Effective heuristics and meta-heuristics for the quadratic assignment problem with tuned parameters and analytical comparisons
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Abstract
Quadratic assignment problem (QAP) is a well-known problem in the facility location and layout. It belongs to the NP-complete class. There are many heuristic and meta-heuristic methods, which are presented for QAP in the literature. In this paper, we applied 2-opt, greedy 2-opt, 3-opt, greedy 3-opt, and VNZ as heuristic methods and tabu search (TS), simulated annealing, and particle swarm optimization as meta-heuristic methods for the QAP. This research is dedicated to compare the relative percentage deviation of these solution qualities from the best known solution which is introduced in QAPLIB. Furthermore, a tuning method is applied for meta-heuristic parameters. Results indicate that TS is the best in 31%of QAPs, and the IFLS method, which is in the literature, is the best in 58 % of QAPs; these two methods are the same in 11 % of test problems. Also, TS has a better computational time among heuristic and meta-heuristic methods.
Keywords
Quadratic assignment problem Heuristics Meta-heuristics Tuning methodBackground
This paper is organized as follows: in the next section, QAP formulation is displayed; afterwards, the heuristic methods are described. After that, the meta-heuristic methods such as SA, TS, and PSO are presented; the tuning method is described next. Computational analyses and comparison results are mentioned in the ‘Results and discussion’ section, and the last section is the ‘Conclusions’ section.
Methods
QAP formulation
where c_{ijkl} is the cost of assigning facility i in location k and simultaneously facility j in location l, and x_{ik} = 1 if location k is assigned to facility i; otherwise, x_{ik} = 0. Also, x_{jl} = 1 if location l is assigned to facility j; otherwise, x_{jl} = 0. The objective function (Equation 1) of this model must be minimized. Each location must be assigned just to one facility, as Equation 2 shows. Equation 3 displays that each facility must be assigned just in one location. The number of facility and location is the same and is equal to n. The variable in this model is binary.
Heuristic methods
Heuristic algorithms do not provide an assurance for optimization of the problem. These methods are an approximation. They have an additional property that worst-case solutions are known. In this section, some heuristic methods as procedures to search the better solution that contains 2-opt, greedy 2-opt, 3-opt, greedy 3-opt, and Vollman, Nugent, Zartler (VNZ) are contemplated.
2-Opt algorithm
Initially, the algorithm considers the transposition of facilities 1 and 2. If the resulting solution's objective function value (OFV) is smaller than that of the initial solution, then it is stored as a candidate for future consideration. Otherwise, it is discarded, and the algorithm considers the transposing of facilities 1 and 3. If this exchange generates a better solution, then it is stored as a candidate for future consideration; otherwise, it is discarded, and so on. Thus, whenever a better solution is found, the algorithm discards the previous best solution. This procedure continues until all the pair-wise exchanges are considered.
For n location in the QAP problem, the 2-opt algorithm consists of three steps:
Step 1. Let S be the initial feasible solution and Z its objective function value; then, set S* = S, Z* = Z, i = 1 and j = i + 1 = 2.
Step 2. Consider the exchange results in a solution S′ that has OFV Z′ < Z*, set Z* = Z′ and S* = S′. If j < n, then repeat step 2; otherwise, set i = i + 1 and j = i + 1. If i < n, repeat step 2; otherwise, proceed to step 3.
Step 3. If S ≠ S*, set S = S*, Z = Z*, i = 1, j = i + 1 = 2 and go to step 2. Otherwise, output S* is selected as the best solution, and the process is terminated.
Greedy 2-opt algorithm
The greedy 2-opt algorithm is a variant of the 2-opt algorithm. The difference between this method and 2-opt is in selecting the best transposition. This method transposes the facility location if the OFV is better than the known OFV and stabilizes this assignment; it then goes to transpose the facility location from the start. It also consists of three steps. Like the 2-opt algorithm, greedy 2-opt also considers pair-wise exchanges. Initially, it considers transposing facilities 1 and 2. If the resulting OFV is less than the previous one, two facilities are immediately transposed. Otherwise, the algorithm will go on to facility 3 and evaluate the exchange, and so on until improvement is found. If facilities 1 and 2 are transposed, then the algorithm will take it as an initial solution and will repeat the algorithm until it is impossible to improve the solution any further. Greedy 2-opt algorithm makes the exchange permanent whenever an improvement is found and thus consumes less computational time than the 2-opt algorithm. On the other hand, greedy 2-opt algorithm produces slightly worse solutions than the 2-opt algorithm.
