obtaining an initial solution for facility layout problem
TRANSCRIPT
Hindawi Publishing CorporationJournal of Industrial MathematicsVolume 2013 Article ID 130251 10 pageshttpdxdoiorg1011552013130251
Research ArticleObtaining an Initial Solution for Facility Layout Problem
Ali Shoja Sangchooli and Mohammad Reza Akbari Jokar
Department of Industrial Engineering Sharif University of Technology Tehran 1458875346 Iran
Correspondence should be addressed to Ali Shoja Sangchooli a shojasangchooliiesharifedu
Received 13 April 2013 Accepted 29 August 2013
Academic Editor Ting Chen
Copyright copy 2013 A Shoja Sangchooli and M R Akbari Jokar This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited
The facility layout approaches can generally be classified into two groups constructive approaches and improvement approachesAll improvement procedures require an initial solution which has a significant impact on final solution In this paper we introducea new technique for accruing an initial placement of facilities on extended plane It is obtained by graph theoretic facility layoutapproaches and graph drawing algorithms To evaluate the performance this initial solution is applied to rectangular facility layoutproblem The solution is improved using an analytical method The approach is then tested on five instances from the literatureTest problems include three large size problems of 50 100 and 125 facilities The results demonstrate effectiveness of the techniqueespecially for large size problems
1 Introduction
The facility layout problem seeks the best positions of facil-ities to optimize some objective The common objective isto reduce material handling costs between the facilities Theproblem has been modeled by a variety of approaches Adetailed review of the different problem formulations can befound in Singh and Sharma [1] The facility layout problem isan optimization problemwhich arises in a variety of problemssuch as placing machines on a factory floor VLSI design andlayout design of hospitals schools
The facility layout approaches can generally be classifiedinto two groups constructive methods and improvementmethods In this paper we consider the placement of facilitieson an extended plane Many improvement approaches havebeen proposed for this problem All improvement proceduresrequire an initial solution Some approaches start from agood but infeasible solution [2ndash4] These models contain apenalty component in their objective function Hence theseapproaches minimize objective function value for feasiblesolutions But some approaches require a feasible initialsolution These approaches use a randomly generated initialsolution [5 6] Mir and Imam [7] have proposed simulatedannealing for a better initial solution They have shown
that a good initial solution has a significant impact on finalsolution
In this paper we introduce a new technique for accruingan initial placement of facilities on an extended plane Thetechnique consists of two stages In the first stage a maximalplanar graph (MPG) is obtained In the second stage thevertices of MPG are drawn on the plane by graph drawingalgorithmsThen vertices are replaced by facilities Hence aninitial solution is obtained
In an MPG the facilities with larger flows are adjacenttogether Hence drawing the MPG on the plane can be agood idea for obtaining an initial solution To evaluate theperformance of the idea this initial solution is applied inrectangular facility layout problemThe solution is improvedby an analytical method by Mir and Imam [7] The approachis then tested on five instances from the literature
The remaining parts of the paper are organized as followsThe next section describes the formulation of the facilitylayout problem chosen for our work Section 3 describesaccruing an initial placement In Section 4 the analyticalmethod is described and the approach is compared to otherapproaches in the literature In Section 5 the proposed initialsolution is compared with random initial solution FinallySection 6 provides a summary and conclusion
2 Journal of Industrial Mathematics
Exterior
F1
F2
F4
F3
F5
F6
Figure 1 An MPG (solid lines) and its correspondent block layout(dashed lines)
1
7
8
5
2
3
4
6
Figure 2 An example of straight line drawing
Step 2
Step 1 Encapsulate facilities
Obtain an MPG
Obtain an initial solution
Improve initial solutionStep 4
Step 3
Figure 3 Steps of the proposed approach
2 Problem Formulation
In this paper we label the facilities 1 2 119873 where 119873 isthe total number of facilities Facilities are assumed to berectangles with fixed shape The notation is given as follows
(119909119894 119910119894) coordinates of the center of facility 119894
119871119894length of facility 119894119882119894width of facility 119894119891119894119895the total cost of flow per unit distance between two
facilities 119894 and 119895
Table 1 Results for test problem 1
Program Cost function valueTOPOPT (Imam and Mir 1989) [5] 794VIP-PLANOPT (2006) [8] 692GOT 7527
Table 2 Results for test problem 2
Program Best designTopopt (Mir and Imam 1989) [5] 132072FLOAT (Imam and Mir 1993) [6] 126494HOT (Imam and Mir 2001) [7] 122540VIP-PLANOPT (2006) [8] 1157GOT 1302
Table 3 Results for test problem 3
Program Cost function valueHOT (Mir and Imam 2001) [7] 8079424VIP-PLANOPT (2006) [8] 782247GOT 768823
Table 4 Results for test problem 4
Program Cost function valueHOT (Mir and Imam 2001) [7] 5585562VIP-PLANOPT (2006) [8] 5381931GOT 5270941
119889119894119895distance between the centers of the facilities 119894 and 119895119889119894119895could be one of the following three distance norms
(1) Euclidean distance
119889119894119895= ((119909
119894minus 119909119895)2
+ (119910119894minus 119910119895)2
)
12
(1)
(2) Squared Euclidean distance
119889119894119895= (119909119894minus 119909119895)2
+ (119910119894minus 119910119895)2
(2)
(3) Rectilinear distance
119889119894119895=10038161003816100381610038161003816119910119894minus 119910119895
10038161003816100381610038161003816+10038161003816100381610038161003816119909119894minus 119909119895
10038161003816100381610038161003816 (3)
The requirement for problem is that the facilities must notoverlap each other The area of overlap is defined as follows
119860119894119895= 120582119894119895(Δ119883119894119895) (Δ119884
119894119895) (4)
where
Δ119883119894119895=
119871119894+ 119871119895
2minus10038161003816100381610038161003816119909119894minus 119909119895
10038161003816100381610038161003816
Δ119884119894119895=
119882119894+119882119895
2minus10038161003816100381610038161003816119910119894minus 119910119895
10038161003816100381610038161003816
120582119894119895= 1 for Δ119883
119894119895ge 0 Δ119884
119894119895ge 0
0 otherwise
(5)
Journal of Industrial Mathematics 3
Table 5 For test problem 4 the coordinates of facilities obtainedbyGOTare given belowThe value of the cost function for this layoutis 5270941
Facility 119883 119884
1 17599 221582 18514 329333 29485 186344 35526 205825 14798 22656 26425 253887 12213 132828 19435 274669 32119 1531910 24743 886211 16231 271112 25533 1762113 14274 2035214 22791 388915 14467 1481216 38012 169417 9735 1127618 24609 2766319 20685 2057820 19866 2257921 15832 842122 8866 3149523 22402 1733524 35551 912525 1695 1937826 36104 253127 16328 2184328 13208 2616129 22439 1473230 14052 1306531 2821 2193432 16436 1228233 24467 1136534 23103 3248935 23283 2496836 19444 1470837 32927 1669938 11708 2885739 1514 2979340 28164 517441 3208 847642 29636 1304943 26373 3112744 8693 1578545 1605 2685946 31965 2609947 29361 958648 33822 12885
Table 5 Continued
Facility 119883 119884
49 19394 2504350 25902 2195451 4234 1653452 19539 3021153 5587 1180154 13399 3283155 28771 1607156 30974 2277457 22453 2211758 1235 2313859 21725 1894460 30419 3185461 24388 1889762 19587 3697463 19943 1283964 24632 1501965 5951 3223366 16918 1434667 23818 640868 19581 1844769 35191 3126370 0927 2134771 32644 1946472 9487 2032573 39749 2178874 18485 162775 15581 3634476 6345 2020377 13484 232178 18527 2073479 11773 2029580 19329 266381 4471 227982 4788 2796183 2751 3661284 26306 1577685 4545 2525786 23545 3589287 10746 3771288 7405 665689 12021 668490 2914 2712391 23367 2985692 1487 1662993 11961 1713794 16126 183495 22426 1607596 19729 1075297 19707 76398 23824 2151199 8257 27817100 9179 24151
4 Journal of Industrial Mathematics
Table 6 Results for test problem 5
Program Cost function valueVIP-PLANOPT (2006) [8] 1084451GOT 1062080
The value of overlap area 119860119894119895is a nonnegative number 119860
119894119895
will be zero only if there is no overlapping between facilities119894 and 119895 The objective is to minimize material handling costsSo the problem can be stated as follows
minimizing cost =119873minus1
sum
119894=1
119873
sum
119895=119894+1
119891119894119895119889119894119895
subject to 119860119894119895= 0 forall119894 119895 119894 lt 119895
(6)
The constraint ensures that facilities do not overlap A similarformulation also can be found in [7]
3 Obtaining an Initial Solution
The initial solution is obtained by graph theoretic facilitylayout approaches (GTFLP) and graph drawing algorithmsThe following subsection describes obtaining an MPGSection 32 describes the drawing of the MPG on the plane
31 Generating aMaximal Planar Graph In GTFLP facilitiesare represented by vertices and flow (adjacency desirability)between them is represented by weighted edges Createdgraph is called adjacency graph Graph theory is particularlyuseful for the facility layout problems because graphs easilyenable us to capture the adjacency information andmodel theproblem A review of graph theory applications to the facilitylayout problem can be found in [9 10] GTFLP consists oftwo stages At the first stage the adjacency graph is convertedto a maximal planar graph (MPG) In the second stage ablock layout is constructed from the MPG The second stageis not our concern here For more details we refer to [11ndash15]Figure 1 shows an MPG and its correspondent block layout
Many heuristic and metaheuristic methods for obtainingan MPG have been suggested [16ndash21] In this paper we usefrom the greedy heuristic [16] It is conceptually simple andcreates high weighted MPGs [16] This heuristic has a simpleinstruction the edges are sorted in nonincreasing order ofweight Each edge is tested in turn and accepted as part of theMPG unless it makes the graph nonplanar So the heuristicneeds planarity testing Boyer and Myrvold [22] developeda simplified 119874(119899) planarity testing algorithm We use thisalgorithm for planarity testing In the worst case119874(1198992) edgesare considered and for each edge the Boyer andMyrvold testis calledHence the approach results in a complexity of119874(1198993)
32 Drawing Maximal Planar Graph on the Plane Graphdrawing as a branch of graph theory applies topologyand geometry to derive two-dimensional representations ofgraphs A graph drawing algorithm reads as input a com-binatorial description of a graph G and produces as output
Table 7 For test problem 5 the coordinates of facilities obtainedby GOT are given below The layout cost is 1062080
Facility 119883 119884
1 28617 195132 25371 220823 27683 320884 6488 135315 1753 343576 10131 169297 29906 252268 28243 364739 14372 3129810 37502 2229911 32586 484612 19349 1324913 3314 2358814 4696 1935415 21953 853916 3413 2553717 33636 32818 3721 1756419 12603 2371320 21763 2142821 16261 3728322 27453 2518523 25213 153624 27111 4184425 21208 2495926 33287 3847127 25882 1912528 4387 1414729 27342 626530 2831 26331 18328 3143232 3833 776133 22131 2703834 7166 267735 18909 4630236 15123 3438337 28974 2260238 18728 465639 33526 865740 12176 3231841 16539 3973242 28383 3948543 1509 2389944 32571 1255945 41221 407846 24335 2784347 2516 306248 37452 25455
Journal of Industrial Mathematics 5
Table 7 Continued
Facility 119883 119884
49 40189 3629450 12672 3682551 34282 2734952 19126 1500553 36395 4132254 31473 4003455 30991 3498356 38702 270357 28176 127358 29219 4521859 25113 1076660 11021 2702961 22158 1423962 22543 1758263 12348 2132264 13684 439665 8569 3614966 8828 3323967 10933 1237368 24203 2604769 12647 707170 29789 2736271 6492 3864272 18394 1799373 4815 1775474 24465 4564675 24704 2444876 48912 3116477 19063 1108878 27843 1637979 35271 2921780 21659 3542981 15303 2808782 32599 310783 3311 2083684 4402 337685 43843 986286 16694 2389987 20628 172388 19088 3556389 48998 2530890 33559 4617691 31162 308692 2482 3660793 44673 2438194 18385 2769495 30086 1085896 33289 1515697 30399 1718898 1552 1907199 36518 32041100 19439 4084101 22932 4334
Table 7 Continued
Facility 119883 119884
102 27279 28155103 31849 19046104 1148 19243105 731 21826106 38088 12744107 22892 30522108 14917 13713109 6995 30065110 40466 30376111 41964 1947112 303 14954113 19953 27768114 27316 2155115 35983 36293116 44374 28853117 22621 41026118 3199 20775119 11309 3988120 41379 24172121 31589 26996122 18031 8178123 18549 22682124 3164 23551125 33452 23294
a drawing of G A graph has infinitely many different draw-ings For a review of various graphs drawing algorithms referto [23] We use algorithm of Chrobak and Payne [24] to forma straight line drawing of the MPG In such a drawing eachedge is drawn using a straight line segment The algorithmdraws vertices in an MPG to integer coordinates in a (2119873 minus4) times (119873 minus 2) grid Figure 2 shows an example of straight linedrawing
For acquiring an initial solution each vertex is replacedby its correspondent facility In a feasible solution facilitieshave no overlaps For this reason the coordinates of facilitiescan be multiplied by maximum dimensions of all facili-ties (width and length) This operation increases distancebetween facilities andmakes the solution feasible For the caseof circular facilities the diameter of circle can be consideredas maximum dimensions
4 Improving Initial Solution and Comparing
To evaluate the performance the initial solution is improvedby an analytical method by Mir and Imam [7] In thismethod the convergence is controlled by carrying out theoptimization using concept of ldquomagnified envelop blocksrdquoThe dimensions of the blocks are determined by multiplyingthe dimensions of the facilities with a ldquomagnification factorrdquoThe optimization is then carried out for these envelopblocks rather than the actual facilitiesThe analytical methodsearches the optimum placements of each envelop block in
6 Journal of Industrial Mathematics
Table 8 The value of cost function in GOT initial solution and thebest value of random placements
119899 GOT initial solution The best random initial solution10 1945 2437311 2122 2758512 2368 3531713 4919 576414 5514 702415 5538 744916 8116 10287717 8468 12207518 10648 1530419 13943 19119320 14265 2009421 17199 22491222 16610 2637023 22128 30749224 25533 3550825 27585 37968426 29929 43803627 36255 5185028 41483 57050529 47543 65109230 57830 73705831 61408 82058432 63687 8643633 59970 94066434 82721 110195235 76220 10494036 92426 124186137 95386 12546838 87532 15006039 93708 146793840 118266 18053841 141363 18391142 104263 195458743 152188 204207344 163529 23925345 166360 237051846 167002 251049347 189027 29008548 226097 30590049 234811 324500350 238855 337198951 264842 358480552 251009 358666753 233805 370262454 298806 404104455 284543 431091956 344755 481656957 371962 4939864
Table 8 Continued
119899 GOT initial solution The best random initial solution58 344044 51097659 375321 503358960 373118 538554961 370817 573555962 457275 628602963 544350 688184664 530408 708518365 502322 690275266 526230 741481167 556568 773325968 639735 867242469 578533 860229870 643592 914437371 589557 893978872 670866 950592573 736669 1058290574 660395 103902375 749795 1115704676 835418 118030077 727648 1157821978 852689 114767679 971135 1273329880 920522 1294614881 882645 142056782 1084711 1500037883 1072241 1436365184 1072132 1606669385 1154018 1552139286 1150925 1756302387 1164220 1608156388 1230008 1726587489 1379479 1913688390 1351210 1970899391 1275975 1895126492 1360771 211423093 1520228 1969614194 1542740 212520095 1581145 2230222796 1640792 2305569397 1486796 2362639498 1645889 2365109399 1607866 24907337100 2073979 26356664
the direction of steepest descent which is opposite to thegradient direction The sizes of the envelop blocks are thenreduced and the optimization process is repeated for thesecond phase The number of optimization phases is equalto the magnification factor number for the envelop blocksIn the last optimization phase the dimensions of the envelopblocks become equal to the actual facilities For more detailwe refer to Mir and Imam [7]
Journal of Industrial Mathematics 7
Table 9 Summary of the results
Problem Number of facilities Best result by other methods GOT Cost reduction1 8 6925 7527 minus6022 20 1157 1302 minus1453 50 782247 768823 134244 100 5381931 5270941 110995 125 1084451 1062080 22371
So the proposed approach for solving a facility layoutproblem can be summarized as follows
Step 1 encapsulating facilities in envelop blocks (mul-tiplying the dimensions of facilities by amagnificationfactor)
Step 2 obtaining an MPG
Step 3 drawing the MPG on the plane and obtainingan initial solution
Step 4 improving initial solution by analyticalmethod
Figure 3 shows summary of these stepsThe proposed approach was coded using the VBNET
programming language in a program named GOT (Graphoptimization technique) Five test problems were run For alltest problems results were obtained on a PCwith Intel T5470processor The results were compared with the previouslypublished papers and commercial software VIP-PLANOPT2006 VIP-PLANOPT is a useful layout software package thatcan generate near-optimal layout [25] Formore details aboutVIP-PLANOPT see Engineering Optimization Software [8]VIP-PLANOPT results were obtained from the softwareuserrsquos manual The results are presented in the followingsections
41 Test Problem 1 This problem of 8 facilities was intro-duced by Imam and Mir [5] Figure 4 shows the steps foraccruing the initial solution Figure 4(a) shows the flowmatrix and dimension of facilities All dimensions and costmatrix elements are integer-valued numbers ranging between1 and 6 There are several pairs of facilities with no flowbetween them Distance norm is squared Euclidean Thegreedy heuristic generates the edges lists of MPG as shownin Figure 4(b)
The straight line drawing algorithm gives the coordinatesof vertices The drawing is shown in Figure 4(c) Then eachvertex is replaced by its correspondent facility The coordi-nates are multiplied by maximum dimensions of facilities(width and length) and finally the initial layout design isshown in Figure 4(d)
The solution is improved by the analytical techniqueFigure 5 shows the final layout The cost function value forthis layout is 7527 and the running time is 04 second Table 1shows the results obtained by the other approaches Thebest solution for this problem is obtained by VIP-PLANOPT2006
