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.
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 [
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 [
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 [
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
In this paper, we label the facilities ( Euclidean distance:
Squared Euclidean distance:
Rectilinear distance:
The requirement for problem is that the facilities must not overlap each other. The area of overlap is defined as follows:
The initial solution is obtained by graph theoretic facility layout approaches (GTFLP) and graph drawing algorithms. The following subsection describes obtaining an MPG. Section
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 [
An MPG (solid lines) and its correspondent block layout (dashed lines).
Many heuristic and metaheuristic methods for obtaining an MPG have been suggested [
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 [
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.
To evaluate the performance, the initial solution is improved by an analytical method by Mir and Imam [
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
Steps of the proposed approach.
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 [
This problem of 8 facilities was introduced by Imam and Mir [
Obtaining an initial solution for test problem #1.
The raw facilities layout data
MPG
Straight line drawing of the MPG
Initial solution
The straight line drawing algorithm gives the coordinates of vertices. The drawing is shown in Figure
The solution is improved by the analytical technique. Figure
Results for test problem #1.
Program | Cost function value |
---|---|
TOPOPT (Imam and Mir, 1989) [ |
794 |
VIP-PLANOPT (2006) [ |
692 |
GOT | 752.7 |
Final layout for test problem #1.
This problem of 20 unequal area facilities was introduced by Imam and Mir [
Results for test problem #2.
Program | Best design |
---|---|
Topopt (Mir and Imam, 1989) [ |
1320.72 |
FLOAT (Imam and Mir, 1993) [ |
1264.94 |
HOT (Imam and Mir, 2001) [ |
1225.40 |
VIP-PLANOPT (2006) [ |
1157 |
GOT | 1302 |
Final layout for test problem #2.
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
Results for test problem #3.
Program | Cost function value |
---|---|
HOT (Mir and Imam, 2001) [ |
80794.24 |
VIP-PLANOPT (2006) [ |
78224.7 |
GOT | 76882.3 |
Final layout for test problem #3.
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
Results for test problem #4.
Program | Cost function value |
---|---|
HOT (Mir and Imam, 2001) [ |
558556.2 |
VIP-PLANOPT (2006) [ |
538193.1 |
GOT | 527094.1 |
For test problem #4, the coordinates of facilities obtained by GOT are given below. The value of the cost function for this layout is 527094.1.
Facility |
|
|
---|---|---|
1 | 17.599 | 22.158 |
