Sraegic Opimizaion of a Transporaion Disribuion Nework K. John Sophabmixay, Sco J. Mason, Manuel D. Rossei Deparmen of Indusrial Engineering Universiy of Arkansas 4207 Bell Engineering Cener Fayeeville, AR 72701, USA Absrac The ransporaion nework of any large manufacuring or reail company is ofen iniially designed and periodically refined o minimize some funcion of cos. In his paper, we develop an ineger linear programming model for he disribuion nework of a large company from a sraegic planning perspecive. Specifically, we are ineresed in fuure nework expansion and conracion opporuniies and how hese phenomena affec nework uilizaion, he company s operaional approaches, and ransporaion asse uilizaion. The model formulaion is presened, followed by some preliminary resuls. Keywords Opimizaion, Transporaion, Disribuion 1. Inroducion Nework-srucured problems arise direcly or indirecly in many pracical scenarios where physical flow ransfers occur beween a se of origin poins (supply nodes) and a se of desinaion poins (demand nodes) [1]. This physical flow can be associaed wih modal ypes used o ransfer a combinaion of goods and/or people via links o and from origin and desinaion poins. The decisions o be made in nework opimizaion problems include he locaion of supply nodes, ransporaion rouing, invenory allocaion, scheduling of flow beween supply and demand nodes, and he design of links wihin he nework [1]. Numerous sudies have been conduced and documened o deermine such decision variables in a disribuion nework. Akyilmaz [2] presens a framework of algorihms ha roue Less-han Truck Load (LTL) shipmens via inermediae erminals. Pirkul and Jayaraman [3] develop a mixed ineger programming (MIP) model for he plan and warehouse locaion problem. The obecive o he model is o minimize oal ransporaion and disribuion coss, including fixed plan opening and operaing coss, as well as warehouse coss. Sharma and Saxena [4] presen a special case of he ransshipmen problem wherein flow occurs sricly beween supplier plans and inermediae faciliies and beween inermediae faciliies and he end cusomer marke. These previous sudies have presened formulaions for he opimal assignmen of eniies in various ypes of disribuion neworks for some single ime period of ineres. In his paper, we expand on he formulaions of previous sudies o include muliple ime periods and o incorporae he refining or updaing componen of ransporaion disribuion nework sraegic planning. In he ransporaion indusry, ransporaion neworks are ofen refined o minimize coss. This refinemen can eiher be an expansion or a conracion wihin he exising nework. Ofen, he complex nework of suppliers, inermediae consolidaion ceners (ICC), and disribuion ceners (DCs) ha exiss hroughou he counry creaes he need for periodic review of boh supplier o ICC and ICC o DC assignmens. Anoher quesion of ineres ofen perains o he choice of ransporaion mode ha should be used o ranspor goods beween ICCs and DCs (i.e., air, highway, rail, ec.). When faced wih growing demand, companies frequenly wish o idenify sraegic locaions for fuure ICCs. Alernaely, companies may also need o deermine which ICC(s) should be closed under low volume condiions. In his paper, we model he disribuion nework of a large company ha is composed of a number of complex produc disribuion subneworks ha are used o ranspor various ypes of goods. In erms of our overall research agenda, we seek o idenify wha ype(s) of sraegic nework updaes/refinemens could be performed o furher minimize disribuion and operaing coss. Some preliminary resuls are presened for a scaled-down version of rue disribuion nework complexiy o promoe undersanding of our model oupus. Secion 2 describes he disribuion nework of ineres, followed by our MIP model formulaion in Secion 3. Afer preliminary
experimenal resuls are discussed in Secion 4, preliminary conclusions and direcions for fuure research are presened in Secion 5. 2. Disribuion Nework Operaions Consider a ransporaion disribuion nework consising of a se of I suppliers, a se of J ICCs, and a se of K DCs (Figure 1). For each load being shipped from ICC J o DC k K, he company uilizes up o hree ypes of ransporaion: rail, full ruckload (TL), and inermodal (i.e., a combinaion of rail and TL). Each load shipped from supplier i I o ICC J is always made via LTL ransporaion. LTL Rail TL Inermodal Suppliers ICCs DCs Figure 1. Transporaion Opions for Disribuion Nework Under Sudy Disribuion operaions wihin he nework are iniiaed when some quaniy of demand, expressed in pounds, for a specific produc ype is generaed by DC k K. Once appropriae suppliers are informed of he demand, hey noify DC k K of heir abiliy o fulfill he demand for he requesed produc. Once supplier i I is insruced o fulfill he DC s demand, supplier i I ships he requesed produc o a nearby ICC via LTL ranspor. Demanded producs ge consolidaed a each ICC J and hen are shipped o he requesing DC by he minimum cos ransporaion opion. 