Journal of Industrial Mathematics

Journal of Industrial Mathematics / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 130251 | 10 pages | https://doi.org/10.1155/2013/130251

Obtaining an Initial Solution for Facility Layout Problem

Academic Editor: Ting Chen
Received13 Apr 2013
Accepted29 Aug 2013
Published24 Oct 2013

Abstract

The facility layout approaches can generally be classified into two groups, constructive approaches and improvement approaches. All improvement procedures require an initial solution which has a significant impact on final solution. In this paper, we introduce a new technique for accruing an initial placement of facilities on extended plane. It is obtained by graph theoretic facility layout approaches and graph drawing algorithms. To evaluate the performance, this initial solution is applied to rectangular facility layout problem. The solution is improved using an analytical method. The approach is then tested on five instances from the literature. Test problems include three large size problems of 50, 100, and 125 facilities. The results demonstrate effectiveness of the technique especially for large size problems.

1. Introduction

The facility layout problem seeks the best positions of facilities to optimize some objective. The common objective is to reduce material handling costs between the facilities. The problem has been modeled by a variety of approaches. A detailed review of the different problem formulations can be found in Singh and Sharma [1]. The facility layout problem is an optimization problem which arises in a variety of problems such as placing machines on a factory floor, VLSI design, and layout design of hospitals, schools.

The facility layout approaches can generally be classified into two groups, constructive methods and improvement methods. In this paper, we consider the placement of facilities on an extended plane. Many improvement approaches have been proposed for this problem. All improvement procedures require an initial solution. Some approaches start from a good but infeasible solution [24]. These models contain a penalty component in their objective function. Hence, these approaches minimize objective function value for feasible solutions. But some approaches require a feasible initial solution. These approaches use a randomly generated initial solution [5, 6]. Mir and Imam [7] have proposed simulated annealing for a better initial solution. They have shown that a good initial solution has a significant impact on final solution.

In this paper, we introduce a new technique for accruing an initial placement of facilities on an extended plane. The technique consists of two stages. In the first stage, a maximal planar graph (MPG) is obtained. In the second stage, the vertices of MPG are drawn on the plane by graph drawing algorithms. Then, vertices are replaced by facilities. Hence, an initial solution is obtained.

In an MPG, the facilities with larger flows are adjacent together. Hence, drawing the MPG on the plane can be a good idea for obtaining an initial solution. To evaluate the performance of the idea, this initial solution is applied in rectangular facility layout problem. The solution is improved by an analytical method by Mir and Imam [7]. The approach is then tested on five instances from the literature.

The remaining parts of the paper are organized as follows. The next section describes the formulation of the facility layout problem chosen for our work. Section 3 describes accruing an initial placement. In Section 4, the analytical method is described, and the approach is compared to other approaches in the literature. In Section 5, the proposed initial solution is compared with random initial solution. Finally, Section 6 provides a summary and conclusion.

2. Problem Formulation

In this paper, we label the facilities , where is the total number of facilities. Facilities are assumed to be rectangles with fixed shape. The notation is given as follows: () coordinates of the center of facility length of facility width of facility the total cost of flow per unit distance between two facilities and  distance between the centers of the facilities and . could be one of the following three distance norms.(1)Euclidean distance: (2)Squared Euclidean distance: (3)Rectilinear distance: The requirement for problem is that the facilities must not overlap each other. The area of overlap is defined as follows: where The value of overlap area is a nonnegative number. will be zero only if there is no overlapping between facilities and . The objective is to minimize material handling costs. So, the problem can be stated as follows: The constraint ensures that facilities do not overlap. A similar formulation also can be found in [7].

3. Obtaining an Initial Solution

The initial solution is obtained by graph theoretic facility layout approaches (GTFLP) and graph drawing algorithms. The following subsection describes obtaining an MPG. Section 3.2 describes the drawing of the MPG on the plane.

3.1. Generating a Maximal Planar Graph

In GTFLP, facilities are represented by vertices, and flow (adjacency desirability) between them is represented by weighted edges. Created graph is called adjacency graph. Graph theory is particularly useful for the facility layout problems, because graphs easily enable us to capture the adjacency information and model the problem. A review of graph theory applications to the facility layout problem can be found in [9, 10]. GTFLP consists of two stages. At the first stage, the adjacency graph is converted to a maximal planar graph (MPG). In the second stage, a block layout is constructed from the MPG. The second stage is not our concern here. For more details, we refer to [1115]. Figure 1 shows an MPG and its correspondent block layout.