2 | 18.514 | 32.933 |
3 | 29.485 | 18.634 |
4 | 35.526 | 20.582 |
5 | 14.798 | 22.65 |
6 | 26.425 | 25.388 |
7 | 12.213 | 13.282 |
8 | 19.435 | 27.466 |
9 | 32.119 | 15.319 |
10 | 24.743 | 8.862 |
11 | 16.231 | 2.711 |
12 | 25.533 | 17.621 |
13 | 14.274 | 20.352 |
14 | 22.791 | 3.889 |
15 | 14.467 | 14.812 |
16 | 38.012 | 16.94 |
17 | 9.735 | 11.276 |
18 | 24.609 | 27.663 |
19 | 20.685 | 20.578 |
20 | 19.866 | 22.579 |
21 | 15.832 | 8.421 |
22 | 8.866 | 31.495 |
23 | 22.402 | 17.335 |
24 | 35.551 | 9.125 |
25 | 16.95 | 19.378 |
26 | 36.104 | 25.31 |
27 | 16.328 | 21.843 |
28 | 13.208 | 26.161 |
29 | 22.439 | 14.732 |
30 | 14.052 | 13.065 |
31 | 28.21 | 21.934 |
32 | 16.436 | 12.282 |
33 | 24.467 | 11.365 |
34 | 23.103 | 32.489 |
35 | 23.283 | 24.968 |
36 | 19.444 | 14.708 |
37 | 32.927 | 16.699 |
38 | 11.708 | 28.857 |
39 | 15.14 | 29.793 |
40 | 28.164 | 5.174 |
41 | 32.08 | 8.476 |
42 | 29.636 | 13.049 |
43 | 26.373 | 31.127 |
44 | 8.693 | 15.785 |
45 | 16.05 | 26.859 |
46 | 31.965 | 26.099 |
47 | 29.361 | 9.586 |
48 | 33.822 | 12.885 |
49 | 19.394 | 25.043 |
50 | 25.902 | 21.954 |
51 | 4.234 | 16.534 |
52 | 19.539 | 30.211 |
53 | 5.587 | 11.801 |
54 | 13.399 | 32.831 |
55 | 28.771 | 16.071 |
56 | 30.974 | 22.774 |
57 | 22.453 | 22.117 |
58 | 12.35 | 23.138 |
59 | 21.725 | 18.944 |
60 | 30.419 | 31.854 |
61 | 24.388 | 18.897 |
62 | 19.587 | 36.974 |
63 | 19.943 | 12.839 |
64 | 24.632 | 15.019 |
65 | 5.951 | 32.233 |
66 | 16.918 | 14.346 |
67 | 23.818 | 6.408 |
68 | 19.581 | 18.447 |
69 | 35.191 | 31.263 |
70 | 0.927 | 21.347 |
71 | 32.644 | 19.464 |
72 | 9.487 | 20.325 |
73 | 39.749 | 21.788 |
74 | 18.485 | 16.27 |
75 | 15.581 | 36.344 |
76 | 6.345 | 20.203 |
77 | 13.484 | 23.21 |
78 | 18.527 | 20.734 |
79 | 11.773 | 20.295 |
80 | 19.329 | 2.663 |
81 | 4.471 | 22.79 |
82 | 4.788 | 27.961 |
83 | 27.51 | 36.612 |
84 | 26.306 | 15.776 |
85 | 4.545 | 25.257 |
86 | 23.545 | 35.892 |
87 | 10.746 | 37.712 |
88 | 7.405 | 6.656 |
89 | 12.021 | 6.684 |
90 | 29.14 | 27.123 |
91 | 23.367 | 29.856 |
92 | 14.87 | 16.629 |
93 | 11.961 | 17.137 |
94 | 16.126 | 18.34 |
95 | 22.426 | 16.075 |
96 | 19.729 | 10.752 |
97 | 19.707 | 7.63 |
98 | 23.824 | 21.511 |
99 | 8.257 | 27.817 |
100 | 9.179 | 24.151 |
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
Results for test problem #5.
Program | Cost function value |
---|---|
VIP-PLANOPT (2006) [ |
1084451 |
GOT | 1062080 |
For test problem #5, the coordinates of facilities obtained by GOT are given below. The layout cost is 1062080.
Facility |
|
|
---|---|---|
1 | 28.617 | 19.513 |
2 | 25.371 | 22.082 |
3 | 27.683 | 32.088 |
4 | 6.488 | 13.531 |
5 | 17.53 | 34.357 |
6 | 10.131 | 16.929 |
7 | 29.906 | 25.226 |
8 | 28.243 | 36.473 |
9 | 14.372 | 31.298 |
10 | 37.502 | 22.299 |
11 | 32.586 | 4.846 |
