22 A Approach to the Facility Layout Desig Optimizatio Rekha Bhowmik Departmet of Computer Sciece, Sam Housto State Uiversity, Hutsville, Texas 7734, USA Summary This paper presets a Iterative Heuristic Algorithm ad Brach ad Boud Algorithm for optimal locatio of clusters o differet levels. The use of cluster aalysis is proposed for groupig highly related departmets for both the methods. The vertical locatio problem is formulated for optimal locatio problem. Results obtaied by both the algorithms are preseted ad compared i terms of travel cost. Key words: Cluster aalysis, itergroup adjacecy matrix, travel cost, Iterative Heuristic Algorithm, Brach ad Boud Algorithm. Itroductio The Facility Layout Problem(FLP) is cocered with locatig a set of departmets, each requirig level area o a give site. The departmet ratigs, based o umber of trips, may be represeted by a adjacecy matrix. The umber of trips betwee pairs of departmets ca be used to decide o the desirability of locatig a departmet ext to each other. Give such order of departmet, the problem is to fid a layout which optimizes a fuctio based o departmet closeess ratigs ad distace. There are 2 sub-problems ivolved i solvig FLP: i) fid a optimal adjacecy matrix subject to certai costraits, ad 2) derive a layout from adjacecy matrix. The three-dimesioal facilities plaig techiques have a umber of shortcomigs. The locatio order calculatios are ot desiged to isolate groups of closely related departmets ad that the locatio process is oly able to optimize the locatio of a departmet with respect to the relative locatios of previously located departmets. Further, the processig time is very high for large datasets ad that departmets ted to split betwee levels, which may ot be acceptable. I this paper a three-step Iterative Heuristic techique is proposed for FLPs. I sectio 2 we preset related work. I sectio 3 we preset a three-step heuristic optimizatio procedure for FLPs. Clusterig Aalysis ad Justificatio is preseted i sectio 4. I sectio 5 the allocatio of departmets to differet levels is formulated followed by Brach ad Boud Algorithm. A iterative heuristic algorithm for FLP problem is discussed i sectio 7 followed by Results ad Coclusios. 2. Related Work Uequal area FLP was formulated first. A give regio L * W is assumed, where L is the legth ad W is the width of the regio. The objective was to partitio the regio ito departmetal sub-regios so as to miimize the total commuicatio cost. Variatio of the Quadratic Assigmet Problem was preseted later. The rectagular layout is split ito blocks of equal-area departmets. This techique reduced the umber of cadidate layouts while allowig departmets to assume differet areas ad differet shapes. Exact Mixed Iteger Programmig formulatio was proposed by []. The model uses a distace-based objective, but is ot based o the discrete represetatio as i the quadratic assigmet problem. [2] formulated a Noliear Layout Techique(NLT) based o three costraits. Two costraits are based o structure of the layout, that is, departmets may ot overlap, ad departmets may ot be located outside the give regio boudaries. The third costrait depeds o the limits of allowable dimesios of each departmet. The departmets have fixed area ad rectagular shapes, but for each departmet the height ad width are optimized usig mathematical model. [3] implemeted a geetic search for uequal area facility layout ad showed how optimal solutios are affected by costraits o permitted departmet shapes, as specified by a maximum allowable aspect ratio for each departmet. [4], [5] proposed a modified Mixed Iteger Programmig-Facilities Layout Plaig model by improvig perimeter costrait ad reported optimal solutios for FLPs with a maximum of eight departmets. Ajos[6] improved NLT method by itroducig Attractor-Repeller. Sherali et al[7] further improved the MIP-FLP model ad provided the approach with up to ie departmets. The approach uses polyhedral outer approximatio of the area costraits that reduces the errors i departmet areas. The first model is based o layout problem ad the secod model is a exact