research Logistics, 2018, 65:291-302.
[22] Sethi S P. A quantity discount lot size model with disposals[J]. International Journal of Production Research, 1984, 22:31-39.
[23] Jucker J V, Rosenblatt M J. Single-period inventory models with demand uncertainty and quantity discounts:behavioral implications and a new solution procedure[J]. Naval Research Logistics, 1985, 32:537-550.
[24] San-José L A, Garcia-Laguna J. Optimal policy for an inventory system with backlogging and all-unit discounts:Application to the composite lot size model[J]. European Journal of Operational Research, 2009, 192:808-823.
[25] Toptal A. Replenishment decisions under an all-units discount shedule and stepwise freight costs[J]. European Journal of Operational Research, 2009, 198:504-510.
[26] Li C L, Ou J, Hsu V N. Dynamic lot sizing with all-unit discount and resales[J]. Naval Research Logistics, 2012, 59:230-243.
[27] Tamjidzard S, Mirmohammadi S H. An optimal (r, Q) policy in a stochastic inventory system with all-units quantity discount and limited sharable resource[J]. European Journal of Operational Research, 2015, 247:93-100.
[28] Li C L, Hsu V N, Xiao W Q. Dynamic lot sizing with batch ordering and truckload discounts[J]. Operations Research, 2004, 52:639-654.
[29] Elhedhli S, Benli Ö. Optimal lot sizing under carload discount schedules[J]. INFOR:Information Systems and Operational Research, 2005, 43:361-370.
[30] Henig M, Gerchak Y, Ernst R, et al. An inventory model embedded in designing a supply contract[J]. Management Science, 1997, 43:184-189.
[31] Lu Y, Song M. Inventory control with a fixed cost and a piecewise linear convex cost[J]. Production and Operations Management, 2014, 23:1966-1984.
[32] Yao D, Chao X, Wu J. Optimal policies for Brownian inventory systems with a piecewise linear ordering cost[J]. IEEE Transactions on Automatic Control, 2017, 62:3235-3248.
[33] Lippman S A. Optimal inventory policy with multiple set-up costs[J]. Management Science, 1969, 16:118-138.
[34] Federgruen A, Lee C-Y. The dynamic lot size model with quantity discount[J]. Naval Research Logistics, 1990, 37:707-713.
[35] Buchanan J M. The theory of monopolistic quantity discounts[J]. The Review of Economic Studies, 1952-1953, 20:199-208.
[36] Garbor A. A note on block tariffs[J]. The Review of Economic Studies, 1955, 23:32-41.
[37] Lal R, Staelin R. An approach for developing an optimal discount policy[J]. Management Science, 1984, 30:1524-1539.
[38] Monahan J P. A quantity discount pricing model to increase vendor profits[J]. Management Sciences, 1984, 30:720-726.
[39] Lee H L, Rosenblatt M J. A generalized quantity discount pricing model to increase supplier's profits[J]. Management Science, 1986, 32:1177-1185.
[40] Zhang J, Chen J. Supplier selection and procurement decisions with uncertain demand, fixed selection costs and quantity discounts[J]. Computers & Operations Research, 2013, 40:2703-2710.
[41] Chen F, Federgruen A, Zheng Y S. Coordiation mechanisms for a distribution system with one supplier and multiple retailers[J]. Management Science, 2001, 47:693-708.
[42] Chen X. Inventory centralization games with price-dependent demand and quantity discount[J]. Operations Research, 2009, 57:1394-1406.
[43] Chen X, Zhang J. A stochastic programming duality approach to inventory centralization games[J]. Operations Research, 2009, 57:840-851.
[44] Benton W C, Park S. A classification of literature on determining the lot size under quantity discounts[J]. European Journal of Operational Research, 1996, 92:219-238.
[45] Munson C L, Rosenblatt M J. Theories and realities of quantity discounts:an exploratory study[J]. Production and Operations Management, 1998, 7:352-369.
[46] Altintas N, Erhun F, Tayur. Quantity discounts under demand uncertainty[J]. Management Science, 2008, 54:777-792.
[47] Fox E J, Metters R, Semple J. Optimal inventory policy with two suppliers[J]. Operations Research, 2006, 54:389-393.
[48] Chen X, Zhang Y, Zhou S X. Preservation of quasi-k-concavity and its applications[J]. Operations Research, 2010, 58:1012-1016.
[49] Wang H, Yu Y, Zhang W, et al. Procurement strategies for lost-sales inventory systems with all-unit dicounts[J]. European Journal of Operational Research, 2019, 272:539-548.
[50] Karlin S. Dynamic inventory policy with varying stochastic demand[J]. Management Science, 1960, 6:231-258.
