Designing a Supply Chain Network Model with Uncertain Demands and Lead Times

Transcription

1 Joural of Ucerta Systems Vol.3, No.2, pp , 2009 Ole at: Desgg a Supply Cha Networ Model wth Ucerta Demads ad Lead Tmes Mohammad Saeed Jabal Amel 1, Nader Azad 1,, Amr Rastpour 2 1 Departmet of Idustral Egeerg, Ira Uversty of Scece ad Techology P.C , Tehra, Ira 2 Idustral Egeerg group, Uversty of Tabrz, P.C , Tabrz, Ira Receved 18 February 2008; Accepted 2 July 2008 Abstract I ths paper, we preset a tegrated stochastc supply cha desg model that optmzes locato, vetory ad trasportato decsos, smultaeously. Customers demads ad dstrbuto ceters lead tmes are geerated radomly ad the obectve s to mmze the locato costs as well as vetory costs at dstrbuto ceters, ad trasportato costs the supply cha. We assume each dstrbuto ceter matas a certa amout of safety stoc order to acheve a certa servce level for the customers t servces. We show that ths problem ca be formulated as a o-lear teger-programmg model. A Tabu search based procedure s developed for solvg the problem. We comprse the Tabu search soluto wth the optmal soluto, ad a smulated aealg algorthm. The results dcate that the method s effcet for a wde varety of problem szes World Academc Press, UK. All rghts reserved. Keywords: faclty locato, vetory, tegrated supply cha desg, ucerta demad, ucerta lead-tme 1 Itroducto All compaes that am to be compettve o the maret have to pay atteto to ther orgazatos related to the etre supply cha. I partcular, compaes have to aalyze the supply cha order to mprove the customer s servce level wthout a ucotrolled growth of cost. I few words, compaes have to crease the effcecy of ther logstcs operatos. Faclty locato problems (FLP), whch are typcally used to desg dstrbuto etwors, volve determg the stes to stall resources, as well as the assgmet of potetal cosumers to those resources. Oe example of FLP, s the locato of maufacturg plats, the assgmet of warehouses to these plats ad fally the assgmet of retalers to each warehouse. For a thorough revew of faclty locato problem see Hamacher ad Drezer [6], Barmel ad Smch-Lev [1], ad Das [2]. There are some papers the lterature, whch cosder the faclty locato problem dstrbuto etwor desg. Klose ad Drexl [8] propose a classfcato for dstrbuto etwor desg problem. Geoffro ad Graves [5] propose a Beder s decomposto approach to solve a capactated sgle source, mult-commodty etwor flow problem. Tragatalergsa et al. [17] cosdered a two-echelo faclty locato problem whch the facltes the frst echelo are ucapactated ad the facltes the secod echelo are capactated. Melo et al. [10] preset a dyamc mult-commodty capactated dstrbuto etwor desg problem but they dd ot propose a soluto algorthm for solvg the problem. Lu ad Bostel [9] preset a faclty locato model for reveres logstcs system ad propose a algorthm based o lagraga relaxato for solvg the model. Recetly some authors have corporated vetory cotrol decsos to the faclty locato problem, ad they show that cost savgs ca be obtaed by cosderg locato ad vetory decsos smultaeously. Smch-Lev [16] cosders a herarchcal plag model for stochastc dstrbuto system, whch the locatos ad demad of customers are determed accordg to some probablty dstrbuto. Jayarama [7] corporates the vetory costs to a faclty locato problem, assumg fxed lot szes ad determstc demads. Nozc ad Turqust [12] Correspodg author. Emal:.azad@aut.ac.r, dr_azad@yahoo.com (N. Azad).

