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1 UC Berkeley Workng Papers Ttle Dscretzaton and Valdaton of the Contnuum Approxmaton Scheme for Termnal System Desgn Permalnk Authors Ouyang, Yanfeng Daganzo, Carlos F. Publcaton Date escholarshp.org Powered by the Calforna Dgtal Lbrary Unversty of Calforna

2 Insttute of Transportaton Studes Unversty of Calforna at Berkeley Dscretzaton and Valdaton of the Contnuum Approxmaton Scheme for Termnal System Desgn Yanfeng Ouyang and Carlos F. Daganzo WORKING PAPER UCB-ITS-WP October 2003 ISSN

3 DISCRETIZATION AND VALIDATION OF THE CONTINUUM APPROXIMATION SCHEME FOR TERMINAL SYSTEM DESIGN Yanfeng Ouyang and Carlos F. Daganzo Insttute of Transportaton Studes Department of Cvl and Envronmental Engneerng Unversty of Calforna at Berkeley, CA Workng Paper August 1st, 2003

4 DISCRETIZATION AND VALIDATION OF THE CONTINUUM APPROXIMATION SCHEME FOR TERMINAL SYSTEM DESIGN Yanfeng Ouyang and Carlos F. Daganzo Insttute of Transportaton Studes and Department of Cvl and Envronmental Engneerng Unversty of Calforna at Berkeley, CA (August 1st, 2003) ABSTRACT Ths paper proposes an algorthm that automatcally translates the "contnuum approxmaton" (CA) recpes for locaton problems nto dscrete desgns. It s appled to termnal systems but can also be used for other logstcs problems. The study also systematcally compares the logstcs costs predcted by the CA approach wth the actual costs for dscrete desgns obtaned wth the automated procedure. Results show that the algorthm systematcally fnds a practcal set of dscrete termnal locatons wth a cost very close to that predcted. The paper also gves condtons under whch the CA cost formulae are a tght lower bound for the exact mnmal costs.

5 1. BACKGROUND THE MODEL AND ALGORITHM A Dsk Model The Algorthm ILLUSTRATIONS Convergence Test Practcal Examples A LOWER BOUND CONCLUSION ACKNOWLEDGEMENTS REFERENCES LIST OF FIGURES... 20

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7 1 DISCRETIZATION AND VALIDATION OF THE CONTINUUM APPROXIMATION SCHEME FOR TERMINAL SYSTEM DESIGN Yanfeng Ouyang and Carlos F. Daganzo Insttute of Transportaton Studes and Department of Cvl and Envronmental Engneerng Unversty of Calforna at Berkeley, CA (August 1st, 2003) 1. BACKGROUND Desgnng a physcal dstrbuton system for mnmal logstcs cost s a complex task. The objectve functon usually ncludes complcated cost expressons for the varous dstrbuton stages,.e., nbound costs for delveres nto the termnals, outbound costs for delveres from termnals to customers, and termnal costs for handlng wthn termnals. Furthermore, the decson varables are usually dscrete and very numerous, ncludng the number of termnals, ther locatons, delvery routes, schedules, and the allocaton of customers to termnals. The paper focuses on the strategc desgn of a termnal system n a contnuous servce area S, where customer demand s dstrbuted wth a spatal densty λ(, x S. The goal s to fnd a set of termnal locatons, x={x 1, x 2, x N }, and a partton of S nto a set of nfluence areas served by these termnals, I={I 1, I 2, I N }, that mnmze the total logstcs cost, Z D (x, I). The number of termnals N s tself a decson varable. The mnmzaton problem s:

