Dynamic Facility Location with Stochastic Demands

Transcription

1 Dynamc Faclty Locaton wth Stochastc Demands Martn Romauch and Rchard F. Hartl Unversty of Venna, Department of Management Scence, Brünner Straße Venna, Austra Abstract. In ths paper, a Stochastc Dynamc Faclty Locaton Problem (SDFLP) s formulated. In the frst part, an exact soluton method based on stochastc dynamc programmng s gven. It s only usable for small nstances. In the second part a Monte Carlo based method for solvng larger nstances s appled, whch s derved from the Sample Average Approxmaton (SAA) method. 1 Introducton The Uncapactated Faclty Locaton Problem (UFLP) has been enhanced nto many drectons. In [9] and [2] you can fnd numerous approaches that consder ether dynamc or stochastc aspects of locaton problems. An exact soluton method to an UFLP wth stochastc demands s dscussed n [6]. The problem consdered there could be nterpreted as a two stage stochastc programm. In [8] you can fnd dynamc (mult perod) aspects as well as the mult commodty aspect. The approach n [10] could be seen as the ntegraton of stochastcs nto the UFLP. In the work n hand a model wll be presented, where the UFLP gets enrched by nventory and randomness n the demand. The UFLP and ts generalzatons are part of the class of NP-hard problems, where no exact effcent soluton methods are known. Frst of all, the am of ths work s the preparaton of tools to develop and nvestgate heurstcs for ths problem type. For ths reason, an exact method for small nstances was developed. Ths makes possble both, to carve out the range of exact solvablty and to compare exact and heurstc solutons. A more detaled descrpton of the problem s now followng. 2 Stochastc Dynamc Warehouse Locaton Problem Our am s to fnd the optmal decsons for producton, nventory and transportaton, to serve the customers durng a certan number of perods, t {1,..., T }. Assume that the company runs a number for the producton stes I = {1, 2,..., n} that have lmted storage capactes, (t). These producton stes need not be used n all perods. When a producton ste s operated at tme

2 2 Martn Romauch and Rchard F. Hartl t, ths s denoted by the bnary varable δ (t) = 1. In ths case the fxed costs o (t) arse. If a locaton s actve, then the exact producton quantty u (t) must be fxed. For each perod, the producton decson s the frst stage of the decson process. It has to be done before the demand of the customers s known. Only the current level of nventory y (t 1) as well as the demand forecasts are known n advance. Demand occurs at varous customer locatons J = {1, 2,..., m}. At any gven perod t the demand d t at customer wll occur wth probablty pt, whereas customer wll not requre any delvery wth probablty 1 p t. Hence, demand can be descrbed by a dchotomous random varable 1 D (τ) (τ t). We also assume that the random varables D (t) are stochastcally ndependent. P(D (t) = d (t) ) = p(t) P(D (t) = 0) = 1 p (t) In the second stage, when the demand s known, we must decde upon the transportaton of approprate quanttes x (t) from the producton stes to the customers. We assume that the tme needed for transportaton can be neglected (.e. the transportaton lead tme s less than one perod). Stockouts (shortages) f (t) are permtted and are penalzed by shortage costs p (t) per unt tme and per unt of the product. We assume that backorderng s not possble and that these potental sales are lost. The perods are lnked by the nventores y (t) the usual nventory balance equatons (1) apply. Here η (t) n perod t at ste. y (t) + η (t) = y (t 1) + u (t) J at the producton stes and denotes the surplus x (t) (1) In ths paper we assume free dsposal, therefore the varable η (t) elmnated by turnng the equalty (1) nto the nequalty (2). can be y (t) y (t 1) + u (t) J x (t) (2) After the completon of the producton and transportaton decsons and after updatng the nventores, the next perod can be consdered. We note here, that for all perods we have to pay attenton to the capacty restrctons (3). 0 u t + y (t 1) (t) (3) In order to have a convenent notaton, we ntroduce the concept of scenaros. A scenaro D t J s a subset of customers where the demand gets realzed. Snce the demands of the dfferent customers are ndependent, the correspondng probablty of a scenaro to occur s gven n formula (4). 1 The embeddng of stochastcs s smlar to the embeddng of stochastcs nto the TSP, see [4].

