Contour line construction for a new rectangular facility in an existing layout with rectangular departments

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1 Contour lne constructon for a new rectangular faclty n an exstng layout wth rectangular departments by Har Kelachankuttu, Raan Batta, Rakesh Nag Department of Industral Engneerng, 438 Bell Hall Unversty at Buffalo (SUNY), Buffalo, NY 460 Submtted March 005; Revsed Jan. 006; Revsed Aprl 006 Abstract In a recent paper, Savas, Batta and Nag [4] consder the optmal placement of a fnte-szed faclty n the presence of arbtrarly-shaped barrers under rectlnear travel. Ther model apples to a layout context, snce barrers can be thought to be exstng departments and the fnte-szed faclty can be vewed as the new department to be placed. In a layout stuaton, the exstng and new departments are typcally rectangular n shape. Ths s a specal case of the Savas et al. paper. However the resultant optmal placement may be nfeasble due to practcal constrants lke asle locatons, electrcal connectons, etc. Hence there s a need for the development of contour lnes,.e. lnes of equal obectve functon value. Wth these contour lnes constructed, one can place the new faclty n the best manner. Ths paper deals wth the problem of constructng contour lnes n ths context. Ths contrbuton can also be vewed as the fnte-sze extenson of the contour lne result of Francs [7]. Keywords: Contour Lne, Faclty Layout, Faclty Locaton. Author for correspondence: nag@buffalo.edu

2 Introducton Accordng to Francs et al. [8] and Bndschedler and Moore [], a facltes layout problem may arse because of a change n the desgn of the product, the addton or deleton of a product from a company s product lne, a sgnfcant ncrease or decrease n the demand for a product, changes n the desgn of the process, etc. Sometmes the layout has to be redesgned to nclude a new faclty such as a sngle machne, cell or a department. Placement of a new faclty n the presence of exstng facltes can be consdered as a restrcted layout problem snce n a plant layout the exstng facltes wll act as barrers where travel and new faclty placement are not permtted. There has been sgnfcant recent work n the area of planar faclty locaton wth barrers. The reader s referred to a recent book by Klamroth [0], a recent chapter by Drezner et al. [6], artcles by Dearng, Hamacher and Klamroth [3], Dearng and Segars [4], Freß, Klamroth and Sprau [9], Dearng, Klamroth and Segars [5], and Nandkonda, Batta and Nag []. Recognzng the practcal relevance of faclty sze consderaton, Savas et al. [4] consder the optmal placement of a fnteszed faclty n the presence of arbtrarly-shaped barrers wth the medan obectve and rectlnear dstance metrc. In a layout context, barrers may be thought of as exstng facltes whch are usually rectangular. Therefore a specal case of ther model n whch the barrers and the faclty are assumed to be rectangular, may be appled to a layout problem where a new rectangular faclty has to be optmally placed n the presence of other rectangular facltes. In a layout context, the optmal ste may not be always sutable for faclty placement. For example the optmal ste may pose concerns due to sharp corners, ogs or narrowng of materal handlng asles. Hence there s a need to fnd a nearby locaton that s usable. Contour lnes, that are lnes of equal obectve functon value, help to evaluate the costs of locatons other than optmal stes. They help to fnd the next best soluton for an exstng layout problem, when the new faclty cannot be placed at ts ntended optmal locaton. Francs [7] has consdered ths problem n the context where facltes are ponts. The fnte-area case s more approprate for facltes layout. The remander of ths paper s organzed as follows. In Secton, we descrbe and defne the problem, whch s defned usng the rectangular metrc. In Secton 3, we brefly vst the grd constructon procedure of Larson and Sadq [] and the Equal Travel Tme Lne concept of Batta et al. []. We then llustrate some new propertes of Equal Travel Tme Lnes n Secton 4. Secton 5 llustrates the

3 contour lne constructon procedure whch s followed by a numercal example n Secton 6. Secton 7 descrbes the complexty of our soluton procedure. Conclusons and drectons for future research are presented n Secton 8. Problem Defnton We are gven a fnte number of rectangular exstng facltes (EFs) n a D plane n whch a rectangular new faclty (NF) has to be placed. The shape of the NF s gven and fxed. We assume that the NF s orented wth ts sdes parallel to the X and Y axes, and one of the four possble orentatons s chosen. The procedure can be repeated for the remanng three orentatons. Each EF s characterzed by ts four corner vertces and has one or more I/O ponts on ts boundary. These EFs are labeled EF,...,EF m. Materal flow between the NF and the EFs takes place through a sngle I/O pont X located on the NF boundary. Let (x, y) represent the coordnates of the I/O pont of the NF, D be the set of all EF I/O ponts, and S be the rectangular space representng the shop floor area. Let d(, : X) be the shortest feasble rectlnear dstance between I/O ponts and (, D X) gventhatx s located at (x, y). Here the term feasble mples a path that does not penetrate the nteror of the NF or EFs, whch serve as mpenetrable barrers to travel. There are two types of materal flows n the problem. One s the flow between the EFs and the NF and the other s the flow between all pars of EFs. All flows takes place through a shortest feasble rectlnear dstance path. In computng the dstance, we permt travel on a EF or NF boundary but not nsde. It certanly makes sense to recognze that the placement of the NF may block shortest path movements between pars of EF I/O ponts, as Fg. llustrates. The same s true for movements between X and an EF I/O pont. At the NF poston n Fg., the NF nterferes wth the flow between I/O ponts -3 and -X. The NF needs to be placed such that: (a) the NF does not ntersect the unon of the nterors of the EFs, and (b) the NF s contaned n S. Placng the NF reduces to locatng the NF I/O pont X. We refer to the set of all feasble locatons for X as F. Note that feasble placements allow the boundares of the NF and EFs to concde. Havng a sngle I/O pont for the NF s a modelng lmtaton made to avod complcatons that can arse due to the flow assgnment choces of EF I/O ponts to the NF I/O ponts. We note here that t would be much harder to consder two or more I/O ponts for the new faclty, snce that makes up a new locaton problem of locatng two or more