8 2 N ( x, D λ( dx, (1) = 1 I Mn Z I) = z ( x, x, I ) D s.t. x S, = 1,2,..., N, I I I = φ for j, U I S, where z D (x, x, I ) s the cost of servng a unt of demand at = j x I va a termnal at x. A smpler verson of (1) s called n the appled mathematcs lterature the optmal resource allocaton problem (Okabe et al., 1992, Du et al., 1999.) These problems also allow pont-lke servce facltes to be located among a contnuum of customers. However, for the problems to be tractable, z D must be a smple functon of a norm, x-x ; e.g. x-x 2. Unfortunately, these smple forms are not realstc for typcal logstcs problems (e.g., ncludng nbound costs). Faclty locaton problems can also be formulated by consderng a fnte number of possble locatons for customers and termnals. Optmal locatons are then selected wth a mxed-nteger program. An extensve lterature also exsts on ths subject; see e.g., Daskn (1995) and Drezner and Hamacher (2002). Ths approach s effectve f the number of canddates s small, but for a problem lke (1), the number of possble choces s so large that a dscrete optmzaton process s not practcal even f done heurstcally. To crcumvent some of those drawbacks and buldng on the work n Newell (1971 and 1973), Daganzo and Newell (1986) proposed a contnuum approxmaton (CA) approach for termnal system desgn. It was argued n ths reference that a near optmum soluton should have nfluence areas as round as possble, wth termnals located near

9 3 ther centers. It was also argued that f n addton λ( vares slowly wth x, and the areas I can be approxmated by a slow-varyng functon of x, A(, such that I A( f x I, then the set functon z D (x, x, I ) n (1) can be approxmated by a smpler functon of two real arguments, z C (x, A(). The functon A s a decson varable representng the desred nfluence area sze for locatons near x. Wth ths approxmaton, (1) can be replaced by Mn ( A) z ( x A( ) Z, λ ( dx, (2) C = S C where Z C (A) s a functonal of A. More detals about the procedure for obtanng z C from z D are gven n Sec. 3, and also n Daganzo (1999). The advantage of (2) s that t can be optmzed pont by pont, by fndng the value of A( that mnmzes z C (x, A() at every x. Ths result s denoted A * (, and the correspondng cost Z * C (A * ). One then looks for a partton of S wth round nfluence areas such that: I * A (, x I, (3) and for a set of centrally located termnals. The hope s that the dscrete soluton so * * dentfed, {x C, I C }, wll satsfy Z ( x, I ) Z ( A ) D C C C. The extent to whch ths happens s explored n ths paper. The paper also proposes a dscretzaton algorthm to obtan the

10 4 soluton {x C, I C }, snce to the authors best knowledge, no systematc procedure has yet been proposed for the dscretzaton step. The closest lterature deals wth surface-fttng problems, and s descrbed under the rubrc locaton optmzaton of observaton ponts for estmatng the total quantty of a contnuous spatal varable n Okabe et al. (1992). Unfortunately, the solutons to these knds of problems (e.g., as n Hor, H. and Nagata, M., 1985) turn out to be Vorono tessellatons, where the parttoned sub-areas may not be round wth proper szes. Thus, ths lterature s not sutable for our purposes. Numercal examples show that the feasble desgns obtaned wth the proposed algorthm ndeed exhbt costs Z x, I ) very close to the CA predcton D( C C * Z C (A * ). The paper also gves suffcent condtons under whch * Z C (A * ) s a tght lower bound for the exact optmal system costs. Snce Z D ( x C, IC ) and Z * C (A * ) are close to each other, the optmalty gap between Z x, I ) and the true optmum should be small under these condtons. D ( C C Ths paper s organzed as follows. Secton 2 develops the dscretzaton algorthm; secton 3 shows the numercal examples; and secton 4 shows the condtons under whch * Z C (A * ) bounds from below the true mnmum. A fnal secton dscusses generalzatons. 2. THE MODEL AND ALGORITHM As dscussed before, a near optmum desgn {x C, I C } should: () satsfy the sze requrement (3), () have nfluence areas as round as possble, and () have termnals