formulatio of facility layout problem. Ajos[8] cosidered oe-dimesioal facility layout problem, which cosists i fidig a optimal placemet of facilities o a straight lie. Lower boud o the optimal value of the space allocatio problem is created. They suggest a heuristic procedure which extracts a feasible solutio to the oe-dimesioal space allocatio problem. Kaufma[9] presets cluster aalysis to fid groups of data. Ajos[0] preseted a framework based for facility layout plaig based o two mathematical models. The first model fids the startig poits for the iterative algorithm. The secod model is a Mauscript received April 5, 2008 Mauscript revised April 20, 2008
23 exact formulatio of the facility layout problem. Layouts with relatively little computatioal effort ca be obtaied. We formulate a FLP based o Iterative heuristic Algorithm ad Brach ad Boud Algorithm to fid optimal layouts. We create clusters first ad the locate the clusters o multilevel. Layouts are the obtaied for each level. 3. Three-Step Iterative Heuristic Techique I [] the justificatio for the use of a three-step procedure for the optimal locatio of departmets o multilevel is preseted. f i = x il = j =, 2,.., where, h kl = absolute value of the vertical distace betwee k th ad l th levels δ kl = 0 for for k k = ( 3 ) The three steps are: (i) Clusterig techique for idetifyig groups of highly related departmets, (ii) Exact or efficiet algorithm(brach ad Boud) for optimizig the itergroup commuicatio cost, ad (iii) Iterative Heuristic techique for layouts of departmets o each level. I this paper, steps i) ad ii) are discussed i detail. 4. Cluster Aalysis Approach Layout techiques have a umber of shortcomigs. The locatio order calculatios are ot desiged to isolate groups of closely related departmets ad that the locatio process is oly able to optimize the locatio of a departmet with respect to the locatios of pre-located departmets. Further, the processig time is high for large problems ad that departmets ted to split betwee levels which may ot be acceptable[7]. 3D layout problem ca be mathematically writte as: Miimize Z= d ij = horizotal distace betwee the locatio of departmets i ad j whe both the departmets are o the same level. d ic ad d jc = horizotal distace from departmet i ad j to the circulatio poit of departmets. This is a mixed iteger oliear programmig problem of very great complexity. The first term of the distace expressio represets the weighted vertical travel, the secod term represets the horizotal travel whe the departmets are located o the same level. The variables for the problem are: xik where R is the prescribed boudary of the level layout. = 0 ( i) ( i) x, y or ( i) ( i) ad ( x, y ) R i =,2,... k =,2,... f i =,2,... = ( h kl ( d 2 ' ic f f x ik l = k = i = j = w subject to i = + + d d ' ij ' jc ) δ kl ) + ( x jl t ij δ kl x il a i A l l =,2,.., f ) ( ) (2) This problem is difficult to solve by ay of the mathematical algorithms. Rewritig the objective fuctio ito 3 parts: Here, the third term represets the sum of itra-level travel costs f x il t ij d' ij x ij 2 l = i= j= The first term represets the iter-level vertical travel cost while the secod term represets the iter-level horizotal travel cost. If the departmets with high iteractios are suitably clustered, it amouts to the vaishig of cotributios
24 from the first two terms. For moderate ad small values of t ij the departmets are ot expected to be located o the same level ad the cotributio from the third term vaishes. Further, for departmets located o differet levels with moderate values of t ij, the first term will domiate the secod. Thus, if the objective fuctio is approximated as the sum of the first ad third terms, igorig the secod term, the problem splits clearly ito a partitioig problem i the domai of quadratic assigmet problem. The use of cluster aalysis techique to maximize the adjacecy withi subsets of departmets while miimizig the travel cost betwee clusters appears to be a extremely good approach. Thus after cluster formatio, the layout problem costitutig the first term of the objective fuctio is miimized ad later the level-wise layout problem is solved by miimizig the third term of objective fuctio with appropriate set of costraits. Thus, a three step procedure is used to solve the multi-level layout problem, i) Use of clusterig techique for idetifyig groups of highly iterrelated departmets, ii) Use of a exact or efficiet algorithm for miimizig iter-group commuicatio cost, ad iii) Use of iterative heuristic algorithms for obtaiig layouts of departmets at each level based o steps ad 2. I the followig sectio, o discussio of step 3 is preseted sice the algorithm has bee described elsewhere. 