Petroleum replenishment and routing problem with variable demands and time windows

Transcription

1 Petroleum replenshment and routng problem wth varable demands and tme wndows Yan Cheng Hsu Jose L. Walteros Rajan Batta Department of Industral and Systems Engneerng, Unversty at Buffalo (SUNY) 34 Bell Hall, Buffalo, NY, USA. Abstract In ths paper we develop a methodologcal framework for desgnng the daly dstrbuton and replenshment operatons of petroleum products over a weekly planng horzon by takng nto account the perspectves of both the transporter and ts customers. The proposed approach consders the possblty of havng late delveres due to the varablty of the customers demands and expected tme wndows. We frst develop an nventory model for the customers to dentfy the optmal order quanttes and tme wndows. Then, we solve a sequence of mxed-nteger optmzaton models for desgnng the dstrbuton routes based on the order quanttes and tme wndows selected by the nventory models. We desgned the optmzaton models so that the late delveres are balanced among the customers n order to mtgate the overall customer dssatsfacton. We test the proposed approach by solvng a test bed of nstances adapted from the lterature. The emprcal results show that the proposed approach can be used for desgnng the dstrbuton plan for delverng petroleum products n condtons where the operatonal capabltes of the transporter are lmted for generatng optmal on-tme plans. Keywords: Petroleum delvery; Inventory routng; Routng and schedulng. 1 Introducton The transportaton ndustry plays a crtcal role n today s global economy fosterng the operatons of nearly all other ndustres around the world. Alone n the U.S., accordng to the U.S. Federal Hghway Admnstraton, about 0 bllon tons of goods, worth more than $10 trllon, were moved across the country just n 01 [47]. Transportaton-related goods and servces represented approxmately 11% of the U.S. gross domestc product n 000, only beng surpassed by the housng, health-care, and food ndustres[0, 4]. Among the total goods transported n the U.S. n 01, more than 1.8 bllon tons corresponded to gasolne, desel and other petroleum-based products, thus becomng the top sxth most transported commodtes n the country [47]. Petroleum products are stll one of the world s most traded commodtes, as they contnue beng the man energy source for the transportaton ndustry. Accordng to the U.S. Natonal Academy of Scences [1], petroleum-based fuels represent about 98% of the energy sources used for moblzng both people and freght n the U.S. In addton to ther use as fuel for transportaton, the need for petroleum products as lubrcants, transmsson and Correspondng author. Emal addresses: yhsu8@buffalo.edu (Yan Cheng Hsu), josewalt@buffalo.edu (Jose L. Walteros), batta@ buffalo.edu (Rajan Batta) 1

