The equilibrium generalized assignment problem and genetic algorithm

The well-known generalized assignment problem (GAP) is to minimize the costs of assigning n jobs to m capacity constrained agents (or machines) such that each job is assigned to exactly one agent. This problem is known to be NP-hard and it is hard from a computational point of view as well. In this paper, follows from practical point of view in real systems, the GAP is extended to the equilibrium generalized assignment problem (EGAP) and the equilibrium constrained generalized assignment problem (ECGAP). A heuristic equilibrium strategy based genetic algorithm (GA) is designed for solving the proposed EGAP. Finally, to verify the computational efficiency of the designed GA, some numerical experiments are performed on some known benchmarks. The test results show that the designed GA is very valid for solving EGAP.     

A cooperative parallel tabu search algorithm for the quadratic assignment problem

In this study, we introduce a cooperative parallel tabu search algorithm (CPTS) for the quadratic assignment problem (QAP). The QAP is an NP-hard combinatorial optimization problem that is widely acknowledged to be computationally demanding. These characteristics make the QAP an ideal candidate for parallel solution techniques. CPTS is a cooperative parallel algorithm in which the processors exchange information throughout the run of the algorithm as opposed to independent concurrent search strategies that aggregate data only at the end of execution. CPTS accomplishes this cooperation by maintaining a global reference set which uses the information exchange to promote both intensification and strategic diversification in a parallel environment. This study demonstrates the benefits that may be obtained from parallel computing in terms of solution quality, computational time and algorithmic flexibility. A set of 41 test problems obtained from QAPLIB were used to analyze the quality of the CPTS algorithm. Additionally, we report results for 60 difficult new test instances. The CPTS algorithm is shown to provide good solution quality for all problems in acceptable computational times. Out of the 41 test instances obtained from QAPLIB, CPTS is shown to meet or exceed the average solution quality of many of the best sequential and parallel approaches from the literature on all but six problems, whereas no other leading method exhibits a performance that is superior to this.     

Lower bounds for the quadratic assignment problem

We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branch-and-bound type algorithm for the quadratic assignment problem.     

An algorithm for the generalized quadratic assignment problem

This paper reports on a new algorithm for the Generalized Quadratic Assignment problem (GQAP). The GQAP describes a broad class of quadratic integer programming problems, wherein M pair-wise related entities are assigned to N destinations constrained by the destinations’ ability to accommodate them. This new algorithm is based on a Reformulation Linearization Technique (RLT) dual ascent procedure. Experimental results show that the runtime of this algorithm is as good or better than other known exact solution methods for problems as large as M=20 and N=15. Current address of P.M. Hahn: 2127 Tryon Street, Philadelphia, PA 19146-1228, USA.     

Elitist genetic algorithm for assignment problem with imprecise goal

The objective of this research paper is to solve a generalized assignment problem with imprecise cost(s)/time(s) instead of precise one by elitist genetic algorithm (GA). Here, the impreciseness of cost(s)/time(s) has been represented by interval valued numbers, as interval valued numbers are the best representation than others like random variable representation with a known probability distribution and fuzzy representation. To solve these types of problems, an elitist GA has been developed with interval valued fitness function. In this developed GA, the existing ideas about the order relations of interval valued numbers have been modified from the point of view of two types of decision making viz., optimistic decision making and pessimistic decision making. This modified approach has been used in the selection process for selecting better chromosomes/individuals for the next generation and in finding the best as well as the worst chromosomes/individuals in each generation. Here two new crossover schemes and two new mutation schemes have been introduced. In order to maintain the feasibility with crossover operations, a repair algorithm has been suggested. Extensive comparative computational studies based on different parameters of our developed algorithm on one illustrative example have also been reported.     

An alternative approach for unbalanced assignment problem via genetic algorithm

This paper presents an alternative approach using genetic algorithm to a new variant of the unbalanced assignment problem that dealing with an additional constraint on the maximum number of jobs that can be assigned to some agent(s). In this approach, genetic algorithm is also improved by introducing newly proposed initialization, crossover and mutation in such a way that the developed algorithm is capable to assign optimally all the jobs to agents. Computational results with comparative performance of the algorithm are reported for four test problems.     