2 124 M.S.J. Amel, N. Azad, ad A. Rastpour: Desgg a Supply Cha Networ Model corporate vetory costs assumg the demads to arrve a Posso maer ad a base stoc vetory polcy. She [13], studed the ot locato-vetory model whch locato, shpmet ad o-lear safety stoc vetory costs are cluded the same model. Erlebacher ad Meller [4] formulate a o-lear teger locatovetory model. They use a cotuous approxmato for solvg the problem. Das et al. [3] apply lagraga relaxato to solve the locato-vetory model. She et al. [15] preset a locato-vetory model that s smlar to the model of She [13] ad use colum geerato for solvg the problem. Mrada ad Garrdo [11] preset a locato-vetory model that s smlar to the model of Das et al. [3], ad they apply lagraga relaxato method for solvg the model. She [14] propose a mult-commodty locato-vetory model ad use colum geerato for solvg the problem. As we metoed the prevous, sgfcat cost savgs ca be obtaed by cosderg locato ad vetory decsos smultaeously stead of the sequetal approach, where locato decsos are made before the vetory decsos. I addto, the demads ad lead tmes the supply cha system are usually ucerta. I ths paper, we preset a tegrated stochastc supply cha desg model that optmzes locato, vetory ad trasportato decsos smultaeously. We assume each customer has ucerta demad that follows a ormal dstrbuto, ad we assume each dstrbuto ceter has ucerta lead-tme that follows a ormal dstrbuto. The goal of our model s to choose a set of dstrbuto ceters to serve the customers ad allocate customers to each ope dstrbuto ceter, ad to determe the vetory polcy based o the formato of customers demads order to mmze the total expected cost of locatg dstrbuto ceters, shpmet, ad vetory costs. A Tabu search based procedure s used for solvg the problem. The remder of ths paper s orgazed as follows. I secto 2, mathematcal formulato of the problem ad aalyss of the model are preseted. Secto 3, dscusses the soluto approach for solvg the problem. Secto 4, dscusses some computatoal results. Fally, secto 5 cotas some coclusos of the paper. 2 Problem Descrpto & Formulato I our model, we assume that each customer has ucerta demad that follows a ormal probablty dstrbuto, ad customers demads are depedet. I addto, we assume that each dstrbuto ceter has ucerta lead tme that follows a ormal probablty dstrbuto, ad dstrbuto ceters lead tmes are depedet. The dstrbuto ceters are assumed to follow a (Q, R) vetory polcy. Now the followg we preset how the problem ca be formulated as a mxed teger olear program Secto 2.1, the, Secto 2.2 we preset some aalyss for solvg ad smplfyg the model. 2.1 Formulato Before presetg the model, let us troduce the otatos that wll be used throughout the paper: Idex sets: K: Set of customers; J: Set of potetal dstrbuto ceters. Parameters ad otatos: µ : Mea daly demad at customer ( K ); 2 σ : Varace of daly demad at customer ( K ); F : Fxed cost for opeg ad operatg dstrbuto ceter ( J ); b : Capacty for the potetal dstrbuto ceter ( J ); h : Ivetory holdg cost per ut of product per year at dstrbuto ceter ( J ); p : Fxed cost per order placed to the suppler by dstrbuto ceter ( J ); L : Mea lead tme for dstrbuto ceter days, ( J ); S : Varace lead tme for dstrbuto ceter days ( J ); 2 g : Fxed cost per shpmet from the suppler to dstrbuto ceter ( J ); a : Cost per ut of a shpmet from the suppler to potetal dstrbuto ceter ( J ); c : Cost per ut of shpmet from dstrbuto ceter to customer ( J, K );

3 Joural of Ucerta Systems, Vol.3, No.2, pp , α : Desred percetage of customer orders satsfed (fll rate); z : α α -percetle of the stadard ormal radom varable Z,.e. p ( Z zα ) = α ; χ : Days per year. Decso varables: 1 f customer s assged to dstrbutoceter Y = (, J); 0 otherwse 1 f dstrbuto ceter s opeed U = ( J); 0 otherwse Q : Order sze at dstrbuto ceter ( J ). Obectve fucto: The obectve fucto mmzes the followg costs: 1. The fxed cost of locatg dstrbuto ceters, gve by term F U. J 2. The aual shpmet cost from dstrbuto ceters to the customers, gve by term χ c µ Y. J 3. The expected aual vetory cost: s the sum of worg vetory ad safety stoc costs. Worg vetory represetg product that has bee ordered from the suppler but ot requested by the customers ad safety stoc cost vetory teded to buffer the system agast stoc outs durg orderg lead tmes. Worg vetory cost cludes the fxed costs of placg orders as well as the shpmet costs from the suppler to the dstrbuto ceters as well a the holdg costs of worg vetory. Let D deote the total aual demad gog through dstrbuto ceter ( χµ ). The the total aual cost of orderg vetory from the suppler to D = Y dstrbuto ceter s gve by D D Q p + ( g + a Q ) + h. Q Q 2 The frst term, represets the total fxed cost of placg orders per year. The secod term represets the delvery cost from the suppler to dstrbuto ceter per year, ad the thrd term s the cost of average worg vetory. We assume that cost of shppg a order of sze M form the suppler to dstrbuto ceter s gve by the term of ( g + a M ). The yearly safety stoc cost at dstrbuto ceter s gve by h z σ σ = L σ + µ S Y. The safety stoc at dstrbuto ceter s h zα ( L σ + µ S ) The problem formulatg s as follows: Y ad α ( ) m : FU + χ c µ Y J J χµ Y hq ( ) χµ α ( σ µ ) + p + g + a Y + + h z L + S Y J Q 2 Subect to: (1) Y = 1, (2) J χµ Y bu, J (3) Y {0,1}, U {0,1}, J, K (4) Q 0, J. (5)

