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A new efficient heuristic algorithm has been developed for the dynamic facility layout problem with budget constraint (DFLPB) using optimization via simulation technique. The heuristic integrates integer programming and discrete event simulation to address DFLPB. In the proposed algorithm, the nonlinear model of the DFLP has been changed to a pure integer programming (PIP) model. Then, the optimal solution of the PIP model has been used in a simulation model that has been designed in a similar manner as the DFLP for determining the probability of assigning a facility to a location. After a sufficient number of runs, the simulation model obtains near optimum solutions. Finally, to test the performance of the algorithm, several test problems have been taken from the literature and solved. The results show that the proposed algorithm is more efficient in terms of speed and accuracy than other heuristic algorithms presented in previous works.

According to [

Rosenblatt [

It should be mentioned that in addition to exact algorithms, many metaheuristic algorithms have been reported in the literature such as a genetic algorithm by [

It should be noticed that most previous researches did not consider the company budget for rearranging the departments. Because these rearrangements are costly activities, it is normal for a company to have a limited budget in this regard. According to the literature, there are just three studies on DFLP with the budget constraints [

In this paper, we first introduce the problem formulation for the DFLPB in Section

The DFLP can be modeled as a modified quadratic assignment problem, similar to the static facility layout problem (SFLP). The notations used in the model are given as follows:

Problem

Problem

Before describing the simulation model for the DFLPB, an idea that was developed by [

Define

Solve Problem

Select a facility that has the maximum value for the assignment, and, if there is a tie, then select a facility randomly. Assume that facility

If

Now,

And the value for NR can be computed based on (

As it will be shown in the computational results.

The value of NR is relatively low in comparison to other developed metaheuristics, which results in quick run times. This is because, the optimal values obtained from Problem

Because the simulation model does not depend on simulation clock, all runs will be completed very quickly (less than 0.01 min in most cases). Therefore, this effective criterion will help the simulation find the near optimum solutions as quickly as possible.

Based on the previous explanations, the heuristic algorithm can be defined as follows.

Initialize: assume that we have a DFLPB with

Calculate the assignment probabilities: solve Problem

Calculate the minimum number of needed runs: according to the results of Step

Run the simulation model: run the simulation model with NR replications. At each replication

Find the best solution: the best solution is determined as

As mentioned before, Enterprise Dynamics 8.1 software has been used for the simulation, Lingo 8.0 for finding the optimal solution of Problem

The computational results with significant level 99% with

Problem number | Budget | Average probabilities | NR | Best solution | Best solution by Sahin et al. [ | % Dev. | Average time | |
---|---|---|---|---|---|---|---|---|