11 5 located near the centers of the nfluence areas. (We assume from now on that dstances are gven by the Eucldean metrc.) 2.1. A Dsk Model To capture () and (), we wll magne that each nfluence area contans a round dsk centered at the termnal, and nstead of {x C, I C } we wll look for a set of N nonoverlappng dsks, where N S * 1 [ A ( ] dx. By sldng the dsks wthn S, dfferent desgns can be obtaned. Two examples are dsplayed n Fgure 1. We use r( ={ r x )} for the set of dsk rad; see dotted arrows. For a good desgn, dsks should jontly cover most of S wthout protrudng outsde t, as shown n Fgure 2(a). Snce each nfluence area must contan one dsk, ths ensures that the nfluence areas are round. In addton, for a good desgn, the area of each dsk should be as close as possble to A(x * );.e., ( * A ( x ) r( x ), =1,2,,N. (4a) π It should be possble to satsfy these two condtons smultaneously snce there always are many ways to cover most of S wth dsks of dfferent szes, as llustrated by Fgure 2(b). Of course, snce dsks cannot tessellate convex Eucldean regons, we cannot expect the equalty n (4a) to be satsfed exactly. Therefore, we look nstead for rad that satsfy * A ( x ) r( x ) =, =1,2,,N, (4b) k

12 6 for k as small as possble. (Gven our defnton for N, k π.) To automate the sldng procedure, we now ntroduce two types of repulsve forces that act on the centers of the dsks. The frst type, termnal force F T, acts along the lne connectng the centers of overlappng dsks. The other type, boundary force F B, acts on dsks touchng the boundary, pontng toward the nteror of S n a drecton normal to the boundary. Sold arrows n Fgure 1(a) depct these forces. Fgure 3 defnes our choces for the magntudes of F T and F B. They depend on r(, vanshng when no dsks overlap or touch the boundary. We use (N+1)f for the magntude of F B, where f s the maxmal value of F T, to ensure that dsks are never pushed out of S. We call a pattern wth zero forces an equlbrum. The dsk centers of an equlbrum gve x C. Ths s suffcent to obtan a soluton snce S can then be easly parttoned nto nfluence areas, I C, that contan the dsks as wll be explaned shortly. Although such equlbrum soluton {x C, I C } may not be unque, t should satsfy the near-optmalty requrements () () The Algorthm The forces defned above are used to slde the dsks wthn S for small dstances, whle r( and the forces themselves are updated. The algorthm stops when all forces vansh. An equlbrum obvously exsts and can be found for a suffcently large k. Conversely, an equlbrum wll not exst f k s too small. Therefore, the algorthm ncreases k by a small ncrement, k, f the current value does not yeld an equlbrum.

13 7 Step szes for dsk movements should not be too large for fast convergence. One could use constant step szes comparable wth the tolerance level ε (n dstance unts) or, even better, gradually decreasng step szes; e.g., µ/m, where µ s an ntal step sze and m s the teraton count. Even for reasonably large k, ths algorthm may not converge to an equlbrum f we encounter sets of degenerate termnal locatons (also called sngular ponts n Okabe et al., 1992). Ths happens for example f ponts are on a straght lne that ntersects the boundares of S orthogonally. In ths case ponts would reman trapped on ths lne, snce all ensung termnal movements would have to be along the lne. Fortunately, such degeneracy s usually unstable, and can be elmnated by small locaton perturbatons. Therefore we add perturbatons of random drecton wth a dsplacement sze δ < ε at each step of the procedure. Once an equlbrum has been obtaned, S s parttoned nto I C wth a weghted- Vorono tessellaton (WVT) that ensures each I contans one entre dsk. The recpe s smple: frst partton S nto very small squares, and then allocate each square to one I wth the rule = arg mn j x x r( x j j ), where x s the center of the square. Ths rule ensures that every dsk s a subset of ts nfluence area. In summary, the steps of the algorthm are: 1) Choose N arbtrary locatons n area S and ntalze all parameters: tolerance ε, ntal step sze µ, perturbaton sze δ, and ncrement for k, k ; set ntal k π and m=1;

