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A Stochastic Transportation Problem

Author

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  • A. C. Williams

    (Research Department, Socony Mobil Oil Company, Inc., Paulsboro, New Jersey)

Abstract

When the market demands for a commodity are not known with certainty, the problem of scheduling shipments to a number of demand points from several supply points is a stochastic transportation problem . If the dynamic aspects of the problem may be neglected, i.e., if the effects of “stock outs” and “inventory carry-overs” can be reflected in terms of linear penalty costs (in the case of undersupply) and linear salvage values (in case of oversupply), one obtains a convex nonlinear programming problem. We show that the author's algorithm for the case of known demands can be generalized to obtain a solution algorithm for this stochastic problem. In case the joint cumulative distribution function for the demand is continuous, however, the algorithm is better described as a special (constructive) case of Dantzig's general method for convex programming. Each of these methods is, in turn, based on the decomposition algorithm. No assumptions are made as to the independence or dependence of the probability distributions of the various demands. If F ((xi) 1 , ..., (xi) n ) is the joint cumulative distribution function for the demands, we assume only that each of the first moments of F exist (are finite), and that F is regular enough for the integration by parts formula to be valid. The analysis here is applicable (subject to a qualification) to the case of indivisible commodities, as well as the case of infinitely divisible ones.

Suggested Citation

  • A. C. Williams, 1963. "A Stochastic Transportation Problem," Operations Research, INFORMS, vol. 11(5), pages 759-770, October.
    • Handle: RePEc:inm:oropre:v:11:y:1963:i:5:p:759-770
    DOI: 10.1287/opre.11.5.759
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    Cited by:

    1. Edwin Romeijn, H. & Zeynep Sargut, F., 2011. "The stochastic transportation problem with single sourcing," European Journal of Operational Research, Elsevier, vol. 214(2), pages 262-272, October.
    2. Soheila Abdi & Fahimeh Baroughi & Behrooz Alizadeh, 2018. "The Minimum Cost Flow Problem of Uncertain Random Network," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(03), pages 1-18, June.
    3. Yolanda Hinojosa & Justo Puerto & Francisco Saldanha-da-Gama, 2014. "A two-stage stochastic transportation problem with fixed handling costs and a priori selection of the distribution channels," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 1123-1147, October.
    4. Qi Feng & Chengzhang Li & Mengshi Lu & Jeyaveerasingam George Shanthikumar, 2022. "Dynamic Substitution for Selling Multiple Products under Supply and Demand Uncertainties," Production and Operations Management, Production and Operations Management Society, vol. 31(4), pages 1645-1662, April.
    5. Killmer, K. A. & Anandalingam, G. & Malcolm, S. A., 2001. "Siting noxious facilities under uncertainty," European Journal of Operational Research, Elsevier, vol. 133(3), pages 596-607, September.
    6. Zhu, Kai & Ji, Kaiyuan & Shen, Jiayu, 2021. "A fixed charge transportation problem with damageable items under uncertain environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 581(C).
    7. Amrit Das & Gyu M. Lee, 2021. "A Multi-Objective Stochastic Solid Transportation Problem with the Supply, Demand, and Conveyance Capacity Following the Weibull Distribution," Mathematics, MDPI, vol. 9(15), pages 1-21, July.
    8. H K Smith & G Laporte & P R Harper, 2009. "Locational analysis: highlights of growth to maturity," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 60(1), pages 140-148, May.
    9. Zhaowei Hao & Long He & Zhenyu Hu & Jun Jiang, 2020. "Robust Vehicle Pre‐Allocation with Uncertain Covariates," Production and Operations Management, Production and Operations Management Society, vol. 29(4), pages 955-972, April.
    10. Subhakanta Dash & S. P. Mohanty, 2018. "Uncertain transportation model with rough unit cost, demand and supply," OPSEARCH, Springer;Operational Research Society of India, vol. 55(1), pages 1-13, March.

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