Many heuristic and metaheuristic methods for obtaining an MPG have been suggested [1621]. In this paper, we use from the greedy heuristic [16]. It is conceptually simple and creates high weighted MPGs [16]. This heuristic has a simple instruction: the edges are sorted in nonincreasing order of weight. Each edge is tested in turn and accepted as part of the MPG unless it makes the graph nonplanar. So, the heuristic needs planarity testing. Boyer and Myrvold [22] developed a simplified planarity testing algorithm. We use this algorithm for planarity testing. In the worst case, edges are considered, and for each edge, the Boyer and Myrvold test is called. Hence, the approach results in a complexity of .

3.2. Drawing Maximal Planar Graph on the Plane

Graph drawing, as a branch of graph theory, applies topology and geometry to derive two-dimensional representations of graphs. A graph drawing algorithm reads as input a combinatorial description of a graph G and produces as output a drawing of G. A graph has infinitely many different drawings. For a review of various graphs drawing algorithms, refer to [23]. We use algorithm of Chrobak and Payne [24] to form a straight line drawing, of the MPG. In such a drawing, each edge is drawn using a straight line segment. The algorithm draws vertices in an MPG to integer coordinates in a grid. Figure 2 shows an example of straight line drawing.

For acquiring an initial solution, each vertex is replaced by its correspondent facility. In a feasible solution, facilities have no overlaps. For this reason, the coordinates of facilities can be multiplied by maximum dimensions of all facilities (width and length). This operation increases distance between facilities and makes the solution feasible. For the case of circular facilities, the diameter of circle can be considered as maximum dimensions.

4. Improving Initial Solution and Comparing

To evaluate the performance, the initial solution is improved by an analytical method by Mir and Imam [7]. In this method, the convergence is controlled by carrying out the optimization using concept of “magnified envelop blocks.” The dimensions of the blocks are determined by multiplying the dimensions of the facilities with a “magnification factor.” The optimization is then carried out for these envelop blocks rather than the actual facilities. The analytical method searches the optimum placements of each envelop block in the direction of steepest descent which is opposite to the gradient direction. The sizes of the envelop blocks are then reduced, and the optimization process is repeated for the second phase. The number of optimization phases is equal to the magnification factor number for the envelop blocks. In the last optimization phase, the dimensions of the envelop blocks become equal to the actual facilities. For more detail, we refer to Mir and Imam [7].

So, the proposed approach for solving a facility layout problem can be summarized as follows. Step 1: encapsulating facilities in envelop blocks (multiplying the dimensions of facilities by a magnification factor). Step 2: obtaining an MPG. Step 3: drawing the MPG on the plane and obtaining an initial solution.Step 4: improving initial solution by analytical method.Figure 3 shows summary of these steps.

The proposed approach was coded using the VB.NET programming language in a program named GOT (Graph optimization technique). Five test problems were run. For all test problems, results were obtained on a PC with Intel T5470 processor. The results were compared with the previously published papers and commercial software VIP-PLANOPT 2006. VIP-PLANOPT is a useful layout software package that can generate near-optimal layout [25]. For more details about VIP-PLANOPT, see Engineering Optimization Software [8]. VIP-PLANOPT results were obtained from the software user’s manual. The results are presented in the following sections.

4.1. Test Problem #1

This problem of 8 facilities was introduced by Imam and Mir [5]. Figure 4 shows the steps for accruing the initial solution. Figure 4(a) shows the flow matrix and dimension of facilities. All dimensions and cost matrix elements are integer-valued numbers ranging between 1 and 6. There are several pairs of facilities with no flow between them. Distance norm is squared Euclidean. The greedy heuristic generates the edges lists of MPG as shown in Figure 4(b).

The straight line drawing algorithm gives the coordinates of vertices. The drawing is shown in Figure 4(c). Then, each vertex is replaced by its correspondent facility. The coordinates are multiplied by maximum dimensions of facilities (width and length), and finally, the initial layout design is shown in Figure 4(d).

