Decision Aiding A visual interactive tolerance approach to sensitivity analysis in MOLP Ana Rosa Pereira Borges a,c, * , Carlos Henggeler Antunes b,c a ISEC, Coimbra Polytechnic Institute, Apartado 10057, Quinta da Nora, 3030-601 Coimbra, Portugal b Department of Electrical Engineering, University of Coimbra, Polo II, 3030-030 Coimbra, Portugal c INESC, Rua Antero de Quental 199, 3000-033 Coimbra, Portugal Received 11 January 2001; accepted 3 August 2001 Abstract This paper deals with the tolerance approach to sensitivity analysis of the weights in multiple objective linear programming (MOLP) problems. A geometric interpretation to compute the results in Hansen et al. [Eur. J. Oper. Res. 38 (1989) 63–69] is presented. Along the same direction, a visual interactive approach based on the analysis of indif- ference regions corresponding to basic efficient solutions is described. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Multiple objective linear programming; Sensitivity analysis; Tolerance approach; Decision analysis; Interactive methods 1. Introduction Sensitivity analysis in mathematical programming problems is aimed at helping to deal with the un- certainty and imprecision which are inherent to the models. In linear programming it generally computes ranges to indicate how much given coefficients can change before the optimal basis changes. However, the ranges obtained by traditional sensitivity analysis are easily computed only when the coefficients are not allowed to change in a simultaneous manner. Wendell (1984, 1985) and Ravi and Wendell (1985, 1989) developed the tolerance approach that enables us to consider the impact of simultaneous and independent changes of more than one coefficient. This approach also enables the use of a priori information about the variability of the coefficients. The tolerance approach to sensitivity analysis in multiple objective linear programming (MOLP) problems proposed by Hansen et al. (1989) is an extension of the tolerance approach in linear pro- gramming. It considers perturbations on the p objective functions weights when using the weighted-sum European Journal of Operational Research 142 (2002) 357–381 www.elsevier.com/locate/dsw * Corresponding author. Tel.: +351-239-851040; fax: +351-239-824692. E-mail address: arborges@isec.pt (A.R. Pereira Borges). 0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII:S0377-2217(01)00305-8