4.. Use of Cluster Aalysis The cluster aalysis procedure is a four stage process: I the first stage, the cost of iteractio(travel) is specified. The ed product of this stage is a dedogram showig the successive fusio of departmets, which culmiates at the stage where all the departmets are i oe group. I the secod stage, departmet areas are itroduced which splits the sigle group ito clusters of closely associated departmets, each of which is small eough to accommodate o a level. We determie the Itergroup Adjacecy Matrix as discussed i sectio 5. I the third stage, vertical layout problem is carried out. This is discussed i sectio 4. The last stage of the process cosists of locatig departmets o differet levels usig a 2D-layout procedure. 4.2. Itergroup Adjacecy Matrix The procedure for clusterig ad determiig iter-group adjacecy matrix ivolves: i) Develop the Adjacecy Matrix betwee pairs of departmets. ii) Fid the largest umber of iteractio(travel) betwee pairs of departmets from the adjacecy matrix. This is the cluster level to start with the cluster aalysis procedure. Choose some cluster level iterval. The pairs of departmets which fall i this cluster level form a cluster ad is desigated by some cluster ame for the purpose of idetificatio. Decrease the cluster level by the cluster level iterval chose. Fid the departmets which fall i this cluster level. We go i for the third ad subsequet cluster levels by further reducig by the cluster level iterval. I this way, the departmets fallig i a particular cluster level are searched ad idetified by cluster ame. iii) Plot the dedogram, ad iv) A search is made i the reverse directio to cosider clusters of desired area. If a cluster has a area less tha the maximum permissible area per level/level, the idetity ad size of the cluster are stored i a table. A check is made for the o-repetitio of a departmet. v) Costruct a Itergroup Adjacecy Matrix represetig the iteractio costs betwee clusters. Thus, if cluster C i is obtaied by groupig of departmets belogig to the set I ad C j is aother cluster represetig the group of departmets belogig to the set J, the elemet T ij of the Iter-group Adjacecy Matrix of clusters is: T ij = t k l k ε I l ε J 4.3. Example A example has bee studied by clusterig usig clusterig algorithm for 3D layout problems. The adjacecy matrix for this example has bee take from [8]. 2 departmets are to be clustered ad the adjacecy matrix cotaiig the commuicatio values betwee departmets is kow. The commuicatio value, t ij betwee departmets i ad j ca be obtaied from the adjacecy matrix. At level of the clusterig procedure, departmets 3 ad 4 are fused to form a cluster, sice t 34 is the largest commuicatio value i the adjacecy matrix. The umber of commuicatio values betwee this ad the remaiig 9 departmets are obtaied. Next largest etry is 82, ad so departmets 2 ad 3 are fused to form a secod group. The ext largest etry is 5 ad so departmets 0 ad are fused to form the third group. All the groups are desigated by cluster ames. Sice the umber of departmets for this example is 2, the first group is amed as 22. Fially, fusio of the groups takes place to form a sigle group cotaiig all the 2 departmets. The dedogram is thus created. Sice the maximum permissible area per level is 9 uits, the groups which have area less tha or equal to 9 uits are listed. Table shows the idetity ad size of clusters for three levels. Table 2 shows the itergroup adjacecy matrix represetig the travel costs betwee clusters.