2 hydraulc fluds, and as raw materals for many producton processes, has made of them ubqutous and vtal for the daly operatons of every country. In order to provde a steady supply of petroleum products that satsfes the growng demand, many supplers face the enormously complex process of plannng the routng strateges for dstrbutng these products to ther customers, whch nclude gasolne retalers (gas statons), producton companes, agrcultural companes, and marne centers, among others. Ths process generally requres the allocaton of multple competng resources whle smultaneously satsfyng many operatve restrctons and regulatory polces (the transportaton of gasolne and other petroleum-based products s hghly regulated by the Federal Motor Carrer Safety Admnstraton (FMCSA) n the U.S. [31]). As a result, snce publcaton of the frst paper on gasolne dstrbuton [7], there has been a contnuous effort to develop quanttatve models that support the decson makng of transportaton companes. Over the last couple of decades numerous soluton approaches have been developed to tackle vehcle routng problems that ncorporate the dstrbuton requrements of many ndustres (e.g., [3, 6, 14, 3, 49, 50]). However, despte the large number of such approaches, there are stll several elements of the dstrbuton logstcs of some products that requre further analyss. One fundamental assumpton that s often consdered by many of these technologes s that there always exst dstrbuton plans that satsfy all the customers requrements (e.g., order quanttes and tme wndows) under the transporter operatonal restrctons. In a real-lfe scenaro, the decsons regardng the product orders and the dstrbuton logstcs are often made ndependently and sequentally by the customers and the transporter wthout any nteracton durng the dstrbuton plannng process. In other words, the order quanttes and expected delvery tme wndows whch are selected by each customer solely based on the customer s own nterests are gven to the transporter n the form of hard constrants. Then, after collectng all the orders, the transporter makes a routng plan amng to fulfll all the requrements whle mnmzng ts own operatonal costs [, 49]. Many mathematcal models assume that there are always feasble routes that satsfy such demands and tme requrements of all the customers. In contrast, n many compettve markets lke the one of products dstrbuton n whch the transportaton decsons must also meet strct regulatory polces fndng feasble solutons that cope wth all customer and governmental requstes s often mpossble. In other words, due to capacty lmtatons and further operatonal requrements, the actual delvery tme may devate from the desred tme wndow for many customers. For the specfc case of petroleum products, there are several reasons why optmally plannng the dstrbuton logstcs s a complex challenge. In addton to the varable nature of the demands, the lmted number of trucks and drvers, the regulatory polces, and the dffcultes posed by the nherent characterstcs of the products (.e., mostly flammable lquds that must be transported n specalzed mult-compartmented trucks and tralers), the heterogenety of the customers and the strct tme requrements make the problem of dentfyng optmal delvery routes especally dffcult. Moreover, because of the replenshment logstcs, most customers request very specfc delvery tme wndows that often overlap among them (e.g., retalers often prefer havng replenshments late at nght when traffc s low, whereas other customers prefer early mornng delveres before any operaton begns). These latter requrements dramatcally mpact the complexty of the dstrbuton plannng, up to the pont that even fndng feasble delvery schedules s smply mpossble. 1.1 Relevant lterature The vehcle routng problem (VRP) s at the cornerstone of most dstrbuton plannng processes. Snce the publcaton of the frst paper n the subject back n the late 50 s [7] (a paper about

3 gasolne dstrbuton), a staggerng number of studes have been developed to tackle many varants of ths problem (for further references see the followng comprehensve surveys [, 9, 49]). Ths problem and ts varatons have contnuously rased the nterest of the academc communty because of ther practcal relevance and nherent dffculty. In fact, several technologcal advancements n the feld of Operatons Research have been dscovered by studyng these partcular problems []. Among all the varatons that can be found n the lterature, the ones that are relevant for ths paper are the VRP wth tme wndows (VRPTW), where each customer must be vsted durng a specfc tme frame [4, 9, 10, 33, 36]; the VRP wth multple compartments (MCVRP), where the vehcles have dfferent capactes and are equpped wth multple compartments that can carry more than one type of product [1, 15, 16, 18, 8, 30, 34, 35, 37, 40, 44]; the VRP wth stochastc demands (SVRP), where the demands are gven by a probablty dstrbuton [7, 8, 17, 48, 38]; the dynamc VRP (DVRP), where the nformaton of some customers becomes avalable durng operaton [43]; the tme wndow assgnment VRP (TWAVRP), whch s a varaton where the tme wndows have to be assgned by the transporter to the customers before the demand of product gven by them s known [45, 46]). In addton to the aforementoned problems regardng vehcle routng, an alternatve type of approach that has been consstently utlzed to plan the dstrbuton logstcs of several products s that of synchronzng the nventory management wth the desgn of the dstrbuton logstcs of the commodtes [, 13, 19, 0, 3, 39]. The resultng nventory routng problems (IRPs) can produce several advantageous strateges f the customers are wllng to delegate the nventory management. In general, the IRP s a product dstrbuton problem n whch one actor the manager s responsble for both transportaton and nventory plannng [19]. In practce, the manager can ether be the producer, the consumer, or the transportaton company dependng on the type of busness. When the managers are the producers or the transportaton companes whch s often the case these ntegrated polces allow them to select the tmng and szes of the delveres, achevng a better utlzaton of ther vehcle fleet and offerng a better servce qualty to ther customers. For these models to be applcable though, the customers must render complete knowledge of ther operatonal needs and full control over ther nventory levels to the manager. In turn, the manager must ensure that the customers wll never out of stock. Nevertheless, n the context of petroleum products, the operatonal decsons of many customers requre them to mantan full control of the nventory levels, whch complcates the use of these latter models. Thus, the man dfference between the problem studed n ths paper and the IRP s that n the latter the supplers control the nventory management of the customers, whereas n the former, the nventory problem s solved by each customer and the results are gven as hard constrants to the transporter. For the specfc context of delverng petroleum products, the frst publcatons that provde specfc applcatons of ths knd date back to the 50 s [7], 80 s [11, 1] and 90 s [5, 50]. Most of the soluton approaches proposed n these papers range between heurstc and exact approaches. For nstance, [3] formulated a fuel delvery problem as a set parttonng model and proposed a branchand-prce algorthm to solve the resultng problem a technque wdely used for solvng VRPs. In addton to that, [41] presented a case study on the delvery networks n Hong Kong, whch contans tanker assgnment and a routng problem wth a heterogeneous fleet of compartmented trucks. A decson support system (DSS) approach was developed to solve the vendor managed nventory (VMI) problem. In [3], an exact algorthm was proposed to tackle the sngle perod and sngle depot case usng an unlmted heterogeneous fleet of compartmented tank trucks petroleum replenshment problem. A heurstc for the mult-perod and sngle depot wth lmted number of trucks was proposed n [4]. The same problem wth tme wndow constrants was tackled further n [6]. More recently, [5] proposed heurstcs for the mult-depot staton replenshment problem consderng tme wndows, n whch the concept of a trp defned by both a route and the truck 3

