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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
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
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
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
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
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
[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
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
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
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
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
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
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
[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
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
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
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
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
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
[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
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
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
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
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
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
[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
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
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
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
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
[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
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
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
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
[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
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
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
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
[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
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
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
[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
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
[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
[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