14 8 2) Calculate the dsk szes wth (4b) and then the forces on every termnal as per Fgure 3; f all the forces equal zero (equlbrum reached), go to step 5); otherwse, move each termnal along the drecton of ts resultant force by a step sze µ/m, and add a random-drecton perturbaton of sze δ. 3) If µ/m < ε, reset m = 0, and ncrease k by k ; 4) m = m+1; go to step 2); 5) Tessellate S wth the WVT recpe. 3. ILLUSTRATIONS 3.1. Convergence Test The algorthm s convergence s llustrated wth a problem that has a known soluton, usng the poly-hexagonal regon S of Fgure 4(a). The sde of each hexagon n S * equals1 3. IfA ( = 3 2 and N = 7, then the partton n Fgure 4(a) s optmal. The ntal locatons are arbtrarly generated and shown n Fgure 4(b). For smplcty a constant step sze µ = 0.01 s used. Fgure 4(c) shows an ntermedate result, and Fgure 4(d) fnds the equlbrum, whch was acheved after 440 teratons. Note that the weghted-vorono tessellaton correspondng to Fgure 4(d) matches n Fgure 4(a). Thus, the algorthm performs as expected Practcal Examples In ths secton we use practcal examples to further llustrate how the algorthm translates A * ( nto dscrete desgns {x C, I C }. The exact costs of the desgn, Z D (x C, I C ), are then compared to the estmated costs Z * C(A * ).

15 9 Daganzo (1999, Secton 5.3.5) gves an example of termnal system desgn, n whch customers are unformly dstrbuted n an L L square area S. They are served wth one transshpment from a depot at one corner of S. Lne-haul vehcles wth nfnte capacty shuttle between the depot and the termnals. Local delvery vehcles have a small capacty vmax, travel full, and vst only one customer per delvery. If we only consder nventory and transportaton costs (both nbound are outbound), and gnore fxed costs such as termnal faclty rents, the formula for z ( x x, I ) (Daganzo, 1999): z D a' b' R( x ) λ( dx I 1 2 av λ( 1.5 v max ( x, x, I ) = s( x, x ) π b max D, n (1) s. (5) Inbound costs Outbound costs In (5), a, b, a, b are cost parameters, R( s the dstance from pont x to the depot, and s(x, x ) s the outbound delvery dstance from x to x. s: On the other hand, the expresson for ( x, A( ) a' b' R( av b z C + λ( A( λ( v 1 max 2 ( x, A( ) = 2 + A ( 1 2 z C n (2), as shown n (Daganzo, 1999), max. (6) Formula (6) s derved from (5) by approxmatng the termnal throughput I λ (dx appearng n the frst term wth λ ( A(, and s(x, x ) wth the average a delvery

16 10 2 dstance A( 3 π n a hypothetcal crcular nfluence area of sze A( I. The dea s to express every tem n (5) as a local property of pont x. Ths local approxmaton devce can be used wth more general forms of (5). Experence shows that t works well when λ( and I vary slowly wth poston, as mentoned n Secton 1. Two scenaros wth dfferent demand densty functons λ( are now used to demonstrate ths dea. The results are then formalzed n Secton 4. Scenaro 1: Consder homogeneous demand λ( =1, x S, and also assume that vmax = b = b = a = b = 1. Then A * ( s obtaned by mnmzng (6), and the result s: 2 ' ' ( ) 1 * vmax a b R x 2 A ( = = 2R b λ Substtutng (7) nto (6) and (6) nto (2), we then fnd: 1 2 (. (7) * C 1 * * 4 ( A ) = ( zc ( x, A ( ) dx = ( R ( x ) S Z λ ) dx. (8) S If we now combne (1) and (5), the result s: Z x, (9) N 1 1 N = ( + ) D (, I) 2R ( x ) I I 1.5 π s( x, x ) dx = 1 = 1 I Our algorthm uses (7) as an nput. The set of dscrete desgns {x C, I C } obtaned wth t, and the assocated values of Z * C (A * ), and Z D (x C, I C ) gven by (8) and (9) for varous L are shown n Fgure 5(a) (d).