4 EF Path EF 3 Shop floor, S NF X I/O ponts Path X EF EF Fgure : Example to llustrate that NF placement can affect the flow dstance between a par of EFs I/O ponts and between the NF I/O pont and an EF I/O pont related ponts. For a gven X, the total weghted travel dstance between EFs and the NF s: D w d(, X : X), where w s the rate of materal flow between the th EF I/O pont and I/O pont of the NF at X. Smlarly, the total weghted travel dstance between all EFs s: D D u d(, : X), where u s the rate of materal flow between I/O ponts and of a par of EFs. The obectve functon s the total materal dstance traveled between pars of EF I/O ponts and between the NF I/O pont and EF I/O ponts, and s gven by: f(x) = w d(, X : X)+ D D u d(, : X). For optmal placement (Savas et al. [4]), the problem s to fnd X F such that f(x ) f(x), X F. In the present paper we are nterested n fndng the set of all ponts that satsfy f(x) =z, wherez s a constant. Such a set of ponts s referred to as a contour lne n Francs [7]. 3 Background D We dvde the plane nto regons where the obectve functon s lnear and establsh that a contour lne s represented by a straght lne n each regon. The slope of the lne potentally changes as we move from one regon to the next. Ths s smlar to the stuaton when EFs and the NF are ponts (refer Francs [7]). Though the method to fnd the slope n a regon remans smlar, the regons are more complex to determne n ths case. Lke the pont case, a grd constructon procedure s employed to determne the regons n a plane. Secton 3. llustrates ths. But n our case, addtonal regons need to be determned because of the effect of fnte-szed NF and EFs on the traversal path 3

5 of materal flow. A fnte-szed NF may nterfere wth the traversal path of flow n some regons of the plane. These nterference regons are dscussed n Secton 3.. In addton, EFs may create alternate traversal paths for materal flow n regons of the layout. Ths s llustrated n Secton Grd constructon and cell formaton The EFs and the I/O ponts of the EFs are needed for the grd constructon process. A grd s formed wth S as follows: Pass lnes parallel to the x and y axes through all EF corners and through all EF I/O ponts, wth each lne termnated at the frst nteror pont of an EF that t encounters or at an edge of S. Fgure llustrates a grd. Let L h denote the set of horzontal traversal lnes and L v denote the set of vertcal traversal lnes. Also, let L = L h Lv be the set of all EF traversal lnes. The EFs and L dvde the regon S m = EF nto a number of cells. A cell s defned as a closed regon n the grd that s not an EF. Cells have the property that a shortest feasble rectlnear path from an EF I/O pont to a pont located n the cell passes through one of the cell corners (see Larson and Sadq []). Furthermore, the functon whch defnes the length of a shortest feasble rectlnear path s concave over the doman of the cell (the varable beng the locaton of the destnaton pont n the cell). Snce we consder rectangular EFs and S s assumed to be rectangular, all the cells formed wll also be rectangular. When a fnte-szed NF s placed, a new set of traversal lnes passng through ts corners and I/O pont are ntroduced. Ths new set of lnes are referred to as NF traversal lnes, L (X), when the NF I/O pont s located at X. We note that t s possble that some lnes are both EF and NF traversal lnes. NF 4 3 EF 3 EF X EF EF 4 NF traversal lnes EF Traversal lnes Fgure : EF and NF traversal lnes 4

6 3. Interference sets Q If a rectangular NF s fully-contaned n a cell, the shortest feasble rectlnear path from ts I/O pont to an EF I/O pont s stll concave. In fact, the dstance functon s dentcal to that of a pont NF located at X. Ths s shown n Lemma and Theorem of Savas et al. [4]. However, when the rectangular NF ntersects one or more EF traversal lnes, dstance measurement from ts I/O pont to an EF I/O pont can be affected. The dsrupton can be characterzed provded the NF ntersects a dstnct set of grd lnes as X s vared. Ths forms the motvaton of nterference sets. Consder a set of grd lnes S L. We seek a maxmal set of locatons Q S such that when the NF I/O locaton X Q S the NF ntersects the grd lnes n the set S and does not ntersect the grd lnes n the set L S. The set Q S s a closed regon whose boundary s composed from two types of segments: (a) locatons X such that the NF boundary ntersects wth some EF boundary, and/or (b) locatons X such that some grd lne(s) n set S concde wth some NF traversal lne(s). An example for the formaton of sets Q s llustrated n Fg. 3. Consder a NF beng moved around a cell such that the NF ntersects the subset of the traversal lnes {h,h +,v }. As llustrated, Q h, Q h+, Q h,v, Q v and Q h+,v are the sets of X locatons such that the NF ntersects the sets of traversal lnes {h }, {h + }, {h,v }, {v } and {h +,v } respectvely. It s noted that each nterference set Q s rectangular and non-ntersectng wth another nterference set. Q h,v s shaded for ease of vsualzaton (whle the others are defned by ther corners). For example, pont 6 of Q h,v s generated when grd lne h s concdent wth NF traversal lne h andgrdlnev s concdent wth NF traversal lne v. v v + v v 0 h + NF h h X h Q = 5 6 v,h h Q = v Q h + Q h = = Q v, h + = 7 8 Fgure 3: Interference Sets Q 5

