×

Production planning and inventories optimization: A backward approach in the convex storage cost case. (English) Zbl 1142.90302

Summary: We study the deterministic optimization problem of a profit-maximizing firm which plans its sales/production schedule. The firm controls both its production and sales rates and knows the revenue associated to a given level of sales, as well as its production and storage costs. The revenue and the production cost are assumed to be respectively concave and convex. In an earlier paper [Nonlinear Anal., Theory Methods Appl. 54A, No. 8, 1365–1395 (2003; Zbl 1064.90013)], we provided an existence result and derive some necessary conditions of optimality. Here, we further assume that the storage cost is convex. This allows us to relate the optimal planning problem to the study of a backward integro-differential equation, from which we obtain an explicit construction of the optimal plan.

MSC:

90B05 Inventory, storage, reservoirs
90B30 Production models
91B38 Production theory, theory of the firm

Citations:

Zbl 1064.90013
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Arrow, K. J.; Karlin, S.; Scarf, H., Studies in the Mathematical Theory of Inventory and Production (1958), Stanford University Press: Stanford University Press Stanford · Zbl 0079.36003
[3] Chazal, M.; Jouini, E.; Tahraoui, R., Production planning and inventories optimization with a general storage cost function, Nonlinear Analysis, 54, 1365-1395 (2003) · Zbl 1064.90013
[4] Feichtinger, G.; Hartl, R., Optimal pricing and production in an inventory model, European Journal of Operation Research, 19, 45-56 (1985) · Zbl 0551.90017
[5] Hale, J. K.; Verduyen Lunel, S. M., Introduction to Functional Differential Equations. Applied Mathematical Sciences, vol. 99 (1993), Springer-Verlag
[6] Holt, C. C.; Modigliani, F.; Muth, J. F.; Simon, H., Planning Production, Inventories and Work Force (1960), Prentice-Hall: Prentice-Hall Englewood Cliffs
[7] Maimon, O.; Khmelnitsky, E.; Kogan, K., Optimal Flow Control in Manufacturing System: Production Planning and Scheduling. Applied Optimization, vol. 18 (1998), Kluwer Academics Publishers: Kluwer Academics Publishers Dordrecht · Zbl 0953.90509
[8] Pekelman, D., Simultaneous price-production decisions, Operations Research, 22, 788-794 (1974) · Zbl 0284.90038
[9] Scarf, H., The optimality of \((s, S)\) policies in the dynamic inventory problem, (Arrow, K. J.; Karlin, S.; Suppes, P., Mathematical Methods in the Social Sciences (1960), Stanford University Press: Stanford University Press Stanford) · Zbl 0203.22102
[10] Sethi, S. P.; Thompson, G. L., Optimal Control Theory: Applications to Management Science. International Series in Management Science/Operations Research (1981), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.