17 11 The dfference between Z * C (A * ), and Z D (x C, I C ) s qute small: 2.4% for L=5, 0.8% for L=7, 0.9% for L=10, and 0.9% for L=25. These relatve dfferences would be even smaller f other fxed costs were also ncluded n our cost expressons. 2 Scenaro 2: Assume now an nhomogeneous demand such that λ ( = R (, x S. All other parameters reman the same. Now we have 1 3 * 4 A ( = 2R (, (10) and Z * C 1 * * 8 ( A ) = ( zc ( x, A ( ) dx = ( R ( x ) S Z λ ) dx, (11) N 1 N x, I) = 2 R ( x + ) λ( dx I π λ( s( x, x ) dx. (12) = 1 = I 1 I 2 D ( S The set of desgns and assocated costs are now shown n Fgure 6(a)-(d). The cost dfferences are 2.6%, 2.3%, 1.6%, and 0.7% respectvely. They are approxmately the same as those n scenaro 1. Ths shows that the cost dfferences are nsenstve to gradual demand varatons. In all the examples the algorthm produced the soluton n less than 30 mnutes on a 1.7 GHz PC wth our choce of parameters. Note too that n both examples, Z * C (A * ) s slghtly smaller than Z D (x C, IC). Ths s not necessarly true n general (Daganzo, 1999), but s qute common. Secton 4 below gves suffcent condtons under whch Z * C (A * ) s a lower bound for the costs of a desgn {x, I}.

18 12 4. A LOWER BOUND We consder n ths secton a generalzaton of (5) of the followng form: z D o ( x, x, I ) = z R( x ), λ ( dx + z ( s( x, x ), λ( ), (12) I Inbound costs Outbound costs where z and z o are ordnary functons of two arguments. For ths case, the local approxmaton devce yelds: z C o 2 ( x, A( ) = z ( R(, ( A( ) + z A(, λ( λ. (13) 3 π We can now prove the followng theorem. Theorem: Z * C (A * ) Z D (x, I), f: (a) Locatons x are centrods of the nfluence areas; (b) the demand densty λ( s a constant, λ, wthn each nfluence area; (c) the nbound transportaton cost s a concave functon of dstance; (d) the outbound transportaton cost s a convex and (e) ncreasng functon of dstance.

19 13 Proof: Consder an arbtrarly shaped nfluence area, I I, wth a termnal located at ts centrod x ; see Fgure 7. Let Z D, (x, I) and Z, (A) represent the parts of (1) and (2) correspondng to nfluence area, and denote s = s(x, x ) for smplcty. Snce the demand densty s constant, substtuton of (12) nto (1) yelds: C Z o z ( R( x ),λ I ) λdx + z ( s, λ ) ( x, ) = λ dx, (14) D, I I I Lkewse, substtuton of (13) nto (2) yelds: o 2 Z C, ( A) = z ( R(, λ A( ) λdx + z A(, λ λdx, (15) 3 π I I If we can prove that Z D, (, I) Z C, ( A ) x, (16) s where A s ( s constraned to be a step functon;.e., As ( = I, f x I, then (16) would establsh that Z ( x, I) Z ( A ) D C s. Ths would prove the theorem snce Z * C (A * ) s * * the optmum of Z C ( A) wthout any constrant; therefore Z ( A ) Z ( A ) Z ( x, I) Note that ( A ) Z, can be expressed as: C s C. C s D