42 Test Problem 2 This problem of 20 unequal areafacilities was introduced by Imam and Mir [6] The dataconsist of only integer valuesThe dimensions of the facilitiesare between 1 and 3 The elements of the cost matrix areintegers between 0 and 5 The distance norm is rectilinearThe final layout obtained by GOT is shown in Figure 6 Thelayout cost is 1302 and the running time is 06 secondTable 2 compares the results obtained byGOTwith the resultsavailable in the literature VIP-PLANOPT2006 has the lowestvalue of the cost function
43 Test Problem 3 This is a problem of 50 facilities ran-domly generated by VIP-PLANOPT 2006 The dimensionsof the facilities are decimal numbers between 1 to 6 Theelements of the cost matrix are all integers between 1 and10 The distance norm is Euclidean The results are shownin Table 3 The best published result has a cost of 782247whereas GOT produces a final layout with a cost of 768823only in 151 seconds Figure 7 shows the final layout
44 Test Problem 4 This is a randomly generated large sizeproblem of 100 facilities The dimensions of the facilities aredecimal numbers between 1 and 6 The cost matrix elementsare integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 4 GOT obtained the costfunction value of 5270941 in 743 secondsThis value is about2 below the cost function value of VIP-PLANOPT 2006The coordinates of the facilities for the layout obtained byGOT are given in Table 5
45 Test Problem 5 This is a large size problem of 125facilities randomly generated by VIP-PLANOPT 2006 Thedimensions of facilities are real numbers between 1 and 6 andelements of the cost matrix are integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 6GOT obtained the cost function value of 1062080 in 1296secondsThis value is about 2 below the cost function valueof VIP-PLANOPT 2006 The coordinates of the facilities forthe layout obtained by GOT are given in Table 7
5 Comparing GOT Initial Solution withRandom Initial Solution
To compare the proposed initial solution (GOT initial solu-tion) with random initial solution a set of test problems (119899 =10 11 12 100) were generated The facilities dimensionswere 1 times 1 and flow matrices were randomly generatedbetween 0 and 10 For acquiring a random initial solution
8 Journal of Industrial Mathematics
Dimensions of the facilities Flow matrix
1
1
1
1 1
1
2
2
2
2
2
2
2 2
2
2
2
2
20
0
0
0
0
0
0
0
0 0
0
0 0
0
0
0 0
0 0 0
0
0
00
3
3
3
3
4
4
4
4
4
4
4
5
5
5
6
6
6
7 8
1
2
2
2
2
23
3
3
3
3
4
4
4
4
44
44
5
5
6
7
8
1
1
2
3 3
4
5
6
7
8
Length Width
(a) The raw facilities layout data
MPGalgorithm
1 5-2
3-2
7-6
8-5
6-3
4-2
3-1
4-3
5-42
3
4
5
6
7
8
9
Number10
11
12
13
14
15
16
17
18
NumberEdge6-4
8-2
8-4
2-1
7-3
8-7
7-2
7-4
7-1
Edge
(b) MPG
Graph drawing algorithm
1
7
8
5
2
3
4
6
(c) Straight line drawing of the MPG
Obtaining initial solution
1
7
8
5
2
3
4
6
(d) Initial solution
Figure 4 Obtaining an initial solution for test problem 1
1
7
85
2
34
6
Figure 5 Final layout for test problem 1
facilities were randomly placed in a (2119899 minus 4) times (119899 minus 2) integergrid For each test problem 20 random placements werefound Table 8 shows the value of cost function in GOT initialsolution and the best value found by random placementsFigures 8 and 9 shows these results graphically The resultsdemonstrated significant improvement in cost function
6 Summary and Conclusion
An initial solution has been presented for the layout designof facilities on a continuous plane The technique consists of
8 4 5
14 11 18 19 3
15
20
13
717
1 2 612
10
9
16
Figure 6 Final layout for test problem 2
two stages In the first stage amaximal planar graph (MPG) isobtained In the second stage the vertices of MPG are drawnon the plane by graph drawing algorithms Then vertices arereplaced by facilities Hence an initial solution is obtainedTo evaluate the performance this initial solution has beenapplied in rectangular facility layout problem and improvedby an analytical method by Mir and Imam [7]
The approach has been tested on five instances fromthe literature Table 6 shows the Summary of the resultsand Figure 8 shows the cost reduction by the technique Forthe large size problems involving 50 100 and 125 facilities
Journal of Industrial Mathematics 9
6 41 22
29
1224 44
4645
1425
5
26
28
10
9
3
2
8
4
30
31
43
47
49
50
40
42
48
36
34
3538
39
33
32
13
18
37
15 17
27 23
20
16
71 19
11
21
Figure 7 Final layout for test problem 3
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Random initial solutionGOT initial solution
Cos
t
05
1
15
2
25
3
35
4
45
5
times106
n
Figure 8 Comparisons of cost function value in GOT initialsolution with the best value of random placements
the layout costs values are better than those obtained by thepreviously published techniques As shown in Table 9 theresults demonstrate effectiveness of the technique especiallyfor large size problems
This paper introduced a simple technique for obtaining agood initial solutionThe technique with somemodificationcan be applied in facility layout approaches that use arandomly generated initial solution In future researches itwould be interesting to analyze the influence of MPG andgraph drawing algorithm on the solution The results can befurther improved by using a metaheuristic such as GRASP[21] and Tabu search [20] for generating a high weightedMPG
0
5
10
15
20
25
8 20 50 100 125
Reduction
times103
minus5
Figure 9 Cost reduction by using GOT
Conflict of Interests
The authors declare that they have no conflict of interests
References
[1] S P Singh and R R K Sharma ldquoA review of differentapproaches to the facility layout problemsrdquo International Jour-nal of Advanced Manufacturing Technology vol 30 no 5-6 pp425ndash433 2006
[2] M F Anjos and A Vannelli ldquoAn attractor-repeller approach tofloorplanningrdquo Mathematical Methods of Operations Researchvol 56 no 1 pp 3ndash27 2002
[3] I Castillo and T Sim ldquoA spring-embedding approach for thefacility layout problemrdquo Journal of the Operational ResearchSociety vol 55 no 1 pp 73ndash81 2004
[4] Z Drezner ldquoDISCON a new method for the layout problemrdquoOperations Research vol 28 no 6 pp 1375ndash1384 1980
[5] M H Imam and M Mir ldquoNonlinear programming approachto automated topology optimizationrdquo Computer-Aided Designvol 21 no 2 pp 107ndash115 1989
[6] M H Imam and M Mir ldquoAutomated layout of facilities ofunequal areasrdquo Computers and Industrial Engineering vol 24no 3 pp 355ndash366 1993
[7] M Mir and M H Imam ldquoHybrid optimization approachfor layout design of unequal-area facilitiesrdquo Computers andIndustrial Engineering vol 39 no 1-2 pp 49ndash63 2001
[8] Engineering Optimization Software VIP-PLANOPT 20062010 httpwwwplanoptcom
[9] L Foulds Graph Theory Applications Springer New York NYUSA 1992
[10] M M D Hassan and G L Hogg ldquoA review of graph theoryapplication to the facilities layout problemrdquo Omega vol 15 no4 pp 291ndash300 1987
[11] M M D Hassan and G L Hogg ldquoOn converting a dualgraph into a block layoutrdquo International Journal of ProductionResearch vol 27 no 7 pp 1149ndash1160 1989
[12] KHWatson and JWGiffin ldquoThevertex splitting algorithm forfacilities layoutrdquo International Journal of Production Researchvol 35 no 9 pp 2477ndash2492 1997
[13] S A Irvine and I Rinsma-Melchert ldquoA new approach tothe block layout problemrdquo International Journal of ProductionResearch vol 35 no 8 pp 2359ndash2376 1997
10 Journal of Industrial Mathematics
[14] P S Welgama P R Gibson and L A R Al-Hakim ldquoFacilitieslayout a knowledge-based approach for converting a dualgraph into a block layoutrdquo International Journal of ProductionEconomics vol 33 no 1ndash3 pp 17ndash30 1994
[15] M A Jokar and A S Sangchooli ldquoConstructing a blocklayout by face areardquo The International Journal of AdvancedManufacturing Technology vol 54 no 5ndash8 pp 801ndash809 2011
[16] L R Foulds P B Gibbons and J W Giffin ldquoFacilities layoutadjacency determination an experimental comparison of threegraph theoretic heuristicsrdquo Operations Research vol 33 no 5pp 1091ndash1106 1985
[17] E G John and J Hammond ldquoMaximally weighted graphtheoretic facilities design planningrdquo International Journal ofProduction Research vol 38 no 16 pp 3845ndash3859 2000
[18] S G Boswell ldquoTESSAmdasha new greedy heuristic for facilitieslayout planningrdquo International Journal of Production Researchvol 30 no 8 pp 1957ndash1968 1992
[19] L R Foulds and D F Robinson ldquoGraph theoretic heuristics forthe plant layout problemrdquo International Journal of ProductionResearch vol 16 no 1 pp 27ndash37 1978
[20] I H Osman ldquoA tabu search procedure based on a randomRoulette diversification for the weighted maximal planar graphproblemrdquoComputers and Operations Research vol 33 no 9 pp2526ndash2546 2006
[21] I H Osman B Al-Ayoubi and M Barake ldquoA greedy randomadaptive search procedure for the weighted maximal planargraph problemrdquo Computers and Industrial Engineering vol 45no 4 pp 635ndash651 2003
[22] J M Boyer and W J Myrvold ldquoOn the cutting edge simplifiedO(n) planarity by edge additionrdquo Journal of Graph Algorithmsand Applications vol 8 no 3 pp 241ndash273 2004
[23] G Di Battista P Eades R Tamassia and I Tollis GraphDrawing Algorithms for the Visualization of Graphs PrenticeHall PTR Upper Saddle River NJ USA 1998
[24] M Chrobak and T H Payne ldquoA linear-time algorithm fordrawing a planar graph on a gridrdquo Information ProcessingLetters vol 54 no 4 pp 241ndash246 1995
[25] S Heragu Facilities Design Iuniverse Inc 2006
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Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Industrial Mathematics
Exterior
F1
F2
F4
F3
F5
F6
Figure 1 An MPG (solid lines) and its correspondent block layout(dashed lines)
1
7
8
5
2
3
4
6
Figure 2 An example of straight line drawing
Step 2
Step 1 Encapsulate facilities
Obtain an MPG
Obtain an initial solution
Improve initial solutionStep 4
Step 3
Figure 3 Steps of the proposed approach
2 Problem Formulation
In this paper we label the facilities 1 2 119873 where 119873 isthe total number of facilities Facilities are assumed to berectangles with fixed shape The notation is given as follows
(119909119894 119910119894) coordinates of the center of facility 119894
119871119894length of facility 119894119882119894width of facility 119894119891119894119895the total cost of flow per unit distance between two
facilities 119894 and 119895
Table 1 Results for test problem 1
Program Cost function valueTOPOPT (Imam and Mir 1989) [5] 794VIP-PLANOPT (2006) [8] 692GOT 7527
Table 2 Results for test problem 2
Program Best designTopopt (Mir and Imam 1989) [5] 132072FLOAT (Imam and Mir 1993) [6] 126494HOT (Imam and Mir 2001) [7] 122540VIP-PLANOPT (2006) [8] 1157GOT 1302
Table 3 Results for test problem 3
Program Cost function valueHOT (Mir and Imam 2001) [7] 8079424VIP-PLANOPT (2006) [8] 782247GOT 768823
Table 4 Results for test problem 4
Program Cost function valueHOT (Mir and Imam 2001) [7] 5585562VIP-PLANOPT (2006) [8] 5381931GOT 5270941
119889119894119895distance between the centers of the facilities 119894 and 119895119889119894119895could be one of the following three distance norms
(1) Euclidean distance
119889119894119895= ((119909
119894minus 119909119895)2
+ (119910119894minus 119910119895)2
)
12
(1)
(2) Squared Euclidean distance
119889119894119895= (119909119894minus 119909119895)2
+ (119910119894minus 119910119895)2
(2)
(3) Rectilinear distance
119889119894119895=10038161003816100381610038161003816119910119894minus 119910119895
10038161003816100381610038161003816+10038161003816100381610038161003816119909119894minus 119909119895
10038161003816100381610038161003816 (3)
The requirement for problem is that the facilities must notoverlap each other The area of overlap is defined as follows
119860119894119895= 120582119894119895(Δ119883119894119895) (Δ119884
119894119895) (4)
where
Δ119883119894119895=
119871119894+ 119871119895
2minus10038161003816100381610038161003816119909119894minus 119909119895
10038161003816100381610038161003816
Δ119884119894119895=
119882119894+119882119895
2minus10038161003816100381610038161003816119910119894minus 119910119895
10038161003816100381610038161003816
120582119894119895= 1 for Δ119883
119894119895ge 0 Δ119884
119894119895ge 0
0 otherwise
(5)
Journal of Industrial Mathematics 3
Table 5 For test problem 4 the coordinates of facilities obtainedbyGOTare given belowThe value of the cost function for this layoutis 5270941
Facility 119883 119884
1 17599 221582 18514 329333 29485 186344 35526 205825 14798 22656 26425 253887 12213 132828 19435 274669 32119 1531910 24743 886211 16231 271112 25533 1762113 14274 2035214 22791 388915 14467 1481216 38012 169417 9735 1127618 24609 2766319 20685 2057820 19866 2257921 15832 842122 8866 3149523 22402 1733524 35551 912525 1695 1937826 36104 253127 16328 2184328 13208 2616129 22439 1473230 14052 1306531 2821 2193432 16436 1228233 24467 1136534 23103 3248935 23283 2496836 19444 1470837 32927 1669938 11708 2885739 1514 2979340 28164 517441 3208 847642 29636 1304943 26373 3112744 8693 1578545 1605 2685946 31965 2609947 29361 958648 33822 12885
Table 5 Continued
Facility 119883 119884
49 19394 2504350 25902 2195451 4234 1653452 19539 3021153 5587 1180154 13399 3283155 28771 1607156 30974 2277457 22453 2211758 1235 2313859 21725 1894460 30419 3185461 24388 1889762 19587 3697463 19943 1283964 24632 1501965 5951 3223366 16918 1434667 23818 640868 19581 1844769 35191 3126370 0927 2134771 32644 1946472 9487 2032573 39749 2178874 18485 162775 15581 3634476 6345 2020377 13484 232178 18527 2073479 11773 2029580 19329 266381 4471 227982 4788 2796183 2751 3661284 26306 1577685 4545 2525786 23545 3589287 10746 3771288 7405 665689 12021 668490 2914 2712391 23367 2985692 1487 1662993 11961 1713794 16126 183495 22426 1607596 19729 1075297 19707 76398 23824 2151199 8257 27817100 9179 24151
4 Journal of Industrial Mathematics
Table 6 Results for test problem 5
Program Cost function valueVIP-PLANOPT (2006) [8] 1084451GOT 1062080
The value of overlap area 119860119894119895is a nonnegative number 119860
119894119895
will be zero only if there is no overlapping between facilities119894 and 119895 The objective is to minimize material handling costsSo the problem can be stated as follows
minimizing cost =119873minus1
sum
119894=1
119873
sum
119895=119894+1
119891119894119895119889119894119895
subject to 119860119894119895= 0 forall119894 119895 119894 lt 119895
(6)
The constraint ensures that facilities do not overlap A similarformulation also can be found in [7]
3 Obtaining an Initial Solution
The initial solution is obtained by graph theoretic facilitylayout approaches (GTFLP) and graph drawing algorithmsThe following subsection describes obtaining an MPGSection 32 describes the drawing of the MPG on the plane
31 Generating aMaximal Planar Graph In GTFLP facilitiesare represented by vertices and flow (adjacency desirability)between them is represented by weighted edges Createdgraph is called adjacency graph Graph theory is particularlyuseful for the facility layout problems because graphs easilyenable us to capture the adjacency information andmodel theproblem A review of graph theory applications to the facilitylayout problem can be found in [9 10] GTFLP consists oftwo stages At the first stage the adjacency graph is convertedto a maximal planar graph (MPG) In the second stage ablock layout is constructed from the MPG The second stageis not our concern here For more details we refer to [11ndash15]Figure 1 shows an MPG and its correspondent block layout
Many heuristic and metaheuristic methods for obtainingan MPG have been suggested [16ndash21] In this paper we usefrom the greedy heuristic [16] It is conceptually simple andcreates high weighted MPGs [16] This heuristic has a simpleinstruction the edges are sorted in nonincreasing order ofweight Each edge is tested in turn and accepted as part of theMPG unless it makes the graph nonplanar So the heuristicneeds planarity testing Boyer and Myrvold [22] developeda simplified 119874(119899) planarity testing algorithm We use thisalgorithm for planarity testing In the worst case119874(1198992) edgesare considered and for each edge the Boyer andMyrvold testis calledHence the approach results in a complexity of119874(1198993)
32 Drawing Maximal Planar Graph on the Plane Graphdrawing as a branch of graph theory applies topologyand geometry to derive two-dimensional representations ofgraphs A graph drawing algorithm reads as input a com-binatorial description of a graph G and produces as output
Table 7 For test problem 5 the coordinates of facilities obtainedby GOT are given below The layout cost is 1062080
Facility 119883 119884
1 28617 195132 25371 220823 27683 320884 6488 135315 1753 343576 10131 169297 29906 252268 28243 364739 14372 3129810 37502 2229911 32586 484612 19349 1324913 3314 2358814 4696 1935415 21953 853916 3413 2553717 33636 32818 3721 1756419 12603 2371320 21763 2142821 16261 3728322 27453 2518523 25213 153624 27111 4184425 21208 2495926 33287 3847127 25882 1912528 4387 1414729 27342 626530 2831 26331 18328 3143232 3833 776133 22131 2703834 7166 267735 18909 4630236 15123 3438337 28974 2260238 18728 465639 33526 865740 12176 3231841 16539 3973242 28383 3948543 1509 2389944 32571 1255945 41221 407846 24335 2784347 2516 306248 37452 25455
Journal of Industrial Mathematics 5
Table 7 Continued
Facility 119883 119884
49 40189 3629450 12672 3682551 34282 2734952 19126 1500553 36395 4132254 31473 4003455 30991 3498356 38702 270357 28176 127358 29219 4521859 25113 1076660 11021 2702961 22158 1423962 22543 1758263 12348 2132264 13684 439665 8569 3614966 8828 3323967 10933 1237368 24203 2604769 12647 707170 29789 2736271 6492 3864272 18394 1799373 4815 1775474 24465 4564675 24704 2444876 48912 3116477 19063 1108878 27843 1637979 35271 2921780 21659 3542981 15303 2808782 32599 310783 3311 2083684 4402 337685 43843 986286 16694 2389987 20628 172388 19088 3556389 48998 2530890 33559 4617691 31162 308692 2482 3660793 44673 2438194 18385 2769495 30086 1085896 33289 1515697 30399 1718898 1552 1907199 36518 32041100 19439 4084101 22932 4334
Table 7 Continued
Facility 119883 119884
102 27279 28155103 31849 19046104 1148 19243105 731 21826106 38088 12744107 22892 30522108 14917 13713109 6995 30065110 40466 30376111 41964 1947112 303 14954113 19953 27768114 27316 2155115 35983 36293116 44374 28853117 22621 41026118 3199 20775119 11309 3988120 41379 24172121 31589 26996122 18031 8178123 18549 22682124 3164 23551125 33452 23294
a drawing of G A graph has infinitely many different draw-ings For a review of various graphs drawing algorithms referto [23] We use algorithm of Chrobak and Payne [24] to forma straight line drawing of the MPG In such a drawing eachedge is drawn using a straight line segment The algorithmdraws vertices in an MPG to integer coordinates in a (2119873 minus4) times (119873 minus 2) grid Figure 2 shows an example of straight linedrawing
For acquiring an initial solution each vertex is replacedby its correspondent facility In a feasible solution facilitieshave no overlaps For this reason the coordinates of facilitiescan be multiplied by maximum dimensions of all facili-ties (width and length) This operation increases distancebetween facilities andmakes the solution feasible For the caseof circular facilities the diameter of circle can be consideredas maximum dimensions