4 used to make delveres n ths route was frstly ntroduced to address ths problem. Instead of generatng possble routes, they ntroduced a method to generate potental feasble trps. Smlarly, as for other VRP varants, the tme wndows gven by the customers are assumed to be fxed and no further consderatons are proposed for the cases where the gven tme wndows render the problem nfeasble. 1. Contrbutons Ths paper ams to develop a methodologcal framework for desgnng the daly dstrbuton operaton for delverng petroleum products that consders the possblty of havng late delveres due to the varablty of the customers demands and expected tme wndows. In addton to maxmzng the dstrbuton profts, the proposed framework attempts to mnmze the dssatsfacton of the customers due to late delveres by balancng the late delveres among the customers over the plannng horzon. The contrbutons of ths paper can be summarzed as follows: We propose a methodologcal framework to solve the daly petroleum dstrbuton problem consderng both transporter s and customers perspectves. We develop an nventory problem that models all the scenaros regardng the delvery tmes for the product orders. Ths model s used to determne the order quanttes and tme wndows for each customer. We propose a sequence of mxed-nteger optmzaton models for desgnng the dstrbuton routes based on the order quanttes and tme wndows selected by the nventory models. We talor the optmzaton models so that the late delveres are balanced among the customers n order to mtgate the overall customers dssatsfacton. We test the proposed approach by solvng a test bed of nstances adapted from the lterature. The emprcal results show that the proposed approach can be used for desgnng the dstrbuton plan for delverng petroleum products n condtons where the operatonal capabltes of the transporter are lmted for generatng optmal on-tme plans. The remander of ths paper s organzed as follows. Secton presents a detaled descrpton of the problem at hand; Secton 3 summarzes of the proposed sequental soluton approach; Secton 4 ntroduces the nventory model that s used to model the gas staton decsons; Secton 5 present the proposed mathematcal formulatons to generate the dstrbuton plan; Secton 6 analyzes the results of the emprcal study; and fnally, Secton 7 provdes the fnal conclusons and further research drectons. Problem descrpton The petroleum replenshment problem deals wth the logstcs of delverng petroleum products to a set of customers n ths case gas statons n such a way that the requrements of such customers are fulflled, under the operatonal capabltes of the transporter, whle maxmzng the total dstrbuton profts yelded by the dstrbuton operaton. From the perspectve of the customers, durng ther daly operatons, gas statons perodcally revew ther underground tanks and place order requests to the supplers when the stock levels of ther products fall below predefned thresholds. The orders typcally consst of a lst of product type requests that nclude the desred quanttes 4