A level-2 reformulation–linearization technique bound for the quadratic assignment problem

This paper studies polyhedral methods for the quadratic assignment problem. Bounds on the objective value are obtained using mixed 0–1 linear representations that result from a reformulation–linearization technique (rlt). The rlt provides different “levels” of representations that give increasing strength. Prior studies have shown that even the weakest level-1 form yields very tight bounds, which in turn lead to improved solution methodologies. This paper focuses on implementing level-2. We compare level-2 with level-1 and other bounding mechanisms, in terms of both overall strength and ease of computation. In so doing, we extend earlier work on level-1 by implementing a Lagrangian relaxation that exploits block-diagonal structure present in the constraints. The bounds are embedded within an enumerative algorithm to devise an exact solution strategy. Our computer results are notable, exhibiting a dramatic reduction in nodes examined in the enumerative phase, and allowing for the exact solution of large instances.     

Lower bounds for the quadratic assignment problem via triangle decompositions

We consider transformations of the (metric) Quadratic Assignment Problem (QAP) that exploit the metric structure of a given instance. We show in particular how the structural properties of rectangular grids can be used to improve a given lower bound. Our work is motivated by previous research of Palubetskes (1988), and it extends a bounding approach proposed by Chakrapani and Skorin-Kapov (1993). Our computational results indicate that the present approach is practical; it has been applied to problems of dimension up ton = 150. Moreover, the new approach yields by far the best lower bounds on most of the instances of metric QAPs that we considered.The authors gratefully acknowledge financial support by the Christian Doppler Laboratorium für Diskrete Optimierung.     

A note on a polynomial time solvable case of the quadratic assignment problem

We identify a class of instances of the Koopmans–Beckmann form of the Quadratic Assignment Problem that are solvable in polynomial time. This class is characterized by a path structure in the flow data and a grid structure in the distance data. Chr18b, one of the test problems in the QAPLIB, is in this class even though this feature of it has not been noticed until now.     

Compounded genetic algorithms for the quadratic assignment problem

We introduce the compounded genetic algorithm. We propose to run a quick genetic algorithm several times as Phase 1, and compile the best solutions in each run to create a starting population for Phase 2. This new approach was tested on the quadratic assignment problem with very good results.     

An efficient cost scaling algorithm for the assignment problem

The cost scaling push-relabel method has been shown to be efficient for solving minimum-cost flow problems. In this paper we apply the method to the assignment problem and investigate implementations of the method that take advantage of assignment's special structure. The results show that the method is very promising for practical use.This author's research was supported in part by ONR Young Investigator Award N00014-91-J-1855, NSF Presidential Young Investigator Grant CCR-8858097 with matching funds from AT&T, DEC and 3M, and a grant from the Powell Foundation.This author's research was supported by the above-mentioned ONR and NSF grants.     

Gilmore-Lawler bound of quadratic assignment problem

The Gilmore-Lawler bound (GLB) is one of the well-known lower bound of quadratic assignment problem (QAP). Checking whether GLB is tight is an NP-complete problem. In this article, based on Xia and Yuan linearization technique, we provide an upper bound of the complexity of this problem, which makes it pseudo-polynomial solvable. We also pseudopolynomially solve a class of QAP whose GLB is equal to the optimal objective function value, which was shown to remain NP-hard.      

Ranking scalar products to improve bounds for the quadratic assignment problem

The eigenvalue bound for the quadratic assignment problem (QAP) is successively improved by considering a set of k-best scalar products, related to the QAP. An efficient procedure is proposedto find such a set of k-best scalar products. A class of QAPs is described for which this procedure in general improves existing lower bounds and at the same time generates good suboptimal solutions. The method leaves the user with a large flexibility in controlling the quality of the bound. However, since the method is sensitive to input data it should only be used in combination with other bounding rules.     

Solving the dynamic capacitated location-routing problem with fuzzy demands by hybrid heuristic algorithm

In this paper, the dynamic capacitated location-routing problem with fuzzy demands (DCLRP-FD) is considered. In the DCLRP-FD, facility location problem and vehicle routing problem are solved on a time horizon. Decisions concerning facility locations are permitted to be made only in the first time period of the planning horizon but, the routing decisions may be changed in each time period. Furthermore, the vehicles and depots have a predefined capacity to serve the customers with altering demands during the time horizon. It is assumed that the demands of customers are fuzzy variables. To model the DCLRP-FD, a fuzzy chance-constrained programming is designed based upon the fuzzy credibility theory. To solve this problem, a hybrid heuristic algorithm (HHA) with four phases including the stochastic simulation and a local search method are proposed. To achieve the best value of two parameters of the model, the dispatcher preference index (DPI) and the assignment preference index (API), and to analyze their influences on the final solution, numerical experiments are carried out. Moreover, the efficiency of the HHA is demonstrated via comparing with the lower bound of solutions and by using a standard benchmark set of test problems. The numerical examples show that the proposed algorithm is robust and could be used in real world problems.     