4 Improving Initial Solution and Comparing
To evaluate the performance the initial solution is improvedby an analytical method by Mir and Imam [7] In thismethod the convergence is controlled by carrying out theoptimization using concept of ldquomagnified envelop blocksrdquoThe dimensions of the blocks are determined by multiplyingthe dimensions of the facilities with a ldquomagnification factorrdquoThe optimization is then carried out for these envelopblocks rather than the actual facilitiesThe analytical methodsearches the optimum placements of each envelop block in
6 Journal of Industrial Mathematics
Table 8 The value of cost function in GOT initial solution and thebest value of random placements
119899 GOT initial solution The best random initial solution10 1945 2437311 2122 2758512 2368 3531713 4919 576414 5514 702415 5538 744916 8116 10287717 8468 12207518 10648 1530419 13943 19119320 14265 2009421 17199 22491222 16610 2637023 22128 30749224 25533 3550825 27585 37968426 29929 43803627 36255 5185028 41483 57050529 47543 65109230 57830 73705831 61408 82058432 63687 8643633 59970 94066434 82721 110195235 76220 10494036 92426 124186137 95386 12546838 87532 15006039 93708 146793840 118266 18053841 141363 18391142 104263 195458743 152188 204207344 163529 23925345 166360 237051846 167002 251049347 189027 29008548 226097 30590049 234811 324500350 238855 337198951 264842 358480552 251009 358666753 233805 370262454 298806 404104455 284543 431091956 344755 481656957 371962 4939864
Table 8 Continued
119899 GOT initial solution The best random initial solution58 344044 51097659 375321 503358960 373118 538554961 370817 573555962 457275 628602963 544350 688184664 530408 708518365 502322 690275266 526230 741481167 556568 773325968 639735 867242469 578533 860229870 643592 914437371 589557 893978872 670866 950592573 736669 1058290574 660395 103902375 749795 1115704676 835418 118030077 727648 1157821978 852689 114767679 971135 1273329880 920522 1294614881 882645 142056782 1084711 1500037883 1072241 1436365184 1072132 1606669385 1154018 1552139286 1150925 1756302387 1164220 1608156388 1230008 1726587489 1379479 1913688390 1351210 1970899391 1275975 1895126492 1360771 211423093 1520228 1969614194 1542740 212520095 1581145 2230222796 1640792 2305569397 1486796 2362639498 1645889 2365109399 1607866 24907337100 2073979 26356664
the direction of steepest descent which is opposite to thegradient direction The sizes of the envelop blocks are thenreduced and the optimization process is repeated for thesecond phase The number of optimization phases is equalto the magnification factor number for the envelop blocksIn the last optimization phase the dimensions of the envelopblocks become equal to the actual facilities For more detailwe refer to Mir and Imam [7]
Journal of Industrial Mathematics 7
Table 9 Summary of the results
Problem Number of facilities Best result by other methods GOT Cost reduction1 8 6925 7527 minus6022 20 1157 1302 minus1453 50 782247 768823 134244 100 5381931 5270941 110995 125 1084451 1062080 22371
So the proposed approach for solving a facility layoutproblem can be summarized as follows
Step 1 encapsulating facilities in envelop blocks (mul-tiplying the dimensions of facilities by amagnificationfactor)
Step 2 obtaining an MPG
Step 3 drawing the MPG on the plane and obtainingan initial solution
Step 4 improving initial solution by analyticalmethod
Figure 3 shows summary of these stepsThe proposed approach was coded using the VBNET
programming language in a program named GOT (Graphoptimization technique) Five test problems were run For alltest problems results were obtained on a PCwith Intel T5470processor The results were compared with the previouslypublished papers and commercial software VIP-PLANOPT2006 VIP-PLANOPT is a useful layout software package thatcan generate near-optimal layout [25] Formore details aboutVIP-PLANOPT see Engineering Optimization Software [8]VIP-PLANOPT results were obtained from the softwareuserrsquos manual The results are presented in the followingsections
41 Test Problem 1 This problem of 8 facilities was intro-duced by Imam and Mir [5] Figure 4 shows the steps foraccruing the initial solution Figure 4(a) shows the flowmatrix and dimension of facilities All dimensions and costmatrix elements are integer-valued numbers ranging between1 and 6 There are several pairs of facilities with no flowbetween them Distance norm is squared Euclidean Thegreedy heuristic generates the edges lists of MPG as shownin Figure 4(b)
The straight line drawing algorithm gives the coordinatesof vertices The drawing is shown in Figure 4(c) Then eachvertex is replaced by its correspondent facility The coordi-nates are multiplied by maximum dimensions of facilities(width and length) and finally the initial layout design isshown in Figure 4(d)
The solution is improved by the analytical techniqueFigure 5 shows the final layout The cost function value forthis layout is 7527 and the running time is 04 second Table 1shows the results obtained by the other approaches Thebest solution for this problem is obtained by VIP-PLANOPT2006
42 Test Problem 2 This problem of 20 unequal areafacilities was introduced by Imam and Mir [6] The dataconsist of only integer valuesThe dimensions of the facilitiesare between 1 and 3 The elements of the cost matrix areintegers between 0 and 5 The distance norm is rectilinearThe final layout obtained by GOT is shown in Figure 6 Thelayout cost is 1302 and the running time is 06 secondTable 2 compares the results obtained byGOTwith the resultsavailable in the literature VIP-PLANOPT2006 has the lowestvalue of the cost function
43 Test Problem 3 This is a problem of 50 facilities ran-domly generated by VIP-PLANOPT 2006 The dimensionsof the facilities are decimal numbers between 1 to 6 Theelements of the cost matrix are all integers between 1 and10 The distance norm is Euclidean The results are shownin Table 3 The best published result has a cost of 782247whereas GOT produces a final layout with a cost of 768823only in 151 seconds Figure 7 shows the final layout
44 Test Problem 4 This is a randomly generated large sizeproblem of 100 facilities The dimensions of the facilities aredecimal numbers between 1 and 6 The cost matrix elementsare integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 4 GOT obtained the costfunction value of 5270941 in 743 secondsThis value is about2 below the cost function value of VIP-PLANOPT 2006The coordinates of the facilities for the layout obtained byGOT are given in Table 5
45 Test Problem 5 This is a large size problem of 125facilities randomly generated by VIP-PLANOPT 2006 Thedimensions of facilities are real numbers between 1 and 6 andelements of the cost matrix are integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 6GOT obtained the cost function value of 1062080 in 1296secondsThis value is about 2 below the cost function valueof VIP-PLANOPT 2006 The coordinates of the facilities forthe layout obtained by GOT are given in Table 7
5 Comparing GOT Initial Solution withRandom Initial Solution
To compare the proposed initial solution (GOT initial solu-tion) with random initial solution a set of test problems (119899 =10 11 12 100) were generated The facilities dimensionswere 1 times 1 and flow matrices were randomly generatedbetween 0 and 10 For acquiring a random initial solution
8 Journal of Industrial Mathematics
Dimensions of the facilities Flow matrix
1
1
1
1 1
1
2
2
2
2
2
2
2 2
2
2
2
2
20
0
0
0
0
0
0
0
0 0
0
0 0
0
0
0 0
0 0 0
0
0
00
3
3
3
3
4
4
4
4
4
4
4
5
5
5
6
6
6
7 8
1
2
2
2
2
23
3
3
3
3
4
4
4
4
44
44
5
5
6
7
8
1
1
2
3 3
4
5
6
7
8
Length Width
(a) The raw facilities layout data
MPGalgorithm
1 5-2
3-2
7-6
8-5
6-3
4-2
3-1
4-3
5-42
3
4
5
6
7
8
9
Number10
11
12
13
14
15
16
17
18
NumberEdge6-4
8-2
8-4
2-1
7-3
8-7
7-2
7-4
7-1
Edge
(b) MPG
Graph drawing algorithm
1
7
8
5
2
3
4
6
(c) Straight line drawing of the MPG
Obtaining initial solution
1
7
8
5
2
3
4
6
(d) Initial solution
Figure 4 Obtaining an initial solution for test problem 1
1
7
85
2
34
6
Figure 5 Final layout for test problem 1
facilities were randomly placed in a (2119899 minus 4) times (119899 minus 2) integergrid For each test problem 20 random placements werefound Table 8 shows the value of cost function in GOT initialsolution and the best value found by random placementsFigures 8 and 9 shows these results graphically The resultsdemonstrated significant improvement in cost function
6 Summary and Conclusion
An initial solution has been presented for the layout designof facilities on a continuous plane The technique consists of
8 4 5
14 11 18 19 3
15
20
13
717
1 2 612
10
9
16
Figure 6 Final layout for test problem 2
two stages In the first stage amaximal planar graph (MPG) isobtained In the second stage the vertices of MPG are drawnon the plane by graph drawing algorithms Then vertices arereplaced by facilities Hence an initial solution is obtainedTo evaluate the performance this initial solution has beenapplied in rectangular facility layout problem and improvedby an analytical method by Mir and Imam [7]
The approach has been tested on five instances fromthe literature Table 6 shows the Summary of the resultsand Figure 8 shows the cost reduction by the technique Forthe large size problems involving 50 100 and 125 facilities
Journal of Industrial Mathematics 9
6 41 22
29
1224 44
4645
1425
5
26
28
10
9
3
2
8
4
30
31
43
47
49
50
40
42
48
36
34
3538
39
33
32
13
18
37
15 17
27 23
20
16
71 19
11
21
Figure 7 Final layout for test problem 3
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Random initial solutionGOT initial solution
Cos
t
05
1
15
2
25
3
35
4
45
5
times106
n
Figure 8 Comparisons of cost function value in GOT initialsolution with the best value of random placements
the layout costs values are better than those obtained by thepreviously published techniques As shown in Table 9 theresults demonstrate effectiveness of the technique especiallyfor large size problems
This paper introduced a simple technique for obtaining agood initial solutionThe technique with somemodificationcan be applied in facility layout approaches that use arandomly generated initial solution In future researches itwould be interesting to analyze the influence of MPG andgraph drawing algorithm on the solution The results can befurther improved by using a metaheuristic such as GRASP[21] and Tabu search [20] for generating a high weightedMPG
0
5
10
15
20
25
8 20 50 100 125
Reduction
times103
minus5
Figure 9 Cost reduction by using GOT
Conflict of Interests
The authors declare that they have no conflict of interests
References
[1] S P Singh and R R K Sharma ldquoA review of differentapproaches to the facility layout problemsrdquo International Jour-nal of Advanced Manufacturing Technology vol 30 no 5-6 pp425ndash433 2006
[2] M F Anjos and A Vannelli ldquoAn attractor-repeller approach tofloorplanningrdquo Mathematical Methods of Operations Researchvol 56 no 1 pp 3ndash27 2002
[3] I Castillo and T Sim ldquoA spring-embedding approach for thefacility layout problemrdquo Journal of the Operational ResearchSociety vol 55 no 1 pp 73ndash81 2004
[4] Z Drezner ldquoDISCON a new method for the layout problemrdquoOperations Research vol 28 no 6 pp 1375ndash1384 1980
[5] M H Imam and M Mir ldquoNonlinear programming approachto automated topology optimizationrdquo Computer-Aided Designvol 21 no 2 pp 107ndash115 1989
[6] M H Imam and M Mir ldquoAutomated layout of facilities ofunequal areasrdquo Computers and Industrial Engineering vol 24no 3 pp 355ndash366 1993
[7] M Mir and M H Imam ldquoHybrid optimization approachfor layout design of unequal-area facilitiesrdquo Computers andIndustrial Engineering vol 39 no 1-2 pp 49ndash63 2001
[8] Engineering Optimization Software VIP-PLANOPT 20062010 httpwwwplanoptcom
[9] L Foulds Graph Theory Applications Springer New York NYUSA 1992
[10] M M D Hassan and G L Hogg ldquoA review of graph theoryapplication to the facilities layout problemrdquo Omega vol 15 no4 pp 291ndash300 1987
[11] M M D Hassan and G L Hogg ldquoOn converting a dualgraph into a block layoutrdquo International Journal of ProductionResearch vol 27 no 7 pp 1149ndash1160 1989
[12] KHWatson and JWGiffin ldquoThevertex splitting algorithm forfacilities layoutrdquo International Journal of Production Researchvol 35 no 9 pp 2477ndash2492 1997
[13] S A Irvine and I Rinsma-Melchert ldquoA new approach tothe block layout problemrdquo International Journal of ProductionResearch vol 35 no 8 pp 2359ndash2376 1997
10 Journal of Industrial Mathematics
[14] P S Welgama P R Gibson and L A R Al-Hakim ldquoFacilitieslayout a knowledge-based approach for converting a dualgraph into a block layoutrdquo International Journal of ProductionEconomics vol 33 no 1ndash3 pp 17ndash30 1994
[15] M A Jokar and A S Sangchooli ldquoConstructing a blocklayout by face areardquo The International Journal of AdvancedManufacturing Technology vol 54 no 5ndash8 pp 801ndash809 2011
[16] L R Foulds P B Gibbons and J W Giffin ldquoFacilities layoutadjacency determination an experimental comparison of threegraph theoretic heuristicsrdquo Operations Research vol 33 no 5pp 1091ndash1106 1985
[17] E G John and J Hammond ldquoMaximally weighted graphtheoretic facilities design planningrdquo International Journal ofProduction Research vol 38 no 16 pp 3845ndash3859 2000
[18] S G Boswell ldquoTESSAmdasha new greedy heuristic for facilitieslayout planningrdquo International Journal of Production Researchvol 30 no 8 pp 1957ndash1968 1992
[19] L R Foulds and D F Robinson ldquoGraph theoretic heuristics forthe plant layout problemrdquo International Journal of ProductionResearch vol 16 no 1 pp 27ndash37 1978
[20] I H Osman ldquoA tabu search procedure based on a randomRoulette diversification for the weighted maximal planar graphproblemrdquoComputers and Operations Research vol 33 no 9 pp2526ndash2546 2006
[21] I H Osman B Al-Ayoubi and M Barake ldquoA greedy randomadaptive search procedure for the weighted maximal planargraph problemrdquo Computers and Industrial Engineering vol 45no 4 pp 635ndash651 2003
[22] J M Boyer and W J Myrvold ldquoOn the cutting edge simplifiedO(n) planarity by edge additionrdquo Journal of Graph Algorithmsand Applications vol 8 no 3 pp 241ndash273 2004
[23] G Di Battista P Eades R Tamassia and I Tollis GraphDrawing Algorithms for the Visualization of Graphs PrenticeHall PTR Upper Saddle River NJ USA 1998
[24] M Chrobak and T H Payne ldquoA linear-time algorithm fordrawing a planar graph on a gridrdquo Information ProcessingLetters vol 54 no 4 pp 241ndash246 1995
[25] S Heragu Facilities Design Iuniverse Inc 2006
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Industrial Mathematics 3
Table 5 For test problem 4 the coordinates of facilities obtainedbyGOTare given belowThe value of the cost function for this layoutis 5270941
Facility 119883 119884
1 17599 221582 18514 329333 29485 186344 35526 205825 14798 22656 26425 253887 12213 132828 19435 274669 32119 1531910 24743 886211 16231 271112 25533 1762113 14274 2035214 22791 388915 14467 1481216 38012 169417 9735 1127618 24609 2766319 20685 2057820 19866 2257921 15832 842122 8866 3149523 22402 1733524 35551 912525 1695 1937826 36104 253127 16328 2184328 13208 2616129 22439 1473230 14052 1306531 2821 2193432 16436 1228233 24467 1136534 23103 3248935 23283 2496836 19444 1470837 32927 1669938 11708 2885739 1514 2979340 28164 517441 3208 847642 29636 1304943 26373 3112744 8693 1578545 1605 2685946 31965 2609947 29361 958648 33822 12885
Table 5 Continued
Facility 119883 119884
49 19394 2504350 25902 2195451 4234 1653452 19539 3021153 5587 1180154 13399 3283155 28771 1607156 30974 2277457 22453 2211758 1235 2313859 21725 1894460 30419 3185461 24388 1889762 19587 3697463 19943 1283964 24632 1501965 5951 3223366 16918 1434667 23818 640868 19581 1844769 35191 3126370 0927 2134771 32644 1946472 9487 2032573 39749 2178874 18485 162775 15581 3634476 6345 2020377 13484 232178 18527 2073479 11773 2029580 19329 266381 4471 227982 4788 2796183 2751 3661284 26306 1577685 4545 2525786 23545 3589287 10746 3771288 7405 665689 12021 668490 2914 2712391 23367 2985692 1487 1662993 11961 1713794 16126 183495 22426 1607596 19729 1075297 19707 76398 23824 2151199 8257 27817100 9179 24151
4 Journal of Industrial Mathematics
Table 6 Results for test problem 5
Program Cost function valueVIP-PLANOPT (2006) [8] 1084451GOT 1062080
The value of overlap area 119860119894119895is a nonnegative number 119860
119894119895
will be zero only if there is no overlapping between facilities119894 and 119895 The objective is to minimize material handling costsSo the problem can be stated as follows
minimizing cost =119873minus1
sum
119894=1
119873
sum
119895=119894+1
119891119894119895119889119894119895
subject to 119860119894119895= 0 forall119894 119895 119894 lt 119895
(6)
The constraint ensures that facilities do not overlap A similarformulation also can be found in [7]
3 Obtaining an Initial Solution
The initial solution is obtained by graph theoretic facilitylayout approaches (GTFLP) and graph drawing algorithmsThe following subsection describes obtaining an MPGSection 32 describes the drawing of the MPG on the plane
31 Generating aMaximal Planar Graph In GTFLP facilitiesare represented by vertices and flow (adjacency desirability)between them is represented by weighted edges Createdgraph is called adjacency graph Graph theory is particularlyuseful for the facility layout problems because graphs easilyenable us to capture the adjacency information andmodel theproblem A review of graph theory applications to the facilitylayout problem can be found in [9 10] GTFLP consists oftwo stages At the first stage the adjacency graph is convertedto a maximal planar graph (MPG) In the second stage ablock layout is constructed from the MPG The second stageis not our concern here For more details we refer to [11ndash15]Figure 1 shows an MPG and its correspondent block layout
Many heuristic and metaheuristic methods for obtainingan MPG have been suggested [16ndash21] In this paper we usefrom the greedy heuristic [16] It is conceptually simple andcreates high weighted MPGs [16] This heuristic has a simpleinstruction the edges are sorted in nonincreasing order ofweight Each edge is tested in turn and accepted as part of theMPG unless it makes the graph nonplanar So the heuristicneeds planarity testing Boyer and Myrvold [22] developeda simplified 119874(119899) planarity testing algorithm We use thisalgorithm for planarity testing In the worst case119874(1198992) edgesare considered and for each edge the Boyer andMyrvold testis calledHence the approach results in a complexity of119874(1198993)
32 Drawing Maximal Planar Graph on the Plane Graphdrawing as a branch of graph theory applies topologyand geometry to derive two-dimensional representations ofgraphs A graph drawing algorithm reads as input a com-binatorial description of a graph G and produces as output
