×

A goal programming procedure for solving problems with multiple fuzzy goals using dynamic programming. (English) Zbl 1012.90074

Summary: This paper describes how the preemptive priority based goal programming (GP) can be used to solve a class of fuzzy programming (FP) problems with the characteristics of dynamic programming (DP).
In the proposed approach, the membership functions of the objective goals of a problem with fuzzy aspiration levels are defined first. Then, under the framework of preemptive priority based GP a multi-stage DP model of the problem for achievement of the highest degree (unity) of each of the membership functions is developed. In the decision process, the goal satisficing philosophy of GP is used recursively to arrive at the most satisfactory solution. Two numerical examples are provided to illustrate the approach.

MSC:

90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
90C29 Multi-objective and goal programming
90C39 Dynamic programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baldwin, J. F.; Pilsworth, B. W., Dynamic programming for fuzzy systems with fuzzy environment, Journal of Mathematical Analysis and Applications, 85, 1-23 (1982) · Zbl 0491.90090
[2] Bellman, R., Dynamic Programming (1957), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0077.13605
[3] Bellman, R. E.; Zadeh, L. A., Decision-making in a fuzzy environment, Management Science, 17, B141-B164 (1970) · Zbl 0224.90032
[4] Bhattacharya, U.; Rao, J. R.; Tiwari, R. N., Fuzzy multi-criteria facility location problem, Fuzzy Sets and Systems, 51, 277-287 (1992) · Zbl 0787.90039
[5] Charnes, A.; Cooper, W. W., Management Models and Industrial Applications of Linear Programming (1961), Wiley: Wiley New York · Zbl 0107.37004
[6] Chen, H. K., A note on a fuzzy goal programming algorithm by Tiwari, Dharmar and Rao, Fuzzy Sets and Systems, 62, 287-290 (1994)
[7] Cohon, J. L., Multiobjective Programming and Planning (1978), Academic Press: Academic Press New York · Zbl 0462.90054
[8] Daellenbach, H. G.; De Kluyver, C. A., Note on multiple objective dynamic programming, Journal of the Operational Research Society, 31, 591-594 (1980) · Zbl 0434.90086
[9] Dauer, J. P.; Krueger, R. J., An iterative approach to goal programming, Operational Research Quarterly, 28, 671-681 (1977) · Zbl 0373.90062
[10] Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press: Academic Press New York · Zbl 0444.94049
[11] Dutta, D.; Rao, J. R.; Tiwari, R. N., Sensitivity analysis in fuzzy linear fractional programming problem, Fuzzy Sets and Systems, 48, 211-216 (1992) · Zbl 0767.90084
[12] Dutta, D.; Rao, J. R.; Tiwari, R. N., Effect of tolerance in fuzzy linear fractional programming, Fuzzy Sets and Systems, 55, 133-142 (1993) · Zbl 0791.90069
[13] Dutta, D.; Tiwari, R. N.; Rao, J. R., Multiple objective linear fractional programming - A fuzzy set theoretic approach, Fuzzy Sets and Systems, 52, 39-45 (1992) · Zbl 0786.90088
[14] Dyson, R. G., Maximin programming, fuzzy linear programming and multicriteria decision making, Journal of the Operational Research Society, 31, 263-267 (1981) · Zbl 0436.90106
[15] Esogbue, A. O., Optimal clustering of fuzzy data via fuzzy dynamic programming, Fuzzy Sets and Systems, 18, 283-298 (1986) · Zbl 0616.62088
[16] Hamacher, H.; Leberling, H.; Zimmermann, H.-J., Sensitivity analysis in fuzzy linear programming, Fuzzy Sets and Systems, 1, 269-281 (1978) · Zbl 0408.90051
[17] Hannan, E. L., On fuzzy goal programming, Decision Sciences, 12, 522-531 (1981)
[18] Hannan, E. L., Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems, 6, 235-248 (1981) · Zbl 0465.90080
[19] Hwang, C. L.; Masud, A. M., Multiple Objective Decision Making - Methods and Applications (1979), Springer: Springer Berlin · Zbl 0397.90001
[20] Ignizio, J. P., Goal Programming and Extensions (1976), D.C. Heath: D.C. Heath Lexington, MA · Zbl 1052.90584