5 and delvery tme wndows as shown n Fgure 1. The suppler collects ths nformaton from all of ts customers to generate: (1) the truck loadng, () the delvery routes, and (3) truck dstrbuton schedule, n order to fulfll these orders. From the perspectve of the suppler, the dstrbuton network s defned as follows. Let G = (N, A) be a drected graph where N = {0, 1,,.., n} s a set of nodes representng the dstrbuton termnal (node 0) and the gas statons (nodes 1,.., n), and A = {(, j) : j and, j N} s the set arcs that represent the road segments connectng the nodes n N. We denote c j (c j = c j and c j > 0) and t j as the travel costs and travel tmes assocated wth the arc (, j), and s as the servce tme of gas staton. The tme wndow [a, b ] specfes the earlest and latest tme lmts for performng the petroleum replenshment,.e., the delvery must occur wthn the gven tme wndow [a, b ]. t 34 s Q ; [a, b ] t 0 s 3 Q 3 ; [a 3, b 3 ] t 03 t 40 s 4 Q 4 ; [a 4, b 4 ] t 1 s 1 Q 1 ; [a 1, b 1 ] t 08 t 70 t 56 s 5 Q 5 ; [a 5, b 5 ] t 81 s 7 Q 7 ; [a 7, b 7 ] t 67 s 8 Q 8 ; [a 8, b 8 ] s 6 Q 6 ; [a 6, b 6 ] Fgure 1: Dstrbuton network. Each truck s dvded nto multple compartments wth known capactes, whch are used to upload dfferent type of products. In other words, two dstnct grades of petroleum must be placed nto two separate compartments to avod nternal contamnaton. Furthermore, the petroleum stored n each compartment must be fully pumped out when fulfllng the dstrbuton servce, as the qualty of any remanng petroleum wll deterorate once t has contact wth ar, due to oxdaton. Therefore, f the underground tanks of a gas staton fal to accommodate a full-compartmented load of petroleum, the remander must be sent back to the termnals, resultng n a send-back cost whch s generally hgh n comparson to other costs. All trucks should begn and end at the termnal and the travel speed of those are consdered to be the same. Thus, the petroleum replenshment problem conssts of determnng: (1) the quantty and tme wndows of delvery for each gas staton; () the loadng of the varous petroleum nto the truck compartments; (3) the delvery routes to the gas statons; (4) the departure tme of each truck from the termnal and the arrval tme at each of ts assgned customers. 5

