Computers & Operations Research 26 (1999) 789 — 798 An interactive bi-objective shortest path approach: searching for unsupported nondominated solutions J.M. Coutinho-Rodrigues *, J.C.N. Clı´maco, J.R. Current Department of Civil Engineering, F.C.T., University of Coimbra, 3049 Coimbra Codex, Portugal Faculty of Economics, University of Coimbra, 3000 Coimbra, Portugal Department of Management Sciences, The Ohio State University, 1775 College Avenue, Columbus, OH 43210, USA Received August 1997; received in revised form October 1998 Abstract In many network routing problems several conflicting objectives must be considered. Even for the bi- objective shortest path problem, generating and presenting the whole set of nondominated solutions (paths) to a decision maker, in general, is not effective because the number of these paths can be very large. Interactive procedures are adequate to overcome these drawbacks. Current et al. [1] proposed an interactive approach based on a NISE-like procedure to search for nondominated supported solutions and using auxiliar con- strained shortest path problems to carry out the search inside the duality gaps. In this paper we propose a new interactive approach to search for unsupported nondominated solutions (lying inside duality gaps) based on a k-shortest path procedure. Both approaches are compared.  1999 Elsevier Science Ltd. All rights reserved. Scope and purpose Network routing problems are generally multidimensional in nature, and in many cases the explicit consideration of multiple objectives is adequate. Objectives related to cost, time, accessibility, environmental impact, reliability and risk are appropriated for selecting the most satisfactory (‘‘best compromise’’) route in many problems. In general there is no single optimal solution in a multiobjective problem but rather, a set of nondominated solutions from which the decision maker must select the most satisfactory. However, generating and presenting the whole set of nondominated paths to a decision maker, in general, is not effective because the number of these paths can be very large. Interactive procedures are adequate to overcome these drawbacks. This paper introduces an interactive procedure to assist the decision maker in identifying the ‘‘best compro- mise’’ solution for the bi-objective shortest path problem. The procedure incorporates an efficient k-shortest path algorithm to identify nondominated solutions lying inside duality gaps. Test problem results indicate that the procedure can be readily executed on a PC for large-scale instances of problems. Keywords: Shortest path; Multiple criteria; Network routing *Corresponding author. Fax: 00351 39 822 511; e-mail: coutinho@dec.uc.pt. 0305-0548/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 8 ) 0 0 0 9 4 - X