4 126 M.S.J. Amel, N. Azad, ad A. Rastpour: Desgg a Supply Cha Networ Model The model mmzes the total expected costs made of: the fxed cost for opeg dstrbuto ceters, the aual shpmet cost from dstrbuto ceters to the customers, ad the expected aual vetory cost. Costrats (2) esure that each customer s assged to exactly oe dstrbuto ceter. Costrats (3) are the capacty costrats assocated wth the dstrbuto ceters. Costrats (4) eforce the tegralty restrctos o the bary varables ad fally costrats (5) eforce the o-egatve restrctos o the correspodg decso varable. 2.2 Aalyss of the Model I the above problem, the decso varable costrats. Therefore, we ca tae the dervatve of the obectve fucto wth respect to optmal value of Q. By tag the dervatve of obectve fucto wth respect to Q s gve by: Q, oly has appeared the obectve fucto ad was ot used the 2( + ) p g χµ Y Q =. h By substtutg (6) to the obectve fucto (1), the problem ca be formulated as follows: Q, ad so we obta the Q, we obta the optmal value of (6) m : FU + χ c µ Y J J h ( P + g ) χµ Y + a χµ Y + h zα ( Lσ + µ S ) Y J (7) Subect to: (2)-(4). Therefore, we ca elmate the Q form ths model. 3 Soluto Approach I ths secto, we propose two approaches for solvg the problem: optmal soluto ad a heurstc method. 3.1 Fdg the Optmal Soluto The two o-lear terms of the obectve fucto (7) (the aual worg vetory cost ad the holdg cost for safety stoc) are the cocave terms, ad the other terms of the obectve fucto are lear, so the obectve fucto (7) s absolutely cocave ad the problem we studed the prevous secto s a cocave teger programmg model. Therefore, f we solve the model wth brach ad boud methods, the resultg solutos are global optma. 3.2 Use of Tabu Search Algorthm for Solvg the Problem The faclty locato problem has bee show to be o-polyomal (NP)-hard problem. Sce ths problem s more complex tha faclty locato problem, so ths problem belogs to the class of NP-hard problems. Therefore, we use Tabu search algorthm for solvg the problem. The overall approach Tabu search algorthm s to avod etramet cycles by forbddg or pealzg moves, whch tae the soluto, the ext terato, to pots the soluto space, prevously vsted. Tabu search uses a local or eghborhood search procedure to teratvely move from a soluto X to a soluto X the eghborhood of X, utl some stoppg crtero has bee satsfed. To explore regos of the search space that would be left uexplored by the local search procedure, tabu search modfes the eghborhood structure of each soluto as the search progresses. The solutos admtted to N * ( X ), the ew eghborhood, are determed usg specal memory structures. The search the progresses by teratvely movg from a soluto X to a soluto X N * ( X ).

5 Joural of Ucerta Systems, Vol.3, No.2, pp , Perhaps the most mportat type of short-term memory to determe the solutos N * ( X ) ; also, the oe that gves ts ame to tabu search, s the use of a tabu lst. Tabu lst cotas the solutos that have bee vsted the recet past (less tha m moves ago, where m s the leth of the tabu lst). The steps of tabu search are show Fg.1 The Tabu search parameters are as follows: G: Number of accepted solutos; r: Couter for the umber of accepted solutos; m: Leth of the tabu lst; X: A feasble soluto; C (X ) : The value of obectve fucto for X. X =φ Select a tal soluto X 0 X = X, X = 0 X 0 r = 0 Whle ( r < G ) Do Geerate soluto X the eghborhood of X, C = C( X ) - C (X ) If the caddate move s the tabu lst, the If C ( X ) < C ( X ) the X = X, = r + 1 X = X, update the tabu lst Else Geerate aother soluto X the eghborhood of X Ed If Else If C 0 the X = X, r = r + 1 Update the tabu lst If C X ) < C X ) the ( r ( X = X Ed If Else Geerate aother soluto Ed If Ed If Ed Whle X the eghborhood of X Fg.1: Tabu search algorthm I the followg sectos, we descrbe the Tabu search algorthm whch we use for solvg the problem Ital Soluto Costructo For obtag the tal soluto, we assg customers to the dstrbuto ceters, radomly. The procedure for obtag the tal soluto s as follows. Step1: Put customers to a set K. Step2: 1- Select a customer from K radomly. 2- Delete the customer from K. Step3: Select a dstrbuto ceter radomly. Step4: If remag capacty of the dstrbuto ceter s greater tha the demad of the customer the assg the customer to the dstrbuto ceter ad go to Step 5 otherwse go to Step 3 for selectg aother dstrbuto ceter. Step5: Is K empty? If yes, stop, otherwse go to Step 2.