6 In addton, the objectve of the dstrbuton problem s twofold: to mnmze the expected total costs for gas statons, and to maxmze the overall dstrbuton profts for the transporters. In ths paper we decouple the petroleum delvery problem nto two parts: the gas staton nventory problem (.e., part (1) of the above lst) and the transporter dstrbuton problem (.e., parts ()-(4)). Frst, for the gas staton nventory problem, we use a model to determne the order quanttes and delvery tme wndow for each gas staton and second, for the transporter s dstrbuton problem we propose a sequence of mxed-nteger formulatons to determne the dstrbuton plan. The descrpton of the proposed framework s gven n the followng secton. 3 Soluton framework The petroleum replenshment problem deals wth both the nventory problem of the gas statons and the dstrbuton problem of the transporter on a daly bass. For each day of the plannng horzon, the order quanttes and requested tme wndows for each gas staton wll be dentfed by an nventory model that s amed to mnmze the expected total costs perceved by the gas statons. Such costs nclude orderng cost, holdng cost, shortage cost, and send-back cost (see Secton 4). The proposed model takes nto consderaton that the exact delvery tmes are not known a prory by the gas statons. Therefore, the order quanttes and desred tme wndows are decded based on an estmaton of the delvery tmes. Once the order quanttes and desred delvery tme wndows are selected by each gas staton usng the proposed nventory model, we then solve a seres of nterrelated mxed-nteger lnear programs that wll determne how to load these demands nto the truck compartments, how to schedule the truck departure and returnng tmes, and how to delver these demands (see Secton 5). By solvng these models, the optmal petroleum dstrbuton plan for each day can be dentfed. In the proposed approach, we solve the replenshment problem sequentally day by day. Thus, the resultng dstrbuton plan obtaned for each day of the planng horzon s then used to calculate the nputs of the nventory models of the subsequent days. Notce that, after solvng the dstrbuton problem of a gven day, the actual delvery tmes for each staton can be used to compute the ntal stock levels for each staton for the day after and thus, the order decson can be made accordngly. The soluton process contnues untl the consdered tme horzon s reached. In ths secton, the steps of the proposed approach, whch are presented n Fgure are summarzed. 6

7 Start 1. Inventory model 3. Truck loadng model 4. Route schedulng 5. Truck routng. Route gererator no Reach the tme perod? yes End Step 1: Inventory model Fgure : Soluton procedure. The frst step of the soluton framework s to generate the product orders of each gas staton. The nformaton requred to generate the orders n ths step comprses the ntal stock levels, the demand rates, and the tank capactes. In addton to the gas staton nformaton, the nventory related costs are also requred to calculate the expected total costs. These costs nclude the unt order cost, fxed order cost, holdng cost, shortage cost, and send-back cost. The nventory model s used to determne the order quanttes and desred delvery tme wndows that mnmzes the gas statons expected total costs based on ther stock levels and petroleum consumpton rates. A full descrpton of ths model s gven n Secton 4. Step : Route generator The delvery routes are generated for the gas statons that placed orders durng the gven day accordng to the nventory model. Typcally, a truck contans four to sx compartments wth dfferent capactes and each gas staton requres one or two compartments to satsfy the petroleum requrement. Therefore, havng routes servng between one and three statons s common n practce. However, we also consder the stuaton where the petroleum dstrbuton serves gas statons wth lower demand rates. For ths case, trucks can vst four to fve statons wthn a route. We descrbe the process used to generate the canddate routes n Secton 5.1. Step 3: Truck loadng capacty check The total delvery quanttes of a route cannot exceed the capactes of the compartments n the truck. The truck loadng model s used to determne the assgnment of the dfferent petroleum products to each of the truck compartments. The route wll be elmnated f the demands of the gas statons of the gven route cannot be loaded nto the truck. In addton, ths model computes the proft of the route, whch s calculated by the revenue receved for delverng the petroleum mnus the travel costs of the route. Step 4: Route schedulng 7

