# zbMATH — the first resource for mathematics

Procurement strategies for lost-sales inventory systems with all-units discounts. (English) Zbl 1403.90060
Summary: Despite the prevalence of all-units discounts in procurement contracts, these discounts pose a technical challenge to analyze procurement strategies due to neither concave nor convex ordering costs. In this paper, we consider the optimal procurement strategies with all-units discounts under the lost-sales setting. By assuming log-concave demands, we find that the optimal procurement strategies have a generalized $$Q$$-jump ($$s, S$$) structure by introducing a new notion of $$Q$$-jump single-crossing. In particular, a sufficient condition is provided for degenerating the optimal procurement strategies from a generalized $$Q$$-jump ($$s, S$$) structure into a $$Q$$-jump ($$s, S$$) structure, which is definitely optimal for the single-period problem. Extensive numerical results suggest that the $$Q$$-jump ($$s, S$$) policy as a heuristic performs considerably well when its optimality sufficient condition is violated. Our results can be extended to systems with multi-break all-units discounts, and systems with all-units discounts on batch ordering.

##### MSC:
 90B05 Inventory, storage, reservoirs
Full Text:
##### References:
 [1] Altintas, N.; Erhun, F.; Tayur, S., Quantity discounts under demand uncertainty, Management Science, 54, 4, 777-792, (2008) · Zbl 1232.90003 [2] Benton, W.; Park, S., A classification of literature on determining the lot size under quantity discounts, European Journal of Operational Research, 92, 2, 219-238, (1996) · Zbl 0916.90084 [3] Buchanan, J. M., The theory of monopolistic quantity discounts, Review of Economic Studies, 20, 3, 199-208, (1952) [4] Caliskan-Demirag, O.; Chen, Y.; Yang, Y., Ordering policies for periodic-review inventory systems with quantity-dependent fixed costs, Operations Research, 60, 4, 785-796, (2012) · Zbl 1260.90010 [5] Chen, X.; Zhang, Y.; Zhou, S. X., Preservation of quasi-K-concavity and its applications, Operations Research, 58, 4, 1012-1016, (2010) · Zbl 1231.90020 [6] Crama, Y.; Torres, A., Optimal procurement decisions in the presence of total quantity discounts and alternative product recipes, European Journal of Operational Research, 159, 2, 364-378, (2004) · Zbl 1065.90047 [7] Dharmadhikari, S.; Joag-Dev, K., Unimodality, convexity, and applications, (1988), Elsevier · Zbl 0646.62008 [8] Dolan, R. J., Quantity discounts: managerial issues and research opportunities, Marketing Science, 6, 1, 1-22, (1987) [9] Drezner, Z.; Wesolowsky, G., Multi-buyer discount pricing, European Journal of Operational Research, 40, 1, 38-42, (1989) · Zbl 0667.90014 [10] Fox, E. J.; Metters, R.; Semple, J., Optimal inventory policy with two suppliers, Operations Research, 54, 2, 389-393, (2006) · Zbl 1167.90325 [11] Gabor, A., A note on block tariffs, Review of Economic Studies, 23, 1, 32-41, (1955) [12] Hu, M.; Pavlin, J. M.; Shi, M., When gray markets have silver linings: all-unit discounts, gray markets, and channel management, Manufacturing & Service Operations Management, 15, 2, 250-262, (2013) [13] Huggins, E. L.; Olsen, T. L., Inventory control with generalized expediting, Operations Research, 58, 1414-1426, (2010) · Zbl 1233.90023 [14] Jucker, J. V.; Rosenblatt, M., Single-period inventory models with demand uncertainty and quantity discounts: behavioral implications and a new solution procedure, Naval Research Logistics, 32, 4, 537-550, (1985) · Zbl 0597.90024 [15] Kolay, S.; Shaffer, G.; Ordover, J. A., All-units discounts in retail contracts, Journal of Economics & Management Strategy, 13, 3, 429-459, (2004) [16] Lal, R.; Staelin, R., An approach for developing an optimal discount pricing policy, Management Science, 30, 12, 1524-1539, (1984) [17] Lee, H. L.; Rosenblatt, M. J., A generalized quantity discount pricing model to increase supplier’s profits, Management Science, 32, 9, 1177-1185, (1986) · Zbl 0605.90022 [18] Li, C.-L.; Ou, J.; Hsu, V., Dynamic lot sizing with all-units discount and resales, Naval Research Logistics, 59, 3-4, 230-243, (2012) [19] Li, Q.; Yu, P., On the quasiconcavity of lost-sales inventory models with fixed costs, Operations Research, 60, 2, 286-291, (2012) · Zbl 1248.90012 [20] Lu, Y.; Song, M., Inventory control with a fixed cost and a piecewise linear convex cost, Production and Operations Management, 23, 11, 1966-1984, (2014) [21] Monahan, J. P., A quantity discount pricing model to increase vendor profits, Management Science, 30, 6, 720-726, (1984) [22] Munson, C. L.; Rosenblatt, M. J., Theories and realities of quantity discounts: an exploratory study, Production and Operations Management, 7, 4, 352-369, (1998) [23] Rosling, K., Inventory cost rate functions with nonlinear shortage costs, Operations Research, 50, 6, 1007-1017, (2002) · Zbl 1163.90346 [24] Rubin, P. A.; Benton, W., A generalized framework for quantity discount pricing schedules, Decision Sciences, 34, 1, 173-188, (2003) [25] San-José, L. A.; García-Laguna, J., Optimal policy for an inventory system with backlogging and all-units discounts: application to the composite lot size model, European Journal of Operational Research, 192, 3, 808-823, (2009) · Zbl 1157.90329 [26] Schoenberg, I., On Pólya frequency functions, Journal d’Analyse Mathématique, 1, 1, 331-374, (1951) · Zbl 0045.37602 [27] Sethi, S. P., A quantity discount lot size model with disposals, International Journal of Production Research, 22, 1, 31-39, (1984) [28] Sohn, K.-I.; Hwang, H., A dynamic quantity discount lot size model with resales, European Journal of Operational Research, 28, 3, 293-297, (1987) · Zbl 0616.90031 [29] Tamjidzad, S.; Mirmohammadi, S. H., An optimal (r, Q) policy in a stochastic inventory system with all-units quantity discount and limited sharable resource, European Journal of Operational Research, 247, 1, 93-100, (2015) · Zbl 1346.90058 [30] Toptal, A., Replenishment decisions under an all-units discount schedule and stepwise freight costs, European Journal of Operational Research, 198, 2, 504-510, (2009) · Zbl 1163.90376 [31] Weng, Z. K., Channel coordination and quantity discounts, Management Science, 41, 9, 1509-1522, (1995) · Zbl 0861.90067 [32] Weng, Z. K., Modeling quantity discounts under general price-sensitive demand functions: optimal policies and relationships, European Journal of Operational Research, 86, 2, 300-314, (1995) · Zbl 0906.90102 [33] Yang, Y.; Chen, Y.; Zhou, Y., Coordinating inventory control and pricing strategies under batch ordering, Operations Research, 62, 1, 25-37, (2014) · Zbl 1291.90027 [34] Zhang, W.; Hua, Z.; Benjaafar, S., Optimal inventory control with dual-sourcing, heterogeneous ordering costs and order size constraints, Production and Operations Management, 21, 3, 564-575, (2012)
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.