Algorithm robust for the bicriteria discrete optimization problem

Abstract

We apply Algorithm Robust to various problems in multiple objective discrete optimization. Algorithm Robust is a general procedure that is designed to solve bicriteria optimization problems. The algorithm performs a weight space search in which the weights are utilized in min-max type subproblems. In this paper, we experiment with Algorithm Robust on the bicriteria knapsack problem, the bicriteria assignment problem, and the bicriteria minimum cost network flow problem. We look at a heuristic variation that is based on controlling the weight space search and has an indirect control on the sample of efficient solutions generated. We then study another heuristic variation which generates samples of the efficient set with quality guarantees. We report results of computational experiments.

Multiple Criteria Decision Making, Multiattribute Utility Theory: Recent Accomplishments and What Lies Ahead

“…Furthermore, problem structuring continues to be of significant interest. See for example Kirkwood (1997) and Ragsdale (2004) Roy (1998), Kouvelis and Sayin (2006), Deb and Gupta (2005), and Wang and Zionts (2006), along with the interesting work on rough sets, based on Pawlak's original idea (Pawlak, 1982; see also Slowinski, 1992, and the discusssion in section 4.2 below). Furthermore, many interval-valued methods provide robust solution approaches, including PAIRS for value tree analysis (Salo and Hämäläinen, 1992) and Robust Portfolio Modeling (Liesiö et al, 2007).…”

On the quality of discrete representations in multiple objective programming

“…At each iteration, the point in the true Pareto set which has the maximum Chebyshev distance from the current representation is selected. Sayin and Kouvelis (2005) and Kouvelis and Sayin (2006) give a two-stage method called Algorithm Robust for generating representations of the Pareto set for discrete BOPs. The coverage, using the measure (7) in (Sayin 2000), is controlled by continuing to refine an interval between two previously generated Pareto points until its length falls below a prespecified value.…”

An exact -constraint method for bi-objective combinatorial optimization problems: Application to the Traveling Salesman Problem with Profits

“…The subtour elimination constraints (SEC) (16) require that each visited vertex v 2 V 0 of a feasible solution be reachable from the depot by two edge-disjoint paths. Constraint (17) forces the depot to be visited and constraints (18) and (19) impose that all variables be 0-1. Note that the model forces all feasible solutions to visit at least three vertices.…”

“…This drawback can be overcome with the TwoPhase Method [30] that finds all supported points of F through a weighted sum scalarization in the first phase, while non-supported points are found during the second phase with problem specific methods. Most algorithms that find the exact Pareto front of MOCO problems are variants of the Two-Phase Method [9], although other parametric approaches based on weighted scalarizations can find the exact Pareto front of BOCO problems [19,24,27].…”