25 Table : Idetity ad size of cluster o three levels Cluster Departmet(s) formig the cluster Cluster Area No., 2, 7 (group umber=38) 8 2 3, 4, 20, 2 (group umber=28) 4 3 9, 0,, 4 (group umber=27) 8 4 2, 3 (group umber=23) 6 5 5 6 6 7 7 3 8 8 3 9 5 3 0 6 5 8 2 9 2 Table 2: Itergroup relatioship matrix No. Area 2 3 4 5 6 7 8 9 0 2 8 0 2 4 0 0 3 8 72 90 0 4 6 60 265 55 0 5 6 25 0 0 6 3 2 2 4 3 0 7 3 2 8 58 65 0 2 0 8 3 3 44 35 64 0 3 56 0 9 3 3 44 35 64 0 3 8 56 0 0 5 2 8 58 65 0 2 62 8 56 0 38 37 3 59 0 3 0 0 3 0 2 2 37 6 20 56 0 0 2 7 7 2 2 0 5. Formulatio of the Problem The vertical layout optimizatio problem ca be writte as: f f C = 2 x ik l = k = i= j= x jl t ij dkl where x ik = if the i th departmet is located o k th level = 0 otherwise x jl = if the j th departmet is located o l th level = 0 otherwise t ij = umber of iteractive trips betwee the departmets i ad j d kl = vertical distace betwee the k th ad l th level = k l = umber of departmets f = umber of levels The costraits are: i = t il ai Al l =,2,.., f () f i = x il = l =, 2,.., ( 2 ) where a i is the area required for departmet i ad A l is the available space o level l. Costait () represets the restrictio of available space o each level while the costrait of type(2) models the coditio that a particular departmet must be located o ay oe of the levels[7]. The above problem is a quadratic assigmet problem. There exists o reliable exact algorithms which ca solve quadratic assigmet problem where the umber of departmets is greater tha 2. However, sice the umber of levels for medium sized problems is usually small ad the umber of clusters is much smaller tha the umber of departmets, it is possible to attempt a exact solutio if a suitable algorithm ca be developed. A iterative heuristic algorithm is discussed i sectio 5 which is simpler ad easier to implemet. 6. Brach ad Boud Algorithm I this sectio, the developmet of a brach ad boud algorithm is described. The brach ad boud algorithm proceeds i a sequece of steps. At each step of the algorithm, a partial layout is at had where a set of departmets are assiged to some locatios. A lower boud, LB, o the cost of all possible completios of this layout is calculated. If LB < cost C* of the available layout so far, proceed to allocate a ew departmet. Otherwise, the partial solutio is fathomed, the last assigmet is prohibited ad a ew assigmet is sought. The method of calculatig the LB by solvig the cadidate problem ad the progress of decisio tree are elaborated below. 6. Calculatio of Lower Boud Assume that the departmets belogig to the set I have already bee assiged. I particular, departmet I is assiged to locatio v i. A lower boud, LB, o the cost of this partial layout is: C + C 2 + C 3 where C = Fixed iteractio cost betwee already assiged departmets C 2 = Lower boud o the iteractio cost betwee uassiged departmets ad assiged departmets C 3 = iteractio cost amog uassiged departmets Cost C is computed as f f l = k= i I x ik j I x jl t ij k l
26 Calculatio of C 2 + C 3 Iitially, cosider the cost of locatig the i th uassiged departmet i v th uoccupied locatio. This is calculated by addig i) fixed cost represetig the travel costs betwee this departmet i the ew locatio ad all the pre-located departmets ad, ii) lower boud o the cost from other uassiged departmets to departmet i. This lower boud is calculated as follows. If K is a uassiged departmet, the lower boud o the travel cost betwee departmets i ad k is give by l k l k mi u ik t v k K where t is selected from the set of possible levels i which the departmet k(which belogs to the set K of departmets ot located) ca be located. Thus, a lower boud o the cost of assigig the i th departmet to v th level is give by: b iv = fiv + uij v v j + lk j I k K where, v j is the level i which the