A Method of Linear Programming with Relaxable Constraints and its Application to a Rural Regional Planning

Abstract The purpose of this paper is to develop a practical systems methodology of multiobjective planning by an extension of linear programming technique. A conventional linear programming model includes only one index to be minimized (or to be maximized) and several constraint inequalities. Among these constraints, some could be relaxed to some extent by planner’s intention. These are called relaxable or soft constraints. Relaxation of the soft constraints usually gives improvement of the objective index value, then it generates a kind of trade-off. In the present method, we start with a one-objective model instead of setting multiobjectives at the outset, and finally find a point of satisfactory compromise between the attainment of the objective and the requirement on the constraints. For this purpose, a new bicriterion problem is defined by introducing, as performance indexes, the original objective index and the total amount of relaxation of the soft constraints. The noninferior solutions of this problem give us informations regarding the components of the constraints whose set values must be relaxed. The preferred set values of the constraints are determined by an interactive procedure utilizing the surrogate worth trade-off technique. The method is applied to a trial of optimum agricultural products and land-use planning in a rural region.