20 14 o 2 Z C, ( As ) = z ( R(, λ I ) λdx + z I, λ λdx. (17) 3 π I I To prove (16) we frst show that the frst term of Z ( x, ) bounds from above the frst term of ( A ) C s D, I Z,. Ths s clear f we compare the frst terms of (14) and (17), because R x ) s the average of R ( by assumpton (a), and Jensen s nequalty suggests ( (assumpton (c)) that: z ( R x ), λ I ) λdx z ( R(, λ I ) ( λ dx. (18) I I Thus, to prove (16) we only have to show that the second term of (14) bounds from above the second term of (17);.e., that o 2 o 2 ( s λ ) λ dx z I, λ λ dx = λ I z I, λ o z I,. (19) 3 π 3 π I Note as a prelmnary step that: z o ( s λ ) λ dx z ( s, λ ), λ dx, (20) o I I

21 15 where s s the average outbound delvery dstance n I. Ths s true, agan, by vrtue of assumpton (d) and Jensen s nequalty. We now defne a pont-to-pont mappng {M: y=m(, x I, y I }, that transforms I nto a round area I wth the same centrod and the same area, and such that s' ( y, x ) s( x, x ) for y=m(; see Fgure 8. [Ths last condton s trvally satsfed by specfyng that all ponts n I I should be fxed ponts;.e., y = x.] We consder now the cost of servng the transformed regon f the demand densty n t s stll λ. Clearly, the nbound costs stay the same. Obvously, s' s, (21) 2 2 where s' = I ' = I s the average outbound delvery dstance n I '. We 3 π 3 π can now wrte: I z o o o o 2 ( s, λ ) λ dx z ( s, λ ) λ dx z ( s', λ ) λ dx = λ I z I, λ I I ' 3 π, (22) where the frst nequalty s (20), the second nequalty follows from (21) and assumpton (e), and the fnal equalty follows from the fact that I = I '. Ths completes the proof. Ths theorem s vald for any N and any partton of S. Of course, t s based on dealzed condtons that are qute unrealstc f strctly enforced--snce the cost

22 16 condtons may not apply n many cases, and demand densty wll rarely be constant n every nfluence area. However, we are often faced wth problems for whch these condtons are approxmately true, such as our examples. In these cases the condtons of the theorem should hold, at least approxmately. Ths s confrmed by the numercal results of Secton 3, whch were not concdental. 5. CONCLUSION Ths paper proposed an automated algorthm to obtan dscrete desgns out of the contnuum approxmaton recpes for locaton problems. It can be easly extended to other logstcs problems. Numercal results show that the algorthm systematcally fnds feasble dscrete termnal desgns wth costs very close to those predcted. The algorthm was llustrated wth Eucldean metrcs and crcular dsks. However, t can easly be extended to other metrcs and/or applcatons that requre elongated nfluence areas. Recall too that our algorthm looks for centrally located termnals. There are systems, however, for whch termnals should not be at the center of ther nfluence areas; e.g. newspaper dstrbuton systems, where t s advantageous to locate drop-off spots on the edge of ther delvery dstrcts (see Daganzo, 1984). In these cases the algorthm should be modfed too. The study also valdates the CA cost predctons, by comparng them wth the costs for actual desgns. The CA predcton s shown to be an approxmate lower bound of the true optmum under certan condtons, and to be qute close to the costs of feasble desgns. In these cases the CA method produces solutons wth a small optmalty gap.

23 17 ACKNOWLEDGEMENTS Ths research s supported n part by a research grant from the Unversty of Calforna Transportaton Center (UCTC).

