Introduction

Multi-objective optimization is an important and complex field in decision making in which many scientific and industrials must cope. Indeed, in many real-world applications, the decision maker is often confronted with several conflicting objectives that should be simultaneously optimized. Hence, the most common purpose is to choose the best trade-offs among all the predefined objectives. A wide variety of resolution methods and techniques for solving combinatorial multi-objective problems have been developed in the literature [1, 2, 3]. Despite the massive number of proposed methods, there are still many open issues in this topic. For example, there is no consideration of uncertainty aspect in the classical multi-objective concepts. However, uncertainty characterizes almost all practical applications in which the big amount of data provides certainly some inevitable imperfections. These imperfections might result from using unreliable information sources caused by inputting data incorrectly, faulty reading instruments or bad analysis of some training data. It may also be the result of poor decision maker opinions due to any lack of its background knowledge or the difficulty of giving a perfect qualification for some costly situations. The classical way to deal with uncertainty is the probabilistic reasoning originated from the middle of the 17th century [4]. Nevertheless, probability theory was considered for a long time as a very good quantitative tool for uncertainty treatment, but as good as it is, this theory is only appropriate when probability distributions are available which is not always the case. Otherwise, there are some situations like the case of total ignorance, which are not well handled and so can make the probabilistic reasoning unsound. In this context, a panoply of non-classical tools for handling uncertainty appears such as fuzzy sets [5] to which we are interested in this work. Moreover, the combination of uncertainty and multi-objectivity aspects has not been deeply studied so far. In particular, this combination leads to specific optimization problems characterized by the necessity of optimizing simultaneously many objectives while considering that some input data are not known beforehand. The challenge of solving these problems lies in their computational complexity. In fact, the effects of unavoidable uncertainties in preference parameters, decision variables, constraints, and/or objective functions could make such problems more complicated and difficult. Unfortunately, almost all existing approaches have been limited to transform the uncertain multi-objective problem into one or more mono-objective problems by using for example aggregation functions [6, 7]. Some other approaches have been focused on treating the problem in its multi-objective context while ignoring the propagation of uncertain inputs to the objectives and obviously to the resulting solutions [8]. Only few studies have been de voted to handle the problem as-is without erasing any of its characteristics by developing an interval-based multi-objective approach [9, 10].