7 3.3 Equal travel tme lnes We now focus on the computaton of dstance from a gven EF I/O pont outsde cell C to a pont X whch belongs to cell C. We know that for a gven pont X C a shortest feasble rectlnear path from pont to X passes through a corner of the cell C (Secton 3.). However, there s no guarantee that the specfc cell corner of C through whch ths path traverses remans the same for dfferent ponts X n the cell C. An Equal Travel Tme Lne (ETTL) nduced by onto cell C allows us to partton C nto two sub-cells such that the cell corner assgnment remans the same wthn each sub-cell. The term ETTL arses from the fact that for all ponts X that le on the ETTL tself two alternate cell corners can be used to reach pont ;.e., they have equal travel tme. The followng materal closely follows the presentaton n Nandkonda et al. []. Consder the cell C shown n Fgure 4. Let the coordnates of the four cell corners of C, labeled,, 3, and 4, be (0, 0), (0,b),(a, b), and (a, 0), respectvely. Let d denote the length of a shortest feasble rectlnear path from pont to cell corner. If d d <bthen wll generate an ETTL va corners and, whch wll be the lne segment (parallel to the edges and3 4ofC) onng (0, ( d d + b)/) and (a, ( d d + b)/). If d d 3 < (a + b), then wll generate an ETTL of a dfferent type through corners and 3, whch wll be the lne segment onng (a ( d d 3 + a + b)/,b)and (a, b ( d d 3 + a + b)/). Ths ETTL s at a 45 o angle wth sdes 3and3 4of the cell. Smlarly, ETTLs can be constructed va corners 3 and 4, and 4, and 4, and and 3. The followng results hold. / ( d d 3 + a + b ) (0,b) (a,b) 3 ETTL va corners and 3 b / ( d d +b ) ETTL va corners and Cell C (0,0) (a,0) 4 a Fgure 4: ETTLs n a cell C 6

8 Result : AnI/Opont generates at most one ETTL n a rectangular cell (Theorem n Nandkonda et al. []). Result : ETTLs between both pars of dagonally opposng cell corners n a rectangular cell cannot exst (Lemma 4.3. n Sarkar et al. [3]). In the tradtonal defnton an ETTL s generated because of an EF. However, f a fnte-szed NF s placed nsde the cell then the shortest rectlnear paths to the I/O pont may change. Ths may cause the ETTL to move to a new locaton. The concept of Movng ETTLs has been descrbed n Sarkar et al. [3]. Result 3: An ETTL may move (from the ETTL poston of the pont NF) f the rectangular NF blocks the cell corners between whch the ETTL s generated (Sarkar et al. [3]). By block we mply nterference to the shortest rectlnear path. Consder the stuaton n Fgure 5 where the I/O pont of an NF, X, s located n cell C. In the absence of the NF, I/O pont of the EF generates an ETTL n cell C va corners c and c 4. Whle accountng for the presence of the NF, the shortest feasble path from to X can be altered or nterfered due to NF. We note that ths can only happen f the NF ntersects at least one grd lne. If S s the set of grd lnes ntersected by the NF, then there exsts an nterference set Q S.Insucha stuaton, the ETTL n cell C can be shfted or moved due to the presence of the NF (refer to Fgure 5).Arguedntheopposteway,fX/ Q, the ETTL wll not move. In addton to ETTLs that are generated due to the flow between EF I/O ponts and the NF I/O pont, there can be ETTLs between pars of EF I/O ponts f the fnte-szed NF causes flow nterference. Specfcally, ths can happen when the NF ntersects grd lnes, or X Q. These ETTLs are analogous to the ones already descrbed n the sense that they partton the regon Q (opposed to cell C) such that the EF I/O pont flows are drected through a specfc corner of Q. To avod repettous exposton we do not elaborate on these further (see also Secton 4). In the next secton we llustrate some new propertes of ETTLs whch are necessary for our analyss. 4 Addtonal Propertes of ETTLs Lke EFs, a rectangular NF can generate ETTLs when the NF nterferes wth the shortest rectlnear path between a par of EF I/O ponts, or between an EF I/O pont and the NF I/O pont. Ths s 7

