James C. Bean

Genetic Algorithms and Random Keys for Sequencing and Optimization

In this paper we present a general genetic algorithm to address a wide variety of sequencing and optimization problems including multiple machine scheduling, resource allocation, and the quadratic assignment problem. When addressing such problems, genetic algorithms typically have difficulty maintaining feasibility from parent to offspring. This is overcome with a robust representation technique called random keys . Computational results are shown for multiple machine scheduling, resource allocation, and quadratic assignment problems. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

BIOMECHANICAL MODEL CALCULATION OF MUSCLE CONTRACTION FORCES: A DOUBLE LINEAR PROGRAMMING METHOD

This paper presents a novel scheme for the use of linear programming to calculate muscle contraction forces in models describing musculoskeletal system biomechanics. Models of this kind are frequently found in the biomechanics literature. In most cases they involve muscle contraction force calculations that are statically indeterminate, and hence use optimization techniques to make those calculations. We present a linear programming optimization technique that solves a two-objective problem with two sequential linear programs. We use the technique here to minimize muscle intensity and joint compression force, since those are commonly used objectives. The two linear program model has the advantages of low computation cost, ready implementation on a micro-computer, and stable solutions. We show how to solve the model analytically in simple cases. We also discuss the use of the dual problem of linear programming to gain understanding of the solution it provides.

A Genetic Algorithm for the Multiple-Choice Integer Program

We present a genetic algorithm for the multiple-choice integer program that finds an optimal solution with probability one though it is typically used as a heuristic. General constraints are relaxed by a nonlinear penalty function for which the corresponding dual problem has weak and strong duality. The relaxed problem is attacked by a genetic algorithm with solution representation special to the multiple-choice structure. Nontraditional reproduction, crossover and mutation operations are employed. Extensive computational tests for dual degenerate problem instances show that suboptimal solutions can be obtained with the genetic algorithm within running times that are shorter than those of the OSL optimization routine.

Matchup Scheduling with Multiple Resources, Release Dates and Disruptions

This paper considers the rescheduling of operations with release dates and multiple resources when disruptions prevent the use of a preplanned schedule. The overall strategy is to follow the preschedule until a disruption occurs. After a disruption, part of the schedule is reconstructed to match up with the preschedule at some future time. Conditions are given for the optimality of this approach. A practical implementation is compared with the alternatives of preplanned static scheduling and myopic dynamic scheduling. A set of practical test problems demonstrates the advantages of the matchup approach. We also explore the solution of the matchup scheduling problem and show the advantages of an integer programming approach for allocating resources to jobs.

An efficient transformation of the generalized traveling salesman problem

AbstractThe Generalized Traveling Salesman Problem (GTSP) is a useful model for problems involving decisions of selection and sequence. The problem is defined on a directed graph in which the nodes have been pregrouped into m mutually exclusive and exhaustive nodesets. Arcs are defined only between nodes belonging to different nodesets with each arc having an associated cost. The GTSP is the problem of finding a minimum cost m-arc directed cycle which includes exactly one node from each nodeset. In this paper, we show how to efficiently transform a GTSP into a standard asymmetric Traveling Salesman Problem (TSP) over the same number of nodes. The transformation allows certain routing problems which involve discrete alternatives to be modeled using the TSP framework. One such problem is the heterogenous Multiple Traveling Salesman Problem (MTSP) for which we provide a GTSP formulation.

Capacity expansion under stochastic demands

We consider the problem of optimally meeting a stochastically growing demand for capacity over an infinite horizon. Under the assumption that demand for product follows either a nonlinear Brownian motion or a non-Markovian birth and death process, we show that this stochastic problem can be transformed into an equivalent deterministic problem. Consistent with earlier work by A. Manne, the equivalent problem is formed by replacing the stochastic demand by its deterministic trend and discounting all costs by a new interest rate that is smaller than the original, in approximate proportion to the uncertainty in the demand.

A Lagrangian Based Approach for the Asymmetric Generalized Traveling Salesman Problem

This paper presents an optimal approach for the asymmetric Generalized Traveling Salesman Problem (GTSP). The GTSP is defined on a directed graph in which the nodes are grouped into m predefined, mutually exclusive and exhaustive sets with the arc set containing no intraset arcs. The problem is to find a minimum cost m-arc directed cycle which includes exactly one node from each set. Our approach employs a Lagrangian relaxation to compute a lower bound on the total cost of an optimal solution. The lower bound and a heuristically determined upper bound are used to identify and remove arcs and nodes which are guaranteed not to be in an optimal solution. Finally, we use an efficient branch-and-bound procedure which exploits the multiple choice structure of the node sets. We present computational results for the optimal approach tested on a series of randomly generated problems. The results show success on a range of problems with up to 104 nodes.

A GENETIC ALGORITHM METHODOLOGY FOR COMPLEX SCHEDULING PROBLEMS

This paper considers the scheduling problem to minimize total tardiness given multiple machines, ready times, sequence dependent setups, machine downtime and scarce tools. We develop a genetic algorithm based on random keys representation, elitist reproduction, Bernoulli crossover and immigration type mutation. Convergence of the algorithm is proved. We present computational results on data sets from the auto industry. To demonstrate robustness of the approach, problems from the literature of different structure are solved by essentially the same algorithm.

Conditions for the existence of planning horizons

We consider the general class of discounted problems involving sequential decision making over an unbounded horizon. Under the assumptions of a finite set of policy alternatives at each decision time and costs that are eventually uniformly bounded by some exponential, existence of an optimal infinite horizon strategy is established. Moreover, it is shown that there is a finite horizon version whose first policy decision agrees with an infinite horizon optimal policy. Under the additional assumption that the infinite horizon optimal strategy is eventually cyclic, the stronger result follows that the infinite horizon optimal solution is unique for almost all interest rates and that therefore a planning horizon exists. Algorithmic implications are explored for a restricted subclass of these problems.

Equipment replacement under technological change

For infinite‐horizon replacement economy problems it is common practice to truncate the problem at some finite horizon. We develop bounds on the error due to such a truncation. These bounds differ from previous results in that they include both revenues and costs. Bounds are illustrated through a numerical example from a real case in vehicle replacement.