Hybrid Algorithm for Solving the Quadratic Assignment Problem

The Quadratic Assignment Problem (QAP) is a combinatorial optimization problem; it belongs to the class of NP-hard problems. This problem is applied in various fields such as hospital layout, scheduling parallel production lines and analyzing chemical reactions for organic compounds. In this paper we propose an application of Golden Ball algorithm mixed with Simulated Annealing (GBSA) to solve QAP. This algorithm is based on different concepts of football. The simulated annealing search can be blocked in a local optimum due to the unacceptable movements; our proposed strategy guides the simulated annealing search to escape from the local optima and to explore in an efficient way the search space. To validate the proposed approach, numerous simulations were conducted on 64 instances of QAPLIB to compare GBSA with existing algorithms in the literature of QAP. The obtained numerical results show that the GBSA produces optimal solutions in reasonable time; it has the better computational time. This work demonstrates that our proposed adaptation is effective in solving the quadratic assignment problem.


I. Introduction
T HE quadratic assignment problem (QAP) is one of the known classical combinatorial optimization problems, in 1976 Sahni and Gonzalez [1] proved that the QAP belongs to the class of NP-hard problems [1].It was introduced for the first time by Koopmans and Beckmann in 1957 [2]; its purpose is to assign n facilities to n fixed locations with a given flow matrix of facilities and distance matrix of locations in order to minimize the total assignment cost.This problem is applied in various fields such as hospital layout [3], scheduling parallel production lines [4] and analyzing chemical reactions for organic compounds [5].
Many recent hybrid approaches have improved performance in solving QAP such as genetic algorithm hybridized with tabu search method [6], ant colony optimization mixed with local search method [7] and ant colony optimization combined with genetic algorithm and local search method [8].Recently the hybrid algorithms are much proposed and used by many researchers to find optimal or near optimal solutions for the QAP.
In this paper we propose a new competitive approach when compared with other existing methods in the literature.The golden ball algorithm mixed with simulated annealing (GBSA) is considered here as a hybrid metaheuristic to apply in the quadratic assignment problem.
The purpose is to assign the facilities to the locations in such a way that the total cost is minimized.Each facility must be placed just at one location.
We consider two n × n matrices, the flow matrix F=f ij and the distance matrix D=d kl .The QAP formulation is given as follows (1): S n is the set of all permutation of n elements {1, 2, …, n}.π(i) and π(j) are respectively locations of facilities i and j, we suppose that π(i)=k and π(j)=l.
is the cost of assigning facility i in location k and facility j in location l.
The objective function (Cost) must be minimized.
Several algorithms are usually used to solve the quadratic assignment problem: • Exact algorithms such as branch and bound algorithm [10] and branch and cut algorithm [11].
In recent year, metaheuristic algorithms are used in solving the QAP more than the exact algorithms which are unable to solve the hard instances of QAP in a reasonable time.Many researchers compared between different metaheuristic algorithms for solving the QAP [20], [21].
In Golden Ball algorithm, groups of solutions are considered as soccer teams which are composed of a fixed number of players, the captain of team plays the rule of the best solution of the group.Each team has a coach who determines the type of training to improve the efficiency of its team.There are two types of training: conventional training and custom training.As shown in Fig. 2, the concept of this method is based on four main phases: initialization phase, training phase, competition phase and transfer phase.
In the initialization phase, we set the value of the number of teams (NT) and the number of players per team (PT).We assign randomly to each team a coach.
In the training phase, all teams must train by following a specific type of training.The conventional training is the daily training of a team.When a team becomes unable to improve its capacities, in this case, it must follow a custom training.
In the competition phase, each team must compete with other team chosen randomly.The winning team receives three points, in the case of equality; both teams receive one point.The accumulated points will be used to order the teams in descending order.
In the transfer phase, we detect three cases of transfer: Season transfer: during the season, all teams must be sorted in the descending order according to the strength value.The strength value is calculated using the following formula (2): q ij is the quality of player i of team j All teams exchange their players in this way: the best player of the first team must be replaced by the worst player of the last team.This worst player will be replaced by this best player.
The best player of the second team must be replaced by the worst player of the penultimate team.This worst player will be replaced by this best player and so forth.Special transfer: When a player of a given team is unable to improve after a custom training, the team must exchange it with a player of another team chosen randomly.
Cessation of coaches: after having ordered all the teams in descending order according to their accumulated point, the weaker teams must change their conventional training by another randomly selected.
The GB algorithm was tested by E.Osaba et al. with four different combinatorial optimization problems [23].The same technique was applied on the flow shop scheduling problem [29] and the job shop scheduling problem [30].