3. Opimizaion Model Formulaion Demand a each DC can only be fulfilled by hose suppliers ha sock he requesed produc ype. The obecive of he proposed opimizaion model is o deermine, for a given demand profile (by DC in pounds), which ICC should service which DC, and by which mode of ransporaion his service should occur, so ha disribuion coss are minimized. This secion deails he opimizaion model formulaion by firs describing model ses and parameers, hen decision variables, followed by he obecive funcion and corresponding problem consrains. Ses and Indices I suppliers ( i = 1... I ) J ICCs ( = 1... J ) K DCs ( k = 1... K ) M modes of ransporaion ( m = 1... M ) Parameers S i T ime periods ( 1... T T k U i V km = ) Disance from supplier i o ICC in miles Disance from ICC o DC k in miles Supplier i o ICC ransporaion cos in $/pound/mile ICC o DC k ransporaion cos by ransporaion mode m in $/pound/mile
N Number of dock doors a ICC in ime period n Number of ICCs (i.e., n = J ) C Capaciy per dock door in ICC in ime period in pounds/door D ik Demand requesed from supplier i by DC k in period in pounds A i Supply ha is available from supplier i in period in pounds Mo Cos of opening ICC in ime period Mc Cos of closing an ICC B Iniial saus of ICC ; B = 1 if ICC is iniially open; B = 0 oherwise Tr lbs Truck capaciy in pounds Ra lbs Rail capaciy in pounds M demand Minimum required demand required o keep an ICC open Variables Vo Vo = 1 when ICC opens in period ; Vo = 0 oherwise Vc Vc = 1 when ICC closes in period ; Vc = 0 oherwise Oc Toal opening and closing cos of ICCs in period Tc Toal ransporaion cos of ICCs in period X Pounds of demand shipped from supplier i hrough ICC o DC k by ransporaion mode m in ime period Y Y = 1 if ICC services DC k in ime period ; Y = 0 oherwise k k Z Z = 1 if ICC is open in ime period ; = 0 k Z oherwise The model s obecive funcion aemps o minimize he oal ransporaion cos and he oal cos of opening and closing ICCs across all ime periods: ( Tc + Oc ) (1) In (1), Tc Oc = i k m X ( U S + V T ) T (2) i i km k = ( Vo Mo + Vc Mc ) T (3) Consrains (2) and (3) calculae he oal ransporaion cos and he oal cos of opening and closing ICCs in period, respecively. Care mus be aken o ensure ha ICCs are no overloaded: X N * C Z J, (4) However, demand a each DC mus be me in every ime period: X D i I, k K, (5) m ik Truck (Consrains 6) and rail (Consrains 7) shipmens mus no exceed he corresponding vehicles payload capaciy in any ime period: X Trlbs J, (6) X Ralbs J, (7)
As he company does no wish o spli a given DC s demand across muliple suppliers (Consrains 8), we mus make sure ha he seleced supplier has a leas he oal pounds demanded from he DC available in he corresponding ime period (Consrains 9): Y = 1 k K, (8) k k m X A i I, (9) i Obviously, any DC s demand should only be assigned o an open ICC: Y n * Z J, (10) k Consrains (11) hrough (14) are inermediae consrains used o ensure ha he cos of opening and closing ICCs is calculaed appropriaely: Vo B J, = 1 (11) Vc Vo Vc B J, = 1 Z 1 J, > 1 Z J, 1 (12) (13) 1 > (14) Finally, an ICC is only allowed o remain open if a minimum amoun of demand flows hrough i: X M * Z i I, (15) demand 4. Verifying he Model: An Example Insance For he sake of clariy and illusraion, le I = 10, J = 3, and K = 6. The number of dock doors per ICC and he capaciy associaed wih each dock door a each ICC = 1... 3 is given for his example disribuion nework insance in Table 1. Iniially, we consider T = 3 ime periods, wih 60 unique DC demands in oal (i.e., 10 demands per DC) being generaed from a discree uniform disribuion over he inerval [0, 100] unis for ime period = 1. In subsequen periods, his demand is increased by 10% in period 2, followed by a 15% increase over period 1 for he period 3 demand. Table 1. ICC Informaion in Example Problem Insance ICC Dock Doors Capaciy/Dock Door 1 3 1000 2 2 2500 3 1 3000 The opimizaion model presened in Secion 3 above was coded in AMPL, hen analyzed using he MIP solver in CPLEX v8.1. Afer running he example problem insance in CPLEX, we observe he ICC assignmens shown in Table 2. For = 1, he model recommends ha ICC #1 be closed, bu he oher wo ICCs (#2 and #3) be open. However, as demand increases in he nex wo ime periods under sudy, model resuls do vary by ime period. The model recommends ha ICCs #1 and #2 be open for = 2, while ICC #3 should be closed during he same ime period. Finally, when = 3, he model suggess ha all demand should flow hrough ICC #1. Observing hese ICC opening and closing schedules resuls in he opimal, minimum cos soluion for he example problem insance under sudy.
