Finding pareto optimal solutions in a supplier-buyer channel

One of the important problem in game theory and/or multicriteria optimization is finding Pareto efficient solutions. A standard technique for generating the set of Pareto optimal solutions is to maximize the weighted sums of different objectives for various settings of the weights. This method is also used routinely in supply chain management. However, it is known that the method returns all Pareto efficient solutions only when the objective functions of each player is a concave with respect decision variables. When this condition is satisfied, the frontier of Pareto optimal solutions in the criteria space tends to be concave. Thus, as we maximize the weighted sum objective function line, it becomes tangent to the frontier at a unique point. When the condition is not met, the Pareto optimal solutions on the non-concave part of the frontier cannot be obtained using the weighted sum single objective method. The standard method is often used in supply chain management even though the concavity condition is often not met. When the condition is not met, an alternate approach called Multiplier method can be used [3]., The multiplier method is based upon the Kuhn-Tucker conditions in Nonlinear programming [1]. In this paper, we illustrate the multiplier method by applying it to cooperative game in a single supplier-single reseller channel.