Table 7 For test problem 5 the coordinates of facilities obtainedby GOT are given below The layout cost is 1062080
Facility 119883 119884
1 28617 195132 25371 220823 27683 320884 6488 135315 1753 343576 10131 169297 29906 252268 28243 364739 14372 3129810 37502 2229911 32586 484612 19349 1324913 3314 2358814 4696 1935415 21953 853916 3413 2553717 33636 32818 3721 1756419 12603 2371320 21763 2142821 16261 3728322 27453 2518523 25213 153624 27111 4184425 21208 2495926 33287 3847127 25882 1912528 4387 1414729 27342 626530 2831 26331 18328 3143232 3833 776133 22131 2703834 7166 267735 18909 4630236 15123 3438337 28974 2260238 18728 465639 33526 865740 12176 3231841 16539 3973242 28383 3948543 1509 2389944 32571 1255945 41221 407846 24335 2784347 2516 306248 37452 25455
Journal of Industrial Mathematics 5
Table 7 Continued
Facility 119883 119884
49 40189 3629450 12672 3682551 34282 2734952 19126 1500553 36395 4132254 31473 4003455 30991 3498356 38702 270357 28176 127358 29219 4521859 25113 1076660 11021 2702961 22158 1423962 22543 1758263 12348 2132264 13684 439665 8569 3614966 8828 3323967 10933 1237368 24203 2604769 12647 707170 29789 2736271 6492 3864272 18394 1799373 4815 1775474 24465 4564675 24704 2444876 48912 3116477 19063 1108878 27843 1637979 35271 2921780 21659 3542981 15303 2808782 32599 310783 3311 2083684 4402 337685 43843 986286 16694 2389987 20628 172388 19088 3556389 48998 2530890 33559 4617691 31162 308692 2482 3660793 44673 2438194 18385 2769495 30086 1085896 33289 1515697 30399 1718898 1552 1907199 36518 32041100 19439 4084101 22932 4334
Table 7 Continued
Facility 119883 119884
102 27279 28155103 31849 19046104 1148 19243105 731 21826106 38088 12744107 22892 30522108 14917 13713109 6995 30065110 40466 30376111 41964 1947112 303 14954113 19953 27768114 27316 2155115 35983 36293116 44374 28853117 22621 41026118 3199 20775119 11309 3988120 41379 24172121 31589 26996122 18031 8178123 18549 22682124 3164 23551125 33452 23294
a drawing of G A graph has infinitely many different draw-ings For a review of various graphs drawing algorithms referto [23] We use algorithm of Chrobak and Payne [24] to forma straight line drawing of the MPG In such a drawing eachedge is drawn using a straight line segment The algorithmdraws vertices in an MPG to integer coordinates in a (2119873 minus4) times (119873 minus 2) grid Figure 2 shows an example of straight linedrawing
For acquiring an initial solution each vertex is replacedby its correspondent facility In a feasible solution facilitieshave no overlaps For this reason the coordinates of facilitiescan be multiplied by maximum dimensions of all facili-ties (width and length) This operation increases distancebetween facilities andmakes the solution feasible For the caseof circular facilities the diameter of circle can be consideredas maximum dimensions
4 Improving Initial Solution and Comparing
To evaluate the performance the initial solution is improvedby an analytical method by Mir and Imam [7] In thismethod the convergence is controlled by carrying out theoptimization using concept of ldquomagnified envelop blocksrdquoThe dimensions of the blocks are determined by multiplyingthe dimensions of the facilities with a ldquomagnification factorrdquoThe optimization is then carried out for these envelopblocks rather than the actual facilitiesThe analytical methodsearches the optimum placements of each envelop block in
6 Journal of Industrial Mathematics
Table 8 The value of cost function in GOT initial solution and thebest value of random placements
119899 GOT initial solution The best random initial solution10 1945 2437311 2122 2758512 2368 3531713 4919 576414 5514 702415 5538 744916 8116 10287717 8468 12207518 10648 1530419 13943 19119320 14265 2009421 17199 22491222 16610 2637023 22128 30749224 25533 3550825 27585 37968426 29929 43803627 36255 5185028 41483 57050529 47543 65109230 57830 73705831 61408 82058432 63687 8643633 59970 94066434 82721 110195235 76220 10494036 92426 124186137 95386 12546838 87532 15006039 93708 146793840 118266 18053841 141363 18391142 104263 195458743 152188 204207344 163529 23925345 166360 237051846 167002 251049347 189027 29008548 226097 30590049 234811 324500350 238855 337198951 264842 358480552 251009 358666753 233805 370262454 298806 404104455 284543 431091956 344755 481656957 371962 4939864
Table 8 Continued
119899 GOT initial solution The best random initial solution58 344044 51097659 375321 503358960 373118 538554961 370817 573555962 457275 628602963 544350 688184664 530408 708518365 502322 690275266 526230 741481167 556568 773325968 639735 867242469 578533 860229870 643592 914437371 589557 893978872 670866 950592573 736669 1058290574 660395 103902375 749795 1115704676 835418 118030077 727648 1157821978 852689 114767679 971135 1273329880 920522 1294614881 882645 142056782 1084711 1500037883 1072241 1436365184 1072132 1606669385 1154018 1552139286 1150925 1756302387 1164220 1608156388 1230008 1726587489 1379479 1913688390 1351210 1970899391 1275975 1895126492 1360771 211423093 1520228 1969614194 1542740 212520095 1581145 2230222796 1640792 2305569397 1486796 2362639498 1645889 2365109399 1607866 24907337100 2073979 26356664
the direction of steepest descent which is opposite to thegradient direction The sizes of the envelop blocks are thenreduced and the optimization process is repeated for thesecond phase The number of optimization phases is equalto the magnification factor number for the envelop blocksIn the last optimization phase the dimensions of the envelopblocks become equal to the actual facilities For more detailwe refer to Mir and Imam [7]
Journal of Industrial Mathematics 7
Table 9 Summary of the results
Problem Number of facilities Best result by other methods GOT Cost reduction1 8 6925 7527 minus6022 20 1157 1302 minus1453 50 782247 768823 134244 100 5381931 5270941 110995 125 1084451 1062080 22371
So the proposed approach for solving a facility layoutproblem can be summarized as follows
Step 1 encapsulating facilities in envelop blocks (mul-tiplying the dimensions of facilities by amagnificationfactor)
Step 2 obtaining an MPG
Step 3 drawing the MPG on the plane and obtainingan initial solution
Step 4 improving initial solution by analyticalmethod
Figure 3 shows summary of these stepsThe proposed approach was coded using the VBNET
programming language in a program named GOT (Graphoptimization technique) Five test problems were run For alltest problems results were obtained on a PCwith Intel T5470processor The results were compared with the previouslypublished papers and commercial software VIP-PLANOPT2006 VIP-PLANOPT is a useful layout software package thatcan generate near-optimal layout [25] Formore details aboutVIP-PLANOPT see Engineering Optimization Software [8]VIP-PLANOPT results were obtained from the softwareuserrsquos manual The results are presented in the followingsections
41 Test Problem 1 This problem of 8 facilities was intro-duced by Imam and Mir [5] Figure 4 shows the steps foraccruing the initial solution Figure 4(a) shows the flowmatrix and dimension of facilities All dimensions and costmatrix elements are integer-valued numbers ranging between1 and 6 There are several pairs of facilities with no flowbetween them Distance norm is squared Euclidean Thegreedy heuristic generates the edges lists of MPG as shownin Figure 4(b)
The straight line drawing algorithm gives the coordinatesof vertices The drawing is shown in Figure 4(c) Then eachvertex is replaced by its correspondent facility The coordi-nates are multiplied by maximum dimensions of facilities(width and length) and finally the initial layout design isshown in Figure 4(d)
The solution is improved by the analytical techniqueFigure 5 shows the final layout The cost function value forthis layout is 7527 and the running time is 04 second Table 1shows the results obtained by the other approaches Thebest solution for this problem is obtained by VIP-PLANOPT2006
42 Test Problem 2 This problem of 20 unequal areafacilities was introduced by Imam and Mir [6] The dataconsist of only integer valuesThe dimensions of the facilitiesare between 1 and 3 The elements of the cost matrix areintegers between 0 and 5 The distance norm is rectilinearThe final layout obtained by GOT is shown in Figure 6 Thelayout cost is 1302 and the running time is 06 secondTable 2 compares the results obtained byGOTwith the resultsavailable in the literature VIP-PLANOPT2006 has the lowestvalue of the cost function
43 Test Problem 3 This is a problem of 50 facilities ran-domly generated by VIP-PLANOPT 2006 The dimensionsof the facilities are decimal numbers between 1 to 6 Theelements of the cost matrix are all integers between 1 and10 The distance norm is Euclidean The results are shownin Table 3 The best published result has a cost of 782247whereas GOT produces a final layout with a cost of 768823only in 151 seconds Figure 7 shows the final layout
44 Test Problem 4 This is a randomly generated large sizeproblem of 100 facilities The dimensions of the facilities aredecimal numbers between 1 and 6 The cost matrix elementsare integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 4 GOT obtained the costfunction value of 5270941 in 743 secondsThis value is about2 below the cost function value of VIP-PLANOPT 2006The coordinates of the facilities for the layout obtained byGOT are given in Table 5
45 Test Problem 5 This is a large size problem of 125facilities randomly generated by VIP-PLANOPT 2006 Thedimensions of facilities are real numbers between 1 and 6 andelements of the cost matrix are integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 6GOT obtained the cost function value of 1062080 in 1296secondsThis value is about 2 below the cost function valueof VIP-PLANOPT 2006 The coordinates of the facilities forthe layout obtained by GOT are given in Table 7
5 Comparing GOT Initial Solution withRandom Initial Solution
To compare the proposed initial solution (GOT initial solu-tion) with random initial solution a set of test problems (119899 =10 11 12 100) were generated The facilities dimensionswere 1 times 1 and flow matrices were randomly generatedbetween 0 and 10 For acquiring a random initial solution
8 Journal of Industrial Mathematics
Dimensions of the facilities Flow matrix
1
1
1
1 1
1
2
2
2
2
2
2
2 2
2
2
2
2
20
0
0
0
0
0
0
0
0 0
0
0 0
0
0
0 0
0 0 0
0
0
00
3
3
3
3
4
4
4
4
4
4
4
5
5
5
6
6
6
7 8
1
2
2
2
2
23
3
3
3
3
4
4
4
4
44
44
5
5
6
7
8
1
1
2
3 3
4
5
6
7
8
Length Width
(a) The raw facilities layout data
MPGalgorithm
1 5-2
3-2
7-6
8-5
6-3
4-2
3-1
4-3
5-42
3
4
5
6
7
8
9
Number10
11
12
13
14
15
16
17
18
NumberEdge6-4
8-2
8-4
2-1
7-3
8-7
7-2
7-4
7-1
Edge
(b) MPG
Graph drawing algorithm
1
7
8
5
2
3
4
6
(c) Straight line drawing of the MPG
Obtaining initial solution
1
7
8
5
2
3
4
6
(d) Initial solution
Figure 4 Obtaining an initial solution for test problem 1
1
7
85
2
34
6
Figure 5 Final layout for test problem 1
facilities were randomly placed in a (2119899 minus 4) times (119899 minus 2) integergrid For each test problem 20 random placements werefound Table 8 shows the value of cost function in GOT initialsolution and the best value found by random placementsFigures 8 and 9 shows these results graphically The resultsdemonstrated significant improvement in cost function
6 Summary and Conclusion
An initial solution has been presented for the layout designof facilities on a continuous plane The technique consists of
8 4 5
14 11 18 19 3
15
20
13
717
1 2 612
10
9
16
Figure 6 Final layout for test problem 2
two stages In the first stage amaximal planar graph (MPG) isobtained In the second stage the vertices of MPG are drawnon the plane by graph drawing algorithms Then vertices arereplaced by facilities Hence an initial solution is obtainedTo evaluate the performance this initial solution has beenapplied in rectangular facility layout problem and improvedby an analytical method by Mir and Imam [7]
The approach has been tested on five instances fromthe literature Table 6 shows the Summary of the resultsand Figure 8 shows the cost reduction by the technique Forthe large size problems involving 50 100 and 125 facilities
Journal of Industrial Mathematics 9
6 41 22
29
1224 44
4645
1425
5
26
28
10
9
3
2
8
4
30
31
43
47
49
50
40
42
48
36
34
3538
39
33
32
13
18
37
15 17
27 23
20
16
71 19
11
21
Figure 7 Final layout for test problem 3
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Random initial solutionGOT initial solution
Cos
t
05
1
15
2
25
3
35
4
45
5
times106
n
Figure 8 Comparisons of cost function value in GOT initialsolution with the best value of random placements
the layout costs values are better than those obtained by thepreviously published techniques As shown in Table 9 theresults demonstrate effectiveness of the technique especiallyfor large size problems
This paper introduced a simple technique for obtaining agood initial solutionThe technique with somemodificationcan be applied in facility layout approaches that use arandomly generated initial solution In future researches itwould be interesting to analyze the influence of MPG andgraph drawing algorithm on the solution The results can befurther improved by using a metaheuristic such as GRASP[21] and Tabu search [20] for generating a high weightedMPG
0
5
10
15
20
25
8 20 50 100 125
Reduction
times103
minus5
Figure 9 Cost reduction by using GOT
Conflict of Interests
The authors declare that they have no conflict of interests
References
[1] S P Singh and R R K Sharma ldquoA review of differentapproaches to the facility layout problemsrdquo International Jour-nal of Advanced Manufacturing Technology vol 30 no 5-6 pp425ndash433 2006
[2] M F Anjos and A Vannelli ldquoAn attractor-repeller approach tofloorplanningrdquo Mathematical Methods of Operations Researchvol 56 no 1 pp 3ndash27 2002
[3] I Castillo and T Sim ldquoA spring-embedding approach for thefacility layout problemrdquo Journal of the Operational ResearchSociety vol 55 no 1 pp 73ndash81 2004
[4] Z Drezner ldquoDISCON a new method for the layout problemrdquoOperations Research vol 28 no 6 pp 1375ndash1384 1980
[5] M H Imam and M Mir ldquoNonlinear programming approachto automated topology optimizationrdquo Computer-Aided Designvol 21 no 2 pp 107ndash115 1989
[6] M H Imam and M Mir ldquoAutomated layout of facilities ofunequal areasrdquo Computers and Industrial Engineering vol 24no 3 pp 355ndash366 1993
[7] M Mir and M H Imam ldquoHybrid optimization approachfor layout design of unequal-area facilitiesrdquo Computers andIndustrial Engineering vol 39 no 1-2 pp 49ndash63 2001
[8] Engineering Optimization Software VIP-PLANOPT 20062010 httpwwwplanoptcom
[9] L Foulds Graph Theory Applications Springer New York NYUSA 1992
[10] M M D Hassan and G L Hogg ldquoA review of graph theoryapplication to the facilities layout problemrdquo Omega vol 15 no4 pp 291ndash300 1987
[11] M M D Hassan and G L Hogg ldquoOn converting a dualgraph into a block layoutrdquo International Journal of ProductionResearch vol 27 no 7 pp 1149ndash1160 1989
[12] KHWatson and JWGiffin ldquoThevertex splitting algorithm forfacilities layoutrdquo International Journal of Production Researchvol 35 no 9 pp 2477ndash2492 1997
[13] S A Irvine and I Rinsma-Melchert ldquoA new approach tothe block layout problemrdquo International Journal of ProductionResearch vol 35 no 8 pp 2359ndash2376 1997
10 Journal of Industrial Mathematics
[14] P S Welgama P R Gibson and L A R Al-Hakim ldquoFacilitieslayout a knowledge-based approach for converting a dualgraph into a block layoutrdquo International Journal of ProductionEconomics vol 33 no 1ndash3 pp 17ndash30 1994
[15] M A Jokar and A S Sangchooli ldquoConstructing a blocklayout by face areardquo The International Journal of AdvancedManufacturing Technology vol 54 no 5ndash8 pp 801ndash809 2011
[16] L R Foulds P B Gibbons and J W Giffin ldquoFacilities layoutadjacency determination an experimental comparison of threegraph theoretic heuristicsrdquo Operations Research vol 33 no 5pp 1091ndash1106 1985
[17] E G John and J Hammond ldquoMaximally weighted graphtheoretic facilities design planningrdquo International Journal ofProduction Research vol 38 no 16 pp 3845ndash3859 2000
[18] S G Boswell ldquoTESSAmdasha new greedy heuristic for facilitieslayout planningrdquo International Journal of Production Researchvol 30 no 8 pp 1957ndash1968 1992
[19] L R Foulds and D F Robinson ldquoGraph theoretic heuristics forthe plant layout problemrdquo International Journal of ProductionResearch vol 16 no 1 pp 27ndash37 1978
[20] I H Osman ldquoA tabu search procedure based on a randomRoulette diversification for the weighted maximal planar graphproblemrdquoComputers and Operations Research vol 33 no 9 pp2526ndash2546 2006
[21] I H Osman B Al-Ayoubi and M Barake ldquoA greedy randomadaptive search procedure for the weighted maximal planargraph problemrdquo Computers and Industrial Engineering vol 45no 4 pp 635ndash651 2003
[22] J M Boyer and W J Myrvold ldquoOn the cutting edge simplifiedO(n) planarity by edge additionrdquo Journal of Graph Algorithmsand Applications vol 8 no 3 pp 241ndash273 2004
[23] G Di Battista P Eades R Tamassia and I Tollis GraphDrawing Algorithms for the Visualization of Graphs PrenticeHall PTR Upper Saddle River NJ USA 1998
[24] M Chrobak and T H Payne ldquoA linear-time algorithm fordrawing a planar graph on a gridrdquo Information ProcessingLetters vol 54 no 4 pp 241ndash246 1995
[25] S Heragu Facilities Design Iuniverse Inc 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Industrial Mathematics
Table 6 Results for test problem 5
Program Cost function valueVIP-PLANOPT (2006) [8] 1084451GOT 1062080
The value of overlap area 119860119894119895is a nonnegative number 119860
119894119895
will be zero only if there is no overlapping between facilities119894 and 119895 The objective is to minimize material handling costsSo the problem can be stated as follows
minimizing cost =119873minus1
sum
119894=1
119873
sum
119895=119894+1
119891119894119895119889119894119895
subject to 119860119894119895= 0 forall119894 119895 119894 lt 119895
(6)
The constraint ensures that facilities do not overlap A similarformulation also can be found in [7]
3 Obtaining an Initial Solution
The initial solution is obtained by graph theoretic facilitylayout approaches (GTFLP) and graph drawing algorithmsThe following subsection describes obtaining an MPGSection 32 describes the drawing of the MPG on the plane
31 Generating aMaximal Planar Graph In GTFLP facilitiesare represented by vertices and flow (adjacency desirability)between them is represented by weighted edges Createdgraph is called adjacency graph Graph theory is particularlyuseful for the facility layout problems because graphs easilyenable us to capture the adjacency information andmodel theproblem A review of graph theory applications to the facilitylayout problem can be found in [9 10] GTFLP consists oftwo stages At the first stage the adjacency graph is convertedto a maximal planar graph (MPG) In the second stage ablock layout is constructed from the MPG The second stageis not our concern here For more details we refer to [11ndash15]Figure 1 shows an MPG and its correspondent block layout
