Decision Support Modified interactive Chebyshev algorithm (MICA) for convex multiobjective programming Mariano Luque a, * , Francisco Ruiz a , Ralph E. Steuer b a University of Málaga, Calle Ejido 6, 29071 Málaga, Spain b University of Georgia, Terry College of Business, Athens, GA 30602-6253, USA article info Article history: Received 6 February 2009 Accepted 7 November 2009 Available online 22 November 2009 Keywords: Multiobjective programming Interactive procedures Tchebycheff method Reference point methods Aspiration criterion vectors Reservation levels abstract In this paper, we describe an interactive procedural algorithm for convex multiobjective programming based upon the Tchebycheff method, Wierzbicki’s reference point approach, and the procedure of Micha- lowski and Szapiro. At each iteration, the decision maker (DM) has the option of expressing his or her objective-function aspirations in the form of a reference criterion vector. Also, the DM has the option of expressing minimally acceptable values for each of the objectives in the form of a reservation vector. Based upon this information, a certain region is defined for examination. In addition, a special set of weights is constructed. Then with the weights, the algorithm of this paper is able to generate a group of efficient solutions that provides for an overall view of the current iteration’s certain region. By modi- fication of the reference and reservation vectors, one can ‘‘steer” the algorithm at each iteration. From a theoretical point of view, we prove that none of the efficient solutions obtained using this scheme impair any reservation value for convex problems. The behavior of the algorithm is illustrated by means of graphical representations and an illustrative numerical example. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction When facing a real decision problem, a decision maker (DM) must often deal with several conflicting objectives. In such cases, the traditional optimization approach, in which a single objective is optimized subject to a given set of constraints, is no longer appli- cable. Instead, a multiobjective model is to be formulated and solved. Because of the rarity of solutions that optimize all objec- tives simultaneously, multiobjective programming utilizes effi- cient solutions. These are solutions from which no objective can be improved without deteriorating at least one of the others. Being ‘‘trade-off efficient” in this way, the set of all efficient solutions is precisely the set of all candidates for optimality. But as for which is to be optimal, this is for the DM to decide, and this often involves a contemplative process. As outlined in Hwang and Masud (1979), procedures for solving multiobjective decision problems can be grouped into three cate- gories depending upon whether preference information is elicited from the DM before, after, or during the solution process. In the ‘‘before” category are a priori methods. In these methods, after elic- iting information from the DM, an optimization problem is solved to compute a solution. A difficulty of a priori methods is that it is hard to know in advance with sufficient accuracy the information required by the optimization problem for it to produce a final solu- tion (an optimal solution or a solution close enough to one to qual- ify in its stead). Also, with these methods, there is the question about being able to recognize a final solution even when con- fronted with one without knowing more about the efficient set. In the ‘‘after” category are a posteriori or, as called by Cohon (1985), generating methods. In these methods, a comprehensive set of efficient solutions (or in the best case the whole efficient set) is generated and shown to the DM. Then, the DM is to choose his or her most preferred solution from the set. The drawback of these methods is that usually a great number of efficient solutions has to be generated, and it can be extremely hard for the DM to manage all of the information. In the ‘‘during” category are interactive procedures. Interactive procedures are designed to overcome the difficulties encountered in a priori and a posteriori methods. In interactive procedures, phases of information elicitation are interleaved with phases of computation. In the beginning, the information exchanged be- tween the DM and procedure is general, but then becomes more lo- cal in character as the procedure continues. In this way, interactive procedures have two main features: (a) they help a DM learn about a problem while solving it, and (b) they put to work iteratively any new insights gained during the solution process to help the DM navigate to a final solution. Prominent interactive procedures include STEM by Benayoun et al. (1971), the Zionts–Wallenius 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.11.011 * Corresponding author. E-mail addresses: mluque@uma.es (M. Luque), rua@uma.es (F. Ruiz), rsteuer@ uga.edu (R.E. Steuer). European Journal of Operational Research 204 (2010) 557–564 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor