Thomas A. Feo

Greedy Randomized Adaptive Search Procedures

Today, a variety of heuristic approaches are available to the operations research practitioner. One methodology that has a strong intuitive appeal, a prominent empirical track record, and is trivial to efficiently implement on parallel processors is GRASP (Greedy Randomized Adaptive Search Procedures). GRASP is an iterative randomized sampling technique in which each iteration provides a solution to the problem at hand. The incumbent solution over all GRASP iterations is kept as the final result. There are two phases within each GRASP iteration: the first intelligently constructs an initial solution via an adaptive randomized greedy function; the second applies a local search procedure to the constructed solution in hope of finding an improvement. In this paper, we define the various components comprising a GRASP and demonstrate, step by step, how to develop such heuristics for combinatorial optimization problems. Intuitive justifications for the observed empirical behavior of the methodology are discussed. The paper concludes with a brief literature review of GRASP implementations and mentions two industrial applications.

A probabilistic heuristic for a computationally difficult set covering problem

An efficient probabilistic set covering heuristic is presented. The heuristic is evaluated on empirically difficult to solve set covering problems that arise from Steiner triple systems. The optimal solution to only a few of these instances is known. The heuristic provides these solutions as well as the best known solutions to all other instances attempted.

A Greedy Randomized Adaptive Search Procedure for Maximum Independent Set

An efficient randomized heuristic for a maximum independent set is presented. The procedure is tested on randomly generated graphs having from 400 to 3,500 vertices and edge probabilities from 0.2 to 0.9. The heuristic can be implemented trivially in parallel and is tested on an MIMD computer with 1, 2, 4 and 8 processors. Computational results indicate that the heuristic frequently finds the optimal or expected optimal solution in a fraction of the time required by implementations of simulated annealing, tabu search, and an exact partial enumeration method.

A GRASP for a difficult single machine scheduling problem

Abstract A greedy randomized adaptive search procedure (GRASP) is presented for an unusually difficult single machine scheduling problem with flow time and earliness penalties. Previous methods reported in the literature provide optimal solutions to problems with up to only 14 jobs. GRASP constructs an optimal solution typically within 10 s of CPU time on a personal computer for 58 out of the 60 problems tested with 30 jobs. For the remaining instances, the method provides a solution extremely close to the optimal.

Machine setup and component placement in printed circuit board assembly

Populating printed circuit boards is one of the most costly and time-consuming steps in electronics assembly. At the beginning of each work order, three decisions are required: (1) a sequence must be specified for placing the individual components on the board; (2) tape reels must be assigned to positions on the magazine rack; and (3) a retrieval plan must be determined should the same component type be assigned to more than one magazine slot. Collectively, these problems can be modeled as a nonlinear integer program. In this paper, we develop a series of algorithms for solving each using an iterative two step approach.Initially, a placement sequence is generated with a weighted, nearest neighbor traveling salesman problem (TSP) heuristic; the two remaining problems are then formulated as a quadratic integer program and solved with a Lagrangian relaxation scheme. As a final step, the current magazine assignments are used to update the placement sequence, and the entire process is repeated.Our ability to deal, at least in part, with simultaneous machine operations represents the major contribution of this work. The methodology was simulated for a set of boards obtained from Texas Instruments and theoretically compared with a heuristic currently in use.

A Greedy Randomized Adaptive Search Procedure for the Two-Partition Problem

We present a greedy randomized adaptive search procedure ( GRASP ) for the network 2-partition problem. The heuristic is empirically compared with the Kernighan-Lin (K&L) method on a wide range of instances. The GRASP approach dominates K&L with respect to solution value on a large percentage of the instances tested. The ability of GRASP to find optimal solutions is assessed by comparing its performance with a general purpose mixed integer programming package.

An Algorithm for the Manufacturing Equipment Selection Problem

Abstract This paper provides a unified framework in which product and process demands can be related to manufacturing system requirements. A nonlinear cost minimization model is developed that can be used by facility planners to guide the analyses underlying the equipment selection problem. The approach extends current work by accounting for machine flexibility. The objective is to determine how many of each machine type to purchase, as well as what fraction of the time each piece of equipment will be configured for a particular type of operation. The resultant problem is solved with a depth-first branch and bound routine that employs a greedy set covering heuristic to find good feasible solutions. This permits early fathoming and greatly contributes to the efficiency of the algorithm. A small example is presented to highlight the computations. This is followed by a discussion of me results for a series of test problems designed to evaluate overall algorithmic performance. We show mat 16 process, 25 machi...

A class of bounded approximation algorithms for graph partitioning

This paper considers the problem of partitioning the nodes of a weighted graph into k disjoint subsets of bounded size, such that the sum of the weights of the edges whose end vertices belong to the same subset is maximized. A class of approximation algorithms based on matching is presented. These algorithms are shown to yield practical worst-case bounds when k is large. Extensive empirical experimentation indicates that the methods produce consistently good solutions to an important VLSI design problem in a fraction of the time required by competing methods.

A GRASP for single machine scheduling with sequence dependent setup costs and linear delay penalties

Abstract The single machine scheduling (SMS) problem with sequence dependent setup costs and linear delay penalties is readily found in industrial environments. The problem is of interest in manufacturing because any increase in the efficiency of a heavily utilized machine translates directly into cost savings. However, finding good solutions to this NP-hard combinatorial optimization problem has challenged researchers for years in industry and academia. A novel methodology, Greedy Randomized Adaptive Search Procedure (GRASP), is used in this paper to develop an efficient heuristic for the SMS problem. The empirical results of other researchers are used to validate the heuristics performance. Previous exact approaches using dynamic programming (DP) with branch and bound provide optimal solutions to SMS instances with up to 20 jobs. A clever implementation of tabu search shows success for problems possessing up to 100 jobs. Our heuristic compares favorably to both the DP and tabu search methods with respect to the solution values obtained and the CPU times required.

The cutting path and tool selection problem in computer aided process planning

Abstract This paper is concerned with minimizing the sum of the tool setup and volume removal times associated with metal cutting operations on a flexible machine. The problem is of interest to most repetitive manufacturers because any reduction in processing time translates directly into cost savings. Findings optimal solutions, though, has proven quite difficult for manufacturing engineers due to the large number of cutting path and tool combinations that may be selected. The problem is modeled an as integer program but transformed by relaxing some of the constraints into one of finding a minimum cut on a simple network. After obtaining a tentative solution at this step, an efficient, but suboptimal, procedure is used to generate alternative process plans. These are seen to speed convergence during branch and bound. Overall performance is judged by examining a wide variety of test problems derived from actual manufacturing data.