On Solving Linguistic Bi-Level Programming Problem Using Dynamic Programming

1. Introduction

This paper contains a bi-level optimization problem and provide a fuzzy rule base structure of this hierarchical class of optimization problem. In classical optimization problem, bi-level programming problem (BLPP) has been used in a vast domain of practice. In the fields of management, it has been used to deal with facility location, environmental regulation, energy policy, etc. In the fields of economic planning, it has been used to deal with oil production, electric power pricing etc. In engineering, to solve optimal design, shape and structure. Decision-makers (DMs) often deal with conflicting objectives in a hierarchical administrative structure. A decision maker has his own objective and decision space at one level and due to other level of hierarchy it may be influenced by the choice of other decision maker. There are two levels with two decision makers in bi-level programming problem. Decision makers of both level controls the variables of its own level. The DM of upper level is called leader and by his decision, the objective function of other level may be affected. Decision maker of lower level is called the follower. Decision makers of both level wants to optimize their objective function with the restriction of decision of one another. The hierarchical structure of the final problem needed an optimal to the follower’s problem first than a solution to the leader’s problem is feasible and then the optimal is selected. Bi-level programming’s solution is slightly difficult to deal because of its built-in non-convexity. The major segment of research in bi-level programming problem is still concerned on the deterministic case. BLPP were initially considered by Candler et. al. (1982) and Fortuny et. al. (1981) as a two player game where the first player can affect the resources available to a second player, this game is known as stackleberg game. For a given move of first player, the other player will maximize a linear program, subject to the available resources.

BLPP by Bard (1991), Bialas and Karwan (1984) and Candler and Townsley (1982) can be formulated as:

Here , , , is an , matrix, is an matrix. is a vector of decisions which can be controlled by the decision makers. is a vector control by leader and by follower, where . denotes the objectives functions of leader while denotes the objectives functions of the follower.