5 | P01 | 1 | 0.7608 | 21 | 106419 | 106419 | 0 | 0.53 |

2 | 0.7984 | 15 | 106419 | 106419 | 0 | 0.38 | ||

3 | 0.8432 | 10 | 106419 | 106419 | 0 | 0.26 | ||

P02 | 1 | 0.7745 | 19 | 105731 | 105731 | 0 | 0.47 | |

2 | 0.8432 | 10 | 105731 | 105731 | 0 | 0.26 | ||

3 | 0.8004 | 15 | 103429 | 104834 | −1.34 | 0.38 | ||

P03 | 1 | 0.8346 | 11 | 103541 | 106011 | −2.33 | 0.28 | |

2 | 0.8343 | 11 | 106049 | 107609 | −1.45 | 0.28 | ||

3 | 0.7954 | 16 | 102092 | 105762 | −3.47 | 0.39 | ||

P04 | 1 | 0.7922 | 16 | 106547 | 106583 | −0.03 | 0.41 | |

2 | 0.7732 | 19 | 107984 | 107984 | 0 | 0.48 | ||

3 | 0.851 | 10 | 106906 | 106906 | 0 | 0.24 | ||

P05 | 1 | 0.8848 | 7 | 104786 | 106328 | −1.45 | 0.18 | |

2 | 0.8679 | 8 | 107870 | 107870 | 0 | 0.21 | ||

3 | 0.821 | 13 | 106285 | 106.328 | −0.04 | 0.31 | ||

P06 | 1 | 0.8847 | 7 | 104315 | 104315 | 0 | 0.18 | |

2 | 0.8771 | 8 | 107698 | 107698 | 0 | 0.19 | ||

3 | 0.8213 | 13 | 104001 | 104262 | −0.25 | 0.31 | ||

P07 | 1 | 0.9235 | 5 | 103582 | 107406 | −3.56 | 0.12 | |

2 | 0.9133 | 5 | 104752 | 108114 | −3.11 | 0.13 | ||

3 | 0.9011 | 6 | 106173 | 106439 | −0.25 | 0.15 | ||

P08 | 1 | 0.8261 | 12 | 107248 | 107248 | 0 | 0.3 | |

2 | 0.8022 | 15 | 107248 | 107248 | 0 | 0.37 | ||

3 | 0.8045 | 15 | 107248 | 107248 | 0 | 0.36 | ||

10 | P09 | 1 | 0.7892 | 17 | 220301 | 220367 | −0.03 | 0.52 |

2 | 0.7954 | 16 | 220776 | 220776 | 0 | 0.49 | ||

3 | 0.8124 | 14 | 217251 | 217251 | 0 | 0.42 | ||

P10 | 1 | 0.8103 | 14 | 216607 | 217106 | −0.23 | 0.43 | |

2 | 0.8092 | 14 | 216767 | 217201 | −0.2 | 0.43 | ||

3 | 0.8674 | 8 | 211837 | 212134 | −0.14 | 0.26 | ||

P11 | 1 | 0.8464 | 10 | 211951 | 214960 | −1.4 | 0.31 | |

2 | 0.8522 | 10 | 206178 | 215622 | −4.38 | 0.3 | ||

3 | 0.8775 | 8 | 215393 | 215393 | 0 | 0.23 | ||

P12 | 1 | 0.8923 | 7 | 216828 | 216828 | 0 | 0.2 | |

2 | 0.8955 | 6 | 216828 | 216828 | 0 | 0.2 | ||

3 | 0.798 | 15 | 216828 | 216828 | 0 | 0.48 | ||

P13 | 1 | 0.8563 | 9 | 205695 | 211620 | −2.8 | 0.28 | |

2 | 0.8543 | 9 | 210958 | 213304 | −1.1 | 0.29 | ||

3 | 0.8439 | 10 | 205060 | 211620 | −3.1 | 0.32 | ||

P14 | 1 | 0.7845 | 17 | 211916 | 212341 | −0.2 | 0.54 | |

2 | 0.798 | 15 | 207966 | 213430 | −2.56 | 0.48 | ||

3 | 0.8238 | 12 | 205335 | 213424 | −3.79 | 0.38 | ||

P15 | 1 | 0.8842 | 7 | 217221 | 217460 | −0.11 | 0.22 | |

2 | 0.8906 | 7 | 218291 | 218794 | −0.23 | 0.21 | ||

3 | 0.8578 | 9 | 214136 | 214823 | −0.32 | 0.28 | ||

P16 | 1 | 0.77 | 20 | 171712 | 220144 | −22 | 0.61 | |

2 | 0.7903 | 16 | 189324 | 220144 | −14 | 0.51 | ||

3 | 0.8431 | 10 | 181917 | 219177 | −17 | 0.32 | ||

Average | 0.8348 | 12 | 157616 | 161343 | −1.89 | 0.33 |

Another important factor regarding the proposed algorithm is the “average CPU time,” which is sufficiently fast for use in these applications. As previously explained, the simulation time depends on many factors such as NR,

The computational results with significant level 99% with

Problem number | Budget | Average probabilities | NR | Best solution | Best solution by Sahin et al. [ | % Dev. | Average time | |
---|---|---|---|---|---|---|---|---|