9 EF Cell C C C C C 3 SC X SC ETTL NF X SC SC 0000 C C EF C C NF ETTL poston when NF s not present or s nfntesmal sze Cell C ETTL Fgure 5: Illustraton of a Movng ETTL possble only when X Qassocated wth the affected rectlnear path. Lemma 4. Consder an EF-NF rectlnear flow whch s nterfered by the NF. If an ETTL s generated t s parallel to the affected traversal path and the edges of Q and the NF whch are parallel to the affected traversal path concde wth each other. Proof: Consder an NF whch nterferes wth the shortest rectlnear path from ts I/O pont to an I/O pont of an EF. Let the NF have length H, wdthw and ts I/O pont at a dstance Y, (Y > H/) on the longer sde as shown n Fg. 6. The NF nterferes wth the traversal path h of the flow n the regon Q, whereq s the unon of sets Q h and Q h,v, assocated wth the affected traversal path h.letxbesuchthat the edges of Q and the NF whch are parallel to the traversal path h concde wth each other. At ths poston the dstance traveled along the path X s Y + H Y + const = H + const and along 4 3 X s H Y + Y + const = H + const. Thss llustrated n Fg. 6a. Therefore at ths poston the flow from X can occur through corners or 4 3 of the NF. Ths remans true for any poston along a lne through ths pont and parallel to the affected traversal path. Therefore, the ETTL generated wll be parallel to the affected traversal path and the edges of Q and the NF whch are parallel to the affected traversal path concde wth each other. The ETTL generated dvdes the regon Q nto Q andq. Insde Q the flow s through corners 4 3 of the NF whereas nsde Q the flow s through corners. Lemma 4. For rectlnear flow between a par of EF I/O ponts whch s nterfered wth by an NF, the ETTL generated s n the nteror of the regon Q assocated wth the affected traversal path and parallel to the path. Proof: Suppose that the NF nterferes wth the shortest rectlnear path between I/O ponts and 8

10 v v Q h 4 3 H Y Y X Y H Y ETTL X Q Q ETTL h Y H/ 4 3 H Y X H/ Q X H Y (a) W NF (b) Fgure 6: Illustratons for Lemmas 4. and 4.. The NF nterferes wth the traversal path h of the flow n the regon Q, whereq s the unon of sets Q h and Q h,v, assocated wth the affected traversal path h. Consder the NF beng placed at the center of Q. Snce X s not at the center (Y > H/), the center of Q wll be offset from the traveral path by (Y H/). At ths poston the dstance travelled along the path s H/+const = H +const and along 4 3 s (H Y +Y H/)+const = H +const. Ths s llustrated n Fg. 6b. Therefore at ths poston the flow from can occur through corners or 4 3 of the NF. Ths remans true for any poston along a lne through ths pont and parallel to the affected traversal path. The lne dvdes the regon Q nto Q andq. Insde Q the flow s through corners 4 3 of the NF whereas nsde Q the flow s through corners. Therefore the ETTL generated wll be at the center of the regon Q assocated wth the affected traversal path and parallel to t. In summary the dfferent types of ETTLs that can be generated are as follows: Regon nsde Q ETTLs due to an EF I/O pont for flow between the EF I/O pont and X (Secton 3.3). ETTLs due to the NF for flow between an EF I/O pont and X (Lemma 4.). ETTLs due to the NF for flow between a par of EF I/O ponts (Lemma 4.). Regon outsde Q (.e., NF fully contaned n a cell) ETTLs due to an EF I/O pont for flow between the EF I/O pont and X. In ths case 9

11 the fnte sze of the faclty wll not cause the ETTL to move from the orgnal pont NF case. We llustrate these types of ETTLs usng the example shown n Fg. 7. There are two EF I/O ponts, and. As llustrated, generates an ETTL for flow between X nsde the cell. The NF also generates ETTLs for flow between X and. Insde the regon abcd the flow from X s through the upper corners m n of the EF.Insdedclk t s through the lower corners p o of the EF.Insdedclk the flow s through ether corners q r or corners t s of the NF. Insde efgh the flow s through the lower corners t s of the NF. However nsde hglk t s through the upper corners q r of the NF. Armed wth the above results we can proceed wth constructng contour lnes. m n EF Q a d e h NF p o k t s q r X b c f g l 3 u v EF x 000w 3 ETTL due to EF for flow between NF ETTL due to NF for flow between NF ETTL due to NF for flow between Fgure 7: Dfferent types of ETTLs 5 Contour Lne Constructon For the functon f(x) the contour lne of value z s represented as L(z)whereL(z) ={X F : f(x) =z}. A contour set whose boundary s a contour lne s the set of all ponts havng values of f(x) z. Francs [7] has shown that for pont-szed NF and EFs, the contour lne s contnuous, wth the correspondng contour set beng convex. However the fnte szes of the NF and the EFs may present complcatons as gven below:. Contour lnes may be ntercepted by an EF and hence could be dsconnected.. The fnte-szed NF may nterfere wth the traversal path of flow between facltes n the regon Q assocated wth the affected traversal path. Therefore ponts nsde Q mayhaveanobectvefuncton value hgher than that of the nearby ponts. Ths can make the contour lne dsconnected. 0