B. Simulated Annealing Method
The simulated annealing algorithm [14] is inspired by the physical annealing process which attempt to improve the quality of the solid by using at the beginning a high temperature T 0 at which the solid is in a liquid state.With the slow decrease of the temperature T (cooling phase) the solid regains its solid form (Fig. 3).Metropolis et al. show how to generate a sequence of successive states of the solid.The new state is accepted if the energy produced by this change of state decreases; otherwise, it is accepted with a probability defined by the following equation (3).The simulated annealing method [31] is one of the oldest algorithms; it is an iterative metaheuristic very used to solve combinatorial optimization problems in the continuous and discrete case.The strong point of this technique is to escape from the local minima and avoid the cyclic behavior.The performance of simulated annealing algorithm depends on a set of parameters which must be controlled.It means that the correct setting of the parameters produces satisfactory results.

IV. Adaptation of GBSA Algorithm
In the initialization phase we generate randomly the initial population of NT×PT solutions.
Each solution is represented in the following manner (Fig. 4): In the training phase, we used the following methods as conventional training functions: 2-opt [32], [33]: this iterative method is a local search algorithm, it repeatedly tries to improve the current assignment by exchanging two facilities.
Insertion method [34]: this method inserts a facility chosen randomly between two facilities.
Swapping mechanism [35]: this method swaps two parts selected randomly; the following figure (Fig. 5) explains the concept of this technique.As a custom training function the proposed adaptation used simulated annealing method [14], [31], it is used when the current solution is blocked in the local minima; it helps to accept some movement and escape from the local optimum.

V. Results and Discussion
The program was run 10 times on different instances of QAPLIB [36].The GBSA algorithm was implemented in C language and compiled using Microsoft Visual Studio 2008, the program code was executed in computer with Genuine Intel( R ) 575 @ 2.00 GHz 2.00 GHz RAM 2,00 Go.
The program uses three parameters: NT (number of groups), PT (number of schedules per group) and T (temperature).I) produce better results during the algorithm run.

The parameters values in the table below (Table
4×5 random solutions are sufficient to obtain good results.At the high temperature, the simulated annealing method becomes unnecessary because proximally 50% of iterations accept decision at the high temperature [37].In this paper we fixed the high temperature at 40 which is considered a symptom of fever in humans.The program stops when the optimal solution is reached or when the execution time exceeds 240 seconds.We take two digits after the comma, for the results shown in the two columns: Average and the Relative Percentage Deviation %RPD.
As Table II shows, the proposed algorithm allows to obtain always the optimal solution of 81,25% of the instances tested in a time not exceeding three seconds.The %RPD of 93,75% of the instances does not exceed 2% and this clearly shows that the GBSA algorithm converges well to the optimal solution.According to the values shown in the Table II, when the value of %RPD is equal to 0.00%, this means that the program reaches exactly the optimal solution at least 8 times per 10 tests and in this case the best and the worst solution are often the same.
Abd El-Nasser et al. [38] presented a comparative study between Meta-heuristic algorithms: Genetic Algorithm (GA), Tabu Search (TS), and Simulated annealing (SA) for solving a real-life (QAP) and analyze their performance in terms of both runtime efficiency and solution quality [38].
The Fig. 6 compares the relative percentage deviation of some instances of QALIB for our proposed algorithm GBSA, GA, TS and SA.The result shows that GBSA has more quality than the other algorithms for solving the QAP.We can deduce that our proposed method has really improved SA's effectiveness in solving these instances which we have chosen as an example for our comparative study.
There exist two sets of problems in QAPLIB that represent a challenge for any proposed algorithm.These problems were introduced by Skorin-Kapov [39] and Taillard [40].
We selected 9 instances from Skorin-Kapov and 7 instances from Taillard.For this list of QAPLIB instances, we compared our proposed method with others recent methods such as: Memetic algorithm (BMA) [41], Breakout local search (BLS) [42] and Cooperative parallel tabu search algorithm (CPTS) [43].The list of instances shown in Table III is a challenge for our algorithm.
We have fixed the maximum execution time of GBSA algorithm at 4 minutes.As the results depict (Table III), the GBSA algorithm needs some improvement to better solve some hard instances of QAP.But in general, the proposed algorithm seems promising to solve the quadratic assignment problem.According to the values of the relative percentage deviation from the best known solution, GBSA algorithm produces results near the global optimum in a reasonable time.

VI. Conclusion
The GBSA algorithm is the result of the hybridization of two methods: golden ball metaheuristic and simulated annealing method.This new hybrid algorithm is based on soccer concepts; it incorporates and guides simulated Annealing technique to escape from the local minima and to find the global optimal solution.This method has never been proposed or tested on QAPLIB instances.In this work we proposed an adaptation of our strategy to solve the QAP.The numerical results indicate the efficiency of the proposed GBSA adaptation and its performance compared to algorithms in literature of QAP.As a result, we deduce that our proposed approach has a high convergence speed.
solution based on the current solution.We used the swap of two random locations S2:=New solution f(S1):= cost of S1 f(S2):= cost of S2 if ( f(S2)<f(S1)) value Repeat all steps until T= 0.

TABLE I .
Parameters values

Table II
Time: Best time per seconds

TABLE II .
Numerical Results of the GBSA Algorithm

TABLE III .
Comparison of GBSA Algorithm with Algorithms in the Literature of the QAP