3 P(D t ) = Lecture Notes n Computer Scence 3 p (t) D t D t ( 1 p (t) Solvng the SDFLP means fndng a strategy that mnmzes the expected costs. Because of the sequencng of the decsons and the uncertan demand, the solutons could be understood as scenaro dependent strateges, where the decsons are dependent on the forecasts and the level of nventory at hand. Fgures 1(a) and 1(b) llustrate the dependency of operatve plannng 2 and the scenaros (realzaton of demand). The left hand sde of Fgure 1(a) shows the producton decsons u (t) and all of the possble subsequent scenaros (8 n number). One of the scenaros s magnfed n the upper part of Fgure 1(b). In each scenaro the decsons for transportaton, nventory and shortage are necessary. ) (4) (a) Scenaros ( I = 2, J = 3) (b) Transton Fg. 1. Sequencng of Decsons In order to complete the model formulaton, we summarze the decson varables and the correspondng costs n Table 1. The decsons δ (t) and u (t) are lnked by formula (5) { δ (t) 1 f u (t) > 0 = 0 f u (t) (5) = 0 whle shortages are defned as f (t) = D (t) I x (t). The total cost F s the sum over all perods of fxed operatng costs, varable producton costs, 2 to keep the fgure as smple as possble shortages and dsposal are not ntegrated.

4 4 Martn Romauch and Rchard F. Hartl δ (t) varable cost {0, 1} o (t) x (t) Z + c (t) y (t) Z + s (t) u (t) Z + m (t) f (t) Z + p (t) Table 1. Varables and Costs descrpton operatng decson and fxed costs transportaton decson and unt transportaton cost nventory level and unt holdng cost producton decson and varable producton cost shortage (lost sales) and unt shortage cost ( T [ F = E t=1 I o (t) δ (t) + m (t) u (t) + s (t) + ] y (t) T t=1 I J c (t) x(t) T + t=1 J Snce all relevant nformaton about the past s contaned n the nventory levels, ths model s well suted to be solved by dynamc programmng. Ths wll be outlned n the next secton. p (t) f (t) 3 Exact Soluton Method 3.1 Stochastc Dynamc Programmng The prncple of dynamc programmng s the recursve estmaton of the value functon. Ths value functon, henceforward denoted by F, contans the aggregate value of the optmal costs n all remanng perods. It can be derved recursvely. It s convenent to frst descrbe the method n general and to apply t to the problem afterwards. Let z R m + be the vector of state varables and u R n + be the vector of decsons. The set of feasble decsons n state z and perod t s denoted by U t (z). The random nfluence n perod t s represented by the random vector r (t) for whch the correspondng dstrbuton s known. It s mportant to note that the random varables {r (t) } have to be stochastcally ndependent. The state transformaton s descrbed by z t+1 = A(z t, u t, r t ) and depends on the current state z t, the random nfluence r t at tme t, and the chosen decson u t. The sngle perod costs n perod t and state z when decson u s taken and random varable r s realzed s denoted by g t (z, u, r). The value functon F t (z) gves the mnmal expected remanng costs when startng n state z n perod t. We now present a varant of the stochastc Bellman equaton (compare Schneeweß[11] S.151 (10.25) or Bertsekas [1] S.16).

5 Lecture Notes n Computer Scence 5 F T (z) = F t (z) = mn u U T (z) mn u U t(z) {E [ g T (z, u, r (T ) ) ]} { E [ g t (z, u, r (t) ) + F t+1 (A t (z, u, r (t) )) ]} t = T 1,..., 1 (6) Recursvely solvng the equaton (6) we get an optmal strategy that balances the cost for mplementng the decson u and the expected resultng remanng costs. Applyng Stochastc Dynamc Programmng (SDP) to the SDFLP s almost straght forward. For solvng the problem we have to teratvely calculate the functons F t. We wll show later that for the SDFLP t s suffcent to consder nteger controls. 3.2 Applcaton to the SDFLP In order to apply the DP equaton (6) to the SDFLP, we frst ntroduce the notaton G ( D, y start, y end, t ) for the sum of nventory holdng costs, shortage costs, and transportaton costs n scenaro D n perod t when startng wth ntal nventory levels y start and where the fnal nventores are requred to be y end. To every gven nventory level y start and scenaro D, the best possble transportaton plan has to be calculated. Ths can be done by solvng a lnear program: G ( D, y start mn x,f J, y end, t ) = I p (t) f + I J s (t) y end + c (t) x (7) s.t. I x + f = d (t) J x + y end y start f, x 0. D I Now the value functon F T of the fnal perod T can be computed. In G, the startng nventory s now gven by y start = u (T ) (T 1) + y whle the termnal nventory must be zero, y end = 0: (T 1) F T (y ) = mn 0 u T + D T J u (T ) 0 1) +y(t (T ) { I δ (T ) o (T ) + u (T ) m (T ) + I P(D T )G ( D T, u (T ) (T 1) + y, 0, T )} (8)

