Tutorial: Heuristic Optimization

Although it is seriously under-represented in most academic programs, heuristic optimization|optimumseeking methods explicitly aimed at good feasible solutions that may not be optimal|comprises most of the optimizationwork actually applied in industrial engineering practice. This tutorial surveys such optimization-based strategies for approximate integer and combinatorial optimization with emphasis on relatively recent developments such as tabu search, simulated annealing and genetic algorithms. INTRODUCTION Heuristic optimization encompasses the variety of optimum-seeking methods explicitly addressed to nding good feasible solutions that may not be optimal. Like exact optimization it operates within a formal framework of decisions, constraints and objective functions, but the solution method need not be just a truncated variant of an exact procedure. Some would say juxtaposing the words \heuristic" and \optimization" produces a contradiction in terms; one is inherently approximate and the other restricted to mathematical optima. However, the overwhelmingmajority of optimization work that is actually applied in industrial engineering practice is implicitly or explicitly heuristic. The models required are usually just too large and complex to be solved to a provably optimal solution. Nearly every successful heuristic takes some advantage of domain-speci c information about the problem form being optimized. Still, there are common themes and patterns su ciently general to be viewed as formal methodologies. This paper provides a brief overview of some of the main heuristic optimization strategies for discrete (integer and combinatorial) optimization models. Emphasis Presented to the Industrial Engineering Research Conference, Nashville, May 1995 is on relatively recent variations on improving search including tabu search, simulated annealing and genetic algorithms (all covered in Reeves [1993]). The development is strongly based on the outline of a graduate course in heuristic optimization which was introduced at Purdue in 1987 by the author and has been taught several times since with considerable success. RELAXATION The one heuristic optimization strategy we will investigate that is explicitly grounded in methods of exact optimization can be termed relaxation. We rst formulate the problem as an Integer Linear Program (ILP) min=max P j cjxj + P k dkyk s:t: P j pi;jxj + P k qi;kyk bi all i xj 0 all j yk 0; integer/0-1 all k (sometimes mildly nonlinear), then form and solve an easier, relaxed version. The relaxation may be the linear program (LP) obtained by dropping integrality requirements, or a Lagrangian relaxation formed by rolling part of the main constraints into the objective function. An approximate optimum is produced by rounding the relaxation optimum in some systematic way to a nearby solution that is feasible in the full model. Rounding can be very di cult, but it is surprisingly easy in many familiar model forms: Shift scheduling uses models of the form min P j cjyj s:t: P j ai;jyj bi all i yj 0; integer all j to choose a collection of work shifts covering needed activities over time. With coe cients ai;j, which show the amount of i coverage provided by shift pattern j, always nonnegative, we may round \up" the LP optimum as d yje to obtain an feasible integer solution.