[21] Ignizio, J. P., A review of goal programming: A tool for multiobjective analysis, Journal of the Operational Research Society, 29, 1109-1119 (1978) · Zbl 0396.90093
[22] Ignizio, J. P., Antenna array beam pattern synthesis via goal programming, European Journal of Operational Research, 6, 286-290 (1981) · Zbl 0447.90089
[23] Ignizio, J. P., The determination of a subset of efficient solutions via goal programming, Computers and Operations Research, 8, 9-16 (1981)
[24] Ignizio, J. P., Linear programming in Single and Multiple-Objective Systems (1982), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0484.90068
[25] Ignizio, J. P., On the (re)discovery of fuzzy goal programming, Decision Sciences, 13, 331-336 (1982)
[26] Ignizio, J. P., Generalized goal programming: An overview, Computers and Operations Research, 10, 277-289 (1983)
[27] Ignizio, J. P., An algorithm for solving the linear goal programming by solving its dual, Journal of the Operational Research Society, 36, 507-515 (1985) · Zbl 0565.90071
[28] Ignizio, J. P., Introduction to Linear Goal Programming (1985), Sage: Sage Beverley Hills, CA · Zbl 0662.90075
[29] Ignizio, J. P., A reply to “Comments on an algorithm for solving the linear goal – programming problem by solving its dual”, Journal of the Operational Research Society, 38, 1149-1154 (1987) · Zbl 0631.90074
[30] Ijiri, Y., Management Goals and Accounting for Control (1965), North-Holland: North-Holland Amsterdam
[31] Kornbluth, J. S.H., A survey of goal programming, Omega, 1, 193-205 (1973)
[32] Kornbluth, J. S.H.; Steuer, R. E., Goal programming with linear fractional criteria, European Journal of Operational Research, 8, 58-65 (1981) · Zbl 0486.90077
[33] Kvanli, A. H., Financial planning using goal programming, Omega, 8, 207-218 (1980)
[34] Lee, S. M., Goal Programming for Decision Analysis (1972), Auerbach Publishers: Auerbach Publishers Philadelphia
[35] Lee, C. S.; Wen, C. G., Fuzzy goal programming approach for water quality management in a river basin, Fuzzy Sets and Systems, 89, 181-192 (1997)
[36] Levary, R. R., Dynamic programming models with goal objectives, International Journal of Systems Science, 15, 309-314 (1984) · Zbl 0537.90088
[37] Lin, W. T., A survey of goal programming applications, Omega, 8, 115-117 (1980)
[38] Luhandjula, M. K., Fuzzy approaches for multiple objective linear fractional optimization, Fuzzy Sets and Systems, 13, 11-23 (1984) · Zbl 0546.90094
[39] Mohamed, R. H., The relationship between goal programming and fuzzy programming, Fuzzy Sets and Systems, 89, 215-222 (1997)
[40] Narasimhan, R., Goal programming in a fuzzy environment, Decision Sciences, 11, 325-336 (1980)
[41] Narasimhan, R., On fuzzy goal programming - Some comments, Decision Sciences, 12, 532-538 (1980)
[42] Nemhauser, G. L., Introduction to Dynamic Programming (1966), Wiley: Wiley New York · Zbl 0139.13202
[43] Ogryczak, W., Symmetric duality theory for linear goal programming, Optimization, 19, 373-396 (1988) · Zbl 0665.90058
[44] Olson, D. L., Comparison of four goal programming algorithms, Journal of the Operational Research Society, 35, 347-354 (1984) · Zbl 0533.90083
[45] Pal, B. B.; Basu, I., A goal programming method for solving fractional programming problems via dynamic programming, Optimization, 35, 145-157 (1995) · Zbl 0839.90120
[46] Pal, B. B.; Basu, I., Selection of appropriate priority structure for optimal land allocation in agricultural planning through goal programming, Indian Journal of Agricultural Economics, 51, 342-354 (1996)
[47] Pal, B. B.; Moitra, B. N., Goal programming approach to fuzzy programming with multiple objectives, (Proceedings of the National Conference on Networking of Machines, Microprocessors, IT and HRD - Need of the Nation in the Next Millennium (1999)), B100-B109
[48] Rao, J. R.; Tiwari, R. N.; Mohanty, B. K., A preference structure on aspiration levels in a goal programming problem - A fuzzy approach, Fuzzy Sets and Systems, 25, 175-182 (1988) · Zbl 0646.90049