Many heuristic and metaheuristic methods for obtainingan MPG have been suggested [16ndash21] In this paper we usefrom the greedy heuristic [16] It is conceptually simple andcreates high weighted MPGs [16] This heuristic has a simpleinstruction the edges are sorted in nonincreasing order ofweight Each edge is tested in turn and accepted as part of theMPG unless it makes the graph nonplanar So the heuristicneeds planarity testing Boyer and Myrvold [22] developeda simplified 119874(119899) planarity testing algorithm We use thisalgorithm for planarity testing In the worst case119874(1198992) edgesare considered and for each edge the Boyer andMyrvold testis calledHence the approach results in a complexity of119874(1198993)
32 Drawing Maximal Planar Graph on the Plane Graphdrawing as a branch of graph theory applies topologyand geometry to derive two-dimensional representations ofgraphs A graph drawing algorithm reads as input a com-binatorial description of a graph G and produces as output
Table 7 For test problem 5 the coordinates of facilities obtainedby GOT are given below The layout cost is 1062080
Facility 119883 119884
1 28617 195132 25371 220823 27683 320884 6488 135315 1753 343576 10131 169297 29906 252268 28243 364739 14372 3129810 37502 2229911 32586 484612 19349 1324913 3314 2358814 4696 1935415 21953 853916 3413 2553717 33636 32818 3721 1756419 12603 2371320 21763 2142821 16261 3728322 27453 2518523 25213 153624 27111 4184425 21208 2495926 33287 3847127 25882 1912528 4387 1414729 27342 626530 2831 26331 18328 3143232 3833 776133 22131 2703834 7166 267735 18909 4630236 15123 3438337 28974 2260238 18728 465639 33526 865740 12176 3231841 16539 3973242 28383 3948543 1509 2389944 32571 1255945 41221 407846 24335 2784347 2516 306248 37452 25455
Journal of Industrial Mathematics 5
Table 7 Continued
Facility 119883 119884
49 40189 3629450 12672 3682551 34282 2734952 19126 1500553 36395 4132254 31473 4003455 30991 3498356 38702 270357 28176 127358 29219 4521859 25113 1076660 11021 2702961 22158 1423962 22543 1758263 12348 2132264 13684 439665 8569 3614966 8828 3323967 10933 1237368 24203 2604769 12647 707170 29789 2736271 6492 3864272 18394 1799373 4815 1775474 24465 4564675 24704 2444876 48912 3116477 19063 1108878 27843 1637979 35271 2921780 21659 3542981 15303 2808782 32599 310783 3311 2083684 4402 337685 43843 986286 16694 2389987 20628 172388 19088 3556389 48998 2530890 33559 4617691 31162 308692 2482 3660793 44673 2438194 18385 2769495 30086 1085896 33289 1515697 30399 1718898 1552 1907199 36518 32041100 19439 4084101 22932 4334
Table 7 Continued
Facility 119883 119884
102 27279 28155103 31849 19046104 1148 19243105 731 21826106 38088 12744107 22892 30522108 14917 13713109 6995 30065110 40466 30376111 41964 1947112 303 14954113 19953 27768114 27316 2155115 35983 36293116 44374 28853117 22621 41026118 3199 20775119 11309 3988120 41379 24172121 31589 26996122 18031 8178123 18549 22682124 3164 23551125 33452 23294
a drawing of G A graph has infinitely many different draw-ings For a review of various graphs drawing algorithms referto [23] We use algorithm of Chrobak and Payne [24] to forma straight line drawing of the MPG In such a drawing eachedge is drawn using a straight line segment The algorithmdraws vertices in an MPG to integer coordinates in a (2119873 minus4) times (119873 minus 2) grid Figure 2 shows an example of straight linedrawing
For acquiring an initial solution each vertex is replacedby its correspondent facility In a feasible solution facilitieshave no overlaps For this reason the coordinates of facilitiescan be multiplied by maximum dimensions of all facili-ties (width and length) This operation increases distancebetween facilities andmakes the solution feasible For the caseof circular facilities the diameter of circle can be consideredas maximum dimensions
4 Improving Initial Solution and Comparing
To evaluate the performance the initial solution is improvedby an analytical method by Mir and Imam [7] In thismethod the convergence is controlled by carrying out theoptimization using concept of ldquomagnified envelop blocksrdquoThe dimensions of the blocks are determined by multiplyingthe dimensions of the facilities with a ldquomagnification factorrdquoThe optimization is then carried out for these envelopblocks rather than the actual facilitiesThe analytical methodsearches the optimum placements of each envelop block in
6 Journal of Industrial Mathematics
Table 8 The value of cost function in GOT initial solution and thebest value of random placements
119899 GOT initial solution The best random initial solution10 1945 2437311 2122 2758512 2368 3531713 4919 576414 5514 702415 5538 744916 8116 10287717 8468 12207518 10648 1530419 13943 19119320 14265 2009421 17199 22491222 16610 2637023 22128 30749224 25533 3550825 27585 37968426 29929 43803627 36255 5185028 41483 57050529 47543 65109230 57830 73705831 61408 82058432 63687 8643633 59970 94066434 82721 110195235 76220 10494036 92426 124186137 95386 12546838 87532 15006039 93708 146793840 118266 18053841 141363 18391142 104263 195458743 152188 204207344 163529 23925345 166360 237051846 167002 251049347 189027 29008548 226097 30590049 234811 324500350 238855 337198951 264842 358480552 251009 358666753 233805 370262454 298806 404104455 284543 431091956 344755 481656957 371962 4939864
Table 8 Continued
119899 GOT initial solution The best random initial solution58 344044 51097659 375321 503358960 373118 538554961 370817 573555962 457275 628602963 544350 688184664 530408 708518365 502322 690275266 526230 741481167 556568 773325968 639735 867242469 578533 860229870 643592 914437371 589557 893978872 670866 950592573 736669 1058290574 660395 103902375 749795 1115704676 835418 118030077 727648 1157821978 852689 114767679 971135 1273329880 920522 1294614881 882645 142056782 1084711 1500037883 1072241 1436365184 1072132 1606669385 1154018 1552139286 1150925 1756302387 1164220 1608156388 1230008 1726587489 1379479 1913688390 1351210 1970899391 1275975 1895126492 1360771 211423093 1520228 1969614194 1542740 212520095 1581145 2230222796 1640792 2305569397 1486796 2362639498 1645889 2365109399 1607866 24907337100 2073979 26356664
the direction of steepest descent which is opposite to thegradient direction The sizes of the envelop blocks are thenreduced and the optimization process is repeated for thesecond phase The number of optimization phases is equalto the magnification factor number for the envelop blocksIn the last optimization phase the dimensions of the envelopblocks become equal to the actual facilities For more detailwe refer to Mir and Imam [7]
Journal of Industrial Mathematics 7
Table 9 Summary of the results
Problem Number of facilities Best result by other methods GOT Cost reduction1 8 6925 7527 minus6022 20 1157 1302 minus1453 50 782247 768823 134244 100 5381931 5270941 110995 125 1084451 1062080 22371
So the proposed approach for solving a facility layoutproblem can be summarized as follows
Step 1 encapsulating facilities in envelop blocks (mul-tiplying the dimensions of facilities by amagnificationfactor)
Step 2 obtaining an MPG
Step 3 drawing the MPG on the plane and obtainingan initial solution
Step 4 improving initial solution by analyticalmethod
Figure 3 shows summary of these stepsThe proposed approach was coded using the VBNET
programming language in a program named GOT (Graphoptimization technique) Five test problems were run For alltest problems results were obtained on a PCwith Intel T5470processor The results were compared with the previouslypublished papers and commercial software VIP-PLANOPT2006 VIP-PLANOPT is a useful layout software package thatcan generate near-optimal layout [25] Formore details aboutVIP-PLANOPT see Engineering Optimization Software [8]VIP-PLANOPT results were obtained from the softwareuserrsquos manual The results are presented in the followingsections
41 Test Problem 1 This problem of 8 facilities was intro-duced by Imam and Mir [5] Figure 4 shows the steps foraccruing the initial solution Figure 4(a) shows the flowmatrix and dimension of facilities All dimensions and costmatrix elements are integer-valued numbers ranging between1 and 6 There are several pairs of facilities with no flowbetween them Distance norm is squared Euclidean Thegreedy heuristic generates the edges lists of MPG as shownin Figure 4(b)
The straight line drawing algorithm gives the coordinatesof vertices The drawing is shown in Figure 4(c) Then eachvertex is replaced by its correspondent facility The coordi-nates are multiplied by maximum dimensions of facilities(width and length) and finally the initial layout design isshown in Figure 4(d)
The solution is improved by the analytical techniqueFigure 5 shows the final layout The cost function value forthis layout is 7527 and the running time is 04 second Table 1shows the results obtained by the other approaches Thebest solution for this problem is obtained by VIP-PLANOPT2006
42 Test Problem 2 This problem of 20 unequal areafacilities was introduced by Imam and Mir [6] The dataconsist of only integer valuesThe dimensions of the facilitiesare between 1 and 3 The elements of the cost matrix areintegers between 0 and 5 The distance norm is rectilinearThe final layout obtained by GOT is shown in Figure 6 Thelayout cost is 1302 and the running time is 06 secondTable 2 compares the results obtained byGOTwith the resultsavailable in the literature VIP-PLANOPT2006 has the lowestvalue of the cost function
43 Test Problem 3 This is a problem of 50 facilities ran-domly generated by VIP-PLANOPT 2006 The dimensionsof the facilities are decimal numbers between 1 to 6 Theelements of the cost matrix are all integers between 1 and10 The distance norm is Euclidean The results are shownin Table 3 The best published result has a cost of 782247whereas GOT produces a final layout with a cost of 768823only in 151 seconds Figure 7 shows the final layout
44 Test Problem 4 This is a randomly generated large sizeproblem of 100 facilities The dimensions of the facilities aredecimal numbers between 1 and 6 The cost matrix elementsare integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 4 GOT obtained the costfunction value of 5270941 in 743 secondsThis value is about2 below the cost function value of VIP-PLANOPT 2006The coordinates of the facilities for the layout obtained byGOT are given in Table 5
45 Test Problem 5 This is a large size problem of 125facilities randomly generated by VIP-PLANOPT 2006 Thedimensions of facilities are real numbers between 1 and 6 andelements of the cost matrix are integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 6GOT obtained the cost function value of 1062080 in 1296secondsThis value is about 2 below the cost function valueof VIP-PLANOPT 2006 The coordinates of the facilities forthe layout obtained by GOT are given in Table 7
5 Comparing GOT Initial Solution withRandom Initial Solution
To compare the proposed initial solution (GOT initial solu-tion) with random initial solution a set of test problems (119899 =10 11 12 100) were generated The facilities dimensionswere 1 times 1 and flow matrices were randomly generatedbetween 0 and 10 For acquiring a random initial solution
8 Journal of Industrial Mathematics
Dimensions of the facilities Flow matrix
1
1
1
1 1
1
2
2
2
2
2
2
2 2
2
2
2
2
20
0
0
0
0
0
0
0
0 0
0
0 0
0
0
0 0
0 0 0
0
0
00
3
3
3
3
4
4
4
4
4
4
4
5
5
5
6
6
6
7 8
1
2
2
2
2
23
3
3
3
3
4
4
4
4
44
44
5
5
6
7
8
1
1
2
3 3
4
5
6
7
8
Length Width
(a) The raw facilities layout data
MPGalgorithm
1 5-2
3-2
7-6
8-5
6-3
4-2
3-1
4-3
5-42
3
4
5
6
7
8
9
Number10
11
12
13
14
15
16
17
18
NumberEdge6-4
8-2
8-4
2-1
7-3
8-7
7-2
7-4
7-1
Edge
(b) MPG
Graph drawing algorithm
1
7
8
5
2
3
4
6
(c) Straight line drawing of the MPG
Obtaining initial solution
1
7
8
5
2
3
4
6
(d) Initial solution
Figure 4 Obtaining an initial solution for test problem 1
1
7
85
2
34
6
Figure 5 Final layout for test problem 1
facilities were randomly placed in a (2119899 minus 4) times (119899 minus 2) integergrid For each test problem 20 random placements werefound Table 8 shows the value of cost function in GOT initialsolution and the best value found by random placementsFigures 8 and 9 shows these results graphically The resultsdemonstrated significant improvement in cost function
6 Summary and Conclusion
An initial solution has been presented for the layout designof facilities on a continuous plane The technique consists of
8 4 5
14 11 18 19 3
15
20
13
717
1 2 612
10
9
16
Figure 6 Final layout for test problem 2
two stages In the first stage amaximal planar graph (MPG) isobtained In the second stage the vertices of MPG are drawnon the plane by graph drawing algorithms Then vertices arereplaced by facilities Hence an initial solution is obtainedTo evaluate the performance this initial solution has beenapplied in rectangular facility layout problem and improvedby an analytical method by Mir and Imam [7]
The approach has been tested on five instances fromthe literature Table 6 shows the Summary of the resultsand Figure 8 shows the cost reduction by the technique Forthe large size problems involving 50 100 and 125 facilities
Journal of Industrial Mathematics 9
6 41 22
29
1224 44
4645
1425
5
26
28
10
9
3
2
8
4
30
31
43
47
49
50
40
42
48
36
34
3538
39
33
32
13
18
37
15 17
27 23
20
16
71 19
11
21
Figure 7 Final layout for test problem 3
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Random initial solutionGOT initial solution
Cos
t
05
1
15
2
25
3
35
4
45
5
times106
n
Figure 8 Comparisons of cost function value in GOT initialsolution with the best value of random placements
the layout costs values are better than those obtained by thepreviously published techniques As shown in Table 9 theresults demonstrate effectiveness of the technique especiallyfor large size problems
This paper introduced a simple technique for obtaining agood initial solutionThe technique with somemodificationcan be applied in facility layout approaches that use arandomly generated initial solution In future researches itwould be interesting to analyze the influence of MPG andgraph drawing algorithm on the solution The results can befurther improved by using a metaheuristic such as GRASP[21] and Tabu search [20] for generating a high weightedMPG
0
5
10
15
20
25
8 20 50 100 125
Reduction
times103
minus5
Figure 9 Cost reduction by using GOT
Conflict of Interests
The authors declare that they have no conflict of interests
References
[1] S P Singh and R R K Sharma ldquoA review of differentapproaches to the facility layout problemsrdquo International Jour-nal of Advanced Manufacturing Technology vol 30 no 5-6 pp425ndash433 2006
[2] M F Anjos and A Vannelli ldquoAn attractor-repeller approach tofloorplanningrdquo Mathematical Methods of Operations Researchvol 56 no 1 pp 3ndash27 2002
[3] I Castillo and T Sim ldquoA spring-embedding approach for thefacility layout problemrdquo Journal of the Operational ResearchSociety vol 55 no 1 pp 73ndash81 2004
[4] Z Drezner ldquoDISCON a new method for the layout problemrdquoOperations Research vol 28 no 6 pp 1375ndash1384 1980
[5] M H Imam and M Mir ldquoNonlinear programming approachto automated topology optimizationrdquo Computer-Aided Designvol 21 no 2 pp 107ndash115 1989
[6] M H Imam and M Mir ldquoAutomated layout of facilities ofunequal areasrdquo Computers and Industrial Engineering vol 24no 3 pp 355ndash366 1993
[7] M Mir and M H Imam ldquoHybrid optimization approachfor layout design of unequal-area facilitiesrdquo Computers andIndustrial Engineering vol 39 no 1-2 pp 49ndash63 2001
[8] Engineering Optimization Software VIP-PLANOPT 20062010 httpwwwplanoptcom
[9] L Foulds Graph Theory Applications Springer New York NYUSA 1992
[10] M M D Hassan and G L Hogg ldquoA review of graph theoryapplication to the facilities layout problemrdquo Omega vol 15 no4 pp 291ndash300 1987
[11] M M D Hassan and G L Hogg ldquoOn converting a dualgraph into a block layoutrdquo International Journal of ProductionResearch vol 27 no 7 pp 1149ndash1160 1989
[12] KHWatson and JWGiffin ldquoThevertex splitting algorithm forfacilities layoutrdquo International Journal of Production Researchvol 35 no 9 pp 2477ndash2492 1997
[13] S A Irvine and I Rinsma-Melchert ldquoA new approach tothe block layout problemrdquo International Journal of ProductionResearch vol 35 no 8 pp 2359ndash2376 1997
10 Journal of Industrial Mathematics
[14] P S Welgama P R Gibson and L A R Al-Hakim ldquoFacilitieslayout a knowledge-based approach for converting a dualgraph into a block layoutrdquo International Journal of ProductionEconomics vol 33 no 1ndash3 pp 17ndash30 1994
[15] M A Jokar and A S Sangchooli ldquoConstructing a blocklayout by face areardquo The International Journal of AdvancedManufacturing Technology vol 54 no 5ndash8 pp 801ndash809 2011
[16] L R Foulds P B Gibbons and J W Giffin ldquoFacilities layoutadjacency determination an experimental comparison of threegraph theoretic heuristicsrdquo Operations Research vol 33 no 5pp 1091ndash1106 1985
[17] E G John and J Hammond ldquoMaximally weighted graphtheoretic facilities design planningrdquo International Journal ofProduction Research vol 38 no 16 pp 3845ndash3859 2000
[18] S G Boswell ldquoTESSAmdasha new greedy heuristic for facilitieslayout planningrdquo International Journal of Production Researchvol 30 no 8 pp 1957ndash1968 1992
[19] L R Foulds and D F Robinson ldquoGraph theoretic heuristics forthe plant layout problemrdquo International Journal of ProductionResearch vol 16 no 1 pp 27ndash37 1978
[20] I H Osman ldquoA tabu search procedure based on a randomRoulette diversification for the weighted maximal planar graphproblemrdquoComputers and Operations Research vol 33 no 9 pp2526ndash2546 2006
[21] I H Osman B Al-Ayoubi and M Barake ldquoA greedy randomadaptive search procedure for the weighted maximal planargraph problemrdquo Computers and Industrial Engineering vol 45no 4 pp 635ndash651 2003
[22] J M Boyer and W J Myrvold ldquoOn the cutting edge simplifiedO(n) planarity by edge additionrdquo Journal of Graph Algorithmsand Applications vol 8 no 3 pp 241ndash273 2004
[23] G Di Battista P Eades R Tamassia and I Tollis GraphDrawing Algorithms for the Visualization of Graphs PrenticeHall PTR Upper Saddle River NJ USA 1998
[24] M Chrobak and T H Payne ldquoA linear-time algorithm fordrawing a planar graph on a gridrdquo Information ProcessingLetters vol 54 no 4 pp 241ndash246 1995
[25] S Heragu Facilities Design Iuniverse Inc 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Industrial Mathematics 5
Table 7 Continued
Facility 119883 119884
