The two-dimensional vector packing problem with piecewise linear cost function

The two-dimensional vector packing problem with piecewise linear cost function (2DVPP-PLC) is a practical problem faced by a manufacturer of children׳s apparel that ships products using courier service. The manufacturer must ship a number of items using standard-sized cartons, where the cost of a carton quoted by the courier is determined by a piecewise linear function of its weight. The cost function is not necessarily convex or concave. The objective is to pack all given items into a set of cartons such that the total delivery cost is minimized while observing both the weight limit and volume capacity constraints. This problem is commonly faced by many manufacturers that ship products using courier service. We formulate the problem as an integer programming model. Since the 2DVPP-PLC generalizes the classical bin packing problem, it is more complex and challenging. Solving it directly using CPLEX is successful only for small instances. We propose a simple heuristic that is extremely fast and produces high-quality solutions for instances of practical size. We develop an iterative local search algorithm to improve the solution quality further. We generate two categories of test data that can serve as benchmark for future research.