12 In the case of pont NF and pont EFs, f one starts wth a locaton of obectve functon value z, then the slope n the cell to whch X belongs wll cause the contour lne to be ncdent on some adacent cells and the process can be repeated untl the contour s closed. In our case, the contour lne and set for a partcular value z can only be constructed after evaluatng all the cells, at least mplctly, to dentfy those cells whch contan an X such that f(x) = z. We start the mplct enumeraton by elmnatng certan cells and thus dentfy a set of canddate cells S z whch potentally contan an X such that f(x) =z. The NF may nterfere wth the materal flow n some cells whle n others t may not. In those cells where the NF nterferes wth the flow t does so only n the regon Q assocated wth the affected traversal path. Therefore the set S z may be classfed as shown n Fg. 8. In our analyss, we consder the obectve functon and the method to calculate the slope of the contour lne n each of them separately. We start by llustratng our methodology through a numercal example where an NF has to be placed n a shop floor havng two EF I/O ponts, and. The weghts of the flow between facltes are as shown n Fg. 9. We then present a general procedure to construct the contour lne for a gven problem nstance. Cells wth no nterferences Set of cells S z Cells wth nterferences Q Outsde Q Fgure 8: Classfcaton of cells 5. Identfcaton of cells Insde a cell, the canddate ponts for the mnma of the obectve functon of a pont NF are the corners of the cell. Let zp represent the obectve functon value of a pont-szed NF located at the } th corner of a cell, where =,, 3, 4. In a cell f the mnmum value of z p,.e. mn {z p,sgreater than z then no pont can be found nsde the cell wth value z snce the placement of a rectangular NF wll only ncrease the obectve functon value. If the NF s not nterferng wth any flow then t may be consdered as a pont nsde the cell. In ths case the maxmum value of z p at the cell corners, }.e. max {z p should be greater than z for the cell to be a canddate. However f the NF nterferes

13 EF 9 0 EF w w = 3/4 = /4 u = u = NF Fgure 9: Numercal Example wth the flow at any of the corners and mn {z p } s less than z, then a pont wth value z may or may not be present n these cells. The presence of such a pont depends on the effect of the NF on the materal flow. Based on the above observatons we present an algorthm to determne the set of canddate cells, S z, whch potentally contan contour lne segments of value z. Algorthm for determnng the set of cells S z : Input: Set of cells from the grd constructon procedure of Secton 3. Intalze: S z = For each cell C: } If mn {z p z If no nterferences } If max {z p z S z S z C Else S z S z C Output: Set of cells S z

14 Applyng ths algorthm to the numercal example, for z = 35, the set of cells s S 35 = {,, 3, 9, 0,,, 3, 4}. 5. Obectve functon In ths secton we analyze the obectve functon and llustrate the method to calculate slope of the lne n each of the classfcatons of set S z. 5.. Cells wth no nterferences Consder a cell C where an NF does not nterfere wth any of the materal flow. Consder a shortest feasble rectlnear path from an I/O pont to the NF as llustrated n Fg. 0. The NF does not affect the length of the path from I/O pont to the NF, and hence, the NF may be consdered as a pont nsde the cell. For a pont NF, rrespectve of ts poston nsde cell C the total weghted travel dstance between a par of EF I/O ponts wll be a constant. However the total weghted travel dstance between EF I/O ponts and X wll vary. The EF I/O ponts may be assgned to approprate corners of the cell. It has to be noted that even though there s no effect of the NF nsde the cell, an EF could generate an ETTL as defned n Secton 3.3. Therefore the assgnment of the I/O ponts may vary nsde the cell. Cell C w 4 Cell C w 3 w w (a) (b) Fgure 0: Cell wth no nterferences Let w,w,w 3,w 4 be the weghts of the I/O ponts assgned to the correspondng corner of a cell/subcell for the purpose of shortest dstance measurement. Note that corners are numbered from the lower left corner and movng counterclockwse. The obectve functon may be wrtten as: f(x, y) =w (x x mn )+w (x max x)+w 3 (x max x)+w 4 (x x mn )+w (y y mn )+w (y y mn )+w 3 (y max y)+w 4 (y max y)+const, 3

15 where (x mn,y mn ), (x max,y mn ), (x max,y max ), (x mn,y max ) are the corners of the cell/subcell. As can be observed the obectve functon s lnear n x and y. Therefore the part of the contour lne nsde ths cell/subcell wll be a lne segment of the followng functonal form: { } y = x w +w 4 w 3 w w +w w 3 w 4 + const. The coeffcent of x yelds the slope of the contour lne where the numerator s the dfference between the sum of the weghts of the I/O ponts served through corners and 4, and the sum of the weghts of the I/O ponts served through corners and 3; and the denomnator s the dfference between the sum of the weghts of the I/O ponts served through corners and, and the sum of the weghts of the I/O ponts served through corners 3 and 4. Ths s smlar to the results for pont-szed NF and EFs as defned n Francs [7]. In the numercal example, cells,, 3,,, 3 and 4 of the set S 35 have no nterferences due to the NF. Insde cell 4, I/O ponts and generate ETTLs for flow to X. Therefore the assgnment of the I/O ponts and to cell corners vares nsde cell 4. However the NF may be consdered as a pont nsde these and the slope may be found usng the results for a pont NF nsde a subcell. 5.. Insde Q Consder cells 9 and 0 n the numercal example where the NF nterferes wth the traversal path h of the flow between X and.. Flow between an EF I/O pont and X: Fgure shows the Q assocated wth the traversal path h. Let the four corners of Q be a, b, c and d. The ETTL for flow between X dvdes the regon Q nto Q andq, referred to as subcells. As shown n Fg. a, nsde Q a shortest rectlnear path from the I/O pont to X s through 4 3 X. Therefore the extra dstance traveled due to the NF s Y h Y h +Y h Y X. The dstance Y h Y X related to path 3 X s a constant. The dstance that vares as we move the NF nsde Q sy h Y h. Further, Y h Y h s equal to Y X Y ab and Y h Y X s equal to Y h Y ab. Therefore the path 4 3 X may be consdered equvalent to the path from through corner a to a pont faclty at X. Hence nsde Q, the rectlnear path from the I/O pont to X can be nterpreted as a path to a pont-szed faclty through the corner of the regon Q. Smlarly n Q, a shortest rectlnear path from the I/O pont to X s through X as shown n Fg. b, whch s equvalent to the path from through corner d to a pont-szed faclty 4