49 40189 3629450 12672 3682551 34282 2734952 19126 1500553 36395 4132254 31473 4003455 30991 3498356 38702 270357 28176 127358 29219 4521859 25113 1076660 11021 2702961 22158 1423962 22543 1758263 12348 2132264 13684 439665 8569 3614966 8828 3323967 10933 1237368 24203 2604769 12647 707170 29789 2736271 6492 3864272 18394 1799373 4815 1775474 24465 4564675 24704 2444876 48912 3116477 19063 1108878 27843 1637979 35271 2921780 21659 3542981 15303 2808782 32599 310783 3311 2083684 4402 337685 43843 986286 16694 2389987 20628 172388 19088 3556389 48998 2530890 33559 4617691 31162 308692 2482 3660793 44673 2438194 18385 2769495 30086 1085896 33289 1515697 30399 1718898 1552 1907199 36518 32041100 19439 4084101 22932 4334
Table 7 Continued
Facility 119883 119884
102 27279 28155103 31849 19046104 1148 19243105 731 21826106 38088 12744107 22892 30522108 14917 13713109 6995 30065110 40466 30376111 41964 1947112 303 14954113 19953 27768114 27316 2155115 35983 36293116 44374 28853117 22621 41026118 3199 20775119 11309 3988120 41379 24172121 31589 26996122 18031 8178123 18549 22682124 3164 23551125 33452 23294
a drawing of G A graph has infinitely many different draw-ings For a review of various graphs drawing algorithms referto [23] We use algorithm of Chrobak and Payne [24] to forma straight line drawing of the MPG In such a drawing eachedge is drawn using a straight line segment The algorithmdraws vertices in an MPG to integer coordinates in a (2119873 minus4) times (119873 minus 2) grid Figure 2 shows an example of straight linedrawing
For acquiring an initial solution each vertex is replacedby its correspondent facility In a feasible solution facilitieshave no overlaps For this reason the coordinates of facilitiescan be multiplied by maximum dimensions of all facili-ties (width and length) This operation increases distancebetween facilities andmakes the solution feasible For the caseof circular facilities the diameter of circle can be consideredas maximum dimensions
4 Improving Initial Solution and Comparing
To evaluate the performance the initial solution is improvedby an analytical method by Mir and Imam [7] In thismethod the convergence is controlled by carrying out theoptimization using concept of ldquomagnified envelop blocksrdquoThe dimensions of the blocks are determined by multiplyingthe dimensions of the facilities with a ldquomagnification factorrdquoThe optimization is then carried out for these envelopblocks rather than the actual facilitiesThe analytical methodsearches the optimum placements of each envelop block in
6 Journal of Industrial Mathematics
Table 8 The value of cost function in GOT initial solution and thebest value of random placements
119899 GOT initial solution The best random initial solution10 1945 2437311 2122 2758512 2368 3531713 4919 576414 5514 702415 5538 744916 8116 10287717 8468 12207518 10648 1530419 13943 19119320 14265 2009421 17199 22491222 16610 2637023 22128 30749224 25533 3550825 27585 37968426 29929 43803627 36255 5185028 41483 57050529 47543 65109230 57830 73705831 61408 82058432 63687 8643633 59970 94066434 82721 110195235 76220 10494036 92426 124186137 95386 12546838 87532 15006039 93708 146793840 118266 18053841 141363 18391142 104263 195458743 152188 204207344 163529 23925345 166360 237051846 167002 251049347 189027 29008548 226097 30590049 234811 324500350 238855 337198951 264842 358480552 251009 358666753 233805 370262454 298806 404104455 284543 431091956 344755 481656957 371962 4939864
Table 8 Continued
119899 GOT initial solution The best random initial solution58 344044 51097659 375321 503358960 373118 538554961 370817 573555962 457275 628602963 544350 688184664 530408 708518365 502322 690275266 526230 741481167 556568 773325968 639735 867242469 578533 860229870 643592 914437371 589557 893978872 670866 950592573 736669 1058290574 660395 103902375 749795 1115704676 835418 118030077 727648 1157821978 852689 114767679 971135 1273329880 920522 1294614881 882645 142056782 1084711 1500037883 1072241 1436365184 1072132 1606669385 1154018 1552139286 1150925 1756302387 1164220 1608156388 1230008 1726587489 1379479 1913688390 1351210 1970899391 1275975 1895126492 1360771 211423093 1520228 1969614194 1542740 212520095 1581145 2230222796 1640792 2305569397 1486796 2362639498 1645889 2365109399 1607866 24907337100 2073979 26356664
the direction of steepest descent which is opposite to thegradient direction The sizes of the envelop blocks are thenreduced and the optimization process is repeated for thesecond phase The number of optimization phases is equalto the magnification factor number for the envelop blocksIn the last optimization phase the dimensions of the envelopblocks become equal to the actual facilities For more detailwe refer to Mir and Imam [7]
Journal of Industrial Mathematics 7
Table 9 Summary of the results
Problem Number of facilities Best result by other methods GOT Cost reduction1 8 6925 7527 minus6022 20 1157 1302 minus1453 50 782247 768823 134244 100 5381931 5270941 110995 125 1084451 1062080 22371
So the proposed approach for solving a facility layoutproblem can be summarized as follows
Step 1 encapsulating facilities in envelop blocks (mul-tiplying the dimensions of facilities by amagnificationfactor)
Step 2 obtaining an MPG
Step 3 drawing the MPG on the plane and obtainingan initial solution
Step 4 improving initial solution by analyticalmethod
Figure 3 shows summary of these stepsThe proposed approach was coded using the VBNET
programming language in a program named GOT (Graphoptimization technique) Five test problems were run For alltest problems results were obtained on a PCwith Intel T5470processor The results were compared with the previouslypublished papers and commercial software VIP-PLANOPT2006 VIP-PLANOPT is a useful layout software package thatcan generate near-optimal layout [25] Formore details aboutVIP-PLANOPT see Engineering Optimization Software [8]VIP-PLANOPT results were obtained from the softwareuserrsquos manual The results are presented in the followingsections
41 Test Problem 1 This problem of 8 facilities was intro-duced by Imam and Mir [5] Figure 4 shows the steps foraccruing the initial solution Figure 4(a) shows the flowmatrix and dimension of facilities All dimensions and costmatrix elements are integer-valued numbers ranging between1 and 6 There are several pairs of facilities with no flowbetween them Distance norm is squared Euclidean Thegreedy heuristic generates the edges lists of MPG as shownin Figure 4(b)
The straight line drawing algorithm gives the coordinatesof vertices The drawing is shown in Figure 4(c) Then eachvertex is replaced by its correspondent facility The coordi-nates are multiplied by maximum dimensions of facilities(width and length) and finally the initial layout design isshown in Figure 4(d)
The solution is improved by the analytical techniqueFigure 5 shows the final layout The cost function value forthis layout is 7527 and the running time is 04 second Table 1shows the results obtained by the other approaches Thebest solution for this problem is obtained by VIP-PLANOPT2006
42 Test Problem 2 This problem of 20 unequal areafacilities was introduced by Imam and Mir [6] The dataconsist of only integer valuesThe dimensions of the facilitiesare between 1 and 3 The elements of the cost matrix areintegers between 0 and 5 The distance norm is rectilinearThe final layout obtained by GOT is shown in Figure 6 Thelayout cost is 1302 and the running time is 06 secondTable 2 compares the results obtained byGOTwith the resultsavailable in the literature VIP-PLANOPT2006 has the lowestvalue of the cost function
43 Test Problem 3 This is a problem of 50 facilities ran-domly generated by VIP-PLANOPT 2006 The dimensionsof the facilities are decimal numbers between 1 to 6 Theelements of the cost matrix are all integers between 1 and10 The distance norm is Euclidean The results are shownin Table 3 The best published result has a cost of 782247whereas GOT produces a final layout with a cost of 768823only in 151 seconds Figure 7 shows the final layout
44 Test Problem 4 This is a randomly generated large sizeproblem of 100 facilities The dimensions of the facilities aredecimal numbers between 1 and 6 The cost matrix elementsare integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 4 GOT obtained the costfunction value of 5270941 in 743 secondsThis value is about2 below the cost function value of VIP-PLANOPT 2006The coordinates of the facilities for the layout obtained byGOT are given in Table 5
45 Test Problem 5 This is a large size problem of 125facilities randomly generated by VIP-PLANOPT 2006 Thedimensions of facilities are real numbers between 1 and 6 andelements of the cost matrix are integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 6GOT obtained the cost function value of 1062080 in 1296secondsThis value is about 2 below the cost function valueof VIP-PLANOPT 2006 The coordinates of the facilities forthe layout obtained by GOT are given in Table 7
5 Comparing GOT Initial Solution withRandom Initial Solution
To compare the proposed initial solution (GOT initial solu-tion) with random initial solution a set of test problems (119899 =10 11 12 100) were generated The facilities dimensionswere 1 times 1 and flow matrices were randomly generatedbetween 0 and 10 For acquiring a random initial solution
8 Journal of Industrial Mathematics
Dimensions of the facilities Flow matrix
1
1
1
1 1
1
2
2
2
2
2
2
2 2
2
2
2
2
20
0
0
0
0
0
0
0
0 0
0
0 0
0
0
0 0
0 0 0
0
0
00
3
3
3
3
4
4
4
4
4
4
4
5
5
5
6
6
6
7 8
1
2
2
2
2
23
3
3
3
3
4
4
4
4
44
44
5
5
6
7
8
1
1
2
3 3
4
5
6
7
8
Length Width
(a) The raw facilities layout data
MPGalgorithm
1 5-2
3-2
7-6
8-5
6-3
4-2
3-1
4-3
5-42
3
4
5
6
7
8
9
Number10
11
12
13
14
15
16
17
18
NumberEdge6-4
8-2
8-4
2-1
7-3
8-7
7-2
7-4
7-1
Edge
(b) MPG
Graph drawing algorithm
1
7
8
5
2
3
4
6
(c) Straight line drawing of the MPG
Obtaining initial solution
1
7
8
5
2
3
4
6
(d) Initial solution
Figure 4 Obtaining an initial solution for test problem 1
1
7
85
2
34
6
Figure 5 Final layout for test problem 1
facilities were randomly placed in a (2119899 minus 4) times (119899 minus 2) integergrid For each test problem 20 random placements werefound Table 8 shows the value of cost function in GOT initialsolution and the best value found by random placementsFigures 8 and 9 shows these results graphically The resultsdemonstrated significant improvement in cost function
6 Summary and Conclusion
An initial solution has been presented for the layout designof facilities on a continuous plane The technique consists of
8 4 5
14 11 18 19 3
15
20
13
717
1 2 612
10
9
16
Figure 6 Final layout for test problem 2
two stages In the first stage amaximal planar graph (MPG) isobtained In the second stage the vertices of MPG are drawnon the plane by graph drawing algorithms Then vertices arereplaced by facilities Hence an initial solution is obtainedTo evaluate the performance this initial solution has beenapplied in rectangular facility layout problem and improvedby an analytical method by Mir and Imam [7]
The approach has been tested on five instances fromthe literature Table 6 shows the Summary of the resultsand Figure 8 shows the cost reduction by the technique Forthe large size problems involving 50 100 and 125 facilities
Journal of Industrial Mathematics 9
6 41 22
29
1224 44
4645
1425
5
26
28
10
9
3
2
8
4
30
31
43
47
49
50
40
42
48
36
34
3538
39
33
32
13
18
37
15 17
27 23
20
16
71 19
11
21
Figure 7 Final layout for test problem 3
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Random initial solutionGOT initial solution
Cos
t
05
1
15
2
25
3
35
4
45
5
times106
n
Figure 8 Comparisons of cost function value in GOT initialsolution with the best value of random placements
the layout costs values are better than those obtained by thepreviously published techniques As shown in Table 9 theresults demonstrate effectiveness of the technique especiallyfor large size problems
This paper introduced a simple technique for obtaining agood initial solutionThe technique with somemodificationcan be applied in facility layout approaches that use arandomly generated initial solution In future researches itwould be interesting to analyze the influence of MPG andgraph drawing algorithm on the solution The results can befurther improved by using a metaheuristic such as GRASP[21] and Tabu search [20] for generating a high weightedMPG
0
5
10
15
20
25
8 20 50 100 125
Reduction
times103
minus5
Figure 9 Cost reduction by using GOT
Conflict of Interests
The authors declare that they have no conflict of interests
References
[1] S P Singh and R R K Sharma ldquoA review of differentapproaches to the facility layout problemsrdquo International Jour-nal of Advanced Manufacturing Technology vol 30 no 5-6 pp425ndash433 2006
[2] M F Anjos and A Vannelli ldquoAn attractor-repeller approach tofloorplanningrdquo Mathematical Methods of Operations Researchvol 56 no 1 pp 3ndash27 2002
[3] I Castillo and T Sim ldquoA spring-embedding approach for thefacility layout problemrdquo Journal of the Operational ResearchSociety vol 55 no 1 pp 73ndash81 2004
[4] Z Drezner ldquoDISCON a new method for the layout problemrdquoOperations Research vol 28 no 6 pp 1375ndash1384 1980
[5] M H Imam and M Mir ldquoNonlinear programming approachto automated topology optimizationrdquo Computer-Aided Designvol 21 no 2 pp 107ndash115 1989
[6] M H Imam and M Mir ldquoAutomated layout of facilities ofunequal areasrdquo Computers and Industrial Engineering vol 24no 3 pp 355ndash366 1993
[7] M Mir and M H Imam ldquoHybrid optimization approachfor layout design of unequal-area facilitiesrdquo Computers andIndustrial Engineering vol 39 no 1-2 pp 49ndash63 2001
[8] Engineering Optimization Software VIP-PLANOPT 20062010 httpwwwplanoptcom
[9] L Foulds Graph Theory Applications Springer New York NYUSA 1992
[10] M M D Hassan and G L Hogg ldquoA review of graph theoryapplication to the facilities layout problemrdquo Omega vol 15 no4 pp 291ndash300 1987
[11] M M D Hassan and G L Hogg ldquoOn converting a dualgraph into a block layoutrdquo International Journal of ProductionResearch vol 27 no 7 pp 1149ndash1160 1989
[12] KHWatson and JWGiffin ldquoThevertex splitting algorithm forfacilities layoutrdquo International Journal of Production Researchvol 35 no 9 pp 2477ndash2492 1997
[13] S A Irvine and I Rinsma-Melchert ldquoA new approach tothe block layout problemrdquo International Journal of ProductionResearch vol 35 no 8 pp 2359ndash2376 1997
10 Journal of Industrial Mathematics
[14] P S Welgama P R Gibson and L A R Al-Hakim ldquoFacilitieslayout a knowledge-based approach for converting a dualgraph into a block layoutrdquo International Journal of ProductionEconomics vol 33 no 1ndash3 pp 17ndash30 1994
[15] M A Jokar and A S Sangchooli ldquoConstructing a blocklayout by face areardquo The International Journal of AdvancedManufacturing Technology vol 54 no 5ndash8 pp 801ndash809 2011
[16] L R Foulds P B Gibbons and J W Giffin ldquoFacilities layoutadjacency determination an experimental comparison of threegraph theoretic heuristicsrdquo Operations Research vol 33 no 5pp 1091ndash1106 1985
[17] E G John and J Hammond ldquoMaximally weighted graphtheoretic facilities design planningrdquo International Journal ofProduction Research vol 38 no 16 pp 3845ndash3859 2000
[18] S G Boswell ldquoTESSAmdasha new greedy heuristic for facilitieslayout planningrdquo International Journal of Production Researchvol 30 no 8 pp 1957ndash1968 1992
[19] L R Foulds and D F Robinson ldquoGraph theoretic heuristics forthe plant layout problemrdquo International Journal of ProductionResearch vol 16 no 1 pp 27ndash37 1978
[20] I H Osman ldquoA tabu search procedure based on a randomRoulette diversification for the weighted maximal planar graphproblemrdquoComputers and Operations Research vol 33 no 9 pp2526ndash2546 2006
[21] I H Osman B Al-Ayoubi and M Barake ldquoA greedy randomadaptive search procedure for the weighted maximal planargraph problemrdquo Computers and Industrial Engineering vol 45no 4 pp 635ndash651 2003
[22] J M Boyer and W J Myrvold ldquoOn the cutting edge simplifiedO(n) planarity by edge additionrdquo Journal of Graph Algorithmsand Applications vol 8 no 3 pp 241ndash273 2004
[23] G Di Battista P Eades R Tamassia and I Tollis GraphDrawing Algorithms for the Visualization of Graphs PrenticeHall PTR Upper Saddle River NJ USA 1998
[24] M Chrobak and T H Payne ldquoA linear-time algorithm fordrawing a planar graph on a gridrdquo Information ProcessingLetters vol 54 no 4 pp 241ndash246 1995
[25] S Heragu Facilities Design Iuniverse Inc 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Industrial Mathematics
Table 8 The value of cost function in GOT initial solution and thebest value of random placements
119899 GOT initial solution The best random initial solution10 1945 2437311 2122 2758512 2368 3531713 4919 576414 5514 702415 5538 744916 8116 10287717 8468 12207518 10648 1530419 13943 19119320 14265 2009421 17199 22491222 16610 2637023 22128 30749224 25533 3550825 27585 37968426 29929 43803627 36255 5185028 41483 57050529 47543 65109230 57830 73705831 61408 82058432 63687 8643633 59970 94066434 82721 110195235 76220 10494036 92426 124186137 95386 12546838 87532 15006039 93708 146793840 118266 18053841 141363 18391142 104263 195458743 152188 204207344 163529 23925345 166360 237051846 167002 251049347 189027 29008548 226097 30590049 234811 324500350 238855 337198951 264842 358480552 251009 358666753 233805 370262454 298806 404104455 284543 431091956 344755 481656957 371962 4939864
Table 8 Continued
119899 GOT initial solution The best random initial solution58 344044 51097659 375321 503358960 373118 538554961 370817 573555962 457275 628602963 544350 688184664 530408 708518365 502322 690275266 526230 741481167 556568 773325968 639735 867242469 578533 860229870 643592 914437371 589557 893978872 670866 950592573 736669 1058290574 660395 103902375 749795 1115704676 835418 118030077 727648 1157821978 852689 114767679 971135 1273329880 920522 1294614881 882645 142056782 1084711 1500037883 1072241 1436365184 1072132 1606669385 1154018 1552139286 1150925 1756302387 1164220 1608156388 1230008 1726587489 1379479 1913688390 1351210 1970899391 1275975 1895126492 1360771 211423093 1520228 1969614194 1542740 212520095 1581145 2230222796 1640792 2305569397 1486796 2362639498 1645889 2365109399 1607866 24907337100 2073979 26356664
the direction of steepest descent which is opposite to thegradient direction The sizes of the envelop blocks are thenreduced and the optimization process is repeated for thesecond phase The number of optimization phases is equalto the magnification factor number for the envelop blocksIn the last optimization phase the dimensions of the envelopblocks become equal to the actual facilities For more detailwe refer to Mir and Imam [7]
Journal of Industrial Mathematics 7
Table 9 Summary of the results
Problem Number of facilities Best result by other methods GOT Cost reduction1 8 6925 7527 minus6022 20 1157 1302 minus1453 50 782247 768823 134244 100 5381931 5270941 110995 125 1084451 1062080 22371
So the proposed approach for solving a facility layoutproblem can be summarized as follows
Step 1 encapsulating facilities in envelop blocks (mul-tiplying the dimensions of facilities by amagnificationfactor)
Step 2 obtaining an MPG
Step 3 drawing the MPG on the plane and obtainingan initial solution