6 128 M.S.J. Amel, N. Azad, ad A. Rastpour: Desgg a Supply Cha Networ Model Improvg the Ital Soluto I ths phase, the ma obectve s to mprove the tal soluto. We apply four dfferet types of move for geeratg a caddate move: mov1, mov2, mov3, mov4. We geerate a caddate move (from X to the caddate 0 soluto X ) usg oe of the four moves radomly. Mov1: Radomly, oe of the opeed dstrbuto ceters ( a ) s closed ad all of the customers are reallocated amog the remag opeed dstrbuto ceters. The procedure of mov1 s as follows: Step1: Select a ope dstrbuto ceter radomly ( a ). Let D be the set of customers that assged to a. Step2: Select a customer () from D, radomly. Step3: Radomly select oe of the opeed dstrbuto ceters that have eough remag capacty for demad of customer, ad assg customer to ths dstrbuto ceter. Step4: Delete the customer from D. Step5: Is D empty? If yes, close a ad stop, otherwse go to Step 2. Mov2: I ths move we select two ope dstrbuto ceters radomly, ( a, a ), ad exchage a ad a. I ths move capactes of a ad a are checed for servg the customers. Mov3: Oe of the opeed dstrbuto ceters ( a ) s closed radomly, ad a closed dstrbuto ceter ( a ) s opeed radomly The we assg all of the customers correspodg to the elmated dstrbuto ceter ( a ) to the ew opeed dstrbuto ceter ( a ). I ths move the capacty of a s checed for servg the customers. Mov4: Select two ope dstrbuto ceters, radomly, ( v, v ). The radomly select a customer ( c ) v ad a customer ( c ) v ad exchage c ad c. I ths move, we must chec the capactes of dstrbuto ceters. Whe the caddate move s geerated, ths feasble soluto s appled to the Tabu search algorthm procedure. Ths process s repeated utl r < G. 4 Computatoal Results The computatoal expermets descrbed ths secto were desged to evaluate the performace of our overall soluto procedure wth respect to a seres of test problems. It was coded vsual basc 6 ad ru o a Petum 4 wth 3 GB processor. The daly demad of the customers was draw from a uform dstrbuto betwee 3 ad 7. The varaces of daly demads of customers were draw from a uform dstrbuto betwee 1 ad 3. I addto, we use the followg parameter values: h s uformly draw from [2, 4]; p s uformly draw from [15, 20]; L s uformly draw from [6, 10]; 2 S s uformly draw from [1, 3]; g s uformly draw from [15, 20]; a s uformly draw from [2, 5]; c s uformly draw from [2, 5]; χ = 250, z = 1.96 (97.5% servce level). α 4.1 Comparso of Optmal ad Tabu Search Solutos For evaluatg the proposed Tabu search algorthm, fftee problems are solved by LINGO 8 software (Table 1). Accordg to Secto 3.1, the soluto, whch we obta from LINGO software, s global optma, ad we ca compare our heurstc soluto by optmal value, whch s obtaed by LINGO software. For each problem, the tug of the parameters s doe by carryg out radom expermets.

