Algorithms for nonlinear integer bicriterion problems

An improved algorithm for solving biobjective integer programs

“…The following result of Bowman (1976), used also in Eswaran, Ravindran, and Moskowitz (1989), was originally stated for the efficient set, but it is useful here to state the equivalent result for the Pareto set.…”

“…In general, the approaches can be classified as exact or heuristic and grouped according to the methodological concepts they use. Among others, the concepts employed in exact algorithms include branch and bound (Aksoy, 1990;Mavrotas and Diakoulaki, 1998;Ramesh, Karwan, and Zionts, 1989, 1990, 1991Shin and Allen, 1994), dynamic programming Karwan, 1981, 1982), implicit enumeration (Klein and Hannan, 1982;Pasternak and Passy, 1973), reference directions (Karaivanova et al, 1995;Narula and Vassilev, 1994), weighted norms Climaco, 1999, 2000;Eswaran, Ravindran, and Moskowitz, 1989;Karaivanova, Narula, and Vassilev, 1993;Neumayer and Schweigert, 1994;Schandl, Klamroth, and Wiecek, 2001;Solanki, 1991;Steuer and Choo, 1983), weighted sums with additional constraints (Chalmet, Lemonidis, and Elzinga, 1986;Ferreira, Climaco, and Paixão, 1994;Neumayer and Schweigert, 1994), and zero-one programming (Bitran, 1977(Bitran, , 1979. Heuristic approaches, such as simulated annealing, tabu search, and evolutionary algorithms, have been proposed for multiobjective integer programs with an underlying combinatorial structure (Ehrgott and Gandibleux, 2002).…”

“…In this paper, we focus on a new algorithm for identifying the complete Pareto set, called the WCN (weighted Chebyshev norm) algorithm, that takes this approach. The WCN algorithm builds on the results of Eswaran, Ravindran, and Moskowitz (1989), who proposed an exact algorithm to compute the complete Pareto set of BIPs based on Chebyshev norms; Solanki (1991), who proposed an approximate algorithm also using Chebyshev norms; and Chalmet, Lemonidis, and Elzinga (1986), who proposed an exact algorithm based on weighted sums.…”

“…Thus, the main factor determining the running time is the number of such subproblems that must be solved. The WCN algorithm is an improvement on the work of Eswaran, Ravindran, and Moskowitz (1989) in the sense that all Pareto outcomes are found by solving only 2|Y E | − 1 subproblems. The number of subproblems solved by Eswaran's algorithms depends on a tolerance parameter and can be much larger (see (8)).…”

Congestion pricing under operational, supply-side uncertainty

“…One can identify efficient paths using several techniques from multiobjective optimization, such as weighted objective functions (Geoffrion, 1968), the -constraining method (Haimes et al, 1971), and a decomposition method using the Chebyshev metric (Eswaran et al, 1989). Since our utility functions are simply linear functions of the path attributes, the weighted objective function method as adapted by Dial (1996) generates these paths efficiently for the bicriterion problem.…”