The solution is improved by the analytical technique. Figure 5 shows the final layout. The cost function value for this layout is 752.7, and the running time is 0.4 second. Table 1 shows the results obtained by the other approaches. The best solution for this problem is obtained by VIP-PLANOPT 2006.


ProgramCost function value

TOPOPT (Imam and Mir, 1989) [5]794
VIP-PLANOPT (2006) [8]692
GOT752.7

4.2. Test Problem #2

This problem of 20 unequal area facilities was introduced by Imam and Mir [6]. The data consist of only integer values. The dimensions of the facilities are between 1 and 3. The elements of the cost matrix are integers between 0 and 5. The distance norm is rectilinear. The final layout obtained by GOT is shown in Figure 6. The layout cost is 1302, and the running time is 0.6 second. Table 2 compares the results obtained by GOT with the results available in the literature. VIP-PLANOPT 2006 has the lowest value of the cost function.


ProgramBest design

Topopt (Mir and Imam, 1989) [5]1320.72
FLOAT (Imam and Mir, 1993) [6]1264.94
HOT (Imam and Mir, 2001) [7]1225.40
VIP-PLANOPT (2006) [8]1157
GOT1302

4.3. Test Problem #3

This is a problem of 50 facilities randomly generated by VIP-PLANOPT 2006. The dimensions of the facilities are decimal numbers between 1 to 6. The elements of the cost matrix are all integers between 1 and 10. The distance norm is Euclidean. The results are shown in Table 3. The best published result has a cost of 78224.7, whereas GOT produces a final layout with a cost of 76882.3 only in 15.1 seconds. Figure 7 shows the final layout.


ProgramCost function value

HOT (Mir and Imam, 2001) [7]80794.24
VIP-PLANOPT (2006) [8]78224.7
GOT76882.3

4.4. Test Problem #4

This is a randomly generated large size problem of 100 facilities. The dimensions of the facilities are decimal numbers between 1 and 6. The cost matrix elements are integers between 1 and 10. The distance norm is rectilinear. The results are shown in Table 4. GOT obtained the cost function value of 527094.1 in 74.3 seconds. This value is about 2% below the cost function value of VIP-PLANOPT 2006. The coordinates of the facilities for the layout obtained by GOT are given in Table 5.


ProgramCost function value

HOT (Mir and Imam, 2001) [7]558556.2
VIP-PLANOPT (2006) [8]538193.1
GOT527094.1


Facility

117.59922.158
218.51432.933
329.48518.634
435.52620.582
514.79822.65
626.42525.388
712.21313.282
819.43527.466
932.11915.319
1024.7438.862
1116.2312.711
1225.53317.621
1314.27420.352
1422.7913.889
1514.46714.812
1638.01216.94
179.73511.276
1824.60927.663
1920.68520.578
2019.86622.579
2115.8328.421
228.86631.495
2322.40217.335
2435.5519.125
2516.9519.378
2636.10425.31
2716.32821.843
2813.20826.161
2922.43914.732
3014.05213.065
3128.2121.934
3216.43612.282
3324.46711.365
3423.10332.489
3523.28324.968
3619.44414.708
3732.92716.699
3811.70828.857
3915.1429.793
4028.1645.174
4132.088.476
4229.63613.049
4326.37331.127
448.69315.785
4516.0526.859
4631.96526.099
4729.3619.586
4833.82212.885
4919.39425.043
5025.90221.954
514.23416.534
5219.53930.211
535.58711.801
5413.39932.831
5528.77116.071
5630.97422.774
5722.45322.117
5812.3523.138
5921.72518.944
6030.41931.854
6124.38818.897
6219.58736.974
6319.94312.839
6424.63215.019
655.95132.233
6616.91814.346
6723.8186.408
6819.58118.447
6935.19131.263
700.92721.347
7132.64419.464
729.48720.325
7339.74921.788
7418.48516.27
7515.58136.344
766.34520.203
7713.48423.21
7818.52720.734
7911.77320.295
8019.3292.663
814.47122.79
824.78827.961
8327.5136.612
8426.30615.776
854.54525.257
8623.54535.892
8710.74637.712
887.4056.656
8912.0216.684
9029.1427.123
9123.36729.856
9214.8716.629
9311.96117.137
9416.12618.34
9522.42616.075
9619.72910.752
9719.7077.63
9823.82421.511
998.25727.817
1009.17924.151

