##
**Hierarchical optimization: A satisfactory solution.**
*(English)*
Zbl 0869.90042

Summary: Hierarchical optimization or multi-level programming techniques are extensions of Stackelberg games for solving decentralized planning problems with multiple decision makers in a hierarchical organization. They become more important for contemporary decentralized organizations where each unit or department seeks its own interests. Traditional approaches include vertex enumeration algorithms and approaches based on Kuhn-Tucker conditions or penalty functions. These are not only technically inefficient, but also lead to a paradox that the follower’s decision power dominates the leader’s. In this study, concepts of memberships of optimalities and degrees of decision powers are proposed to solve the above problems efficiently. In the proposed hierarchy decision process, the leader first sets memberships of optimalities of his/her possible objective values and decisions, as well as his/her decision power; and then asks the follower for his/her optima calculated in isolation under given constraints. The follower’s decision with the corresponding levels of optimalities and decision powers are submitted to and modified by the leader with considerations of overall benefit for the organization and distribution of decision power until a best preferred solution is reached.

### MSC:

90B50 | Management decision making, including multiple objectives |

91A65 | Hierarchical games (including Stackelberg games) |

91B06 | Decision theory |

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

### Keywords:

hierarchical optimization; multi-level programming; Stackelberg games; decentralized planning; multiple decision makers; memberships of optimalities; degrees of decision powers
Full Text:
DOI

### References:

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

[2] | (Anandalingam, G.; Friesz, T. L., Hierarchical Optimization (1992), J.C. Baltzer AC: J.C. Baltzer AC Basel) · Zbl 0751.90067 |

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

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

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

[6] | Bard, J. F., Optimality conditions for the bilevel programming problem, Naval Res. Logistics Quart., 31, 13-26 (1984) · Zbl 0537.90087 |

[7] | Bard, J. F.; Falk, J. E., An explicit solution to the multilevel programming problem, Comput. Oper. Res., 9, 77-100 (1982) |

[8] | Basar, T.; Olsder, G. J., Dynamic Noncooperative Game Theory (1982), Academic Press: Academic Press London · Zbl 0479.90085 |

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

[10] | Ben-Ayed, O., Bilevel linear programming, Comput. Oper. Res., 20, 485-501 (1993) · Zbl 0783.90068 |

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

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

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

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

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

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

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

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

[19] | Simaan, M.; Cruz, J. B., On the Stackelberg strategy in nonzero-sum games, J. Optim. Theory Appl., 11, 533-555 (1973) · Zbl 0243.90056 |

[20] | Simon, H. A., A behavioral model of rational choice, Quart. J. Econom., 69, 99-114 (1955) |

[21] | Wang, Q.; Parlar, M., Static game theory models and their applications in management science, European J. Oper. Res., 42, 1-21 (1989) · Zbl 0669.90107 |

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

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

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

[25] | Zimmermann, H.-J., Fuzzy Sets, Decision Making and Expert Systems (1987), Kluwer Academic: Kluwer Academic Norwell |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.