8 The truck schedule of each feasble route generated s then determned n ths step. The objectves are to fnd the truck departure and returnng tmes, the delvery tmes for each gas staton n the route, and a set of penaltes for those routes falng to satsfy the tme wndow constrants. A route that fals to satsfy the tme wndow constrants s not elmnated n ths step. Instead, we add a penalty that accounts for the total tme the truck followng the gven route arrves before or after the gas staton tme wndows. Consequently, the generated routes are dvded nto two sets: feasble (on tme) routes and nfeasble (late) routes and ther costs are updated wth the correspondng penaltes. Step 5: Truck routng In the truck routng problem, the truck assgnment wll be decded by maxmzng the total profts of delverng the petroleum to all gas statons. The objectve calculaton ncludes the route proft found n Step 3, as well as the penaltes for late delveres obtaned n Step 4. Step 6: Termnaton condton As mentoned before, the petroleum dstrbuton problem s solved sequentally on a daly bass. The results of route schedulng and truck routng for day t are nputs for day t + 1. If the consdered tme horzon s not reached, then return to Step 1 to resolve the problem for the subsequent day. If the tme horzon s reached, we termnate the process. 4 The nventory model for the gas statons The nventory model seeks to determne the desred delvery tme wndows [a, b ] over the tme horzon T and the order quanttes q for each customer N so that the expected customer s total cost s mnmzed. Let l be the actual delvery tme for customer N. The total cost of each customer gven delvery tme l s named C (l ) and comprses: (1) the orderng cost P (q ), whch represents the cost of orderng product to the suppler; () the holdng cost H (l, a, b, q ), whch s the opportunty cost of havng nventory; (3) the shortage cost S (l, a, b, q ), whch s the cost the customer ncurs f at some pont t runs out of nventory (e.g., for the case of a retaler, the equty cost assocated wth losng potental sales); and (4) the send-back cost B (l, a, b, q ), whch s a monetary penalty payed to the transporter for orderng product n excess (.e., more than what the customer can accommodate at delvery tme). Notce that the order quanttes and tme wndows are selected by each customer before the transporter decdes upon the dstrbuton planng. Hence, from the perspectve of the customer, the exact delvery tme l s not known when solvng ts nventory problem. Thus, the total cost for customer gven as a functon of the delvery tme l s: C (l ) = mn P (q ) + H (l, a, b, q ) + S (l, a, b, q ) + B (l, a, b, q ). (1) [a,b ] T,q 0 The orderng cost P s often gven by a functon ncludng a fxed orderng cost k plus the product between the order quantty q and a untary prce per gallon F. We wll assume that the order quantty s made so that upon arrval, the transporter delvers enough gasolne to fll the underground tanks of the gas staton. Therefore, such an order quantty depends on the ntal stock r, the demand rate d, and the expected delvery tme. Ideally, the gas staton would expect the product to be delvered at the mdpont (a + b )/ of the tme wndow, as ths would mnmze possble shortages or send-backs. Nevertheless, snce the gas staton does not know n advance the actual delvery tme l as ths s decded by the transporter after all the orders are collected the gas staton must then base ts decsons on the dea that the delvery truck wll arrve at any 8

9 tme wthn the tme wndow [a, b ]. For ths reason, from the perspectve of the gas statons, the delvery tme l U[a, b ] s a random varable unformly dstrbuted over the nterval gven by a and b. Furthermore, snce the expected delvery tme s the mdpont (a +b )/, the order quantty s therefore equal to the capacty of the underground tank mnus the amount of left at (a + b )/. In general, shortage costs are generally more expensve than the holdng costs. Thus, the customers am to select tme wndows so that the tme to empty T e, also referred to as the stock out tme (.e., the expected tme n whch the customer runs out of nventory), occurs after the wndow upper bound b. To properly model the above nventory problem, there are several cases that must be consdered dependng on the values of a, b, q, l, and T e. Partcularly, f for the gven values of [a, b ], the tme to empty T e falls ether: (1) before a, () between a and b or, (3) after b. The followng assumptons are made for the gas staton nventory model: for each demanded product, the gas staton solves an nventory model to determne [a, b ] and q ; the delvery tme l s assumed unformly dstrbuted over the nterval [a, b ]; the tme horzon s dscretzed hourly; 1 workng hours are consdered for one workng day; no backloggng of demand s allowed (.e., sales not met due to shortages are lost); the demand rate of each gas staton s assumed to be constant and known; nventory s contnuously revewed and all replenshment decsons are made at the begnnng of each tme perod; each gas staton has a known startng level r and that nformaton s not dsclosed to the suppler; once the orders are placed, those cannot be changed; the order quantty q s equal to the capacty of the underground tank mnus the amount left at the expected arrval tme; the stock out tme of each staton T e s known (snce the demand rate and the ntal stock are known, the tme to empty can be calculated n advance); the possble tme wndow choces are dscrete. All the mathematcal notaton for the nventory model are gven n Table 1. 9