5 | P17 | 1 | 0.9306 | 11 | 481675 | 481675 | 0 | 0.48 |

2 | 0.88 | 29 | 480208 | 481682 | −0.31 | 1.25 | ||

3 | 0.7845 | 173 | 494401 | 480453 | 2.9 | 7.45 | ||

P18 | 1 | 0.8758 | 31 | 468932 | 484799 | −3.27 | 1.35 | |

2 | 0.7849 | 172 | 483921 | 490290 | −1.3 | 7.39 | ||

3 | 0.782 | 182 | 478213 | 486726 | −1.75 | 7.82 | ||

P19 | 1 | 0.8221 | 85 | 474661 | 489583 | −3.05 | 3.64 | |

2 | 0.8762 | 31 | 492274 | 493018 | −0.15 | 1.34 | ||

3 | 0.9911 | 2 | 489450 | 489450 | 0 | 0.1 | ||

P20 | 1 | 0.9317 | 11 | 477414 | 484876 | −1.54 | 0.47 | |

2 | 0.9618 | 6 | 484856 | 489912 | −1.03 | 0.24 | ||

3 | 0.8198 | 88 | 470294 | 484954 | −3.02 | 3.8 | ||

P21 | 1 | 0.8261 | 79 | 475885 | 488262 | −2.54 | 3.38 | |

2 | 0.811 | 104 | 476112 | 487935 | −2.42 | 4.49 | ||

3 | 0.8148 | 97 | 469153 | 487822 | −3.83 | 4.18 | ||

P22 | 1 | 0.9153 | 15 | 473,148 | 486493 | −2.74 | 0.64 | |

2 | 0.8581 | 43 | 473392 | 488199 | −3.03 | 1.87 | ||

3 | 0.9523 | 7 | 485532 | 487360 | −0.37 | 0.3 | ||

P23 | 1 | 0.9302 | 11 | 458388 | 478000 | −4.1 | 0.48 | |

2 | 0.8334 | 69 | 466110 | 487007 | −4.29 | 2.95 | ||

3 | 0.8044 | 118 | 467295 | 486801 | −4.01 | 5.08 | ||

P24 | 1 | 0.8628 | 40 | 480468 | 491080 | −2.16 | 1.71 | |

2 | 0.9481 | 8 | 489292 | 494369 | −1.03 | 0.33 | ||

3 | 0.9867 | 3 | 476618 | 491237 | −2.98 | 0.12 | ||

10 | P25 | 1 | 0.9476 | 8 | 939786 | 981531 | −4.25 | 0.38 |

2 | 0.8936 | 23 | 985031 | 985031 | 0 | 1.1 | ||

3 | 0.9821 | 3 | 979638 | 979638 | 0 | 0.16 | ||

P26 | 1 | 0.906 | 18 | 979655 | 979655 | 0 | 0.87 | |

2 | 0.7792 | 192 | 955783 | 981478 | −2.62 | 9.41 | ||

3 | 0.954 | 7 | 952918 | 977462 | −2.51 | 0.33 | ||

P27 | 1 | 0.9215 | 13 | 955190 | 984103 | −2.94 | 0.65 | |

2 | 0.9272 | 12 | 972096 | 993049 | −2.11 | 0.58 | ||

3 | 0.8726 | 33 | 960196 | 983112 | −2.33 | 1.63 | ||

P28 | 1 | 0.9512 | 7 | 950604 | 971759 | −2.18 | 0.35 | |

2 | 0.8484 | 52 | 974385 | 974385 | 0 | 2.54 | ||

3 | 0.9101 | 17 | 973223 | 974792 | −0.16 | 0.81 | ||

P29 | 1 | 0.7854 | 170 | 936480 | 978456 | −4.29 | 8.34 | |

2 | 0.9871 | 3 | 980346 | 980346 | 0 | 0.13 | ||

3 | 0.7638 | 260 | 947673 | 978748 | −3.18 | 12.73 | ||

P30 | 1 | 0.782 | 182 | 949566 | 970024 | −2.11 | 8.91 | |

2 | 0.793 | 147 | 972765 | 972765 | 0 | 7.2 | ||

3 | 0.7929 | 147 | 969998 | 970435 | −0.04 | 7.22 | ||

P31 | 1 | 0.7887 | 160 | 962403 | 978549 | −1.65 | 7.83 | |

2 | 0.8457 | 55 | 990976 | 990976 | 0 | 2.67 | ||

3 | 0.7747 | 210 | 979339 | 979339 | 0 | 10.27 | ||

P32 | 1 | 0.8746 | 32 | 971053 | 985001 | −1.42 | 1.57 | |

2 | 0.8432 | 57 | 958486 | 986493 | −2.84 | 2.8 | ||

3 | 0.9894 | 2 | 977270 | 985817 | −0.87 | 0.12 | ||

Average | 0.8729 | 67 | 721720 | 733644 | −1.7 | 3.19 |

The computational results with significant level 99% with

Problem number | Budget | Average probabilities | NR | Best solution | Best solution by Sahin et al. [ | % Dev. | Average time | |
---|---|---|---|---|---|---|---|---|