departmet j is located. Hece, C 2 + C 3 is obtaied by solvig the followig iteger programmig problem. C2 + C3 f = mi biv i I v= subject to f x iv = v = ad A * i xiv Av i = xiv where, v =, 2,, f f = umber of levels = umber of departmets x iv = 0 or where A v * is the available area o v th level at this step of optimizatio. The first group of costraits represet the coditio that each u-located departmet must fid a locatio while the secod group of costraits stipulate the level space restrictio o each level. This problem is a zero-oe liear programmig problem ad is fairly easy to solve, from which the lower boud, LB= C + C 2 + C 3 ca be calculated. A brach ad boud scheme ca ow be employed. A algorithm for solvig the problem is required which has bee outlied above. A decisio rule for brachig from the lowest boud has bee employed. 6.2 Brach ad Boud Algorithm The followig otatios will be used for brach ad boud algorithm. IOP = Iitial Optimizatio Problem Z o = Curret least upper boud o the optimal solutio (CP) i = Curret cadidate problem beig explored. Cadidate list = active sub-problems (that are still cadidate to be explored). The brach ad boud algorithm ca be summarized as follows:. Iitialize Z o to be a large positive costat. Set K=. Cosider (IOP) to be (CP) i 2. Set i = 3. Solve the optimizatio problem (CPR) i. If (CPR) i has o feasible solutio, either does (CP) i. The miimum value of (CPR) i is ot less tha the miimum value of (CP) i 4. Boud (CP) i. Apply a appropriate algorithm to (CPR) i to boud all solutios emaatig from (CP) i 4 If (CPR) i reveals a feasible solutio of IOP, go to step 6, otherwise, go to step 8. 5 If (CP) i < Z o go to step, otherwise go to step 0 6 Set Z o equal to the solutio value of (CP) i ad go to step 0 7 If the boud calculated i step 4 is less tha Z o, go to step 9, otherwise, go to step 0. 8 Add (CP) i to cadidate list ad go to step 0. 9 If all the problems created by the last brach have bee explored(bouded ad aalyzed), go to step 2, otherwise, go to step. That is, if i=k, go to step 2, otherwise, go to step. 0 Explore the ext sub-problem amog those created by the last brach. That is, let i = i + ad go to step 3. 2 If the cadidate is empty, go to step 5, otherwise, go to step 3. 3. Remove a problem from the cadidate list for brachig. Label the problem CP. Decisio rules to brach from the lowest boud or brach from the ewest active boud or a combiatio of the two are geerally used. 4. Brach o CP. Partitio CP ito K ew sub-problems (CP) i, i=, 2,, K ad go to step 2. 5. If a feasible solutio has ot bee reached, go to step 7, otherwise, go to step 6. 6. The best feasible solutio to date is a optimal solutio for IOP. Go to step 8.
27 7. No feasible solutio for IOP exists, go to step 8. 8. Stop. Program Developmet A program is developed for the brach ad boud algorithm. The iput cosists of: i) umber of clusters ii) umber of levels iii) maximum umber of odes iv) umber of variables(=umber of clusters * umber of levels) v) adjacecy matrix cotaiig the iteractio trips betwee clusters vi) maximum area permitted per level, ad vii) area assiged to each cluster. Calculate Lower Boud, that is, the iteractio cost amog already assiged clusters, iteractio cost amog uassiged clusters, ad iteractio cost from uassiged clusters to assiged clusters. The Brach ad Boud algorithm proceeds as show i Fig.. 6.3 Tree diagram A example with umber of clusters as 6, ad umber of levels as 3 is cosidered for geeratig the tree diagram. The maximum umber of odes is 00 ad the area permitted per level is uits. Fig 2 shows the part of decisio tree for this problem. The circles are called odes ad represet the set of all possible feasible solutios that ca be reached from the ode. The umber i each circle is the ode umber ad sequetially represets the