Case Of Quadratic Assignment Problem Research Articles

A new hybrid-heuristic for large-scale combinatorial optimization: A case of quadratic assignment problem

Real world combinatorial optimization problems arise in very-large-scale. Solving such problems to optimality is impractical. In the past several decades, different heuristics and meta-heuristics have been proposed as solution procedure. In particular, to tackle the large-scale real applications, researchers have intelligently combined key components of competing methodologies to create superior solution procedures. In this study, we design a hybrid-heuristic (HH) using high-level solution algorithm framework that blends key components of three well-known meta-heuristics, scatter search, critical even tabu search, and genetic algorithm random keys, and applies to large-scale quadratic assignment problem (QAP). The HH has the potential to go beyond the goals of usual hybrid-heuristics, it potentially provides a framework for a self-adapted heuristic that can be applied to a wide range of problems with minimal inclusion of problem specific information. Extensive computational experiments applied to large-scale QAP instances from database of benchmark problems e.g., QAPLIB and taixxeyy collections, show that the proposed hybrid-heuristic (HH) provides an efficient approach to the large-scale QAP. The algorithm provides new best solutions for 61 out of 70 taixxeyy instances of large-scale QAP and finds the same best known solutions for QAPLIB and the 9 remaining instances from taixxeyy collections.

New special cases of the Quadratic Assignment Problem with diagonally structured coefficient matrices

We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time.

Embedding signed graphs in the line

Signed graphs are graphs with an assignment of a positive or a negative sign to each edge. These graphs are helpful to represent different types of networks. For instance, they have been used in social networks, where a positive sign in an edge represents friendship between the two endpoints of that edge, while a negative sign represents enmity. Given a signed graph, an important question is how to embed such a graph in a metric space so that in the embedding every vertex is closer to its positive neighbors than to its negative ones. This problem is known as Sitting Arrangement (SA) problem and it was introduced by Kermarrec et al. (Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 388---399, 2011). Cygan et al. (Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), 2012) proved that the decision version of SA problem is NP-Complete when the signed graph has to be embedded into the Euclidean line. In this work, we study the minimization version of SA (MinSA) problem in the Euclidean line. We relate MinSA problem to the well known quadratic assignment (QA) problem. We establish such a relation by proving that local minimums in MinSA problem are equivalent to local minimums in a particular case of QA problem. In this document, we design two heuristics based on the combinatorial structure of MinSA problem. We experimentally compare their performances against heuristics designed for QA problem. This comparison favors the proposed heuristics.

Efficiently solvable cases of quadratic assignment problem with generalized monotonic and incomplete anti-monge matrices

The paper describes new sufficient conditions that provide optimum to a quadratic assignment problem for a given substitution. The conditions separate out sets of matrices in the matrix space. The sets are formed by incomplete Anti-Monge matrices, which permit opposite ordering of rows and columns, and by additive monotone matrices, which are outside the class of Toeplitz matrices.

Quadratic Assignment Formulation for the Multifactor Facilities Layout Problem

The multifactor facilities layout problem is one of the important cases of quadratic assignment problem (QAP), which deals with the problem of optimum assignment of n facilities to n locations and optimizing multifactor cost objective. The authors present an alternate mathematical model for QAP that combines multiple qualitative and quantitative factors separately, treating them in a different manner for the facilities layout problem. The composite factor is obtained by combining the weighted combined qualitative and quantitative factors. The distances between the locations weigh the composite factor. The objective function is to minimize the total weighted workflow distance between the facilities. The results of the proposed model are compared with that of an existing model that handles multiple qualitative and quantitative factors in the same manner.

Quadratic Assignment Formulation for the Multifactor Facilities Layout Problem

The multifactor facilities layout problem is one of the important cases of quadratic assignment problem (QAP), which deals with the problem of optimum assignment of n facilities to n locations and optimizing multifactor cost objective. The authors present an alternate mathematical model for QAP that combines multiple qualitative and quantitative factors separately, treating them in a different manner for the facilities layout problem. The composite factor is obtained by combining the weighted combined qualitative and quantitative factors. The distances between the locations weigh the composite factor. The objective function is to minimize the total weighted workflow distance between the facilities. The results of the proposed model are compared with that of an existing model that handles multiple qualitative and quantitative factors in the same manner.

Combined computer-aided approach for the facilities design problem and estimation of the distribution parameter in the case of multigoal optimization

The multigoal facilities design problem, a case of quadratic assignment problem, deals with the problem of optimum assignment of n facilities to n locations, optimizing multigoal cost objectives. These objectives are classified into two categories, conflicting and congruent. Conflicting objectives aim at minimization of total flow cost and maximization of total closeness rating, whereas congruent objectives aim at maximization of distance weighted cost of several attributes, vis. flow, closeness rating, etc. The technique available for dealing with the congruent objective optimization does not consider the weightage to these attributes. The optimality of a suboptimal solution should also be checked at all levels of weightages in order to arrive at a correct solution, with the help of estimation of the distribution parameter, in the case of the multigoal facilities design problem. This paper presents a combined computer-aided approach which determines the solution of the multigoal facilities design problem at different weightages to attributes, finds the parameter of distribution viz. mean, median, mode, standard deviation, checks the optimality of solution, plots the histogram and gives the nature of distribution.