4.5. Test Problem #5

This is a large size problem of 125 facilities randomly generated by VIP-PLANOPT 2006. The dimensions of facilities are real numbers between 1 and 6, and elements of the cost matrix are integers between 1 and 10. The distance norm is rectilinear. The results are shown in Table 6. GOT obtained the cost function value of 1062080 in 129.6 seconds. This value is about 2% below the cost function value of VIP-PLANOPT 2006. The coordinates of the facilities for the layout obtained by GOT are given in Table 7.


ProgramCost function value

VIP-PLANOPT (2006) [8]1084451
GOT1062080


Facility

128.61719.513
225.37122.082
327.68332.088
46.48813.531
517.5334.357
610.13116.929
729.90625.226
828.24336.473
914.37231.298
1037.50222.299
1132.5864.846
1219.34913.249
133.31423.588
1446.9619.354
1521.9538.539
1634.1325.537
1733.63632.8
1837.2117.564
1912.60323.713
2021.76321.428
2116.26137.283
2227.45325.185
2325.21315.36
2427.11141.844
2521.20824.959
2633.28738.471
2725.88219.125
2843.8714.147
2927.3426.265
302.83126.3
3118.32831.432
3238.337.761
3322.13127.038
347.16626.77
3518.90946.302
3615.12334.383
3728.97422.602
3818.7284.656
3933.5268.657
4012.17632.318
4116.53939.732
4228.38339.485
4315.0923.899
4432.57112.559
4541.22140.78
4624.33527.843
472.51630.62
4837.45225.455
4940.18936.294
5012.67236.825
5134.28227.349
5219.12615.005
5336.39541.322
5431.47340.034
5530.99134.983
5638.70227.03
5728.17612.73
5829.21945.218
5925.11310.766
6011.02127.029
6122.15814.239
6222.54317.582
6312.34821.322
6413.68443.96
658.56936.149
668.82833.239
6710.93312.373
6824.20326.047
6912.6477.071
7029.78927.362
716.49238.642
7218.39417.993
734.81517.754
7424.46545.646
7524.70424.448
7648.91231.164
7719.06311.088
7827.84316.379
7935.27129.217
8021.65935.429
8115.30328.087
8232.59931.07
833.31120.836
8444.0233.76
8543.8439.862
8616.69423.899
8720.62817.23
8819.08835.563
8948.99825.308
9033.55946.176
9131.16230.86
9224.8236.607
9344.67324.381
9418.38527.694
9530.08610.858
9633.28915.156
9730.39917.188
9815.5219.071
9936.51832.041
10019.43940.84
10122.9324.334
10227.27928.155
10331.84919.046
10411.4819.243
1057.3121.826
10638.08812.744
10722.89230.522
10814.91713.713
1096.99530.065
11040.46630.376
11141.96419.47
11230.314.954
11319.95327.768
11427.3162.155
11535.98336.293
11644.37428.853
11722.62141.026
11831.9920.775
11911.30939.88
12041.37924.172
12131.58926.996
12218.0318.178
12318.54922.682
12431.6423.551
12533.45223.294

5. Comparing GOT Initial Solution with Random Initial Solution

To compare the proposed initial solution (GOT initial solution) with random initial solution, a set of test problems were generated. The facilities dimensions were , and flow matrices were randomly generated between 0 and 10. For acquiring a random initial solution, facilities were randomly placed in a ( − 4) × ( − 2) integer grid. For each test problem, 20 random placements were found. Table 8 shows the value of cost function in GOT initial solution and the best value found by random placements. Figures 8 and 9 shows these results graphically. The results demonstrated significant improvement in cost function.


