An algorithmic framework for improving heuristic solutions - Part 1. A deterministic discount coupon traveling salesman problem

Abstract

The efforts to develop solution algorithms for combinatorial optimization problems can be classified broadly into two categories, heuristic and rigorous approaches. These two approaches have ever-conflicting advantages and disadvantages in their computational load and solution quality. Due to the computational infeasibility of the rigorous approach for many practical combinatorial applications problem-specific heuristics are prevalent, even though they cannot provide any guarantee on the solution's quality. In this paper, we propose a novel way to improve upon a solution obtained from heuristics by applying dynamic programming to the subset of the states visited during simulation of the heuristics. The method represents a way to take a family of solutions and patch them together as an improved solution. However, 'patching' is accomplished in state space rather than in solution space. We develop and apply the approach to a new traveling salesman problem (TSP) variant, which illustrates the important notions of optional and conditional tasks in planning and scheduling applications, to examine the degree of improvement that can be obtained by the method. For a small problem, we compare the quality of the solution with the globally optimal one. The proposed method can be generalized to other planning and scheduling problems as long as a reasonable set of heuristics exist for their solution. (C) 2003 Elsevier Ltd. All rights reserved.