The continuous modular design problem with linear separable side constraints. (English) Zbl 0746.90055
Summary: This paper describes a heuristic for designing a single module in the presence of linear side constraints and nonlinear requirements constraints. The objective is to minimize the total parts cost to satisfy demand for a set of end products. We model the problem as a nonconvex programming problem and show that the problem can be transformed into an equivalent convex problem. Due to the size of the potential applications we develop a heuristic procedure that is an extension of algorithm of an T. L. Shaftel and G. L. Thompson [Nav. Res. Logist. Q. 30, 199-215 (1983; Zbl 0536.90054)]. Closed form expressions are derived for determining feasible movement directions and step lengths. Computational results are promising since the CPU time required to run the heuristic is small and the solutions compared favorably with solutions generated by standard nonlinear programming software.
MSC:
| 90C26 | Nonconvex programming, global optimization |
| 90B30 | Production models |
| 90-08 | Computational methods for problems pertaining to operations research and mathematical programming |
| 90C25 | Convex programming |
| 90C30 | Nonlinear programming |
| 90C90 | Applications of mathematical programming |
Keywords:
design; manufacturing industries; heuristic; linear side constraints; nonlinear requirements constraintsCitations:
Zbl 0536.90054Software:
NPSOLReferences:
| [1] | Bazaraa, M. S.; Shetty, C. M., Nonlinear Programming: Theory and Algorithms (1979), Wiley: Wiley New York · Zbl 0476.90035 |
| [2] | Evans, D., Modular design — A special case of nonlinear programming, Operations Research, 11, 637-647 (1963) · Zbl 0113.35801 |
| [3] | Evans, D., A note on modular design — A special case of nonlinear programming, Operations Research, 18, 562-564 (1970) |
| [4] | Gill, P.; Murray, W.; Saunders, M.; Wright, M., User’s guide for npsol (Version 4.0): A Fortran package for nonlinear programming, (Technical Report SOL 86-2 (1986), Systems Optimization Laboratory, Department of Operations Research, Stanford University) |
| [5] | Goldberg, J., The modular design problem with linear separable side constraints: Heuristics and applications (1984), The University of Michigan, Unpublished Ph.D. Dissertation |
| [6] | Goldberg, J., Integer modular design with linear separable side constraints, (Working Paper #86-022 (1986), Department of Systems and Industrial Engineering, University of Arizona) · Zbl 0746.90055 |
| [7] | Karmarkar, U. S.; Kubat, P., Modular product design and product support, European Journal of Operational Research, 29, 74-82 (1987) · Zbl 0609.90051 |
| [8] | Mangasarian, O. L., Nonlinear Programming (1969), McGraw-Hill: McGraw-Hill New York · Zbl 0194.20201 |
| [9] | Moscato, D. R., An economic theory of modular production (1972), Columbia University, Unpublished Ph.D. Dissertation |
| [10] | Murtagh, B. A.; Saunders, M. A., A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints, Mathematical Programming Study, 16, 84-117 (1982) · Zbl 0477.90069 |
| [11] | O’Reilly, G. P., The modular design problem with production and inventory considerations (1975), Columbia University, Unpublished Ph.D. Dissertation |
| [12] | Passy, U., Modular design: An application of structured geometric programming, Operations Research, 18, 441-453 (1970) · Zbl 0218.90057 |
| [13] | Robinson, S. M., A quadratically convergent algorithm for general nonlinear programming problems, Mathematical Programming, 3, 415-429 (1972) |
| [14] | Shaftel, T., An integer approach to modular design, Operations Research, 19, 130-134 (1971) · Zbl 0216.54502 |
| [15] | Shaftel, T., How modular design reduces production costs, The Arizona Business Review, 21, 1-5 (1972) |
| [16] | Shaftel, T.; Thompson, G., A simplex-like algorithm for the continuous modular design problem, Operations Research, 25, 788-805 (1977) · Zbl 0383.90080 |
| [17] | Shaftel, T.; Thompson, G., The continuous multiple-modular design problem, Naval Research Logistics quarterly, 30, 199-215 (1983) · Zbl 0536.90054 |
| [18] | Smeers, Y., A monotonic Passy-like algorithm for the modular design problem, Mathematical Programming, 7, 46-55 (1974) · Zbl 0314.90084 |
| [19] | Srinivasan, V.; Thompson, G. L., Accelerated algorithms for labeling and relabeling trees, Journal of the Association for Computing Machinery, 19, 712-726 (1972) · Zbl 0255.90071 |
| [20] | Starr, M. K., Modular production — A new concept, 43, 131-142 (1965), November-December |
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