16 at X. d a Q 4 3 X c h h X b h Q ETTL d a Q 4 3 X h c h X h b (a) (b) Fgure : Shortest rectlnear paths n Q and Q In summary, when Q s splt nto subcells by an ETTL generated due to I/O pont, the shortest rectlnear dstance from to X n a subcell can be measured from the subcell corner that s closer to and at the other sde of the ETTL from X. Letw,w,w 3,w 4 be the weghts of the I/O ponts not affected by the NF and assgned to the correspondng corner of a cell/subcell for the purpose of shortest dstance measurement. Then the total weghted travel dstance between EF-NF nsde Q (and Q) s: fq EF NF (x, y) =w (x x mn )+w (y y mn )+w (x max x)+w (y y mn )+w 3 (x max x)+ w 3 (y max y)+w 4 (x x mn )+w 4 (y max y)+w (x x mn )+w (y y mn )+const. = x(w + w + w 4 w 3 w )+y(w + w + w w 3 w 4 )+const. fq EF NF (x, y) =w (x x mn )+w (y y mn )+w (x max x)+w (y y mn )+w 3 (x max x)+ w 3 (y max y)+w 4 (x x mn )+w 4 (y max y)+w (x x mn )+w (y max y)+const. = x(w + w + w 4 w 3 w )+y(w + w w 3 w 4 w )+const.. Flow between a par of EF I/O ponts: Fgure shows the ETTL for flow between n the regon Q assocated wth the traversal path h. It dvdes the regon Q nto Q andq. Insde Q a shortest rectlnear path from the I/O pont to the pont s through 4 3. Therefore the extra dstance traveled due to the NF s (Y h Y h ). As shown n Fg. a, Y h Y h s equal to Y X Y ab. Smlarly n Q, the shortest rectlnear path from the I/O pont to s through 5

17 . Therefore the extra dstance travelled due to the NF s (Y h Y h ). As shown n Fg. b, Y h Y h s equal to Y cd Y X. d a Q 4 3 c h h X b h Q ETTL d a Q 4 3 h c h X h b (a) (b) Fgure : Shortest rectlnear paths n Q and Q Then the total weghted travel dstance between EF-EF nsde Q (and Q) s: fq EF EF (x, y) =u (y y mn )+u (y y mn )+const = y(u +u )+const. fq EF EF (x, y) =u (y max y)+u (y max y)+const = y( u u )+const. 3. Obectve functon: Insde Q the ETTLs for flow between and X and are at the same locaton as shown n Fg. 3. ETTLs dvde Q nto Q andq. Consderng Q, the obectve functon may be wrtten as: f Q (x, y) =x(w + w + w 4 w 3 w )+y(w + w + w w 3 w 4 )+y(u +u )+const. Once agan, the obectve functon s lnear n x and y. Therefore the part of the contour lne nsde subcell Q wll be a lne segment of the followng functonal form: { } y = x w +w +w 4 w 3 w w +w +u +u w 3 w 4 + const. The coeffcent of x gves the slope of the contour lne nsde Q. The slope of the lne nsde Q may be found n a smlar manner. 6

18 d c Q a 0 Q Q b ETTL, ETTL X Fgure 3: Subcells 5..3 Outsde Q Consder the regon outsde Q n cell 9. As llustrated n Fg. 4, the NF does not nterfere wth any of the materal flow n the regon outsde Q and hence t may be consdered as a pont nsde the regon. For a pont-szed faclty, rrespectve of ts poston nsde the regon, the total weghted travel dstance between a par of EF I/O ponts s a constant. However the total weghted travel dstance between an EF I/O pont and X wll vary and the weghts of the I/O ponts may be assgned to the corners of the regon. Regon outsde Q may also be consdered as a subcell. In ths example, may be assgned to the lower left corner and to the lower rght corner of the subcell. The slope may then be found by applyng the results dscussed n Secton 5.. for a pont-szed NF. 4 3 d OutsdeQ 9 9 c d c Q a b a b Fgure 4: Outsde Q 5.3 Contour lne constructon method We summarze the contour lne constructon procedure as follows: Consder startng at a feasble pont X F whch has an obectve functon value of z. 7