7 Joural of Ucerta Systems, Vol.3, No.2, pp , Table 1: Comparso of optmal ad Tabu search solutos Optmal Value Tabu Search Soluto NO. # Customers # DCs Cost CPU tme Cost CPU tme Gap(%) hours lmt hours lmt Gap: Gap from optmal soluto (%). It ca be see that the solutos of the Tabu search algorthm are optmal (or ear optmal) dfferet problems (Table 1). The average CPU tme are less tha or equal to 262 secods for Tabu search algorthm (CPU tmes are the secods). However, the maxmal average CPU tme for obtag the optmal solutos s equal to secods, ad for staces 14 ad 15 by a 3 hours tme lmt, LINGO ca ot fd the optmal soluto, ad the Tabu search algorthm ths stace s better tha the solutos that are obtaed by LINGO. 4.2 Comparso of Tabu Search based Approach ad Smulated Aealg based Approach I ths secto, we compare our Tabu search method wth smulated aealg (SA) method. The procedure for obtag tal soluto ad caddate move, we use SA method, are the same to the procedure of obtag tal soluto ad caddate move Tabu search algorthm. The comparso of Tabu search algorthm ad SA algorthm are show Table 2. It ca be see that the soluto qualty Tabu search algorthm s better tha the soluto qualty SA algorthm. Table 2: Comparso of Tabu search algorthm ad SA algorthm Tabu search algorthm SA algorthm NO. # Customers # DCs Cost CPU Tme Cost CPU Tme

8 130 M.S.J. Amel, N. Azad, ad A. Rastpour: Desgg a Supply Cha Networ Model 5 Coclusos I ths paper, we have outled a tegrated stochastc supply cha desg model that optmzes locato, vetory ad trasportato decsos smultaeously. We assumed each customer has ucerta demad wth a ormal dstrbuto, also, we assumed each dstrbuto ceter has ucerta lead tme wth a ormal dstrbuto. A Tabu search based procedure was developed for solvg the problem. We comprsed the Tabu search based procedure wth the optmal soluto, ad a Smulated aealg algorthm. The results of extesve computatoal tests dcate that the method s effcet for a wde varety of problem szes ad structures. Refereces [1] Bramel, J., ad D. Smch-Lev, The Logc of Logstcs, Sprger-Verlag, New Yor, [2] Das, M.S., Networ ad Dscrete Locato: Models Algorthms ad Applcatos, Wley-Iterscece, New Yor, [3] Das, M., C. Coullard, ad Z.J. She, A vetory-locato model: Formulato, soluto algorthm ad computatoal results, Aals of Operatos Research, vol.110, pp , [4] Erlebacher, S.J., ad R.D. Meller, The teracto of locato ad vetory desgg dstrbuto systems, IIE Trasactos, vol.32, pp , [5] Geoffro, A.M., ad G.W. Graves, Multcommodty dstrbuto system desg by Beder s decomposto, Maagemet Scece, vol.20, pp , [6] Hamacher, H., ad Z. Drezer, Faclty Locato: Applcatos ad Theory, Sprger-Verlag, Berl, [7] Jayarama, V., Trasportato, faclty locato ad vetory ssues dstrbuto etwor desg, Iteratoal Joural of Physcal Dstrbuto ad Logstcs Maagemet, vol.18, o.5, pp , [8] Klose, A., ad A. Drexl, Faclty locato models for dstrbuto system desg, Europea Joural of Operatoal Research, vol.50, pp , [9] Lu, Z., ad N. Bostel, A faclty locato model for logstcs systems cludg reverse flows: the case of remaufacturg actvtes, Computers & Operatos Research, vol.34, pp , [10] Melo, M.T., S. Ncel, ad S. Saldaha da Gamma, Dyamc mult-commodty capactated faclty locato: a mathematcal modelg framewor for strategc supply cha plag, Computers & Operatos Research, vol.33, pp , [11] Mrada, P.A., ad R.A. Garrdo, Icorporatg vetory cotrol decsos to a strategc dstrbuto etwor desg model wth stochastc demad, Trasportato Research, vol.40, o.3, pp , [12] Nozc, L.K., ad M. Turqust, Itegratg vetory mpacts to a fxed-charge model for locatg dstrbuto ceters, Trasportato Research, Part E, Logstcs ad Trasportatos Revew, vol.34, o.3, pp , [13] She, Z.J., Effcet algorthms for varous supply cha problems, Ph.D. Dssertato, Northwester Uversty, Evasto, IL, [14] She. Z.J., A mult-commodty supply cha desg problem, IIE Trasactos, vol.37, pp , [15] She, Z.J., C. Coullard, ad M. Das, A ot locato-vetory model, Trasportato Scece, vol.37, pp.40 55, [16] Smch-Lev, D., Herarchcal desg for probablstc dstrbuto systems eucldea spaces, Maagemet Scece, vol.38, pp , [17] Tragatalergsa, S., J. Holt, ad M. Roqvst, A exact method for the two-echelo, sgle-source, capactated faclty locato problem, Europea Joural of Operatoal Research, vol.123, pp , 2000.