24 18 REFERENCES 1. Okabe, A., Boots, B. and Sughara, K. (1992). Spatal Tessellatons: Concepts and Applcatons of Vorono Dagrams. Wley, Chchester, UK. 2. Du, Q., Faber, V. and Gunzburger, M. (1999) Centrodal Vorono tessellatons: applcatons and algorthms, SIAM Revew, 41(4): Daskn, M.S. (1995) Network and Dscrete Locaton: Models, Algorthms and Applcatons, Wley, New York, USA. 4. Drezner, Z. and Hamacher, H.W. (2002) Faclty Locaton: Applcatons and Theory. Sprnger, Berln, Germany. 5. Newell, G.F. (1971) Dspatchng polces for a transportaton route, Transportaton Scence 5, Newell, G.F. (1973) Schedulng, locaton, transportaton and contnuum mechancs: some smple approxmatons to optmzaton problems, SIAM J. Appl. Math. 25(3): Daganzo, C.F. and Newell, G.F. (1986). Confguraton of physcal dstrbuton networks, Networks, 16: Daganzo, C.F. (1999). Logstcs System Analyss, 3 rd Edton. Sprnger, Berln, Germany. 9. Hor, H. and Nagata, M. (1985) Examples of optmzaton methods for envronment montorng systems, Report B-266-R-53-2, Envronmental Scences, Mnstry of Educaton, Japan, (n Japanese).

25 Daganzo, C.F. (1984). The dstance traveled to vst N ponts wth a maxmum of C stops per vehcle: an analytc model and an applcaton, Transportaton Scence, 18(4):

26 20 LIST OF FIGURES FIGURE 1. Dsks and termnals: (a) an nfeasble overlappng pattern; (b) a feasble nonoverlappng pattern. FIGURE 2. Two possble layouts of 7 dsks n a hexagon: (a) homogeneous pattern; (b) nhomogeneous pattern. FIGURE 3. Possble defntons of forces: (a) repulsve force for termnal par (, j); (b) boundary force for termnal. FIGURE 4. Verfcaton of convergence: (a) area S: (b) ntal locatons; (c) locatons after 200 teratons; (d) equlbrum locatons after 440 teratons. FIGURE 5. Termnal desgns for homogeneous customer demand: (a) L=5; (b) L=7; (c) L=10; (d) L=25. FIGURE 6. Termnal desgns for nhomogeneous customer demand: (a) L=5; (b) L=7; (c) L=10; (d) L=25. FIGURE 7. Logstc operatons n I. FIGURE 8. Mappng ponts from I nto a round area I.

27 21 F T x j r(x j ) S x r(x ) x j r(x j ) S x r(x ) x k r(x k ) F B x k r(x k ) F T (a) (b) FIGURE 1 Dsks and termnals: (a) an nfeasble overlappng pattern; (b) a feasble non-overlappng pattern.

28 22 (a) (b) FIGURE 2 Two possble layouts of 7 dsks n a hexagon: (a) homogeneous pattern; (b) nhomogeneous pattern.

29 23 Termnal force F T Boundary force F B f (N+1)f 0 r(x )+r(x j ) x x j (a) 0 r(x ) (b) Dstance to boundary FIGURE 3 Possble defntons of forces: (a) repulsve force for termnal par (, j); (b) boundary force for termnal.

30 24 N=7, Intal Locatons (a) (b) (c) (d) FIGURE 4 Verfcaton of convergence: (a) area S; (b) ntal locatons; (c) locatons after 200 teratons; (d) equlbrum locatons after 440 teratons.

31 25 Z C Z D Z C Z D (a) (b) Z C Z D Z C Z D (c) (d) FIGURE 5 Termnal desgns for homogeneous customer demand: (a) L=5; (b) L=7; (c) L=10; (d) L=25.

32 26 Z C Z D Z C Z D (a) (b) Z C Z D Z C Z D (c) (d) FIGURE 6 Termnal desgns for nhomogeneous customer demand: (a) L=5; (b) L=7; (c) L=10; (d) L=25.

33 27 I Termnal at x s(x, x ) λ dx R(x ) R( Depot FIGURE 7 Logstc operatons n I.

34 28 x s s < s y=m( y= x s = s Orgnal area I Vrtual round area I Demand ponts Mappng ponts FIGURE 8 Mappng ponts from I nto a round area I.