6 6 Martn Romauch and Rchard F. Hartl Fg. 2. Comparson of Dfferent Solutons Gong back n tme, we have to turn to the general case n perod t < T. Now we have to take nto account the remanng costs n perods t + 1,..., T when makng the decson n perod t. The startng nventory s now gven by y start = u (t) + y (t 1) whle the nventory at the end of perod t s y end = y (t). When determnng G(D t, u (t) + y (t 1), y (t), t) agan a lnear program has to be solved. The recurson for the value functon becomes: F t (y (t 1) ) = mn + D t J P(D t ) 0 u (t) 0 y (t) u (t) 0 +y (t 1) mn (t+1) (t) { I { Gt ( Dt, u (t) δ (t) o (t) + y (t 1) + u (t) m (t) +, y (t), t ) + F t+1 (y (t) ) }} (9) In the SDFLP the data and the controls are assumed to be nteger. In the problem (7) we therefore have to solve an nteger lnear program. It turns out to be a mn cost flow problem and therefore t s totally unmodular, such that usng the Smplex method for solvng the relaxed lnear program results n nteger solutons for the transportaton quanttes x and the shortages f. The computatonal effort of ths exact algorthm s ncreasng exponentally wth the capacty at the locatons and the number of customers. The addtonal effort that emerges from addng addtonal perods to the problem s lnear. Ths DP formulaton s only applcable for small problem nstances and for larger problem nstances heurstc approaches are necessary. Ths s consdered n the next secton.

7 Lecture Notes n Computer Scence 7 4 Heurstc Approach A heurstc desgned to solve stochastc combnatoral optmzaton problems s the Sample Average Approxmaton Method (SAA); see Kleywegt et al. [5]. Our model deals wth a mult stage problem and that s the reason why ths method s not drectly applcable. In what follows we frst present the classcal SAA for solvng statc stochastc combnatoral optmzaton problems. Afterwards, we wll explan how ths method can be modfed n order to be applcable to our problem. Consder the followng stochastc combnatoral optmzaton problem (10) n whch W s a random vector wth known dstrbuton P, and S s the fnte set of feasble solutons. v = mn x S g(x), g(x) := E P G(x, W ) (10) The man dea of the SAA method s to replace the expected value E P G(x, W ) = G(x, w)p (dw) (whch s usually very tme consumng) by the average of a sample. The followng substtute problem (11) s an estmator of the orgnal problem (10). mn ĝn(x), x S ĝ N (x) := 1 N The SAA method works n three steps N G(x, W ) (11) =1 1. Generate a set of ndependent dentcally dstrbuted samples: {W 1,..., W N } M =1 of the random varable W. 2. Solve the correspondng optmzaton problems,.e. optmze: ˆv = mn x S ĝ(x), ĝ (x) = 1 N N G(x, W ). =1 3. Estmate the soluton qualty. Ths s done by frst computng mean and varance of the sample: ˆv = 1 M M ˆv, ˆσ 2 1 = M(M 1) =1 M (ˆv ˆv) 2. =1 Then a soluton x s chosen (e.g. we can take the soluton wth the smallest ˆv ) and ts obectve value s estmated more accurately by generatng a larger sample {W 1,..., W N } (N >> N) N ṽ = 1 N G( x, W ), σ 2 = =1 1 N (N 1) (G( x, W ) ṽ) 2. N =1

8 8 Martn Romauch and Rchard F. Hartl Calculate the value gap and σ 2 gap: gap = ṽ ˆv, σ 2 gap = σ 2 + ˆσ 2 Snce E(ˆv) v E(ṽ) the values ṽ and ˆv can be nterpreted as bounds on v : let x S denote an optmal solutons of (10) then the frst nequalty E(ˆv) v comes from takng the expected value on the followng nequalty: whch results n ˆv ĝ (x ) E(ˆv ) E(ĝ (x )) = v. After completng Step 3, we have to nspect the values of gap and σ 2 gap. If these values are too large, one must repeat the procedure wth ncreased values of N, M and N. In [10] ths method s appled for a Supply-Chan Management problem that ncludes locaton decsons. Because of the mult-perod structure of the SDFLP, the above SAA procedure has to be adapted. In partcular, one must pay specal attenton to the way how the samplng s done. The samplng s done ndependently n every stage and every state of the SDP spmly by modfyng formulas (8) and (9), where for every perod and nventory level we only sum over a small randomly chosen sets of scenaros. To be more specfc, the expected value n formula (9) passes over nto (12) where {D } denotes the sample chosen n stage t and state y (t). 1 N N =1 5 Results 0 y (t) mn (t+1) { Gt ( D, u (t) + y (t 1), y (t), t ) + F t+1 (y (t) ) }} (12) The mplementaton was done n C++ and an addtonal lbrary from the GNU Lnear Programmng Kt (GLPK) [7] was used. For small examples where the product I (t) s small enough (say 50) t s possble to fnd the exact soluton usng the dynamc programmng approach descrbed n Secton 3. In Fgure 2, we can see the cost dstrbutons for the exact and the heurstc approach n a small example (3 perods, 3 customers, 2 facltes and capacty = 3). Here one can see that the shapes are qute dfferent. The nstance consdered has very uncertan demand (every customer has probablty 50%). Hence the two peaks n the optmal soluton are not very surprsng. It s nterestng to observe that the heurstc soluton does not show these twn peaks. Ths second peak dmnshes f the probablty s close to 0 or 1. An experment was made where the probablty for the demand vared from 0 to 1 for all customers,.e.: (p (t) = p {0, 0.01, 0.02,..., 1}). The result s depcted n Fgure 3 where grey areas represent postve probabltes that these cost values occur. Every vertcal lne (p fxed) corresponds to a dstrbuton functon. For nstance,