GOT initial solutionThe best random initial solution

1019452437.3
1121222758.5
1223683531.7
1349195764
1455147024
1555387449
16811610287.7
17846812207.5
181064815304
191394319119.3
201426520094
211719922491.2
221661026370
232212830749.2
242553335508
252758537968.4
262992943803.6
273625551850
284148357050.5
294754365109.2
305783073705.8
316140882058.4
326368786436
335997094066.4
3482721110195.2
3576220104940
3692426124186.1
3795386125468
3887532150060
3993708146793.8
40118266180538
41141363183911
42104263195458.7
43152188204207.3
44163529239253
45166360237051.8
46167002251049.3
47189027290085
48226097305900
49234811324500.3
50238855337198.9
51264842358480.5
52251009358666.7
53233805370262.4
54298806404104.4
55284543431091.9
56344755481656.9
57371962493986.4
58344044510976
59375321503358.9
60373118538554.9
61370817573555.9
62457275628602.9
63544350688184.6
64530408708518.3
65502322690275.2
66526230741481.1
67556568773325.9
68639735867242.4
69578533860229.8
70643592914437.3
71589557893978.8
72670866950592.5
737366691058290.5
746603951039023
757497951115704.6
768354181180300
777276481157821.9
788526891147676
799711351273329.8
809205221294614.8
818826451420567
8210847111500037.8
8310722411436365.1
8410721321606669.3
8511540181552139.2
8611509251756302.3
8711642201608156.3
8812300081726587.4
8913794791913688.3
9013512101970899.3
9112759751895126.4
9213607712114230
9315202281969614.1
9415427402125200
9515811452230222.7
9616407922305569.3
9714867962362639.4
9816458892365109.3
9916078662490733.7
10020739792635666.4

6. Summary and Conclusion

An initial solution has been presented for the layout design of facilities on a continuous plane. The technique consists of two stages. In the first stage, a maximal planar graph (MPG) is obtained. In the second stage, the vertices of MPG are drawn on the plane by graph drawing algorithms. Then, vertices are replaced by facilities. Hence, an initial solution is obtained. To evaluate the performance, this initial solution has been applied in rectangular facility layout problem and improved by an analytical method by Mir and Imam [7].

The approach has been tested on five instances from the literature. Table 6 shows the Summary of the results, and Figure 8 shows the cost reduction by the technique. For the large size problems involving 50, 100, and 125 facilities, the layout costs values are better than those obtained by the previously published techniques. As shown in Table 9, the results demonstrate effectiveness of the technique, especially for large size problems.


ProblemNumber of facilitiesBest result by other methodsGOTCost reduction

#18692.5752.7−60.2
#22011571302−145
#35078224.776882.31342.4
#4100538193.1527094.111099
#51251084451106208022371

This paper introduced a simple technique for obtaining a good initial solution. The technique, with some modification, can be applied in facility layout approaches that use a randomly generated initial solution. In future researches, it would be interesting to analyze the influence of MPG and graph drawing algorithm on the solution. The results can be further improved by using a metaheuristic such as GRASP [21] and Tabu search [20] for generating a high weighted MPG.

Conflict of Interests

The authors declare that they have no conflict of interests.