Step 4 improving initial solution by analyticalmethod
Figure 3 shows summary of these stepsThe proposed approach was coded using the VBNET
programming language in a program named GOT (Graphoptimization technique) Five test problems were run For alltest problems results were obtained on a PCwith Intel T5470processor The results were compared with the previouslypublished papers and commercial software VIP-PLANOPT2006 VIP-PLANOPT is a useful layout software package thatcan generate near-optimal layout [25] Formore details aboutVIP-PLANOPT see Engineering Optimization Software [8]VIP-PLANOPT results were obtained from the softwareuserrsquos manual The results are presented in the followingsections
41 Test Problem 1 This problem of 8 facilities was intro-duced by Imam and Mir [5] Figure 4 shows the steps foraccruing the initial solution Figure 4(a) shows the flowmatrix and dimension of facilities All dimensions and costmatrix elements are integer-valued numbers ranging between1 and 6 There are several pairs of facilities with no flowbetween them Distance norm is squared Euclidean Thegreedy heuristic generates the edges lists of MPG as shownin Figure 4(b)
The straight line drawing algorithm gives the coordinatesof vertices The drawing is shown in Figure 4(c) Then eachvertex is replaced by its correspondent facility The coordi-nates are multiplied by maximum dimensions of facilities(width and length) and finally the initial layout design isshown in Figure 4(d)
The solution is improved by the analytical techniqueFigure 5 shows the final layout The cost function value forthis layout is 7527 and the running time is 04 second Table 1shows the results obtained by the other approaches Thebest solution for this problem is obtained by VIP-PLANOPT2006
42 Test Problem 2 This problem of 20 unequal areafacilities was introduced by Imam and Mir [6] The dataconsist of only integer valuesThe dimensions of the facilitiesare between 1 and 3 The elements of the cost matrix areintegers between 0 and 5 The distance norm is rectilinearThe final layout obtained by GOT is shown in Figure 6 Thelayout cost is 1302 and the running time is 06 secondTable 2 compares the results obtained byGOTwith the resultsavailable in the literature VIP-PLANOPT2006 has the lowestvalue of the cost function
43 Test Problem 3 This is a problem of 50 facilities ran-domly generated by VIP-PLANOPT 2006 The dimensionsof the facilities are decimal numbers between 1 to 6 Theelements of the cost matrix are all integers between 1 and10 The distance norm is Euclidean The results are shownin Table 3 The best published result has a cost of 782247whereas GOT produces a final layout with a cost of 768823only in 151 seconds Figure 7 shows the final layout
44 Test Problem 4 This is a randomly generated large sizeproblem of 100 facilities The dimensions of the facilities aredecimal numbers between 1 and 6 The cost matrix elementsare integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 4 GOT obtained the costfunction value of 5270941 in 743 secondsThis value is about2 below the cost function value of VIP-PLANOPT 2006The coordinates of the facilities for the layout obtained byGOT are given in Table 5
45 Test Problem 5 This is a large size problem of 125facilities randomly generated by VIP-PLANOPT 2006 Thedimensions of facilities are real numbers between 1 and 6 andelements of the cost matrix are integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 6GOT obtained the cost function value of 1062080 in 1296secondsThis value is about 2 below the cost function valueof VIP-PLANOPT 2006 The coordinates of the facilities forthe layout obtained by GOT are given in Table 7
5 Comparing GOT Initial Solution withRandom Initial Solution
To compare the proposed initial solution (GOT initial solu-tion) with random initial solution a set of test problems (119899 =10 11 12 100) were generated The facilities dimensionswere 1 times 1 and flow matrices were randomly generatedbetween 0 and 10 For acquiring a random initial solution
8 Journal of Industrial Mathematics
Dimensions of the facilities Flow matrix
1
1
1
1 1
1
2
2
2
2
2
2
2 2
2
2
2
2
20
0
0
0
0
0
0
0
0 0
0
0 0
0
0
0 0
0 0 0
0
0
00
3
3
3
3
4
4
4
4
4
4
4
5
5
5
6
6
6
7 8
1
2
2
2
2
23
3
3
3
3
4
4
4
4
44
44
5
5
6
7
8
1
1
2
3 3
4
5
6
7
8
Length Width
(a) The raw facilities layout data
MPGalgorithm
1 5-2
3-2
7-6
8-5
6-3
4-2
3-1
4-3
5-42
3
4
5
6
7
8
9
Number10
11
12
13
14
15
16
17
18
NumberEdge6-4
8-2
8-4
2-1
7-3
8-7
7-2
7-4
7-1
Edge
(b) MPG
Graph drawing algorithm
1
7
8
5
2
3
4
6
(c) Straight line drawing of the MPG
Obtaining initial solution
1
7
8
5
2
3
4
6
(d) Initial solution
Figure 4 Obtaining an initial solution for test problem 1
1
7
85
2
34
6
Figure 5 Final layout for test problem 1
facilities were randomly placed in a (2119899 minus 4) times (119899 minus 2) integergrid For each test problem 20 random placements werefound Table 8 shows the value of cost function in GOT initialsolution and the best value found by random placementsFigures 8 and 9 shows these results graphically The resultsdemonstrated significant improvement in cost function
6 Summary and Conclusion
An initial solution has been presented for the layout designof facilities on a continuous plane The technique consists of
8 4 5
14 11 18 19 3
15
20
13
717
1 2 612
10
9
16
Figure 6 Final layout for test problem 2
two stages In the first stage amaximal planar graph (MPG) isobtained In the second stage the vertices of MPG are drawnon the plane by graph drawing algorithms Then vertices arereplaced by facilities Hence an initial solution is obtainedTo evaluate the performance this initial solution has beenapplied in rectangular facility layout problem and improvedby an analytical method by Mir and Imam [7]
The approach has been tested on five instances fromthe literature Table 6 shows the Summary of the resultsand Figure 8 shows the cost reduction by the technique Forthe large size problems involving 50 100 and 125 facilities
Journal of Industrial Mathematics 9
6 41 22
29
1224 44
4645
1425
5
26
28
10
9
3
2
8
4
30
31
43
47
49
50
40
42
48
36
34
3538
39
33
32
13
18
37
15 17
27 23
20
16
71 19
11
21
Figure 7 Final layout for test problem 3
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Random initial solutionGOT initial solution
Cos
t
05
1
15
2
25
3
35
4
45
5
times106
n
Figure 8 Comparisons of cost function value in GOT initialsolution with the best value of random placements
the layout costs values are better than those obtained by thepreviously published techniques As shown in Table 9 theresults demonstrate effectiveness of the technique especiallyfor large size problems
This paper introduced a simple technique for obtaining agood initial solutionThe technique with somemodificationcan be applied in facility layout approaches that use arandomly generated initial solution In future researches itwould be interesting to analyze the influence of MPG andgraph drawing algorithm on the solution The results can befurther improved by using a metaheuristic such as GRASP[21] and Tabu search [20] for generating a high weightedMPG
0
5
10
15
20
25
8 20 50 100 125
Reduction
times103
minus5
Figure 9 Cost reduction by using GOT
Conflict of Interests
The authors declare that they have no conflict of interests
References
[1] S P Singh and R R K Sharma ldquoA review of differentapproaches to the facility layout problemsrdquo International Jour-nal of Advanced Manufacturing Technology vol 30 no 5-6 pp425ndash433 2006
[2] M F Anjos and A Vannelli ldquoAn attractor-repeller approach tofloorplanningrdquo Mathematical Methods of Operations Researchvol 56 no 1 pp 3ndash27 2002
[3] I Castillo and T Sim ldquoA spring-embedding approach for thefacility layout problemrdquo Journal of the Operational ResearchSociety vol 55 no 1 pp 73ndash81 2004
[4] Z Drezner ldquoDISCON a new method for the layout problemrdquoOperations Research vol 28 no 6 pp 1375ndash1384 1980
[5] M H Imam and M Mir ldquoNonlinear programming approachto automated topology optimizationrdquo Computer-Aided Designvol 21 no 2 pp 107ndash115 1989
[6] M H Imam and M Mir ldquoAutomated layout of facilities ofunequal areasrdquo Computers and Industrial Engineering vol 24no 3 pp 355ndash366 1993
[7] M Mir and M H Imam ldquoHybrid optimization approachfor layout design of unequal-area facilitiesrdquo Computers andIndustrial Engineering vol 39 no 1-2 pp 49ndash63 2001
[8] Engineering Optimization Software VIP-PLANOPT 20062010 httpwwwplanoptcom
[9] L Foulds Graph Theory Applications Springer New York NYUSA 1992
[10] M M D Hassan and G L Hogg ldquoA review of graph theoryapplication to the facilities layout problemrdquo Omega vol 15 no4 pp 291ndash300 1987
[11] M M D Hassan and G L Hogg ldquoOn converting a dualgraph into a block layoutrdquo International Journal of ProductionResearch vol 27 no 7 pp 1149ndash1160 1989
[12] KHWatson and JWGiffin ldquoThevertex splitting algorithm forfacilities layoutrdquo International Journal of Production Researchvol 35 no 9 pp 2477ndash2492 1997
[13] S A Irvine and I Rinsma-Melchert ldquoA new approach tothe block layout problemrdquo International Journal of ProductionResearch vol 35 no 8 pp 2359ndash2376 1997
10 Journal of Industrial Mathematics
[14] P S Welgama P R Gibson and L A R Al-Hakim ldquoFacilitieslayout a knowledge-based approach for converting a dualgraph into a block layoutrdquo International Journal of ProductionEconomics vol 33 no 1ndash3 pp 17ndash30 1994
[15] M A Jokar and A S Sangchooli ldquoConstructing a blocklayout by face areardquo The International Journal of AdvancedManufacturing Technology vol 54 no 5ndash8 pp 801ndash809 2011
[16] L R Foulds P B Gibbons and J W Giffin ldquoFacilities layoutadjacency determination an experimental comparison of threegraph theoretic heuristicsrdquo Operations Research vol 33 no 5pp 1091ndash1106 1985
[17] E G John and J Hammond ldquoMaximally weighted graphtheoretic facilities design planningrdquo International Journal ofProduction Research vol 38 no 16 pp 3845ndash3859 2000
[18] S G Boswell ldquoTESSAmdasha new greedy heuristic for facilitieslayout planningrdquo International Journal of Production Researchvol 30 no 8 pp 1957ndash1968 1992
[19] L R Foulds and D F Robinson ldquoGraph theoretic heuristics forthe plant layout problemrdquo International Journal of ProductionResearch vol 16 no 1 pp 27ndash37 1978
[20] I H Osman ldquoA tabu search procedure based on a randomRoulette diversification for the weighted maximal planar graphproblemrdquoComputers and Operations Research vol 33 no 9 pp2526ndash2546 2006
[21] I H Osman B Al-Ayoubi and M Barake ldquoA greedy randomadaptive search procedure for the weighted maximal planargraph problemrdquo Computers and Industrial Engineering vol 45no 4 pp 635ndash651 2003
[22] J M Boyer and W J Myrvold ldquoOn the cutting edge simplifiedO(n) planarity by edge additionrdquo Journal of Graph Algorithmsand Applications vol 8 no 3 pp 241ndash273 2004
[23] G Di Battista P Eades R Tamassia and I Tollis GraphDrawing Algorithms for the Visualization of Graphs PrenticeHall PTR Upper Saddle River NJ USA 1998
[24] M Chrobak and T H Payne ldquoA linear-time algorithm fordrawing a planar graph on a gridrdquo Information ProcessingLetters vol 54 no 4 pp 241ndash246 1995
[25] S Heragu Facilities Design Iuniverse Inc 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Industrial Mathematics 7
Table 9 Summary of the results
Problem Number of facilities Best result by other methods GOT Cost reduction1 8 6925 7527 minus6022 20 1157 1302 minus1453 50 782247 768823 134244 100 5381931 5270941 110995 125 1084451 1062080 22371
So the proposed approach for solving a facility layoutproblem can be summarized as follows
Step 1 encapsulating facilities in envelop blocks (mul-tiplying the dimensions of facilities by amagnificationfactor)
Step 2 obtaining an MPG
Step 3 drawing the MPG on the plane and obtainingan initial solution
Step 4 improving initial solution by analyticalmethod
Figure 3 shows summary of these stepsThe proposed approach was coded using the VBNET
programming language in a program named GOT (Graphoptimization technique) Five test problems were run For alltest problems results were obtained on a PCwith Intel T5470processor The results were compared with the previouslypublished papers and commercial software VIP-PLANOPT2006 VIP-PLANOPT is a useful layout software package thatcan generate near-optimal layout [25] Formore details aboutVIP-PLANOPT see Engineering Optimization Software [8]VIP-PLANOPT results were obtained from the softwareuserrsquos manual The results are presented in the followingsections
41 Test Problem 1 This problem of 8 facilities was intro-duced by Imam and Mir [5] Figure 4 shows the steps foraccruing the initial solution Figure 4(a) shows the flowmatrix and dimension of facilities All dimensions and costmatrix elements are integer-valued numbers ranging between1 and 6 There are several pairs of facilities with no flowbetween them Distance norm is squared Euclidean Thegreedy heuristic generates the edges lists of MPG as shownin Figure 4(b)
The straight line drawing algorithm gives the coordinatesof vertices The drawing is shown in Figure 4(c) Then eachvertex is replaced by its correspondent facility The coordi-nates are multiplied by maximum dimensions of facilities(width and length) and finally the initial layout design isshown in Figure 4(d)
The solution is improved by the analytical techniqueFigure 5 shows the final layout The cost function value forthis layout is 7527 and the running time is 04 second Table 1shows the results obtained by the other approaches Thebest solution for this problem is obtained by VIP-PLANOPT2006
42 Test Problem 2 This problem of 20 unequal areafacilities was introduced by Imam and Mir [6] The dataconsist of only integer valuesThe dimensions of the facilitiesare between 1 and 3 The elements of the cost matrix areintegers between 0 and 5 The distance norm is rectilinearThe final layout obtained by GOT is shown in Figure 6 Thelayout cost is 1302 and the running time is 06 secondTable 2 compares the results obtained byGOTwith the resultsavailable in the literature VIP-PLANOPT2006 has the lowestvalue of the cost function
43 Test Problem 3 This is a problem of 50 facilities ran-domly generated by VIP-PLANOPT 2006 The dimensionsof the facilities are decimal numbers between 1 to 6 Theelements of the cost matrix are all integers between 1 and10 The distance norm is Euclidean The results are shownin Table 3 The best published result has a cost of 782247whereas GOT produces a final layout with a cost of 768823only in 151 seconds Figure 7 shows the final layout
44 Test Problem 4 This is a randomly generated large sizeproblem of 100 facilities The dimensions of the facilities aredecimal numbers between 1 and 6 The cost matrix elementsare integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 4 GOT obtained the costfunction value of 5270941 in 743 secondsThis value is about2 below the cost function value of VIP-PLANOPT 2006The coordinates of the facilities for the layout obtained byGOT are given in Table 5
45 Test Problem 5 This is a large size problem of 125facilities randomly generated by VIP-PLANOPT 2006 Thedimensions of facilities are real numbers between 1 and 6 andelements of the cost matrix are integers between 1 and 10Thedistance norm is rectilinearThe results are shown in Table 6GOT obtained the cost function value of 1062080 in 1296secondsThis value is about 2 below the cost function valueof VIP-PLANOPT 2006 The coordinates of the facilities forthe layout obtained by GOT are given in Table 7
5 Comparing GOT Initial Solution withRandom Initial Solution
To compare the proposed initial solution (GOT initial solu-tion) with random initial solution a set of test problems (119899 =10 11 12 100) were generated The facilities dimensionswere 1 times 1 and flow matrices were randomly generatedbetween 0 and 10 For acquiring a random initial solution
8 Journal of Industrial Mathematics
Dimensions of the facilities Flow matrix
1
1
1
1 1
1
2
2
2
2
2
2
2 2
2
2
2
2
20
0
0
0
0
0
0
0
0 0
0
0 0
0
0
0 0
0 0 0
0
0
00
3
3
3
3
4
4
4
4
4
4
4
5
5
5
6
6
6
7 8
1
2
2
2
2
23
3
3
3
3
4
4
4
4
44
44
5
5
6
7
8
1
1
2
3 3
4
5
6
7
8
Length Width
(a) The raw facilities layout data
MPGalgorithm
1 5-2
3-2
7-6
8-5
6-3
4-2
3-1
4-3
5-42
3
4
5
6
7
8
9
Number10
11
12
13
14
15
16
17
18
NumberEdge6-4
8-2
8-4
2-1
7-3
8-7
7-2
7-4
7-1
Edge
(b) MPG
Graph drawing algorithm
1
7
8
5
2
3
4
6
(c) Straight line drawing of the MPG
Obtaining initial solution
1
7
8
5
2
3
4
6
(d) Initial solution
Figure 4 Obtaining an initial solution for test problem 1
1
7
85
2
34
6
Figure 5 Final layout for test problem 1
facilities were randomly placed in a (2119899 minus 4) times (119899 minus 2) integergrid For each test problem 20 random placements werefound Table 8 shows the value of cost function in GOT initialsolution and the best value found by random placementsFigures 8 and 9 shows these results graphically The resultsdemonstrated significant improvement in cost function
6 Summary and Conclusion
An initial solution has been presented for the layout designof facilities on a continuous plane The technique consists of
8 4 5
14 11 18 19 3
15
20
13
717
1 2 612
10
9
16
Figure 6 Final layout for test problem 2
two stages In the first stage amaximal planar graph (MPG) isobtained In the second stage the vertices of MPG are drawnon the plane by graph drawing algorithms Then vertices arereplaced by facilities Hence an initial solution is obtainedTo evaluate the performance this initial solution has beenapplied in rectangular facility layout problem and improvedby an analytical method by Mir and Imam [7]
The approach has been tested on five instances fromthe literature Table 6 shows the Summary of the resultsand Figure 8 shows the cost reduction by the technique Forthe large size problems involving 50 100 and 125 facilities
Journal of Industrial Mathematics 9
6 41 22
29
1224 44
4645
1425
5
26
28
10
9
3
2
8
4
30
31
43
47
49
50
40
42
48
36
34
3538
39
33
32
13
18
37
15 17
27 23
20
16
71 19
11
21
Figure 7 Final layout for test problem 3
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Random initial solutionGOT initial solution
Cos
t
05
1
15
2
25
3
35
4
45
5
times106
n
Figure 8 Comparisons of cost function value in GOT initialsolution with the best value of random placements
the layout costs values are better than those obtained by thepreviously published techniques As shown in Table 9 theresults demonstrate effectiveness of the technique especiallyfor large size problems
This paper introduced a simple technique for obtaining agood initial solutionThe technique with somemodificationcan be applied in facility layout approaches that use arandomly generated initial solution In future researches itwould be interesting to analyze the influence of MPG andgraph drawing algorithm on the solution The results can befurther improved by using a metaheuristic such as GRASP[21] and Tabu search [20] for generating a high weightedMPG
0
5
10
15
20
25
8 20 50 100 125
Reduction
times103
minus5