order i which the brach ad boud algorithm is carried out. At each ode, the approximate problem is solved to obtai the lower boud ad this value is also show i the decisio tree. The lies coectig the odes are the braches. Nodes 56-58 are termial odes at this poit because braches do ot emaate from them. The brachig is doe as follows: Departmet is allotted to level ad the resultig sub-problem has zero as the lower boud. Similarly departmet is allotted to level 2(ode 3) ad level 3(ode 4). Sice ode 3 has the smallest lower boud, it is selected first for further brachig. With the costrait that departmet is a cadidate for all the three levels, departmet 2 is allotted to all the three levels. Termial odes 2, 4, 5, 6, 7 are still cadidates for the optimal solutio. The miimum lower boud value amog all the cadidate odes provides a lower boud for the problem, 55 at odes 2 or 4. Brach from ode 2 or 4 sice it has a smaller lower boud tha ay other cadidate ode. Assig departmet to levels, 2, 3. Node 4 has the least boud. Brach from ode 4 ad proceed as above. Fig. Boud ad Boud Techique Fig. 2 shows that odes 67(ot show) ad 68 yield a feasible solutio with a value of 395 ad that o other ode has a smaller value tha this. Thus, the solutios correspodig to odes 67 ad 68 must be optimal. Departmets 3 ad 4 are allocated to middle level. Departmets, 2 remai together ad ca be allocated to either first or third level ad departmets 5, 6 ca also be allocated to either first or third level. Departmets/ Levels Departmets/ 5,6 Third,2 3,4 Secod 3,4,2 First 5,6 Miimum cost=395 uits
28 Compute the commuicatio cost ad prit the layout. Step 8. Perform Steps 2 to 7 util a sufficiet umber of alterate layouts are available. Step 9. Select the least cost layout. Fig. 2 Tree-Brach ad Boud Algorithm(part-of) Number of levels=3 Number of departmets=6 Area/level= uits 7. Iterative Heuristic Algorithm for Multilevel Problem We discuss below a iterative algorithm for allocatio of clusters to miimize vertical commuicatio costs. I each iteratio a hierarchical procedure for 2-D locatio problem is made use of. The iterative algorithm is described below. Step. Costruct a iter-group adjacecy matrix represetig the commuicatio costs betwee clusters. Set i =. Set all d ij s to uity. Step 2. Compute the travel cost matrix(tij) wherei each elemet of the matrix is give by t ij * d ij = T ij Step 3. Rak the clusters for locatio o the basis of travel cost. The cluster havig the maximum travel cost with other clusters should be raked first. The cluster havig the largest travel cost with the previously located cluster should be raked ext for locatio. Thus at ay step, the cluster havig the maximum sum of travel costs with all the previously located clusters will be raked ext. Hece a complete orderig of clusters ca be established. Step 4. Locate the first cluster at the middle level(mth limitatio o the umber of levels ad available area o each level. Step 5. As i Step 4, complete the (m+)th level. Step 6. Repeat Steps 4 ad 5 util all the clusters are located. Step 7. Calculate the vertical distace represetig the weighted distace betwee clusters(or betwee levels) takig the middle level as datum. 8. Results A program is developed for FLP based o Brach ad Boud algorithm ad the Iterative Heuristic algorithm. The iput cosists of umber of levels, adjacecy matrix cotaiig umber of travel trips betwee clusters, area(i uits) of each cluster, ad maximum area permitted per level. The program requires to determie pairs of departmets formig a cluster, checkig if a cluster has bee cosidered for a particular level, ad also allocatig departmets o differet levels. A optimal layout desig is thus obtaied. Tables 3a, 3b ad 3c show the allocatio of departmets for three, four ad five level examples. To test cluster