Abstract
Das (Oper Res 25(5):835–850, 1977) considered the optimization of a cost function associated with an (S − 1, S) inventory model assuming the parameters to be the initial number of items in the stock and the service rate. A similar optimization problem associated with an M/E k /1 queueing system with parameters being the number of servers and the service rate was considered by Kumin (Manag Sci 20:126–129, 1973). Both carried out case-dependent computations and indicated the difficulty of finding general convexity and optimization results for functions with both integer and real variables. In this paper, generalized mixed convexity and computational optimization results for the cost function associated with the (S − 1, S) inventory system suggested by Das are provided. The generalized convexity results determine the convexity region of the cost function, and therefore the region of possible minimal values of the cost function in the domain. In addition, algorithms to determine the generalized convexity and computational optimization results for the cost function are given.
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References
Al-Yakoob SM, Sherali HD, Al-Jazzaf M (2010) A mixed-integer mathematical modeling approach to exam timetabling. Comput Manag Sci 7: 19–46
Andersson J, Melchiors P (2001) A two-echelon inventory model with lost sales. Int J Prod Econ 69: 307–315
Benders JF (2005) Partitioning procedures for solving mixed-variables programming problems. Comput Manag Sci 2: 3–19
Das C (1977) The (S − 1, S) Inventory model under time limit on backorders. Oper Res 25(5): 835–850
Feeney G, Sherbrooke C (1966) The (S − 1, S) Inventory policy under compound poisson demand. Manag Sci 12(5): 391–411
Galliher H, Morse P, Simond M (1959) Dynamics of two classes of continuous-review inventory systems. Oper Res 7: 362–384
Gross D, Harris CM (1973) Continuous-review (s, S) inventory models with state-dependent lead times. Manag Sci 19: 567–574
Gümüş ZH, Floudas CA (2005) Global optimization of mixed-integer bilevel programming problems. Comput Manag Sci 2: 181–212
Hadley G, Whitin T (1963) Analysis of inventory systems. Prentice-Hall, Englewood Cliffs
Hayya JC, Bagchi U, Ramasesh R (2011) Cost relationships in stochastic inventory systems: A simulation study of the (S, S − 1, t=1) model. Int J Prod Econ 130: 196–202
Kumin H (1973) On characterizing the extrema of a function of two variables, one of which is discrete. Manag Sci 20: 126–129
Liu L, Cheung KL (1997) Service constrained inventory models with random lifetimes and lead times. J Oper Res Soc 48(10): 1022–1028
Moinzadeh K (1989) Operating characteristics of the (S–1, S) inventory system with partial backorders and constant resupply times. Manag Sci 35(4): 472–477
Moinzadeh K, Schmidt CP (1991) An (S–1, S) inventory system with emergency orders. Oper Res 39(3): 308–321
Smith SA (1977) Optimal inventories for (S–1, S) system with no backorders. Manag Sci 23(5): 522–528
Tokgöz E (2009) Algorithms for mixed convexity and optimization of 2-smooth mixed functions. Int J Pure Appl Math 57(1): 103–110
Tokgöz E, Maalouf M, Kumin H (2009) A Hessian matrix for functions with integer and continuous variables. Int J Pure Appl Math 57(2): 209–218
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Tokgöz, E., Kumin, H. Mixed convexity and optimization results for an (S − 1, S) inventory model under a time limit on backorders. Comput Manag Sci 9, 417–440 (2012). https://doi.org/10.1007/s10287-011-0134-y
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DOI: https://doi.org/10.1007/s10287-011-0134-y