3-Opt algorithm
Greedy 3-opt algorithm
Greedy 3-opt algorithm is also similar to the greedy 2-opt algorithm, but it makes the three facility exchange permanent whenever its resulting OFV is better than the current OFV and repeats the algorithm with the new transposition as the initial solution. The transposition in this method is similar to that in 3-opt.
VNZ method
The VNZ method was introduced by Vollman et al. ([1968]). This method is using less storage space than 2-opt.
There is not any randomization, and also, these methods cannot orient the current solution to the optimum solution in a limited time. However, the meta-heuristic methods contain a good search approach with a reasonable time. These methods are considered in the next subsection.
Meta-heuristic methods
In the original definition, meta-heuristics are solution methods that manage an interaction between the local improvement procedures and higher level strategies to create a process capable of escaping from local optimum solution and performing a good search of solution space. These methods have also come to include any procedures that employ strategies for overcoming the trap of local optimality in complex solution spaces, especially those procedures that take advantage of one or more neighborhood structures as a means of defining acceptable moves to transformation from one solution to another. In this research, TS, SA, and PSO are applied for the QAP, and their comparison has been done for the selected data sets.
Tabu search
Classical methods often face great difficulty when confronted with hard optimization problems present in real situations. Tabu search (TS), for the first time, was proposed by Glover ([1989, 1990]). This meta-heuristic approach is, in a theatrical manner, changing the ability of solving problems of practical significance. The pseudo code of TS, which is applied in this research, is as follows:
Step 1. Let S be the initial feasible solution and Z its objective function value; then, set S* = S, Z* = Z, max short-term memory (STM) = 5, and max iteration = 1,000; iter = 1. Best O value = O value.
Step 2. Random (i, j) = rand/Long-term memory (LTM) (i, j), (n 1, n 2) = the indices of maximum value in random.
Step 3. If there is none (n 1, n 2) in STM matrix, change n 1 and n 2 locations; otherwise, repeat step 2.
Step 4. Insert n 1 and n 2 in STM and release the last indices from STM (e.g., m 1, m 2); and LTM(m 1, m 2) = LTM(m 1, m 2) + 1.
Step 5. Calculate the objective function value (Z) of the new permutation.
Step 6. If Z ≤ Z*, then Z* = Z, S* = S, and iter = iter + 1.
Step 7. If iter ≤ max iteration, then repeat step 2; otherwise, print Z* and S*.
Simulated annealing
where OFV and OFV_{B} are the objective function values for this iteration and are the best computed one until this iteration. T is the temperature of the algorithm in the iteration, and P is the probability of acceptance for each move in the annealing process. The proposed SA pseudo code for QAP is as follows:
Step 1. Let S be the initial feasible solution and Z its objective function value; then, set S* = S, Z* = Z. T = 100, T_{0} = 0.1, r = 0.95, n limit max = 5 and n over max = 10.
Step 2. n limit = 0 and n over = 0.
Step 3. Transpose two facilities in the current layout randomly and calculate the objective function value (Z).
Step 4. If Z ≤ Z*, then accept the transposition; it means that Z* = Z, S* = S, and then n limit = n limit + 1, n over = n over + 1; if n over = n over max or n limit = n limit max, then proceed to step 6.Otherwise, repeat step3; if Z > Z*, then proceed to step 5.
Step 5. Calculate Equation 3; if P ≥ rand (0, 1), then Z* = Z, S* = S, and then n limit = n limit + 1, n over = n over + 1; if n over = n over max or n limit = n limit max, then proceed to step 6; otherwise, repeat step 3.If P < rand(0, 1), then n over = n over + 1; if n over = n over max, then proceed to step 6; otherwise, repeat step 3.
Step 6. T = r × T, where r is the rate of cooling. If T ≤ T_{0}, then proceed to step 7; otherwise, repeat step 2.
Step 7. Print S* and Z*.
Partial swarm optimization
where i is the index of the particle, t is the index of an iteration, v_{i} is the vector of velocity, x_{i} is the position, w is the weight of current velocity, b_{1} is the weight of difference between personal best and current positions, b_{2} is the weight of difference between global best and current positions, and rand is used for randomization. The PSO pseudo code, which is presented for QAP in this research, is as follows:
Step 1. max iteration = 100, number of particle = 15, and w = n − 1, b_{1} = n/2 and b_{2} = (n/2) + 2, where n is the dimension of the problem. Make 15 permutations as the initial solutions and Z* = min (OFV), S* = S, and iter = 1.
Step 2. For i = 1 to w, transpose two facility. Do this step for each particle.
Step 3. Calculate objective function value (Z) for each new particle; find the personal best OFV for each particle (p best), and find the global best OFV (g best).
Step 4. Generate a random discrete number between 0 and b_{1}, and for i = 1 to this random number, simulate each particle to p best.
Step 5. Generate a random discrete number between 0 and b_{2}, and for i = 1 to this random number, simulate each particle to g best.
Step 6. iter = iter + 1; if iter < max iteration, then repeat step 2;otherwise, proceed to step 7.
Tuning method
Results and discussion
Computational analyses
Heuristic computational results for the test problems
Names of instances | n | 2-Opt | Greedy 2-opt | 3-Opt | Greedy 3-opt | VNZ | Exact | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Best OFV | Time (s) | Best OFV | Time (s) | Best OFV | Time (s) | Best OFV | Time (s) | Best OFV | Time (s) | Best OFV | ||
Nug12 | 12 | 630 | 0.035 | 720 | 0.070 | 612 | 0.060 | 662 | 0.038 | 630 | 0.115 | 578 |
Nug14 | 14 | 1040 | 0.033 | 1270 | 0.031 | 1076 | 0.120 | 1122 | 0.037 | 1094 | 0.079 | 1014 |
Nug15 | 15 | 1168 | 0.080 | 1410 | 0.031 | 1194 | 0.158 | 1266 | 0.081 | 1210 | 0.094 | 1150 |
Nug16a | 16 | 1636 | 0.036 | 2032 | 0.061 | 1710 | 0.150 | 1744 | 0.092 | 1708 | 0.160 | 1610 |
Nug16b | 16 | 1312 | 0.062 | 1478 | 0.039 | 1304 | 0.175 | 1342 | 0.047 | 1338 | 0.092 | 1240 |
Nug17 | 17 | 1764 | 0.043 | 2204 | 0.051 | 1824 | 0.183 | 1900 | 0.052 | 1812 | 0.094 | 1732 |
Nug18 | 18 | 1988 | 0.033 | 2438 | 0.038 | 2002 | 0.325 | 2102 | 0.078 | 2084 | 0.104 | 1930 |
Nug20 | 20 | 2676 | 0.095 | 3070 | 0.037 | 2700 | 0.300 | 2780 | 0.089 | 2762 | 0.076 | 2570 |
Nug21 | 21 | 2560 | 0.055 | 3224 | 0.073 | 2528 | 0.482 | 2612 | 0.145 | 2568 | 0.096 | 2438 |
Nug22 | 22 | 3836 | 0.050 | 4580 | 0.043 | 3706 | 0.702 | 3850 | 0.070 | 4002 | 0.172 | 3596 |
Nug24 | 24 | 3670 | 0.051 | 4550 | 0.052 | 3580 | 0.889 | 3776 | 0.121 | 3826 | 0.143 | 3488 |
Nug25 | 25 | 3816 | 0.056 | 4746 | 0.053 | 3850 | 0.999 | 4014 | 0.194 | 3946 | 0.186 | 3744 |
Nug27 | 27 | 5582 | 0.069 | 6436 | 0.037 | 5622 | 1.580 | 5932 | 0.214 | 5792 | 0.138 | 5234 |
Nug30 | 30 | 6180 | 0.081 | 7450 | 0.052 | 6282 | 3.476 | 6670 | 0.260 | 6648 | 0.238 | 6124 |
Bur26a | 26 | 5536606 | 0.104 | 5686514 | 0.079 | 5450887 | 5.318 | 5499462 | 0.390 | 5642823 | 0.269 | 5426670 |
Bur26b | 26 | 3865109 | 0.201 | 4078658 | 0.082 | 3825543 | 6.888 | 3921950 | 0.390 | 3898585 | 0.340 | 3817852 |
Bur26c | 26 | 5559892 | 0.158 | 5691003 | 0.089 | 5432628 | 6.066 | 5488940 | 0.377 | 5560520 | 0.300 | 5426795 |
Bur26d | 26 | 3955005 | 0.160 | 4032499 | 0.079 | 3822905 | 5.472 | 3845665 | 0.408 | 3999274 | 0.324 | 3821225 |
Bur26e | 26 | 5457341 | 0.131 | 5656466 | 0.082 | 5387936 | 6.510 | 5441312 | 0.382 | 5610566 | 0.351 | 5386879 |
Bur26f | 26 | 3882393 | 0.149 | 4038741 | 0.079 | 3783547 | 4.828 | 3818354 | 0.390 | 3982126 | 0.282 | 3782044 |
Bur26g | 26 | 10178342 | 0.148 | 10668053 | 0.080 | 10119845 | 5.332 | 10243826 | 0.389 | 10405839 | 0.281 | 10117172 |
Bur26h | 26 | 7246369 | 0.101 | 7644484 | 0.085 | 7100009 | 5.651 | 7216120 | 0.391 | 7286571 | 0.281 | 7098658 |
Solution qualities with CPU time of small-size test problems
Names of instances | n | SA | TS | PSO | IFLS with local search (OXPM) | Exact | ||||
---|---|---|---|---|---|---|---|---|---|---|
RPD | Time | RPD | Time | RPD | Time | RPD | Time | OFV | ||
(s) | (s) | (s) | (s) | |||||||
Nug12 | 12 | 0.00 | 0.16 | 0.00 | 0.12 | 7.61 | 0.27 | 1.38 | 0.70 | 578 |
Nug14 | 14 | 1.78 | 0.19 | 0.39 | 0.17 | 1.38 | 0.29 | 1.58 | 1.28 | 1,014 |
Nug15 | 15 | 3.30 | 0.16 | 0.87 | 0.15 | 2.26 | 0.27 | 0.17 | 1.69 | 1,150 |
Nug16a | 16 | 0.75 | 0.16 | 1.37 | 0.14 | 0.75 | 0.31 | 0.00 | 2.17 | 1,610 |
Nug16b | 16 | 0.65 | 0.17 | 0.00 | 0.14 | 0.00 | 0.32 | 0.00 | 2.14 | 1,240 |
Nug17 | 17 | 1.04 | 0.17 | 0.69 | 0.15 | 3.00 | 0.32 | 1.27 | 2.77 | 1,732 |
Nug18 | 18 | 2.69 | 0.18 | 1.04 | 0.14 | 3.01 | 0.34 | 0.93 | 3.42 | 1,930 |
Nug20 | 20 | 2.49 | 0.20 | 1.56 | 0.16 | 1.95 | 0.36 | 0.70 | 5.28 | 2,570 |
Nug21 | 21 | 1.72 | 0.20 | 0.25 | 0.17 | 2.71 | 0.40 | 0.25 | 6.89 | 2,438 |
Nug22 | 22 | 1.78 | 0.20 | 0.50 | 0.17 | 1.33 | 0.41 | 1.33 | 7.77 | 3,596 |
Nug24 | 24 | 4.59 | 0.21 | 1.15 | 0.18 | 3.96 | 0.45 | 2.87 | 10.91 | 3,488 |
Solution qualities with CPU time of medium-size test problems
Names of instances | n | SA | TS | PSO | IFLS with local search (OXPM) | Exact | ||||
---|---|---|---|---|---|---|---|---|---|---|
RPD | Time | RPD | Time | RPD | Time | RPD | Time | OFV | ||
(s) | (s) | (s) | (s) | |||||||
Nug25 | 25 | 3.69 | 0.24 | 1.55 | 0.18 | 1.92 | 0.48 | 0.21 | 12.97 | 3,744 |
Bur26a | 26 | 0.10 | 0.62 | 0.09 | 0.53 | 0.20 | 1.32 | 0.09 | 15.86 | 5,426,670 |
Bur26b | 26 | 0.67 | 0.61 | 0.19 | 0.54 | 0.42 | 1.30 | 0.17 | 15.56 | 3,817,852 |
Bur26c | 26 | 0.06 | 0.60 | 0.26 | 0.55 | 0.30 | 1.34 | 0.00 | 15.41 | 5,426,795 |
Bur26d | 26 | 0.12 | 0.60 | 0.02 | 0.54 | 0.05 | 1.30 | 0.01 | 15.38 | 3,821,225 |
Bur26e | 26 | 0.01 | 0.60 | 0.03 | 0.51 | 0.05 | 1.33 | 0.26 | 15.16 | 5,386,879 |
Bur26f | 26 | 0.26 | 0.61 | 0.05 | 0.54 | 0.33 | 1.29 | 0.00 | 15.55 | 3,782,044 |
Bur26g | 26 | 0.06 | 0.59 | 0.01 | 0.54 | 0.42 | 1.34 | 0.01 | 15.28 | 10,117,172 |
Bur26h | 26 | 0.03 | 0.60 | 0.01 | 0.52 | 0.35 | 1.32 | 0.00 | 14.88 | 7,098,658 |
Nug27 | 27 | 2.67 | 0.26 | 1.38 | 0.22 | 4.43 | 0.48 | 2.79 | 17.42 | 5,234 |
Nug30 | 30 | 5.94 | 0.27 | 2.65 | 0.22 | 6.01 | 0.54 | 1.32 | 22.45 | 6,124 |
Tai30a | 30 | 4.70 | 0.27 | 4.75 | 0.25 | 5.90 | 0.85 | 2.46 | 18.28 | 1,818,146 |
Tai30b | 30 | 3.55 | 0.26 | 1.62 | 0.25 | 5.18 | 0.87 | 2.33 | 18.23 | 637,117,113 |
Tai40a | 40 | 5.31 | 0.37 | 6.12 | 0.32 | 5.53 | 1.16 | 3.16 | 62.66 | 3,139,370 |
Tai40b | 40 | 5.00 | 0.34 | 3.07 | 0.29 | 3.84 | 1.07 | 4.18 | 60.44 | 637,250,948 |
Tai50a | 50 | 5.90 | 0.47 | 6.49 | 0.39 | 7.19 | 1.36 | 2.90 | 158.94 | 4,938,796 |
Tai50b | 50 | 3.86 | 0.45 | 5.15 | 0.40 | 7.23 | 1.35 | 1.87 | 161.27 | 458,821,517 |
Solution qualities with CPU time of large size test problems
Names of instances | n | SA | TS | PSO | IFLS with local search (OXPM) | Exact | ||||
---|---|---|---|---|---|---|---|---|---|---|
RPD | Time | RPD | Time | RPD | Time | RPD | Time | OFV | ||
(s) | (s) | (s) | (s) | |||||||
Lipa80a | 80 | 0.25 | 0.89 | 0.15 | 0.79 | 40.70 | 2.39 | 0.72 | 1734.11 | 253,195 |
Lipa80b | 80 | 23.46 | 0.94 | 23.55 | 0.80 | 23.93 | 2.38 | 20.65 | 1889.81 | 7,763,962 |
Tai80a | 80 | 6.13 | 0.94 | 6.66 | 0.79 | 7.01 | 2.37 | 2.87 | 1390.58 | 13,527,910 |
Tai80b | 80 | 2.93 | 0.96 | 3.91 | 0.82 | 4.55 | 2.48 | 0.71 | 1122.56 | 841,223,593 |
Lipa90a | 90 | 2.01 | 1.12 | 0.49 | 0.93 | 1.03 | 3.07 | 0.67 | 2964.41 | 360,630 |
Lipa90b | 90 | 13.55 | 1.17 | 14.59 | 0.96 | 14.71 | 3.35 | 0.00 | 3200.88 | 12,490,441 |
Tai100a | 100 | 4.66 | 1.69 | 4.11 | 1.20 | 4.33 | 4.82 | 2.69 | 3560.03 | 21,090,402 |
Tai100b | 100 | 3.06 | 2.15 | 2.02 | 1.73 | 4.17 | 5.21 | 0.93 | 3595.17 | 1.186E + 09 |
The performances of our proposed meta-heuristic approaches and IFLS with local search are studied by analysis of variance (ANOVA) test.
Conclusions
This research considers the heuristic and meta-heuristic solution methods for QAP, and the comparison among them has been done. In addition, a tuning method is declared. The comparison has been executed for the selected data set which was extracted from QAPLIB. The results show that 3-opt has better results than the other heuristic methods. Moreover, our meta-heuristic methods are better than the heuristic methods in solution quality. In this paper, some small, medium, and large test problems are used for comparing our meta-heuristic methods to a method from literature (i.e., IFLS with local search). ANOVA test is run for the results, and it showed that our methods are not considerably different from IFLS. TS is the most excellent method in computational time. Comparisons between the selected solution methods for more instances from the QAPLIB and comparisons of these algorithms with other meta-heuristics and hybrid algorithms can be conducted in future research.
Notes
Acknowledgments
The authors would like to thank the anonymous referees for their constructive comments on earlier version of this work.
Supplementary material
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