References

  1. S. P. Singh and R. R. K. Sharma, “A review of different approaches to the facility layout problems,” International Journal of Advanced Manufacturing Technology, vol. 30, no. 5-6, pp. 425–433, 2006. View at: Publisher Site | Google Scholar
  2. M. F. Anjos and A. Vannelli, “An attractor-repeller approach to floorplanning,” Mathematical Methods of Operations Research, vol. 56, no. 1, pp. 3–27, 2002. View at: Publisher Site | Google Scholar
  3. I. Castillo and T. Sim, “A spring-embedding approach for the facility layout problem,” Journal of the Operational Research Society, vol. 55, no. 1, pp. 73–81, 2004. View at: Publisher Site | Google Scholar
  4. Z. Drezner, “DISCON: a new method for the layout problem,” Operations Research, vol. 28, no. 6, pp. 1375–1384, 1980. View at: Google Scholar
  5. M. H. Imam and M. Mir, “Nonlinear programming approach to automated topology optimization,” Computer-Aided Design, vol. 21, no. 2, pp. 107–115, 1989. View at: Google Scholar
  6. M. H. Imam and M. Mir, “Automated layout of facilities of unequal areas,” Computers and Industrial Engineering, vol. 24, no. 3, pp. 355–366, 1993. View at: Google Scholar
  7. M. Mir and M. H. Imam, “Hybrid optimization approach for layout design of unequal-area facilities,” Computers and Industrial Engineering, vol. 39, no. 1-2, pp. 49–63, 2001. View at: Publisher Site | Google Scholar
  8. Engineering Optimization Software, VIP-PLANOPT, 2006, 2010, http://www.planopt.com/.
  9. L. Foulds, Graph Theory Applications, Springer, New York, NY, USA, 1992.
  10. M. M. D. Hassan and G. L. Hogg, “A review of graph theory application to the facilities layout problem,” Omega, vol. 15, no. 4, pp. 291–300, 1987. View at: Google Scholar
  11. M. M. D. Hassan and G. L. Hogg, “On converting a dual graph into a block layout,” International Journal of Production Research, vol. 27, no. 7, pp. 1149–1160, 1989. View at: Google Scholar
  12. K. H. Watson and J. W. Giffin, “The vertex splitting algorithm for facilities layout,” International Journal of Production Research, vol. 35, no. 9, pp. 2477–2492, 1997. View at: Google Scholar
  13. S. A. Irvine and I. Rinsma-Melchert, “A new approach to the block layout problem,” International Journal of Production Research, vol. 35, no. 8, pp. 2359–2376, 1997. View at: Google Scholar
  14. P. S. Welgama, P. R. Gibson, and L. A. R. Al-Hakim, “Facilities layout: a knowledge-based approach for converting a dual graph into a block layout,” International Journal of Production Economics, vol. 33, no. 1–3, pp. 17–30, 1994. View at: Google Scholar
  15. M. A. Jokar and A. S. Sangchooli, “Constructing a block layout by face area,” The International Journal of Advanced Manufacturing Technology, vol. 54, no. 5–8, pp. 801–809, 2011. View at: Publisher Site | Google Scholar
  16. L. R. Foulds, P. B. Gibbons, and J. W. Giffin, “Facilities layout adjacency determination: an experimental comparison of three graph theoretic heuristics,” Operations Research, vol. 33, no. 5, pp. 1091–1106, 1985. View at: Google Scholar
  17. E. G. John and J. Hammond, “Maximally weighted graph theoretic facilities design planning,” International Journal of Production Research, vol. 38, no. 16, pp. 3845–3859, 2000. View at: Publisher Site | Google Scholar
  18. S. G. Boswell, “TESSA—a new greedy heuristic for facilities layout planning,” International Journal of Production Research, vol. 30, no. 8, pp. 1957–1968, 1992. View at: Google Scholar
  19. L. R. Foulds and D. F. Robinson, “Graph theoretic heuristics for the plant layout problem,” International Journal of Production Research, vol. 16, no. 1, pp. 27–37, 1978. View at: Google Scholar
  20. I. H. Osman, “A tabu search procedure based on a random Roulette diversification for the weighted maximal planar graph problem,” Computers and Operations Research, vol. 33, no. 9, pp. 2526–2546, 2006. View at: Publisher Site | Google Scholar
  21. I. H. Osman, B. Al-Ayoubi, and M. Barake, “A greedy random adaptive search procedure for the weighted maximal planar graph problem,” Computers and Industrial Engineering, vol. 45, no. 4, pp. 635–651, 2003. View at: Publisher Site | Google Scholar
  22. J. M. Boyer and W. J. Myrvold, “On the cutting edge: simplified O(n) planarity by edge addition,” Journal of Graph Algorithms and Applications, vol. 8, no. 3, pp. 241–273, 2004. View at: Google Scholar
  23. G. Di Battista, P. Eades, R. Tamassia, and I. Tollis, Graph Drawing: Algorithms for the Visualization of Graphs, Prentice Hall PTR, Upper Saddle River, NJ, USA, 1998.
  24. M. Chrobak and T. H. Payne, “A linear-time algorithm for drawing a planar graph on a grid,” Information Processing Letters, vol. 54, no. 4, pp. 241–246, 1995. View at: Google Scholar
  25. S. Heragu, Facilities Design, Iuniverse Inc., 2006.

Copyright © 2013 Ali Shoja Sangchooli and Mohammad Reza Akbari Jokar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

3898 Views | 616 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder