An extended goal programming approach with piecewise penalty functions for uncertain supplier-material selection problem in cardboard box manufacturing systems.

In this study a real case multi-objective material and supplier selection problem in cardboard box production industries is studied. This problem for the first time optimizes the objective functions such as total wastage amounts remained from all raw sheets, total costs of the system including purchasing cost and transportation cost (including fixed and variable costs) of the raw sheets, and total overplus of produced cardboard boxes. To be closer to the real situations, as a novelty, the problem is formulated in belief-degree-based uncertain environment with normal distribution where this type of uncertainty applies the ideas of experts. A solution approach including two steps is proposed to solve the problem. In the first step, the proposed uncertain formulation is converted to a crisp form using a typical chance constrained programming scheme. In the second step, a new goal programming approach containing a piecewise penalty function is developed in order to solve the obtained multi-objective crisp formulation. In this approach, based on the ideas of experts, multiple goals are considered with different penalty values. A case study from cardboard box industries is considered to evaluate the proposed formulations and solution approach. According to the obtained results, the proposed solution approach is compared to similar approaches of the literature and its efficiency is studied.