[49] Rehman, T.; Romero, C., Goal programming with penalty functions and livestock ration formulation, Agricultural Systems, 23, 117-132 (1987)
[50] Romero, C., A survey of generalized goal programming, European Journal of Operational Research, 25, 183-191 (1986) · Zbl 0586.90057
[51] Rommelfanger, H.; Hanuscheck, R.; Wolf, J., Linear programming with fuzzy objectives, Fuzzy Sets and Systems, 29, 31-48 (1989) · Zbl 0662.90045
[52] Rubin, P. A.; Narasimhan, R., Fuzzy goal programming with nested priorities, Fuzzy Sets and Systems, 14, 115-129 (1984) · Zbl 0546.90092
[53] Saaty, T. L., Exploring the interface between hierarchies, multiple objectives and fuzzy sets, Fuzzy Sets and Systems, 1, 57-68 (1978) · Zbl 0378.94001
[54] Sakawa, M.; Yano, H.; Yumine, T., An interactive fuzzy satisficing method for multiobjective linear programming problems and its application, IEEE Transactions on Systems, Man and Cybernatics, 17, 654-661 (1987)
[55] Schaible, S.; Karuppan, C. M., Designing a quality control system in a service organization: A goal programming case study, European Journal of Operational Research, 81, 249-258 (1995) · Zbl 0927.90078
[56] Schroeder, R. G., A survey of management science in university operations, Management Science, 19, 895-906 (1973)
[57] Slowinski, R., A multicriteria fuzzy linear programming method for water supply system development planning, Fuzzy Sets and Systems, 19, 217-237 (1986) · Zbl 0626.90085
[58] Stein, W. E., Optimal stopping in a fuzzy environment, Fuzzy Sets and Systems, 3, 253-259 (1980) · Zbl 0435.90105
[59] Steuer, R. E., Multiple Criteria Optimization: Theory, Computation and Application (1986), Wiley: Wiley New York · Zbl 0663.90085
[60] Tanaka, H.; Okuda, T.; Asai, K., On fuzzy mathematical programming, Journal of Cybernatics, 3, 37-46 (1974) · Zbl 0297.90098
[61] Taylor III, B. W.; Moore, L. T.; Clayton, E. R., R&D project selection and man power allocation with integer nonlinear goal programming, Management Science, 28, 1149-1158 (1982)
[62] Tiwari, R. N.; Dharmar, S.; Rao, J. R., Priority structure in fuzzy goal programming, Fuzzy Sets and Systems, 19, 251-259 (1986) · Zbl 0602.90078
[63] Tiwari, R. N.; Dharmar, S.; Rao, J. R., Fuzzy goal programming - An additive model, Fuzzy Sets and Systems, 24, 27-34 (1987) · Zbl 0627.90073
[64] Trzaskalik, T., Multiple criteria discrete dynamic programming, Mathematics Today, XII - A, 173-199 (1994)
[65] Trzaskalik, T., Dynamic goal programming models, (Caballero, R.; Ruiz, F.; Steuer, R. E., Advances in Multiple Objective and Goal Programming (1997), Springer: Springer Berlin), 111-119 · Zbl 1162.90554
[66] Vira, J., Fuzzy expectation values in multistage optimization problems, Fuzzy Sets and Systems, 6, 161-167 (1981) · Zbl 0466.90081
[67] Weeda, P. J., A dynamic programming formulation for the one machine sequencing problem, European Journal of Operational Research, 2, 298-300 (1978) · Zbl 0381.90054
[68] Wiedey, G.; Zimmermann, H.-J., Media selection and fuzzy linear programming, Journal of the Operational Research Society, 29, 1071-1084 (1978) · Zbl 0388.90087
[69] Yang, T.; Ignizio, J. P.; Kim, H. J., Fuzzy programming with nonlinear membership functions: Piecewise linear approximation, Fuzzy Sets and Systems, 41, 39-53 (1991) · Zbl 0743.90115
[70] Zadeh, L. A., Fuzzy sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606
[71] Zeleney, M., Multiple Criteria Decision Making (1982), McGraw-Hill: McGraw-Hill New York · Zbl 0588.90019
[72] Zimmermann, H.-J., Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1, 45-55 (1978) · Zbl 0364.90065
[73] Zimmermann, H.-J., Fuzzy mathematical programming, Computers and Operations Research, 10, 291-298 (1983)
[74] Zimmermann, H.-J., Applications of fuzzy set theory to mathematical programming, Information Sciences, 36, 29-58 (1985) · Zbl 0578.90095
[75] Zimmermann, H.-J., Fuzzy Sets, Decision Making, and Expert Systems (1987), Kluwer Academic Publishers: Kluwer Academic Publishers Boston
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.