12 | 19.349 | 13.249 |
13 | 3.314 | 23.588 |
14 | 46.96 | 19.354 |
15 | 21.953 | 8.539 |
16 | 34.13 | 25.537 |
17 | 33.636 | 32.8 |
18 | 37.21 | 17.564 |
19 | 12.603 | 23.713 |
20 | 21.763 | 21.428 |
21 | 16.261 | 37.283 |
22 | 27.453 | 25.185 |
23 | 25.213 | 15.36 |
24 | 27.111 | 41.844 |
25 | 21.208 | 24.959 |
26 | 33.287 | 38.471 |
27 | 25.882 | 19.125 |
28 | 43.87 | 14.147 |
29 | 27.342 | 6.265 |
30 | 2.831 | 26.3 |
31 | 18.328 | 31.432 |
32 | 38.33 | 7.761 |
33 | 22.131 | 27.038 |
34 | 7.166 | 26.77 |
35 | 18.909 | 46.302 |
36 | 15.123 | 34.383 |
37 | 28.974 | 22.602 |
38 | 18.728 | 4.656 |
39 | 33.526 | 8.657 |
40 | 12.176 | 32.318 |
41 | 16.539 | 39.732 |
42 | 28.383 | 39.485 |
43 | 15.09 | 23.899 |
44 | 32.571 | 12.559 |
45 | 41.221 | 40.78 |
46 | 24.335 | 27.843 |
47 | 2.516 | 30.62 |
48 | 37.452 | 25.455 |
49 | 40.189 | 36.294 |
50 | 12.672 | 36.825 |
51 | 34.282 | 27.349 |
52 | 19.126 | 15.005 |
53 | 36.395 | 41.322 |
54 | 31.473 | 40.034 |
55 | 30.991 | 34.983 |
56 | 38.702 | 27.03 |
57 | 28.176 | 12.73 |
58 | 29.219 | 45.218 |
59 | 25.113 | 10.766 |
60 | 11.021 | 27.029 |
61 | 22.158 | 14.239 |
62 | 22.543 | 17.582 |
63 | 12.348 | 21.322 |
64 | 13.684 | 43.96 |
65 | 8.569 | 36.149 |
66 | 8.828 | 33.239 |
67 | 10.933 | 12.373 |
68 | 24.203 | 26.047 |
69 | 12.647 | 7.071 |
70 | 29.789 | 27.362 |
71 | 6.492 | 38.642 |
72 | 18.394 | 17.993 |
73 | 4.815 | 17.754 |
74 | 24.465 | 45.646 |
75 | 24.704 | 24.448 |
76 | 48.912 | 31.164 |
77 | 19.063 | 11.088 |
78 | 27.843 | 16.379 |
79 | 35.271 | 29.217 |
80 | 21.659 | 35.429 |
81 | 15.303 | 28.087 |
82 | 32.599 | 31.07 |
83 | 3.311 | 20.836 |
84 | 44.02 | 33.76 |
85 | 43.843 | 9.862 |
86 | 16.694 | 23.899 |
87 | 20.628 | 17.23 |
88 | 19.088 | 35.563 |
89 | 48.998 | 25.308 |
90 | 33.559 | 46.176 |
91 | 31.162 | 30.86 |
92 | 24.82 | 36.607 |
93 | 44.673 | 24.381 |
94 | 18.385 | 27.694 |
95 | 30.086 | 10.858 |
96 | 33.289 | 15.156 |
97 | 30.399 | 17.188 |
98 | 15.52 | 19.071 |
99 | 36.518 | 32.041 |
100 | 19.439 | 40.84 |
101 | 22.932 | 4.334 |
102 | 27.279 | 28.155 |
103 | 31.849 | 19.046 |
104 | 11.48 | 19.243 |
105 | 7.31 | 21.826 |
106 | 38.088 | 12.744 |
107 | 22.892 | 30.522 |
108 | 14.917 | 13.713 |
109 | 6.995 | 30.065 |
110 | 40.466 | 30.376 |
111 | 41.964 | 19.47 |
112 | 30.3 | 14.954 |
113 | 19.953 | 27.768 |
114 | 27.316 | 2.155 |
115 | 35.983 | 36.293 |
116 | 44.374 | 28.853 |
117 | 22.621 | 41.026 |
118 | 31.99 | 20.775 |
119 | 11.309 | 39.88 |
120 | 41.379 | 24.172 |
121 | 31.589 | 26.996 |
122 | 18.031 | 8.178 |
123 | 18.549 | 22.682 |
124 | 31.64 | 23.551 |
125 | 33.452 | 23.294 |
To compare the proposed initial solution (GOT initial solution) with random initial solution, a set of test problems
The value of cost function in GOT initial solution and the best value of random placements.
