PROJECT Method for Multiobjective Optimization Based on Gradient Projection and Reference Points

In this paper, we propose a new interactive method for multiobjective programming (MOP) called the PROJECT method. Interactive methods in MOP are techniques that can help the decision maker (DM) to generate the most preferred solution from a set of efficient solutions. An interactive method should be capable of capturing the preferences of the DM in a pragmatic and comprehensive way. In certain decision situations, it may be easier and more reliable for DMs to follow an interactive process for providing local tradeoffs than other kinds of preferential information like aspiration levels, objective function classification, etc. The proposed PROJECT method belongs to the class of interactive local tradeoff methods. It is based on the projection of utility function gradients onto the tangent hyperplane of an efficient set and on a new local search procedure that inherits the advantages of the reference-point method to search for the best compromise solution within a local region. Most of the interactive methods based on local tradeoffs assume convexity conditions in a MOP problem, which is too restrictive in many real-life applications. The use of a reference-point procedure makes it possible to generate any efficient solutions, even the nonsupported solutions or efficient solutions located in the nonconvex part of the efficient frontier of a nonconvex MOP problem. The convergence of the proposed method is investigated. A nonlinear example is examined using the new method, as well as a case study on efficiency analysis with value judgements. The proposed PROJECT method is coded in Microsoft Visual C++ and incorporated into the software PROMOIN (Interactive MOP).