5 | P33 | 1 | 0.887 | 166 | 576451 | 577086 | −0.11 | 10.27 |

2 | 0.9611 | 13 | 579704 | 579704 | 0 | 0.79 | ||

3 | 0.9256 | 44 | 577493 | 577493 | 0 | 2.76 | ||

P34 | 1 | 0.9588 | 14 | 551951 | 571846 | −3.48 | 0.86 | |

2 | 0.995 | 2 | 559139 | 572396 | −2.32 | 0.15 | ||

3 | 0.9009 | 103 | 556359 | 570,537 | −2.49 | 6.39 | ||

P35 | 1 | 0.8387 | 899 | 566291 | 579113 | −2.21 | 55.76 | |

2 | 0.8735 | 264 | 556438 | 579406 | −3.96 | 16.37 | ||

3 | 0.887 | 166 | 566301 | 574225 | −1.38 | 10.28 | ||

P36 | 1 | 0.9824 | 5 | 557872 | 572964 | −2.63 | 0.32 | |

2 | 0.9731 | 8 | 554936 | 578631 | −4.09 | 0.49 | ||

3 | 0.9262 | 44 | 545506 | 569880 | −4.28 | 2.7 | ||

P37 | 1 | 0.9224 | 50 | 552347 | 559934 | −1.35 | 3.08 | |

2 | 0.9568 | 15 | 551905 | 559078 | −1.28 | 0.92 | ||

3 | 0.9379 | 29 | 555069 | 559506 | −0.79 | 1.81 | ||

P38 | 1 | 0.9888 | 4 | 544879 | 569457 | −4.32 | 0.23 | |

2 | 0.941 | 26 | 559640 | 567166 | −1.33 | 1.62 | ||

3 | 0.8689 | 310 | 546839 | 567749 | −3.68 | 19.2 | ||

P39 | 1 | 0.8817 | 199 | 569470 | 569470 | 0 | 12.33 | |

2 | 0.7843 | 6740 | 570521 | 570521 | 0 | 417.85 | ||

3 | 0.9374 | 30 | 563648 | 569382 | −1.01 | 1.84 | ||

P40 | 1 | 0.992 | 3 | 556582 | 579411 | −3.94 | 0.19 | |

2 | 0.927 | 42 | 565906 | 586310 | −3.48 | 2.63 | ||

3 | 0.9413 | 26 | 560792 | 577719 | −2.93 | 1.61 | ||

10 | P41 | 1 | 0.9291 | 39 | 1133743 | 1171634 | −3.23 | 2.88 |

2 | 0.986 | 4 | 1155647 | 1172520 | −1.44 | 0.32 | ||

3 | 0.9198 | 54 | 1129068 | 1171500 | −3.62 | 3.96 | ||

P42 | 1 | 0.9475 | 21 | 1166613 | 1174896 | −0.71 | 1.52 | |

2 | 0.966 | 11 | 1137578 | 1175998 | −3.27 | 0.77 | ||

3 | 0.9457 | 22 | 1162838 | 1177009 | −1.2 | 1.62 | ||

P43 | 1 | 0.9223 | 50 | 1169208 | 1169208 | 0 | 3.63 | |

2 | 0.8739 | 260 | 1179660 | 1179660 | 0 | 19.01 | ||

3 | 0.8867 | 167 | 1134677 | 1164129 | −2.53 | 12.23 | ||

P44 | 1 | 0.9085 | 80 | 1140598 | 1151468 | −0.94 | 5.81 | |

2 | 0.7854 | 6462 | 1152874 | 1152874 | 0 | 471.72 | ||

3 | 0.9123 | 70 | 1122006 | 1147234 | −2.2 | 5.11 | ||

P45 | 1 | 0.9703 | 9 | 1114861 | 1127044 | −1.08 | 0.65 | |

2 | 0.858 | 453 | 1141881 | 1141881 | 0 | 33.09 | ||

3 | 0.8781 | 225 | 1128472 | 1129703 | −0.11 | 16.44 | ||

P46 | 1 | 0.8762 | 240 | 1132099 | 1146000 | −1.21 | 17.55 | |

2 | 0.7867 | 6149 | 1154691 | 1154691 | 0 | 448.88 | ||

3 | 0.8583 | 449 | 1145044 | 1145858 | −0.07 | 32.75 | ||

P47 | 1 | 0.778 | 8585 | 1210573 | 1210573 | 0 | 626.71 | |

2 | 0.9444 | 23 | 1210573 | 1210573 | 0 | 1.7 | ||

3 | 0.8937 | 132 | 1210573 | 1210573 | 0 | 9.62 | ||

P48 | 1 | 0.7786 | 8389 | 1199048 | 1189154 | 0.83 | 612.37 | |