|
GOT initial solution | The best random initial solution |
---|---|---|
10 | 1945 | 2437.3 |
11 | 2122 | 2758.5 |
12 | 2368 | 3531.7 |
13 | 4919 | 5764 |
14 | 5514 | 7024 |
15 | 5538 | 7449 |
16 | 8116 | 10287.7 |
17 | 8468 | 12207.5 |
18 | 10648 | 15304 |
19 | 13943 | 19119.3 |
20 | 14265 | 20094 |
21 | 17199 | 22491.2 |
22 | 16610 | 26370 |
23 | 22128 | 30749.2 |
24 | 25533 | 35508 |
25 | 27585 | 37968.4 |
26 | 29929 | 43803.6 |
27 | 36255 | 51850 |
28 | 41483 | 57050.5 |
29 | 47543 | 65109.2 |
30 | 57830 | 73705.8 |
31 | 61408 | 82058.4 |
32 | 63687 | 86436 |
33 | 59970 | 94066.4 |
34 | 82721 | 110195.2 |
35 | 76220 | 104940 |
36 | 92426 | 124186.1 |
37 | 95386 | 125468 |
38 | 87532 | 150060 |
39 | 93708 | 146793.8 |
40 | 118266 | 180538 |
41 | 141363 | 183911 |
42 | 104263 | 195458.7 |
43 | 152188 | 204207.3 |
44 | 163529 | 239253 |
45 | 166360 | 237051.8 |
46 | 167002 | 251049.3 |
47 | 189027 | 290085 |
48 | 226097 | 305900 |
49 | 234811 | 324500.3 |
50 | 238855 | 337198.9 |
51 | 264842 | 358480.5 |
52 | 251009 | 358666.7 |
53 | 233805 | 370262.4 |
54 | 298806 | 404104.4 |
55 | 284543 | 431091.9 |
56 | 344755 | 481656.9 |
57 | 371962 | 493986.4 |
58 | 344044 | 510976 |
59 | 375321 | 503358.9 |
60 | 373118 | 538554.9 |
61 | 370817 | 573555.9 |
62 | 457275 | 628602.9 |
63 | 544350 | 688184.6 |
64 | 530408 | 708518.3 |
65 | 502322 | 690275.2 |
66 | 526230 | 741481.1 |
67 | 556568 | 773325.9 |
68 | 639735 | 867242.4 |
69 | 578533 | 860229.8 |
70 | 643592 | 914437.3 |
71 | 589557 | 893978.8 |
72 | 670866 | 950592.5 |
73 | 736669 | 1058290.5 |
74 | 660395 | 1039023 |
75 | 749795 | 1115704.6 |
76 | 835418 | 1180300 |
77 | 727648 | 1157821.9 |
78 | 852689 | 1147676 |
79 | 971135 | 1273329.8 |
80 | 920522 | 1294614.8 |
81 | 882645 | 1420567 |
82 | 1084711 | 1500037.8 |
83 | 1072241 | 1436365.1 |
84 | 1072132 | 1606669.3 |
85 | 1154018 | 1552139.2 |
86 | 1150925 | 1756302.3 |
87 | 1164220 | 1608156.3 |
88 | 1230008 | 1726587.4 |
89 | 1379479 | 1913688.3 |
90 | 1351210 | 1970899.3 |
91 | 1275975 | 1895126.4 |
92 | 1360771 | 2114230 |
93 | 1520228 | 1969614.1 |
94 | 1542740 | 2125200 |
95 | 1581145 | 2230222.7 |
96 | 1640792 | 2305569.3 |
97 | 1486796 | 2362639.4 |
98 | 1645889 | 2365109.3 |
99 | 1607866 | 2490733.7 |
100 | 2073979 | 2635666.4 |
Comparisons of cost function value in GOT initial solution with the best value of random placements.
Cost reduction by using GOT.
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 [
The approach has been tested on five instances from the literature. Table
Summary of the results.
Problem | Number of facilities | Best result by other methods | GOT | Cost reduction |
---|---|---|---|---|
#1 | 8 | 692.5 | 752.7 | −60.2 |
#2 | 20 | 1157 | 1302 | −145 |
#3 | 50 | 78224.7 | 76882.3 | 1342.4 |
#4 | 100 | 538193.1 | 527094.1 | 11099 |
#5 | 125 | 1084451 | 1062080 | 22371 |
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 [
The authors declare that they have no conflict of interests.