aalysis program, layouts were obtaied for each level. Table 3a presets the first example with the umber of departmets as 2 which are located o three levels. The optimal departmetive(cost) of 2094 uits ad 235 uits was obtaied usig Brach ad Boud algorithm ad Iterative Heuristic Algorithm respectively. The secod example ivolved the allocatio of 3 clusters to four levels. The cost of locatig departmets is 3495 uits ad is show i table 3b. The third example ivolved the allocatio of 3 clusters o five levels. This is show i Table 3c. Table 3a: Represetatio of Departmet locatio o three levels Number of clusters= 2 Maximum area permitted/ level = 9 sq uits Level Iterative Heuristic Brach ad Boud Third,6, 9, 0, 5, 6, 0,, 2 Secod 2, 4,7, 8,, 2 2, 4, 7, 8, 9 First 3, 5 3 Note: represets the clusters Iterative Heuristic Algorithm, total cost = 235 uits Brach ad Boud Algorithm, total cost = 2094 uits Table 3b: Represetatio of Departmet locatio o four levels Number of clusters= 3 Maximum area permitted/ level = 4 sq uits Level Iterative Heuristic Brach ad Boud Fourth 2,6, 3, Third 3, 3 2 4,7, 2
29 Secod 4,5, 9, 2 5, 8, 0,, 3 First 7, 8, 0, 3, 6 Note: represets the clusters Iterative Heuristic Algorithm, total cost = 3570 uits Brach ad Boud Algorithm, total cost = 3495 uits Table 3c: Represetatio of departmet locatio o five levels Number of clusters= 3 Maximum area permitted/ level = 2 sq uits Level Iterative Heuristic Brach ad Boud Fifth Fourth 2,7, 8, 3, 8, 9, 3, Third 4,, 2 4,, 2 Secod 3 3 First 5, 6, 9, 0 2,5, 6, 7, 0 Note: represets the clusters Iterative Heuristic Algorithm, total cost = 3570 uits Brach ad Boud Algorithm, total cost = 3822 uits Table 4 presets the compariso of results obtaied by Iterative Heuristic ad Brach ad Boud Algorithm. The results show that the results obtaied by Brach ad Boud Algorithm are superior to those obtaied by Iterative Heuristic Algorithm. The cost of allocatig clusters o differet levels is show i uits. Both the methods start by locatig the clusters i the middle first, that is, the m th floor. Table 4: Compariso of Brach ad Boud Algorithm ad Iterative Heuristic Algorithm Number of clusters Algorithm 2 3 3 Number of clusters 3 4 5 Brach ad Boud 2094.00 3495.00 3822.00 Iterative Heuristic 235.00 3570.00 3938.00 Tables 5a ad 5b compare both Iterative Heuristic algorithm ad Brach ad Brach algorithm i terms of total cost for locatig clusters o three ad four level examples. Total cost is the sum of itra-level travel cost, iter-level horizotal cost, iter-level vertical travel cost, ad weighted vertical travel cost. For table 5a, the itra-level travel cost is 442.9 uits ad the iter-level horizotal cost is 556.4 uits(displayed i Fig. 2). Table 5b shows the total cost which is the sum of itra-level travel cost(=2776.28) uits, the iter-level horizotal cost (= 593.) uits, iter-level vertical travel cost, ad the weighted vertical travel cost. The use of Iterative Heuristic Algorithm for horizotal ad vertical movemet gives good results as compared to the Brach ad Boud Algorithm(which is accurate). I both the cases, the layouts obtaied are very practical. Table 5a: Cost of allocatig clusters for three-level example Number of clusters= 2 Maximum area permitted/ level = 9 sq uits Factor Vertical Weighted Total cost travel vertical cost travel cost I II I II I II 2.0 235 2094 4270 488 4243.59 4304.9 4.0 235 2094 8540 8376 853.59 8492.9 8.0 235 2094 7080 6752 27053.59 26868.9 I is the Iterative Heuristic Techique for vertical travel cost. II is the Brach ad Boud method for vertical travel cost. Table 5b: Cost of allocatig clusters for four-level example Number of clusters= 3 Maximum area permitted/ level = 4 sq uits Factor Vertical Weighted Total cost travel vertical cost travel cost I II I II I II 2.0 3570 3495 740 6990 