19 . Draw the EF traversal lnes and form the grd.. Identfy the cell C whch contans X. 3. Identfy the subcells n cell C. (a) Identfy ETTLs formed due to EF I/O ponts, for flow between EF I/O ponts and X n cell C and determne ts subcells. Note that the NF s fnte-szed and could cause the ETTL to move from the pont NF case (Result 3). (b) Identfy the nterference sets Qs, where the NF nterferes wth any of the materal flows. Identfy the ETTLs formed, due to the NF for materal flow between a par of EF I/O ponts or an EF I/O pont and X, nsde cell C and hence determne the subcells. 4. For cell C and ts subcells, assgn weghts of the unaffected EF I/O pont to X materal flow to the approprate cell corner. For affected materal flow between a par of EF I/O ponts and between an EF I/O pont and X, assgn weghts to the corners as descrbed n Secton For cell C and ts subcells, the slope of the contour lne s determned as follows: (a) From step 4, let, w be the sum of the weghts of the materal flow between EF I/O ponts and X assgned to corner, u v and u hk be the sum of the weghts of the materal flow between pars of EF I/O ponts affected along the vertcal and horzontal path(s), let,, 3, 4 be the corners of the cell/subcell startng from the lower left corner and movng antclockwse, then, a =(w + w 4 +u v +u v4 ) (w + w 3 +u v +u v3 ), and b =(w + w +u h +u h ) (w 3 + w 4 +u h3 +u h4 ) (b) Slope of the contour lne = a b 6. Proceed to construct the contour lne that passes through X, usng the slope computed n Step 5. If ths lne termnates (n ether drecton) at the boundary of an adacent cell/subcell, t dentfes a pont n ths adacent cell/subcell that has an obectve functon value of z. Inths 8

20 case, recompute the slope usng Step 5 and then proceed to draw the contnuaton of ths contour lne. It s also possble that the lne termnates at the boundary of the feasble regon F.Inths case, examne cells n set S z (determned usng the algorthm of Secton 5.) through whch ths contour lne has not yet passed, and attempt to fnd a new startng pont that has an obectve functon value z. To do ths one can take advantage of the fact that nsde a cell/subcell the assgnment of EF I/O ponts to the corners of the cell/subcell remans unchanged. Hence the determnaton of a pont of obectve functon value z can be set up as a lnear program wth no obectve, and wth constrants that lmts the locaton of X to be wthn the cell/subcell as well as a constrant that specfes that the obectve functon value for the chosen X s z. Ifths lnear program s nfeasble, the cell/subcell contans no pont wth obectve functon value z. Otherwse, t returns one such pont. The procedure s then restarted wth ths new locaton and ts assocated cell C. 6 Numercal Example Fgure 5 shows the contour lnes constructed for z = 3, 35, 4 and 43 usng the above procedure. As can be observed the contour lnes for z = 3 and 35 are ntercepted by the EFs makng them dsconnected. To see the applcaton of contour lne constructon to a layout context, consder the stuaton where EF and EF are exstng departments and that a new department ndcated by NF needs to be placed. The dffculty s that the NF s a machne that produces sgnfcant vbraton and has to be set nto a concrete pad that needs to be bult nto the floor. The floor areas where ths concrete pad can be placed s ndcated n Fgure 5. (In general, there mght be other such dsont areas avalable throughout the faclty.) We mmedately recognze that gven ths practcal constrant on the NF placement, the optmum soluton gven by the contour lne labeled 3 wll not work. The soluton to ths constraned problem s to locate the NF at E (as shown n the fgure) yeldng an obectve value of 35. Asde from solvng the constraned problem, contour lnes dramatcally mprove our ntuton regardng the obectve space. The vsual pcture whch s akn to an elevaton map allows us to readly compare soluton alternatves. Havng the contour lnes n the entre feasble regon can gve the desgner a holstc perspectve for evaluatng constructon costs and materal handlng costs along wth other practcal ssues that the formulaton does not address drectly. 9

21 EF EF 3 35 E Area where foundaton strengthenng can be accomplshed NF Infeasble Regon Fgure 5: Contour lnes 7 Soluton Complexty Our soluton methodology proceeds n two steps. Frst the number of cells n whch obectve functon value z s potentally present has to be dentfed usng the algorthm of Secton 5.. Each EF generates horzontal and vertcal traversal lnes from ts boundares. Hence f there are m EFs, m horzontal and m vertcal lnes wll be produced. Smlarly n I/O ponts can generate at most n horzontal and n vertcal lnes. Hence the maxmum number of cells generated s O((m + n) ). The contour lne s then constructed by calculatng the slope of the lne n each of the dentfed cells. Here the complexty depends on the number of subcells generated nsde each cell. Result states that ETTLs between dagonally-opposte corners for rectangular EFs are not possble. Nandkonda et al. [] have shown that an I/O pont of an EF wll generate at most one ETTL. Hence n I/O ponts of EFs can generate n ETTLs, n the worst case generatng O(n ) subcells per cell. In addton to ths, subcells may be generated due to ETTLs for materal flow between a par of EF I/O ponts and between an EF I/O pont and X. If, n a cell, the NF nterferes the materal flow wth k (k n) EF I/O ponts, then the maxmum number of ETTLs formed nsde the cell for flow between EF I/O 0