[51] Sobel M J. Production smoothing with stochastic demand I:finite horizon case[J]. Management Science, 1969, 16:195-207.
[52] Sobel M J. Production smoothing with stochastic demand II:infinite horizon case[J]. Management Science, 1971, 17:724-735.
[53] Chen Y, Sethi S P, Zhang H. A production-inventory problem for an energy buy-back program[J]. IEEE Transactions on Automatic Control, 2007, 4:395-406.
[54] Ha Y A, Tong S, Zhang H. Sharing demand information in competing supply chains with production diseconomies[J]. Management Science, 2011, 57:566-581.
[55] Porteus E. Stochastic inventory theory[M]//Handbooks in Operations Research and Management Science. North-Holland:Elsevier, 1990:605-652.
[56] Zhang W, Hua Z, Benjaafar S. Optimal inventory control with dual-sourcing heterogeneous ordering cost and order size constraints[J]. Production and Operations Management, 2012, 21:564-575.
[57] Lu Y, Song M, Yang Y. Approximation approaches for inventory systems with general production/ordering cost structures[J]. Production and Operations Management, 2018, 27:417-432.
[58] Beyer D, Sethi S P. A proof of the EOQ formula using quasi-variational inequalities[J]. International Journal of Systems Science, 1998, 29:1295-1299.
[59] Perera S, Janakiraman G, Niu S-C. Optimality of (s, S) policies in EOQ models with general cost structures[J]. International Journal of Production Economics, 2017, 187:216-228.
[60] Yao D, Chao X, Wu J. Optimal control policy for a Brownian inventory system with concave ordering cost[J]. Journal of Applied Probability, 2015, 52:900-925.
[61] Helmes K L, Stockbridge R H, Zhu C. Continuous inventory models of diffusion type:long-term average criterion[J]. The Annals of Applied Probability, 2017, 27:1831-1885.
[62] Helmes K L, Stockbridge R H, Zhu C. A weak convergence approach to inventory control using a long-term average criterion[J]. Advance of Applied Probability, 2018, 50:1032-1074.
[63] Wei B, Yao D. Ergodic inventory control with diffusion demand and general ordering costs[J]. Operations Research Letters, 2021, 49:578-585.
[64] Helmes K L, Stockbridge R H, Zhu C. A measure approach for continuous inventory models:discounted cost criterion[J]. SIAM Journal on Control and Optimization, 2015, 53:2100-2140.
[65] Xu F, Yao D, Zhang H. Impulse control with discountinuous setup costs:discounted cost criterion[J]. SIAM Journal on Control and Optimization, 2021, 59:267-295.
[66] Perera S, Janakiraman G, Niu S-C. Optimality of (s, S) inventory policies under renewal demand and general cost structures[J]. Production and Operations Management, 2018, 27:368-383.
[67] Xin L, Goldberg D A. Optimality gap of constant-order policies decays exponentially in the leadtime for lost sales models[J]. Operations Research, 2016, 64:1556-1565.
[68] Bu J, Gong X, Yao D. Constant-order policies for lost-sales inventory models with random supply functions:Asymptotics and heuristic[J]. Operations Research, 2020, 68:1063-1073.
[69] Xin L. Understanding the performance of capped base-stock policies in lost-sales inventory models[J]. Operations Research, 2021, 69(1):61-70.
[70] Xin L, Goldberg D A. Asymptotic optimality of tailored base-surge policies in dual-sourcing inventory systems[J]. Management Science, 2018, 64:437-452.
[71] Bensoussan A, Liu R H, Sethi S P. Optimality of an (s, S) policy with compound Poisson and diffustion demands:A quasi-variational inequalities approach[J]. SIAM Journal on Control and Optimization, 2005, 44:1650-1676.
[72] Van Foreest N D, Wijngaard J. On optimal policies for production-inventory systems with compound Poisson demand and setup costs[J]. Mathematics of Operations Research, 2014, 39:517-532.
[73] Cohen M A, Lee H L. Designing the right global supply chain network[J]. Manufacturing & Service Operations Management, 2020, 22:15-24.
[74] Mišić V V, Perakis G. Data analytics in operations management:a review[J]. Manufacturing & Service Operations Management, 2020, 22:158-169.
[75] Baardman L, Levin I, Perakis G, et al. Leveraging comparables for new product sales forecasting[J]. Production and Operations Management, 2018, 27:2340-2343.
[76] Ban G-Y, Rudin C. The big data newsvendor:Practical insights from machine learning[J]. Operations Research, 2019, 67:90-108.
[77] Bertsimas D, Kallus N. From predictive to prescriptive analytics[J]. Management Science, 2020, 66:1025-1044.
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