9 Lecture Notes n Computer Scence 9 Fg. 3. Dstrbutons of the Optmal Soluton to Instances wth Dfferent Levels of Probablty (p (t) = p {0, 0.01, 0.02,..., 1}) at p = 0.1 fve peaks occur. When the probablty p ncreases, the number of peaks n the dstrbuton functon decreases. The key decson to make the heurstc work well s to choose the rght sample sze N and the rght number of samples M. In Fgure 4 the statstcal lower bound ˆv (calculated n step 3) s depcted for dfferent values of N and M. Choosng a sample sze N that s large enough seems to be more mportant than a large number of samples. In Fgure 4(b) the regon N > 70 of Fgure 4) s magnfed to see the effect of a choosng the number of samples more clearly. One also can see that the statstcal gap stays postve f at least 11 samples of sze 71 are chosen. It s also nterestng to note that for small values of N and suffcent large M the correspondng bounds are qute good, although the correspondng ndvdual solutons are qute bad. Ths stuaton s depcted n Fgure 5. 6 Concluson and Further Research In ths paper a stochastc dynamc faclty locaton problem was proposed and exact and heurstc soluton methods were presented. The examples that can be solved to optmalty are qute small and therefore of mnor practcal nterest. But the comparson of the SAA results and the exact soluton method shows the applcablty of the proposed method for larger nstances of the SDFLP. To get more nsght nto ths method, t wll be necessary to make a transfer of the theoretcal results known for the SAA method (see [5] for statstcal bounds). For comparson purposes t would be nterestng to adopt metaheurstc concepts: e.g. by usng the varable-sample approach (for references see [3]). In our further research we also want to consder other exact soluton technques consderng the SDFLP as a multstage stochastc program.

10 10 Martn Romauch and Rchard F. Hartl costs sol M=1 M=81 M=101 optmal M= sample sze (N) (a) (b) Detal Fg. 4. Choce of sample sze N and the number of samples M. opt SAA (N=5, M=7)... average of 7 solutons SAA (N=5)... evaluaton of one soluton costs states Fg. 5. Expected costs to dfferent levels of nventory y (1 : [0, 0]; 2 : [0, 1];...; 4 : [0, 3]; 5 : [1, 0];...; 12 : [2, 3]).

11 Lecture Notes n Computer Scence 11 References 1. Bertsekas, D.P.: Dynamc Programmng and Optmal Control, Athena Scentfc, Belmont, MA (1995) 2. Hammer, P.L.: Annals of Operatons Research: Recent Developments n the Theory and Applcatons of Locaton Models 1/2, Kluwer Academc Publshers (2002) 3. Homem-de-Mello, T.: Varable-Sample Methods for Stochastc Optmmzaton, ACM Trans. on Modelng and Computer Smulaton (2003) 4. Jallet, P.: Probablstc Travellng Salesman Problems, Ph.D. thess, MIT (1985) 5. Kleywegt, A.,Shapro,A., Hohem-de-Mello, T.: The Sample Average Method for Stochastc Dscrete Optmzaton, SIAM J. Optm. 12 (2001/02) 6. Laporte, G., Louveaux, F., Van Hamme, L.: Exact Soluton to a Locaton Problem wth Stochatc Demands, Transportaton Scence, Vol. 28,(1994) 7. Makhorn, A.:GNU Lnear Programmng Kt - Reference Manual - Verson 4.1, Department for Appled Informatcs, Moscow Avaton Insttute, Moscow, Russa (2003) 8. Melo, M.T., Nckel, S., Saldanha da Gama, F.: Large-Scale Models for Dynamc Mult-Commodty Capactated Faclty Locaton, Berchtsrehe des Fraunhofer Inststtuts für Techno- und Wrtschaftsmathematk (ITWM), ISSN , Kaserslautern (2003) 9. Hesse Owen, S., Daskn, M.S.: Strategc Faclty Locaton: A Revew, Eropean Journal of Operatonal Research, Vol. 111, (1998) 10. Santoso, T., Ahmed, S., Goetschalckx, M., Shapro, J.: A Stochastc Programmng Approach for Supply Chan Network Desgn under Uncertanty, The Stochastc Programmng E-Prnt Seres (SPEPS), (2003) 11. Schneeweß, C.: Dynamsches Programmeren, Physca-Verlag, (1974)