Figure 9 Cost reduction by using GOT
Conflict of Interests
The authors declare that they have no conflict of interests
References
[1] S P Singh and R R K Sharma ldquoA review of differentapproaches to the facility layout problemsrdquo International Jour-nal of Advanced Manufacturing Technology vol 30 no 5-6 pp425ndash433 2006
[2] M F Anjos and A Vannelli ldquoAn attractor-repeller approach tofloorplanningrdquo Mathematical Methods of Operations Researchvol 56 no 1 pp 3ndash27 2002
[3] I Castillo and T Sim ldquoA spring-embedding approach for thefacility layout problemrdquo Journal of the Operational ResearchSociety vol 55 no 1 pp 73ndash81 2004
[4] Z Drezner ldquoDISCON a new method for the layout problemrdquoOperations Research vol 28 no 6 pp 1375ndash1384 1980
[5] M H Imam and M Mir ldquoNonlinear programming approachto automated topology optimizationrdquo Computer-Aided Designvol 21 no 2 pp 107ndash115 1989
[6] M H Imam and M Mir ldquoAutomated layout of facilities ofunequal areasrdquo Computers and Industrial Engineering vol 24no 3 pp 355ndash366 1993
[7] M Mir and M H Imam ldquoHybrid optimization approachfor layout design of unequal-area facilitiesrdquo Computers andIndustrial Engineering vol 39 no 1-2 pp 49ndash63 2001
[8] Engineering Optimization Software VIP-PLANOPT 20062010 httpwwwplanoptcom
[9] L Foulds Graph Theory Applications Springer New York NYUSA 1992
[10] M M D Hassan and G L Hogg ldquoA review of graph theoryapplication to the facilities layout problemrdquo Omega vol 15 no4 pp 291ndash300 1987
[11] M M D Hassan and G L Hogg ldquoOn converting a dualgraph into a block layoutrdquo International Journal of ProductionResearch vol 27 no 7 pp 1149ndash1160 1989
[12] KHWatson and JWGiffin ldquoThevertex splitting algorithm forfacilities layoutrdquo International Journal of Production Researchvol 35 no 9 pp 2477ndash2492 1997
[13] S A Irvine and I Rinsma-Melchert ldquoA new approach tothe block layout problemrdquo International Journal of ProductionResearch vol 35 no 8 pp 2359ndash2376 1997
10 Journal of Industrial Mathematics
[14] P S Welgama P R Gibson and L A R Al-Hakim ldquoFacilitieslayout a knowledge-based approach for converting a dualgraph into a block layoutrdquo International Journal of ProductionEconomics vol 33 no 1ndash3 pp 17ndash30 1994
[15] M A Jokar and A S Sangchooli ldquoConstructing a blocklayout by face areardquo The International Journal of AdvancedManufacturing Technology vol 54 no 5ndash8 pp 801ndash809 2011
[16] L R Foulds P B Gibbons and J W Giffin ldquoFacilities layoutadjacency determination an experimental comparison of threegraph theoretic heuristicsrdquo Operations Research vol 33 no 5pp 1091ndash1106 1985
[17] E G John and J Hammond ldquoMaximally weighted graphtheoretic facilities design planningrdquo International Journal ofProduction Research vol 38 no 16 pp 3845ndash3859 2000
[18] S G Boswell ldquoTESSAmdasha new greedy heuristic for facilitieslayout planningrdquo International Journal of Production Researchvol 30 no 8 pp 1957ndash1968 1992
[19] L R Foulds and D F Robinson ldquoGraph theoretic heuristics forthe plant layout problemrdquo International Journal of ProductionResearch vol 16 no 1 pp 27ndash37 1978
[20] I H Osman ldquoA tabu search procedure based on a randomRoulette diversification for the weighted maximal planar graphproblemrdquoComputers and Operations Research vol 33 no 9 pp2526ndash2546 2006
[21] I H Osman B Al-Ayoubi and M Barake ldquoA greedy randomadaptive search procedure for the weighted maximal planargraph problemrdquo Computers and Industrial Engineering vol 45no 4 pp 635ndash651 2003
[22] J M Boyer and W J Myrvold ldquoOn the cutting edge simplifiedO(n) planarity by edge additionrdquo Journal of Graph Algorithmsand Applications vol 8 no 3 pp 241ndash273 2004
[23] G Di Battista P Eades R Tamassia and I Tollis GraphDrawing Algorithms for the Visualization of Graphs PrenticeHall PTR Upper Saddle River NJ USA 1998
[24] M Chrobak and T H Payne ldquoA linear-time algorithm fordrawing a planar graph on a gridrdquo Information ProcessingLetters vol 54 no 4 pp 241ndash246 1995
[25] S Heragu Facilities Design Iuniverse Inc 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Industrial Mathematics
Dimensions of the facilities Flow matrix
1
1
1
1 1
1
2
2
2
2
2
2
2 2
2
2
2
2
20
0
0
0
0
0
0
0
0 0
0
0 0
0
0
0 0
0 0 0
0
0
00
3
3
3
3
4
4
4
4
4
4
4
5
5
5
6
6
6
7 8
1
2
2
2
2
23
3
3
3
3
4
4
4
4
44
44
5
5
6
7
8
1
1
2
3 3
4
5
6
7
8
Length Width
(a) The raw facilities layout data
MPGalgorithm
1 5-2
3-2
7-6
8-5
6-3
4-2
3-1
4-3
5-42
3
4
5
6
7
8
9
Number10
11
12
13
14
15
16
17
18
NumberEdge6-4
8-2
8-4
2-1
7-3
8-7
7-2
7-4
7-1
Edge
(b) MPG
Graph drawing algorithm
1
7
8
5
2
3
4
6
(c) Straight line drawing of the MPG
Obtaining initial solution
1
7
8
5
2
3
4
6
(d) Initial solution
Figure 4 Obtaining an initial solution for test problem 1
1
7
85
2
34
6
Figure 5 Final layout for test problem 1
facilities were randomly placed in a (2119899 minus 4) times (119899 minus 2) integergrid For each test problem 20 random placements werefound Table 8 shows the value of cost function in GOT initialsolution and the best value found by random placementsFigures 8 and 9 shows these results graphically The resultsdemonstrated significant improvement in cost function
6 Summary and Conclusion
An initial solution has been presented for the layout designof facilities on a continuous plane The technique consists of
8 4 5
14 11 18 19 3
15
20
13
717
1 2 612
10
9
16
Figure 6 Final layout for test problem 2
two stages In the first stage amaximal planar graph (MPG) isobtained In the second stage the vertices of MPG are drawnon the plane by graph drawing algorithms Then vertices arereplaced by facilities Hence an initial solution is obtainedTo evaluate the performance this initial solution has beenapplied in rectangular facility layout problem and improvedby an analytical method by Mir and Imam [7]
The approach has been tested on five instances fromthe literature Table 6 shows the Summary of the resultsand Figure 8 shows the cost reduction by the technique Forthe large size problems involving 50 100 and 125 facilities
Journal of Industrial Mathematics 9
6 41 22
29
1224 44
4645
1425
5
26
28
10
9
3
2
8
4
30
31
43
47
49
50
40
42
48
36
34
3538
39
33
32
13
18
37
15 17
27 23
20
16
71 19
11
21
Figure 7 Final layout for test problem 3
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Random initial solutionGOT initial solution
Cos
t
05
1
15
2
25
3
35
4
45
5
times106
n
Figure 8 Comparisons of cost function value in GOT initialsolution with the best value of random placements
the layout costs values are better than those obtained by thepreviously published techniques As shown in Table 9 theresults demonstrate effectiveness of the technique especiallyfor large size problems
This paper introduced a simple technique for obtaining agood initial solutionThe technique with somemodificationcan be applied in facility layout approaches that use arandomly generated initial solution In future researches itwould be interesting to analyze the influence of MPG andgraph drawing algorithm on the solution The results can befurther improved by using a metaheuristic such as GRASP[21] and Tabu search [20] for generating a high weightedMPG
0
5
10
15
20
25
8 20 50 100 125
Reduction
times103
minus5
Figure 9 Cost reduction by using GOT
Conflict of Interests
The authors declare that they have no conflict of interests
References
[1] S P Singh and R R K Sharma ldquoA review of differentapproaches to the facility layout problemsrdquo International Jour-nal of Advanced Manufacturing Technology vol 30 no 5-6 pp425ndash433 2006
[2] M F Anjos and A Vannelli ldquoAn attractor-repeller approach tofloorplanningrdquo Mathematical Methods of Operations Researchvol 56 no 1 pp 3ndash27 2002
[3] I Castillo and T Sim ldquoA spring-embedding approach for thefacility layout problemrdquo Journal of the Operational ResearchSociety vol 55 no 1 pp 73ndash81 2004
[4] Z Drezner ldquoDISCON a new method for the layout problemrdquoOperations Research vol 28 no 6 pp 1375ndash1384 1980
[5] M H Imam and M Mir ldquoNonlinear programming approachto automated topology optimizationrdquo Computer-Aided Designvol 21 no 2 pp 107ndash115 1989
[6] M H Imam and M Mir ldquoAutomated layout of facilities ofunequal areasrdquo Computers and Industrial Engineering vol 24no 3 pp 355ndash366 1993
[7] M Mir and M H Imam ldquoHybrid optimization approachfor layout design of unequal-area facilitiesrdquo Computers andIndustrial Engineering vol 39 no 1-2 pp 49ndash63 2001
[8] Engineering Optimization Software VIP-PLANOPT 20062010 httpwwwplanoptcom
[9] L Foulds Graph Theory Applications Springer New York NYUSA 1992
[10] M M D Hassan and G L Hogg ldquoA review of graph theoryapplication to the facilities layout problemrdquo Omega vol 15 no4 pp 291ndash300 1987
[11] M M D Hassan and G L Hogg ldquoOn converting a dualgraph into a block layoutrdquo International Journal of ProductionResearch vol 27 no 7 pp 1149ndash1160 1989
[12] KHWatson and JWGiffin ldquoThevertex splitting algorithm forfacilities layoutrdquo International Journal of Production Researchvol 35 no 9 pp 2477ndash2492 1997
[13] S A Irvine and I Rinsma-Melchert ldquoA new approach tothe block layout problemrdquo International Journal of ProductionResearch vol 35 no 8 pp 2359ndash2376 1997
10 Journal of Industrial Mathematics
[14] P S Welgama P R Gibson and L A R Al-Hakim ldquoFacilitieslayout a knowledge-based approach for converting a dualgraph into a block layoutrdquo International Journal of ProductionEconomics vol 33 no 1ndash3 pp 17ndash30 1994
[15] M A Jokar and A S Sangchooli ldquoConstructing a blocklayout by face areardquo The International Journal of AdvancedManufacturing Technology vol 54 no 5ndash8 pp 801ndash809 2011
[16] L R Foulds P B Gibbons and J W Giffin ldquoFacilities layoutadjacency determination an experimental comparison of threegraph theoretic heuristicsrdquo Operations Research vol 33 no 5pp 1091ndash1106 1985
[17] E G John and J Hammond ldquoMaximally weighted graphtheoretic facilities design planningrdquo International Journal ofProduction Research vol 38 no 16 pp 3845ndash3859 2000
[18] S G Boswell ldquoTESSAmdasha new greedy heuristic for facilitieslayout planningrdquo International Journal of Production Researchvol 30 no 8 pp 1957ndash1968 1992
[19] L R Foulds and D F Robinson ldquoGraph theoretic heuristics forthe plant layout problemrdquo International Journal of ProductionResearch vol 16 no 1 pp 27ndash37 1978
[20] I H Osman ldquoA tabu search procedure based on a randomRoulette diversification for the weighted maximal planar graphproblemrdquoComputers and Operations Research vol 33 no 9 pp2526ndash2546 2006
[21] I H Osman B Al-Ayoubi and M Barake ldquoA greedy randomadaptive search procedure for the weighted maximal planargraph problemrdquo Computers and Industrial Engineering vol 45no 4 pp 635ndash651 2003
[22] J M Boyer and W J Myrvold ldquoOn the cutting edge simplifiedO(n) planarity by edge additionrdquo Journal of Graph Algorithmsand Applications vol 8 no 3 pp 241ndash273 2004
[23] G Di Battista P Eades R Tamassia and I Tollis GraphDrawing Algorithms for the Visualization of Graphs PrenticeHall PTR Upper Saddle River NJ USA 1998
[24] M Chrobak and T H Payne ldquoA linear-time algorithm fordrawing a planar graph on a gridrdquo Information ProcessingLetters vol 54 no 4 pp 241ndash246 1995
[25] S Heragu Facilities Design Iuniverse Inc 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Industrial Mathematics 9
6 41 22
29
1224 44
4645
1425
5
26
28
10
9
3
2
8
4
30
31
43
47
49
50
40
42
48
36
34
3538
39
33
32
13
18
37
15 17
27 23
20
16
71 19
11
21
Figure 7 Final layout for test problem 3
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Random initial solutionGOT initial solution
Cos
t
05
1
15
2
25
3
35
4
45
5
times106
n
Figure 8 Comparisons of cost function value in GOT initialsolution with the best value of random placements
the layout costs values are better than those obtained by thepreviously published techniques As shown in Table 9 theresults demonstrate effectiveness of the technique especiallyfor large size problems
This paper introduced a simple technique for obtaining agood initial solutionThe technique with somemodificationcan be applied in facility layout approaches that use arandomly generated initial solution In future researches itwould be interesting to analyze the influence of MPG andgraph drawing algorithm on the solution The results can befurther improved by using a metaheuristic such as GRASP[21] and Tabu search [20] for generating a high weightedMPG
0
5
10
15
20
25
8 20 50 100 125
Reduction
times103
minus5
Figure 9 Cost reduction by using GOT
Conflict of Interests
The authors declare that they have no conflict of interests
References
[1] S P Singh and R R K Sharma ldquoA review of differentapproaches to the facility layout problemsrdquo International Jour-nal of Advanced Manufacturing Technology vol 30 no 5-6 pp425ndash433 2006
[2] M F Anjos and A Vannelli ldquoAn attractor-repeller approach tofloorplanningrdquo Mathematical Methods of Operations Researchvol 56 no 1 pp 3ndash27 2002
[3] I Castillo and T Sim ldquoA spring-embedding approach for thefacility layout problemrdquo Journal of the Operational ResearchSociety vol 55 no 1 pp 73ndash81 2004
[4] Z Drezner ldquoDISCON a new method for the layout problemrdquoOperations Research vol 28 no 6 pp 1375ndash1384 1980
[5] M H Imam and M Mir ldquoNonlinear programming approachto automated topology optimizationrdquo Computer-Aided Designvol 21 no 2 pp 107ndash115 1989
[6] M H Imam and M Mir ldquoAutomated layout of facilities ofunequal areasrdquo Computers and Industrial Engineering vol 24no 3 pp 355ndash366 1993
[7] M Mir and M H Imam ldquoHybrid optimization approachfor layout design of unequal-area facilitiesrdquo Computers andIndustrial Engineering vol 39 no 1-2 pp 49ndash63 2001
[8] Engineering Optimization Software VIP-PLANOPT 20062010 httpwwwplanoptcom
[9] L Foulds Graph Theory Applications Springer New York NYUSA 1992
[10] M M D Hassan and G L Hogg ldquoA review of graph theoryapplication to the facilities layout problemrdquo Omega vol 15 no4 pp 291ndash300 1987
[11] M M D Hassan and G L Hogg ldquoOn converting a dualgraph into a block layoutrdquo International Journal of ProductionResearch vol 27 no 7 pp 1149ndash1160 1989
[12] KHWatson and JWGiffin ldquoThevertex splitting algorithm forfacilities layoutrdquo International Journal of Production Researchvol 35 no 9 pp 2477ndash2492 1997
[13] S A Irvine and I Rinsma-Melchert ldquoA new approach tothe block layout problemrdquo International Journal of ProductionResearch vol 35 no 8 pp 2359ndash2376 1997
10 Journal of Industrial Mathematics
[14] P S Welgama P R Gibson and L A R Al-Hakim ldquoFacilitieslayout a knowledge-based approach for converting a dualgraph into a block layoutrdquo International Journal of ProductionEconomics vol 33 no 1ndash3 pp 17ndash30 1994
[15] M A Jokar and A S Sangchooli ldquoConstructing a blocklayout by face areardquo The International Journal of AdvancedManufacturing Technology vol 54 no 5ndash8 pp 801ndash809 2011
[16] L R Foulds P B Gibbons and J W Giffin ldquoFacilities layoutadjacency determination an experimental comparison of threegraph theoretic heuristicsrdquo Operations Research vol 33 no 5pp 1091ndash1106 1985
[17] E G John and J Hammond ldquoMaximally weighted graphtheoretic facilities design planningrdquo International Journal ofProduction Research vol 38 no 16 pp 3845ndash3859 2000
[18] S G Boswell ldquoTESSAmdasha new greedy heuristic for facilitieslayout planningrdquo International Journal of Production Researchvol 30 no 8 pp 1957ndash1968 1992
[19] L R Foulds and D F Robinson ldquoGraph theoretic heuristics forthe plant layout problemrdquo International Journal of ProductionResearch vol 16 no 1 pp 27ndash37 1978
[20] I H Osman ldquoA tabu search procedure based on a randomRoulette diversification for the weighted maximal planar graphproblemrdquoComputers and Operations Research vol 33 no 9 pp2526ndash2546 2006
[21] I H Osman B Al-Ayoubi and M Barake ldquoA greedy randomadaptive search procedure for the weighted maximal planargraph problemrdquo Computers and Industrial Engineering vol 45no 4 pp 635ndash651 2003
[22] J M Boyer and W J Myrvold ldquoOn the cutting edge simplifiedO(n) planarity by edge additionrdquo Journal of Graph Algorithmsand Applications vol 8 no 3 pp 241ndash273 2004
[23] G Di Battista P Eades R Tamassia and I Tollis GraphDrawing Algorithms for the Visualization of Graphs PrenticeHall PTR Upper Saddle River NJ USA 1998
[24] M Chrobak and T H Payne ldquoA linear-time algorithm fordrawing a planar graph on a gridrdquo Information ProcessingLetters vol 54 no 4 pp 241ndash246 1995
[25] S Heragu Facilities Design Iuniverse Inc 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Industrial Mathematics
[14] P S Welgama P R Gibson and L A R Al-Hakim ldquoFacilitieslayout a knowledge-based approach for converting a dualgraph into a block layoutrdquo International Journal of ProductionEconomics vol 33 no 1ndash3 pp 17ndash30 1994
[15] M A Jokar and A S Sangchooli ldquoConstructing a blocklayout by face areardquo The International Journal of AdvancedManufacturing Technology vol 54 no 5ndash8 pp 801ndash809 2011
[16] L R Foulds P B Gibbons and J W Giffin ldquoFacilities layoutadjacency determination an experimental comparison of threegraph theoretic heuristicsrdquo Operations Research vol 33 no 5pp 1091ndash1106 1985
[17] E G John and J Hammond ldquoMaximally weighted graphtheoretic facilities design planningrdquo International Journal ofProduction Research vol 38 no 16 pp 3845ndash3859 2000
[18] S G Boswell ldquoTESSAmdasha new greedy heuristic for facilitieslayout planningrdquo International Journal of Production Researchvol 30 no 8 pp 1957ndash1968 1992
[19] L R Foulds and D F Robinson ldquoGraph theoretic heuristics forthe plant layout problemrdquo International Journal of ProductionResearch vol 16 no 1 pp 27ndash37 1978
[20] I H Osman ldquoA tabu search procedure based on a randomRoulette diversification for the weighted maximal planar graphproblemrdquoComputers and Operations Research vol 33 no 9 pp2526ndash2546 2006
[21] I H Osman B Al-Ayoubi and M Barake ldquoA greedy randomadaptive search procedure for the weighted maximal planargraph problemrdquo Computers and Industrial Engineering vol 45no 4 pp 635ndash651 2003
[22] J M Boyer and W J Myrvold ldquoOn the cutting edge simplifiedO(n) planarity by edge additionrdquo Journal of Graph Algorithmsand Applications vol 8 no 3 pp 241ndash273 2004
[23] G Di Battista P Eades R Tamassia and I Tollis GraphDrawing Algorithms for the Visualization of Graphs PrenticeHall PTR Upper Saddle River NJ USA 1998
[24] M Chrobak and T H Payne ldquoA linear-time algorithm fordrawing a planar graph on a gridrdquo Information ProcessingLetters vol 54 no 4 pp 241ndash246 1995
[25] S Heragu Facilities Design Iuniverse Inc 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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