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**Fuzzy approach for multi-level programming problems.**
*(English)*
Zbl 0838.90140

Summary: Multi-level programming techniques are developed to solve descentralized planning problems with multiple decision makers in a hierarchical organization. These become more important for contemporary decentralized organizations where each unit or department seeks its own interests. Traditional approaches include vertex enumeration and transformation approaches. The former is in search of a compromise vertex based on adjusting the control variable(s) of the higher level and thus is rather inefficient. The latter transfers the lower-level programming problem to be the constraints of the higher level by its Kuhn-Tucker conditions or penalty function; the corresponding auxiliary problem becomes nonlinear and the decision information is also implicit.

In this study, we use the concepts of tolerance membership functions and multiple objective optimization to develop a fuzzy approach for solving the above problems. The upper-level decision maker defines his or her objective and decisions with possible tolerances which are described by membership functions of fuzzy set theory. This information then constrains the lower-level decision maker’s feasible space. A solution search relies on the change of membership functions instead of vertex enumeration and no higher order constraints are generated. Thus, the proposed approach will not increase the complexities of original problems and will usually solve a multi-level programming problem in a single iteration. To demonstrate our concept, we have solved numerical examples and compared their solutions with classical solutions.

In this study, we use the concepts of tolerance membership functions and multiple objective optimization to develop a fuzzy approach for solving the above problems. The upper-level decision maker defines his or her objective and decisions with possible tolerances which are described by membership functions of fuzzy set theory. This information then constrains the lower-level decision maker’s feasible space. A solution search relies on the change of membership functions instead of vertex enumeration and no higher order constraints are generated. Thus, the proposed approach will not increase the complexities of original problems and will usually solve a multi-level programming problem in a single iteration. To demonstrate our concept, we have solved numerical examples and compared their solutions with classical solutions.

### MSC:

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

90B70 | Theory of organizations, manpower planning in operations research |

90C29 | Multi-objective and goal programming |

### Keywords:

multi-level programming techniques; descentralized planning; multiple decision makers; hierarchical organization
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\textit{H.-S. Shih} et al., Comput. Oper. Res. 23, No. 1, 73--91 (1996; Zbl 0838.90140)

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### References:

[1] | Bard, J. F., Coordination of a multidivisional organization through two levels of management, Omega, 11, 457-468 (1983) |

[2] | Burton, R. M., The multilevel approach to organizational issues of the firm—a critical review, Omega, 5, 395-414 (1977) |

[3] | Ben-Ayed, O.; Blair, C. E., Computational difficulties of bilevel linear programming, Ops Res., 38, 556-560 (1990) · Zbl 0708.90052 |

[5] | Bard, J. F., An efficient point algorithm for a linear two-stage optimization problem, Ops Res., 31, 670-684 (1983) · Zbl 0525.90086 |

[6] | Wen, U. P.; Hsu, S. T., A note on a linear bilevel programming algorithm based on bicriteria programming, Computers Ops Res., 16, 79-83 (1989) · Zbl 0659.90080 |

[7] | Wen, U. P.; Hsu, S. T., Efficient solutions for the linear bilevel programming problem, Eur J. Opl Res., 62, 354-362 (1991) · Zbl 0765.90083 |

[8] | Wen, U. P.; Hsu, S. T., Linear bi-level programming problems—a review, J. Opl Res. Soc., 42, 125-133 (1991) · Zbl 0722.90046 |

[9] | Ben-Ayed, O., Bilevel linear programming, Computers Ops Res., 20, 485-501 (1993) · Zbl 0783.90068 |

[10] | Bialas, W. F.; Karwan, M. H., On two-level optimization, IEEE Trans. Autom. Control, AC-27, 211-214 (1982) · Zbl 0487.90005 |

[11] | Bialas, W. F.; Karwan, M. H., Two-level linear programming, Mgmt Sci., 30, 1004-1020 (1984) · Zbl 0559.90053 |

[12] | Candler, W.; Townsley, R., A linear two-level programming problem, Computers Ops Res., 9, 59-76 (1982) |

[13] | Haurie, A.; Savard, G.; White, D. J., A note on: an efficient point algorithm for a linear two-stage optimization problem, Ops Res., 38, 553-555 (1990) · Zbl 0708.90051 |

[14] | Bard, J. F., Optimality conditions for the bilevel programming problem, Nav. Res. Logist. Q., 31, 13-26 (1984) · Zbl 0537.90087 |

[15] | Clark, P. A.; Westerberg, A. W., A note on the optimality conditions for the bilevel programming problem, Nav. Res. Logist. Q., 35, 413-418 (1988) · Zbl 0653.90045 |

[16] | Ünlü, G., A linear bilevel programming algorithm based on bicriteria programming, Computers Ops Res., 14, 173-179 (1987) · Zbl 0626.90086 |

[17] | Candler, W., A linear bilevel programming algorithm: a comment, Computers Ops Res., 15, 297-298 (1988) · Zbl 0641.90050 |

[18] | Marcotte, P.; Savard, G., A note on the Pareto optimality of solutions to the linear bilevel programming problem, Computers Ops Res., 18, 355-359 (1991) · Zbl 0717.90045 |

[19] | Fortuny-Amat, J.; McCarl, B., A representation and economic interpretation of a two-level programming problem, J. Opl Res. Soc., 32, 783-792 (1981) · Zbl 0459.90067 |

[20] | Bard, J. F.; Falk, J. E., An explicit solution to the multi-level programming problem, Computers Ops Res., 9, 77-100 (1982) |

[21] | Bard, J. F.; Moore, J. T., A branch and bound algorithm for the bilevel programming problem, SIAM J. Sci. Stat. Comput., 11, 281-292 (1990) · Zbl 0702.65060 |

[22] | Wen, U. P.; Yang, Y. H., Algorithms for solving the mixed integer two-level linear programming problem, Computers Ops Res., 17, 133-142 (1990) · Zbl 0683.90055 |

[23] | Anandalingam, G.; White, D. J., A solution for the linear static Stackelberg problem using penalty functions, IEEE Trans. Autom. Control, 35, 1170-1173 (1990) · Zbl 0721.90098 |

[24] | Basar, T.; Olsder, G. J., Dynamic Noncooperative Game theory (1982), Academic Press: Academic Press New York · Zbl 0479.90085 |

[25] | Aubin, J.-P., Optima and Equilibria: An Introduction to Nonlinear Analysis (1993), Springer-Verlag: Springer-Verlag Berlin |

[26] | Lai, Y. J.; Hwang, C. L., Fuzzy Muthematical Programming—Methods and Applications (1993), Springer-Verlag: Springer-Verlag Berlin |

[27] | Bellman, R.; Zadeh, L. A., Decision-making in a fuzzy environment, Mgmt Sci., 17, B141-164 (1970) · Zbl 0224.90032 |

[28] | Lai, Y. J.; Hwang, C. L., Fuzzy Multiple Objective Decision Making—Methods and Applications (1994), Springer-Verlag: Springer-Verlag Berlin |

[29] | Anandalingam, G., A mathematical programming model of decentralized multi-level systems, J. Opl Res. Soc., 39, 1021-1033 (1988) · Zbl 0657.90061 |

[30] | Anandalingam, G.; Friesz, T. L., Hierarchical optimization: an introduction, (Anandalingam, G.; Friesz, T. L., Annals of Operations Research, Vol. 34 (1992)), 1-11 · Zbl 0751.90067 |

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