A new multi-interval weights approach in fuzzy goal programming for a multi-criteria problem

Goal programming (GP) is an effective method to solve linear multi-objective problems. However, the determination of weights for GP is still of much concern. Hence, this study presents a new insight into multi-interval weights to solve linear multi-objective fuzzy GP problems. In the proposed approach, multi-interval weights that enable fuzzy goals to achieve their aspired levels based on their relative importance are considered in an uncertain environment. In formulating the model of the problem, the membership functions for each fuzzy goal are initially defined. Then, these functions are transformed into membership goals by assigning the highest membership value (optimal weight) and introducing under- and over-deviational variables to each. In the solution process, the interval weights (derived from a pairwise interval judgment matrix) associated with unwanted deviational variables are introduced to the goal achievement function with the objective of minimisation, and thus, realise the aspired goal levels of the problem. To illustrate the proposed approach, a numerical example is solved with real weights, which were collected through questionnaires. A better result is obtained with regard to optimal solution when the interval is divided into sub intervals, which also shows better result than that of the main interval under any matrix comparing weights.