509.38 7067.5 4.0 3570 3495 4280 3980 22249.38 24057.05 8.0 3570 3495 28560 27960 36529.38 38037.05 I is the Iterative Heuristic Techique for vertical travel cost. II is the Brach ad Boud method for vertical travel cost. Figure a ad b show the least cost for three-level example usig cluster aalysis approach ad Brach ad Boud algorithm respectively. Comparig the total cost by the algorithms ivolvig clusterig techique ad the correspodig costs by the 3-D algorithm, it is clear that the clusterig techique is far superior particularly whe the factor for vertical movemet is high. Thus the treatmet of the 3D commuicatio cost miimizatio problem as a K-partitio problem is essetial. 9. Coclusio It is iterestig to ote that the algorithm is extremely efficiet ad easy to implemet. It yields comparable results at a fractio of computig cost. It is ideally suited for solvig reasoably medium-sized problems. The Iterative Heuristic algorithm gives reasoably good results at egligible computig cost. For large size problems, it gives very efficiet layouts. Because of the closeess of these results, it is postulated that the deviatio of these solutios from the exact optimum i large problems will be margial. Use of Iterative Heuristic Algorithm after clusterig for both horizotal ad vertical travel cost yields good
220 results. The Brach ad Boud Algorithm i covetio with the ew procedure described for the determiatio of Lower Boud of the cadidate problem is suitable for medium-sized assigmet problems i facilities plaig. The Iterative Heuristic Algorithm gives very good results, close to the exact solutio obtaied by Brach ad Boud Algorithm. a) Brach ad Boud Techique Total Cost = 4298 uits Operatioal Research Vol. 57, pp. 74-89(999) [3] D.M. Tate ad A.E. Smith, Uequal-area facility layout by geetic search, IIE Tras-actios, Vol. 27, pp. 465-472(995). [4] R. D. Meller, K.-Y. Gau, The facility layout problem: Recet ad emergig treds ad perspectives, Joural of Maufacturig Systems, Vol. 5, No. 5, pp. 35(996). [5] R. D. Meller, V. Narayaa, ad P. H. Vace, Optimal facility layout desig, Operatio Research. Letters, Vol. 23, p. 7(998). [6] M. F. Ajos, ad A. Vaelli, A attractor-repeller approach to floorplaig, Mathmetical Methods of Operatios Research. Vol. 56, No., pp. 3-27(2002). [7] H. D. Sherali, B. M. P. Fraticelli, ad R. D. Meller, Ehaced model formulatios for optimal facility layout, Operatios Research, Vol. 5, No. 4, pp. 629 644(2003). [8] M. F. Ajos, A. Keigs, ad A. Vaelli, A Semidefiite Optimizatio Approach for the Sigle-Row Layout Problem with Uequal Dimesios., Discrete Optimizatio, Vol. 2, No. 2, pp. 3-22(2005). [9] L. Kaufma ad P. J. Rousseeuw, Fidig Groups i Data: A Itroductio to Cluster Aalysis, New York, Joh Wiley(2005) [0] M. F. Ajos ad A. Vaelli, A New Mathematical Programmig Framework for Facility Layout Desig, INFORMS Joural o Computig, Vol. 8, No., pp.-8(2006) [] R.Bhowmik, Allocatig usig Hierarchical Clusterig Techique, Proceedigs of Hawaii Iteratioal Coferece o Computer Scieces, Hawaii, 2004. a) Iterative Heuristic Techique Total Cost = 442.9 uits Fig.3 Layouts obtaied by Brach ad Boud ad Iterative Heuristic Techique Number of levels = 3 Max area permitted/level=9 uits Refereces Rekha Bhowmik received her B.E., M.E., ad PhD degrees, from Idia Istitute of Techology, Idia. Her research iterest icludes Data Miig, Database Security, Spatial Database, Text Miig, Optimizatio Techiques. She is a member of IEEE, ACM, ISCA. [] B. Motreuil, A modelig framework for itegratig layout desig ad flow etwork desig, I J.A. White ad I.W. Pece, editors, Progress i Material Hadlig ad Logistics, Spriger-Verlag, Vol. 2, pp. 95-6(99). [2] D. J. Va Camp, M. W. Carter, ad A. Vaelli, A Noliear Optimizatio Approach for Solvig Facility Layout Problems, Europea Joural of