Table 2. Assignmen of ICCs as Recommended by Opimizaion Model Oupu Period ICC Z* 1 1 0 1 2 1 1 3 1 2 1 1 2 2 1 2 3 0 3 1 1 3 2 0 3 3 0 Table 3 displays represenaive model oupu wih respec o ransporaion model decisions when = 2. Table 3 shows oal pounds shipped from which supplier hrough which ICC o which DC by wha mode of ransporaion for each shipmen. The decision variables for he oher wo ime periods show comparable resuls/rends. Table 3. Transporaion Decisions Summary for Time Period 2 Pounds Supplier ICC DC Mode 68 1 1 1 LTL 104 1 1 3 LTL 81 1 1 5 LTL 81 9 1 1 LTL 49 9 1 3 LTL 88 9 1 5 LTL 19 3 1 1 LTL 67 3 1 2 LTL 106 3 1 5 LTL 41 7 1 1 LTL 84 7 1 3 LTL 1370 7 1 5 LTL 96 6 1 1 LTL 67 6 1 4 Rail 104 6 1 4 Rail 40 1 1 4 Rail 44 9 1 4 Rail 404 7 1 4 Rail 27 6 1 6 Rail 104 1 1 6 Truck 38 9 1 6 Truck 92 3 1 6 Truck 107 7 1 6 Truck 19 6 2 4 Truck 35 3 2 2 Truck 30 1 2 2 Truck 2706 9 2 2 Truck 85 3 2 2 Truck 83 7 2 2 Truck 61 6 2 2 Truck Taking ino consideraion he high variabiliy of demand, we examine he model s sensiiviy o demand variaion over ime. As menioned in he model formulaion above, coss are incurred each ime an ICC is opened and/or
closed. Furher, an ICC can remain open only if some minimum amoun of demand (e.g., 3,000 pounds) flows hrough i for a single ime period. Preliminary experimens validaed he inuiion ha ICCs frequenly oscillae beween open and closed saes under variable demand. An imporan par of validaing he ICC opening and closing componens of he model is o mee wih company personnel o deermine if hey wish o resric o some accepable level he number of ICC openings and/or closings per ime period. 5. Conclusions and Fuure Research In his paper, we presen a mixed-ineger programming model for analyzing he disribuion nework of a large company from a sraegic planning perspecive. Using he proposed model, opimal ransporaion roue assignmens can be made wih minimum oal cos. In addiion, he saus of each ICC in he nework can be deermined in each ime period in erms of wheher or no i is/should be open for business. The example problem insance presened in his paper was seleced only for illusraive purposes. An exension of he proposed model would be o include real daa from he company, and hen analyze he enire nework disribuion problem for he company. In he fuure, we will also coninue o embellish and validae he opimizaion model presened in his paper. In parallel, he research eam will analyze he dynamic ramificaions of he opimizaion model s soluion in erms of invenory build-up and required order cycle ime via Mone Carlo simulaion and/or discree even simulaion. References 1. Subramanian, S., 1999. Opimizaion Models and Analysis of Rouing, Locaion, Disribuion, and Design Problems on Neworks, Docoral Disseraion, Virginia Polyechnic Insiue and Sae Universiy. 2. Akyilmaz, M.O., 1994. An Algorihmic Framework for Rouing LTL Shipmens. Journal of Operaional Research Sociey 45(5), 529-538. 3. Pirkul, H., Jayaraman V., 1996. Producion, Transporaion, and Disribuion Planning In A Muli Commodiy Tri-Echelon Sysem. Transporaion Science 30(4), 291-302. 4. Sharma, R. R. K., Saxena, A., 2000. Dual Based Procedures For The Special Case Of Transporaion Problem. European Journal of Operaional Research 122(3), 37-51.