Two Reformulations for the Dynamic Quadratic Assignment Problem

Abstract: Problem statement: The Dynamic Quadratic Assignment Problem (DQAP), an NP-hard problem, is outlined and reformulated in two altern ative models: Linearized model and logic-based model. Approach: The solution methods for both models based on comb inatorial methods (Benders’ Decomposition and Approximate Dynamic Programming) and constraint logic programming, respectively, are proposed. Results: Proofs of model equivalence and solution methodolo gy are presented. Conclusion: Both proposed models are more simplified leading t o possible hybrid adaptations of existing techniques for more practic l approaches.


INTRODUCTION
A Dynamic Quadratic Assignment Problem (DQAP), mathematically formulated as a modified QAP, is defined as follows: given n 4 ×t cost coefficients c ijklt (i, j, k, l = 1, 2,…, n and t = 1, 2,.., T), determine an n 2 ×t solution matrix X = || x ijt || so as to: Minimize: n n n n T n n n T 1 ijklt ijt klt ijlt ijt il(t 1) i 1 j 1 k 1 l 1 t 1 i 1 j 1 l 1 t 1 C X X R X X = Represents the cost of assigning facility i to location j and facility k to location l at period t. f ist = The workflow cost from facility i to facility k at period t d jlt = The distance from location j to location l at period t R ijlt = Represents the rearranging cost when facility i located on location j at period t is moved to location l at period (t+1) X ijt = 1, if facility i is assigned to location j at period t. Otherwise, X ijt is 0 Since the DQAP is NP-hard problem that is difficult to deal with in case of solving directly for an optimal solution. The objective of this study is to reformulate the DQAP into two alternative forms: Linearized model and logic-based model and to propose solution methods for both models.

RESULTS AND DISCUSSION
Extending the theorem and proof in Lawler (1963), it is possible to demonstrate that the linearization of DQAP is equivalent to DQAP. Let the DQAP defined in (1-4) be designated problem Q and the MILP defined in (5-11) be designated problem L. The following theorem assures the equivalence of Q and L for any given set of cost coefficients. Proof: It is sufficient to show that the constraints of problem L are such that for any given permutation matrix X (L) at a given period t, Y at period t and V from period t to t+1 are determined uniquely by the relations: ijklt ijt klt Y X X = and: Since all of the variables are restricted to the value of 0 and 1, these relations are equivalent to: In order to prove the converse, let ijt klt X X 1 = = .
Then, from the constraints: Since the objective function is to minimize:  (12) and (13). Therefore, it follows that: The proposed solution method: For solving large scale MILP, Benders' Decomposition (BD) technique, which is presented by Benders (1962), can be applied. The algorithm solves a MILP problem via structure exploitation by decomposing a MILP into two problems-an integer master problem and a linear programming sub-problem-which are solved iteratively.
Note that the solution of MILP in the master problem in this approach can further be approximated by the round-up of the solution from the relaxed linear assignment problem using the original Hungarian method. Dynamic Programming (DP) technique based on Rosenblatt (1986) can also be applied to the method in order to determine the sub-optimal solution and accelerate the convergence rate. BD generates a database of a subset of feasible solutions for DP to determine an approximate optimal solution. In order to accelerate BD, a trust-region constraint can be implemented into the master problem with a successive adaptation procedure (Muenvanichakul and Charnsethikul, 2009) to improve its performance.
Implementation of BD to linearized DQAP (5-11) lead to: A linear programming sub-problem (dual problem): for given * ijklt U and ijl( t 1) V + . The solution procedure starts by solving the subproblem (14-16) from an initial layout * ijt X either from an initial guess value in the first iteration or a solution from the previous step (from the master problem) and then solving the master problem (17-21) from the solution of the sub-problem * ijklt U and ijl( t 1) V + . The procedure repeats until the different between the upper bound UB, the minimum of the current upper bound and the sub-problem objective value and the lower bound LB, the maximum of the current lower bound and the master problem objective value , is less than a given tolerance δ i.e.:

UB-LB≤δ BD
The cost of the total layout is the sub-problem objective value and the layout is ijt X from the master problem.
Note that the constraint (18) in the master problem is corresponding to the optimality cut of the linearized DQAP. Theorem 2 assures that there exists no feasibility cut in BD of linearized DQAP (5-11); thereby the sub-problem (14-16) can be determined from (22) and (23) given below.
Theorem 2: There exists no feasibility cut in BD of the linearization DQAP.
Proof: It is sufficient to show that given ijt klt X ,X for the corresponding the sub problem (14-16), to maximize the objective function (14) and: Consequently, the problem is always feasible with no feasibility cut. In this form, the constraints with (n×T) real variables consist of entirely checkable constraints and search variables. A logic-based method can treat the alldifferent constraint directly without converting it to inequalities as well as constraint satisfaction seeks a feasible solution to a set of constraints.

Logic-based model
It is possible to demonstrate that the logic-based model of DQAP is equivalent to DQAP. Let the DQAP defined in (5-11) be designated problem Q and the logic-based model of DQAP defined in (24-25) be designated problem L. The following theorem assures the equivalence of Q and L for any given set of cost coefficients. Proof: It is sufficient to show that the constraints of problem L are such that for any given permutation matrix Y (L) at a given period t, Y at period t are determined uniquely by the relations:  (Hooker, 2000) is a way of implementing the constraint satisfaction since integer programming methods cannot deal directly with an all-different constraint, {Y 11 ,Y 21 ,…, Y n1 ,…, Y nt }. It uses a programming language to specify at least the outline how the problem is to be solved. In this problem, ECLiPSe is a possible approach to solve the corresponding logic-based model. A tree search method in ECLiPSe is directly the way to find the solution without adding special inference. Especially, constraint propagation enforcing arcconsistency for general array expression, developed by Brand (2001), can be applied as the array constraint library to the program. Detail implementation of the method and result are presented in Muenvanichakul (2009).

CONCLUSION
This study reformulated DQAP in two alternative forms: Linearized DQAP Model and Logic-Based DQAP Model. Both approaches leading to more simplified models can help developing more possible solution methods.