22 ponts and X wll be k. Addtonally the flow between k I/O ponts and the rest of the I/O ponts may be affected by the NF. Ths can generate a maxmum of k(n k) ETTLs for the flows between pars of EF I/O ponts. In addton to the ETTLs the boundares of Qs wll also dvde the cell nto subcells. Hence the number of subcells that can be generated nsde a cell s O(n ). Therefore the total number of subcells that can be generated s O(m n + n 4 ), whch s polynomally bound. 8 Summary Ths paper addresses the contour lne constructon procedure for a fnte-szed rectangular new faclty to be placed n a layout havng other exstng rectangular facltes. Optmal placement of a fnteszed new faclty n the presence of other facltes has been studed by Savas et al. [4]. However due to other consderatons the optmal ste may not be always sutable for placement of the new faclty. Ths necesstates the new faclty to be placed at alternate locatons and provdes the motvaton for ths paper. Contour lnes helps to place the I/O pont of the new faclty at locatons other than the optmal ste wth characterzed ncrease n the obectve functon value. To draw the contour lnes, we dvde the plane nto regons and formulate a method to calculate the slope of the contour lne n each of these regons. Our work can be vewed as an extenson to the work by Francs [7] who developed the contour lne procedure for locatng a pont NF n the presence of pont EFs. Francs [7] also use the concept of dvdng the plane nto regons and calculatng the slope of the contour lne n each of those regons. Ths work s applcable for a facltes locaton problem. To extend ths to a layout context we have to assume fnte sze for the new faclty and the presence of other fnte-szed exstng facltes. However, the fnte sze of the new faclty and the presence of other facltes makes the problem complex to solve snce travel s not permtted n these regons and the rectlnear metrc gets destroyed. By a systematc dvson of the plane nto subregons where the slope of the lne remans constant, we develop a procedure for constructng layout contour lnes. Unlke the locaton case, layout contour lnes are non-convex and dsconnected. Complexty analyss establshes that the methodology developed s polynomally bound. It needs to be mentoned that from a faclty locaton pont of vew the I/O ponts are ndeed exstng facltes and the rectangular regons of the EFs are barrers. Wthn ths context, the paper deals wth a planar sngle medan faclty locaton problem wth the rectangular metrc n the presence of rectangular barrers, where one of the barrers belongs to the new faclty. Snce the man purpose

23 of ths paper was to develop contour lnes for ths problem and demonstrate ts potental applcablty n layout analyss (by generalzng Francs [7]), we used he terms I/O ponts, exstng facltes and new faclty. Acknowledgment Ths work s supported by the Natonal Scence Foundaton va grant number DMI The authors also wsh to acknowledge the help of two anonymous referees, whose comments sgnfcantly mproved the exposton of ths paper. References [] R. Batta, A. Ghose, and U. Palekar. Locatng facltes on the manhattan metrc wth arbtrarly shaped barrers and convex forbdden regons. Transportaton Scence, 3():6 36, 989. [] A.E. Bndschedler and J.M. Moore. Optmum locaton of new machnes n exstng plant layouts. Journal of Industral Engneerng, :4 48, 96. [3] P. Dearng, H.W. Hamacher, and K. Klamroth. Domnatng sets for rectlnear center locaton problems wth polyhedral barrers. Naval Research Logstcs, 49(7): , 00. [4] P. Dearng and R. Segars Jr. An equvalence result for sngle faclty planar locaton problems wth rectlnear dstance and barrers. Annals of Operatons Research, :89 0, 00. [5] P.M. Dearng, K. Klamroth, and R. Segars Jr. Planar locaton problems wth block dstance and barrers. Annals of Operatons Research, 36:7 43, 005. [6] Z. Drezner, K. Klamroth, A. Schöbel, and G.O. Wesolowsky. The weber problem. In Z. Drezner and H.W. Hamacher, edtors, Faclty Locaton: Applcatons and Theory, chapter, pages 36. Sprnger-Verlag, second edton, 004. [7] R. L. Francs. Note on the optmum locaton of new machnes n exstng plant layouts. Journal of Industral Engneerng, 4():57 59, January-February 963. [8]R.L.Francs,L.F.McGnns,andJ.A.Whte. Faclty Layout and Locaton: An Analytcal Approach. Prentce Hall, Englewood Clffs, NJ, 99.

24 [9] L. Freß, K. Klamroth, and M. Sprau. A wavefront approach to center locaton problems wth barrers. Annals of Operatons Research, 36:35 48, 005. [0] K. Klamroth. Sngle Faclty Locaton Problems wth Barrers. Sprnger-Verlag, 00. [] R.C. Larson and G. Sadq. Faclty locatons wth the manhattan metrc n the presence of barrers to travel. Operatons Research, 3(4):65 669, January 983. [] P. Nandkonda, R. Batta, and R. Nag. Locatng a -center on a manhattan plane wth arbtrarly shaped barrers. Annals of Operatons Research, 3:57 7, 003. [3] A. Sarkar, R. Batta, and R. Nag. Placng a fnte sze faclty wth a center obectve on a rectlnear plane wth barrers. European Journal of Operatonal Research, to appear 005. [4] S. Savas, R. Batta, and R. Nag. Fnte-sze faclty placement n the presence of barrers to rectlnear travel. Operatons Research, 50(6):08 03, November-December 00. 3