2 | 0.872 | 278 | 1152896 | 1201885 | −4.08 | 20.3 | ||

3 | 0.9003 | 105 | 1181360 | 1181360 | 0 | 7.68 | ||

Average | 0.9076 | 864 | 858596 | 870759 | −1.58 | 63.02 |

As listed in Table

In Table

To sum up, the proposed algorithm provides good solution quality in comparison to the algorithms developed in previous researches. It was able to improve the optimal solution for a known data-set by 1.72% on average. Regarding the run time, the algorithm has reasonable run time in comparison to previous researches.

In this research, a new heuristic algorithm has been developed to address the dynamic facility layout problem with budget constraints using optimization via simulation technique. The proposed heuristic algorithm integrates mathematical programming and simulation methods. The optimization via simulation approach was selected and used to show the efficiency of simulation technique to solve even such a large scale combinatorial problem. However, the simulation technique has a vast range of benefits in real world applications, but we tried to show that, simulation is a powerful technique among its so-called competitors such as genetic algorithm, ant colony Optimization, and other evolutionary algorithms. The first contribution of the current study is that it defines the optimal solution of the linear programming model in terms of empirical distributions for a simulation model. This idea can decrease the number of replications required in the simulation model, leading to better speed. The performance of the proposed algorithm was tested over a wide range of test problems taken from the literature. The proposed algorithm improved the objective function of the problem by 1.72% on average, whereas the time required for the largest problem with

The proposed algorithm not only avoids uncommon issues in metaheuristic algorithms such as premature events, parameter tuning, and trapping in local optimums but also uses a simulation technique that produces feasible solutions without the use of any specific nonrealistic assumptions. Regarding the constraints, inherent in this kind of research, we think that if we use the new version of Lingo software and run the algorithm on a faster computer (in particular, one with a faster CPU) the results will be further improved. Finally, for the future works, we strongly suggest concentrating on a cost sensitivity process (including the rearrangement and material handling costs), which will occur in future periods and have a great influence on the optimal solution. As a suggestion, fuzzy costs may be useful under uncertainty conditions, or at least the time value of the monetary investment must be considered.