Abstract
This paper considers a new class of stochastic resource allocation problems that requires simultaneously determining the customers that a capacitated resource must serve and the stock levels of multiple items that may be used in meeting these customers’ demands. Our model considers a reward (revenue) for serving each assigned customer, a variable cost for allocating each item to the resource, and a shortage cost for each unit of unsatisfied customer demand in a single-period context. The model maximizes the expected profit resulting from the assignment of customers and items to the resource while obeying the resource capacity constraint. We provide an exact solution method for this mixed integer nonlinear optimization problem using a Generalized Benders Decomposition approach. This decomposition approach uses Lagrangian relaxation to solve a constrained multi-item newsvendor subproblem and uses CPLEX to solve a mixed-integer linear master problem. We generate Benders cuts for the master problem by obtaining a series of subgradients of the subproblem’s convex objective function. In addition, we present a family of heuristic solution approaches and compare our methods with several MINLP (Mixed-Integer Nonlinear Programming) commercial solvers in order to benchmark their efficiency and quality.
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References
Ağralı, S., & Geunes, J. (2009). A single-resource allocation problem with Poisson resource requirements. Optimization Letters, 3(4), 559–571.
Akçay, H., Li, H., & Xu, S. H. (2007). Greedy algorithm for the general multidimensional knapsack problem. Annals of Operation Research, 150(1), 17–29.
Barnhart, C., & Cohn, A. M. (1998). The stochastic knapsack problem with random weights: a heuristic approach to robust transportation planning. In Proceedings of Tristan III, Puerto Rico, 17–23 June.
Benjaafar, S., Elhafsi, M., & Vericourt, F. D. (2004). Demand allocation in multiple-product, multiple-facility, make-to-stock systems. Management Science, 50(10), 1431–1448.
Dean, B. C., Goemans, M. X., & Vondrak, J. (2004). Approximating the stochastic knapsack problem: the benefit of adaptivity. In Proceedings of the 45th annual IEEE symposium on foundations of computer science.
Federgruen, A., & Zipkin, P. H. (1984). A combined vehicle routing and inventory allocation problem. Operations Research, 32(5), 1019–1037.
Federgruen, A., Prastacos, G., & Zipkin, P. H. (1986). An allocation and distribution model for perishable products. Operations Research, 34(1), 75–82.
Geoffrion, A. M. (1972). Generalized Benders decomposition. Journal of Optimization Theory and Applications, 10(4), 237–260.
Geromel, J. C., & Belloni, M. R. (1986). Nonlinear programs with complicating variables: theoretical analysis and numerical experience. IEEE Transactions on Systems, Man, and Cybernetics, 16(2), 231–239.
Geunes, J., Shen, Z.-J., & Romeijn, H. E. (2004). Economic ordering decisions with market choice flexibility. Naval Research Logistics, 51(1), 117–136.
Gorman, M. F., & Ahire, S. (2006). A major appliance manufacturer rethinks its inventory policies for service vehicles. Interfaces, 36(5), 407–419.
Grotschel, M. L., Lovasz, L., & Schrijver, A. (1981). The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1, 169–197.
Hadley, G., & Whitin, T. M. (1963). Analysis of inventory systems. Prentice-Hall: Englewood Cliffs.
Heyman, D. P., & Sobel, M. J. (1984). Stochastic models in operations research, vol. 2: stochastic optimization. New York: McGraw-Hill.
Iwata, S. L., Fleischer, L., & Fujishige, S. (2000). A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. In Proceedings of the 32nd annual ACM symposium on theory of computing, Portland, Oregon (pp. 97–106).
Kellerer, H., Pferschy, U., & Pisinger, D. (2004). Knapsack problems. Berlin: Springer.
Kleywegt, A., & Papastavrou, J. D. (2001). The dynamic and stochastic knapsack problem with random sized items. Operations Research, 49(1), 26–41.
Kosuch, S., & Lisser, A. (2010). Upper bounds for the 0-1 stochastic knapsack problem and a B&B algorithm. Annals of Operation Research, 176(1), 77–93.
Merzifonluoğlu, Y., Geunes, J., & Romeijn, H. E. The static stochastic knapsack problem with normally distributed item sizes. Mathematical Programming (2011, forthcoming).
Schrijver, A. (2000). A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory. Series B, 80(2), 346–355.
Shen, Z.-J., Coullard, C., & Daskin, M. S. (2003). A joint location-inventory model. Transportation Science, 37(1), 40–55.
Smith, S. A., Chambers, J. C., & Shlifer, E. (1980). Optimal inventories based on job completion rate for repairs requiring multiple items. Management Science, 26(8), 849–852.
Taaffe, K., Geunes, J., & Romeijn, H. E. (2008). Target market selection and marketing effort under uncertainty: the selective newsvendor. European Journal of Operational Research, 189(3), 987–1003.
Teunter, R. H., & Haneveld, W. K. (2002a). Inventory control of service parts in the final phase. European Journal of Operational Research, 137(3), 497–511.
Teunter, R. H., & Haneveld, W. K. (2002b). Inventory control of service parts in the final phase: a central depot and repair kits. European Journal of Operational Research, 138(1), 76–86.
Vasquez, M., & Vimont, Y. (2005). Improved results on the 0-1 multidimensional knapsack problem. European Journal of Operational Research, 165, 70–81.
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Chen, S., Geunes, J. Optimal allocation of stock levels and stochastic customer demands to a capacitated resource. Ann Oper Res 203, 33–54 (2013). https://doi.org/10.1007/s10479-011-0871-x
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DOI: https://doi.org/10.1007/s10479-011-0871-x