A new exact discrete linear reformulation of the quadratic assignment problem

The quadratic assignment problem (QAP) is a challenging combinatorial problem. The problem is NP-hard and in addition, it is considered practically intractable to solve large QAP instances, to proven optimality, within reasonable time limits. In this paper we present an attractive mixed integer linear programming (MILP) formulation of the QAP. We first introduce a useful non-linear formulation of the problem and then a method of how to reformulate it to a new exact, compact discrete linear model. This reformulation is efficient for QAP instances with few unique elements in the flow or distance matrices. Finally, we present optimal results, obtained with the discrete linear reformulation, for some previously unsolved instances (with the size n = 32 and 64), from the quadratic assignment problem library, QAPLIB.     

On the robust assignment problem under a fixed number of cost scenarios

We investigate the complexity of the min-max assignment problem under a fixed number of scenarios. We prove that this problem is polynomial-time equivalent to the exact perfect matching problem in bipartite graphs, an infamous combinatorial optimization problem of unknown computational complexity.     

A simplex-based labelling algorithm for the linear fractional assignment problem

This paper constructs an algorithm to solve the fractional assignment problem. Algorithms that are currently used are mostly based on parametric approaches and must solve a sequence of optimization procedures. They also neglect the difficulties caused by degeneracy. The proposed algorithm performs optimization once and overcomes degeneracy. The main features of the algorithm are an effective initial heuristic approach, a simple labelling procedure and an implicit primal-dual schema. A numerical example is presented and demonstrates that the proposed algorithm is easy to apply. Computational results are compared with those from other developed methods. The results show that the proposed algorithm is efficient.     

Autocorrelation measures for the quadratic assignment problem

In this article we provide an exact expression for computing the autocorrelation coefficient ξ and the autocorrelation length ? of any arbitrary instance of the Quadratic Assignment Problem (QAP) in polynomial time using its elementary landscape decomposition. We also provide empirical evidence of the autocorrelation length conjecture in QAP and compute the parameters ξ and ? for the 137 instances of the QAPLIB. Our goal is to better characterize the difficulty of this important class of problems to ease the future definition of new optimization methods. Also, the advance that this represents helps to consolidate QAP as an interesting and now better understood problem.     

A labeling algorithm for the sensitivity ranges of the assignment problem

In view of the simplex-type algorithm, the assignment problem is inherently highly degenerate. It may be the optimal basis has changed, but the optimal assignment is unchanged when parameter variation occurs. Degeneracy then makes sensitivity analysis difficult, as well as makes the classical Type I range, which identifies the range the optimal basis unchanged, impractical. In this paper, a labeling algorithm is proposed to identify two other sensitivity ranges – Type II range and Type III range. The algorithm uses the reduced cost matrix, provided in the final results of most solution algorithms for AP, to determine the Type II range which reflects the stability of the current optimal assignment. Thus, the algorithm generates a streamlined situation from searching the optimal solution until performing the sensitivity analysis of the assignment problem. The Type III range, reflecting the flexibility of optimal decision making, can be obtained immediately after the Type II range is determined. Numerical examples are presented to demonstrate the algorithm.     

Upper and lower bounding procedures for the multiple knapsack assignment problem

We formulate the multiple knapsack assignment problem (MKAP) as an extension of the multiple knapsack problem (MKP), as well as of the assignment problem. Except for small instances, MKAP is hard to solve to optimality. We present a heuristic algorithm to solve this problem approximately but very quickly. We first discuss three approaches to evaluate its upper bound, and prove that these methods compute an identical upper bound. In this process, reference capacities are derived, which enables us to decompose the problem into mutually independent MKPs. These MKPs are solved euristically, and in total give an approximate solution to MKAP. Through numerical experiments, we evaluate the performance of our algorithm. Although the algorithm is weak for small instances, we find it prospective for large instances. Indeed, for instances with more than a few thousand items we usually obtain solutions with relative errors less than 0.1% within one CPU second.