We consider a multi-objective linear programming model with type-2 fuzzy objectives. The considered model has the flexibility for the user to specify the more general membership functions for objectives to reflect the inherent fuzziness, while being simple and practical. We develop two solution strategies with reasonable computing costs. The additional cost, as compared to the type-1 fuzzy model, is indeed insignificant. These two algorithms compute Pareto optimal solutions of the type-2 problems, one being based on a maxmin approach and the other on aggregating the objectives. Finally, applying the proposed algorithms, we work out two illustrative examples.
In crisp operation research, it is assumed that we know the objective function f(x), whose values present what is best for the decision maker(s). Often, uncertainties have also been introduced in mathematical programming problems; in these cases, we cannot predict the exact outcome of objective function of a decision x, and outcome depends on the unknown factors. If experts can provide narrow intervals that contain x with certain degree of confidence α, then we can use type-1 fuzzy objective functions [37]. For definitions and solution methods for such problems, see [28]. However, sometime because of degree of uncertain information, experts cannot present fuzzy objective with deterministic membership grades [58]. Moreover, type-2 fuzzy sets offer a higher dimension to the problem so that more means would be available to accommodate for the inherent uncertainties than the type-1 fuzzy sets, having crisp membership functions (see [32] and [39]). In fact, when determination of exact membership functions may not be possible, type-2 fuzzy sets may turn to effective [48]. For example, Hisdal [20] believes that “increasing fuzziness in a description means increased ability to handle inexact information in logically correct manner”, or John [23] asserts that “type-2 fuzzy sets allow for linguistic grades of membership, thus assigning in knowledge representation, and they also offer improvement of inference in constraint of type-1 sets”.
Different approaches are developed for tackle more uncertainty than what can be modeled by type-1 fuzzy in mathematical optimization models. For example, various studies used interval type-2 fuzzy to model parameters of mathematical optimization models [3] and [22].
In this study, we present a multi-objective model with crisp parameters and type-2 membership functions for the objectives and then discuss the solving approaches. As discussed before type-2 fuzzy can be used when experts are not certain about type-1 fuzzy membership of objective functions. To the best knowledge of the authors, this study is the first work that considers type-2 memberships functions for objectives.
A drawback for the use of type-2 fuzzy sets in multi-objective models may be due to the extra computations needed for computing solutions (see [46]), but this should not be a deterrence as the extra capability offered by these models turns to be compensatory. In fact, it is a usual practice to accept the complications caused by considering fuzzy weights for multi-objective programming problems in order to attain more practical models. Moreover, the proposed approaches have reasonable computing costs and the additional cost, as compared to the type-1 fuzzy model, is indeed insignificant.
Next, we present some preliminaries and necessary notations and definitions about multi-objective linear models and type-2 fuzzy, with a brief review of type-2 fuzzy linear programming approaches in Section 2. In Section 3, we propose the type-2 fuzzy multi-objective linear models and solution methods. We work out two illustrative examples in Section 4 and conclude in Section 5.
We considered the adaptation of type-2 fuzzy sets for modeling and solving multi-objective linear programming problems with fuzzy objectives. Using type-2 membership functions can effectively provide a natural mechanism to present inherent uncertainties in real problems not being appropriately handled by type-1 fuzzy sets. Type-2 fuzzy sets provide us additional degrees of freedom to represent the uncertainties and the fuzziness in modeling mathematical optimization problems. Here, we first proposed how to construct type-2 fuzzy membership functions for the objectives. To solve the problem, we converted each type-2 objective function to three type-1 objective functions. Two algorithms were proposed for finding a Pareto optimal solution, one being based on a maxmin approach and the other being based on aggregation of the objectives. For both algorithms, we analytically characterized the Pareto optimal solutions and compared their commuting cost with the ones due to type-1 fuzzy models. Two numerical examples were worked out to illustrate the proposed approaches.