10 Table 1: Mathematcal notaton for the nventory model. Notaton Q T e d l r F k h s p a b q Defnton The underground tank capacty of staton Stock out tme of staton Demand rate of staton Delvery lead tme whch s unknown by the customers as the routng schedule s generated after all the orders are known and t can be anywhere between a and b. It s assumed to follow a unform dstrbuton wth lmts a and b Intal stock level of underground tank of staton The order cost per gallon Fxed cost per delvery Holdng cost (per unt per unt tme) Shortage cost (per unt per unt tme) Send-back cost (per unt per unt tme) Earlest tme wndow of staton Latest tme wndow of staton Order quantty of gas staton When gas statons receve the orders, three scenaros may occur: Scenaro 1: the stock out tme of a staton T e occurs pror to a. Scenaro : the stock out tme of a staton T e occurs wthn tme wndow [a, b ]. Ths scenaro comprses two cases: (1) both the lead tme l and stock out tme T e occur before the mdpont (a + b )/, () both the lead tme l and stock out tme T e occur after the mdpont (a + b )/. In other words, we have the possbltes a l T e (a + b )/ b and a T e l (a + b )/ b n case 1; and a (a + b )/ l T e b and a (a + b )/ l T e b n case. Scenaro 3: the stock out tme of the staton T e occurs after b, whch comprses two cases as well: (1) the petroleum s receved before the mdpont of the tme wndow (l (a + b )/) or, () the petroluem s receved after the mdpont of the tme wndow (l (a + b )/). These three scenaros are summarzed n Table and the correspondng expected total cost calculatons are presented afterwards. Table : Summary of scenaros. Scenaro 1 Scenaro Scenaro 3 Stock out tme T e Case n scenaro < a a T e b Case 1a: a l T e (a + b )/ b Case 1b: a T e l (a + b )/ b Case a: a (a + b )/ l T e b Case b: a (a + b )/ T e l b b < T e Case 1: l (a + b )/ Case : l (a + b )/ 10

11 Scenaro 1: T e a In scenaro 1, the stock out tme of a staton T e occurs pror to a. Ths mples that there wll be a cost assocated wth a shortage of petroleum. Consequently, the send-back cost s 0 and the order quantty q s equal to the underground tank capacty Q. Fgure 3 provdes a graphcal representaton of the nventory of scenaro 1. Q Q -(4-l )d r 0 e T a b 4 l Fgure 3: The nventory model of scenaro 1. The orderng cost s gven by fxed cost plus varable cost, then k + F Q, where k s the fxed cost and F s the unt order cost per gallon. The expected daly holdng volume can be calculated from the nventory functon depcted n Fgure 3. Thus, the holdng volume before the underground tank becomes empty s (r T e )/ and the expected daly holdng volume after order arrval s Q (4 l ) [(4 l ) d ]/. Therefore, we obtan the expected daly holdng cost [ r T e + Q (4 l ) (4 l ) ] d h The total shortage cost s gven by [ (l T e ) d ] s whch s average shortage volume tmes the shortage cost. The send-back cost s 0 because the underground tank wll become empty pror to recevng the ordered petroleum and thus t can accommodate the full order. Hence, the total cost for the scenaro 1 denoted by C S1 (l ) s then: C S1 (l ) = orderng cost + holdng cost + shortage cost + send-back cost 11

12 [ r T e = (k + F Q ) + + Q (4 l ) (4 l ) ] d h + [ (l T e ) d ] s () To calculate the expected cost, we take the ntegral over a and b for l. Thus, the expected total cost of scenaro 1, E[C S1 ], s gven by: E[C S1 ] = b a C S1 1 dl b a = 1 6 [ 3( 48 + a + b )hq + 6(k + F Q + 3hr T e + d [ (178 + a + a ( 7 + b ) + ( 7 + b )b )h + s(a + a b + b 3(a + b )T e + 3(T e ) ))] (3) Scenaro : a T e b In scenaro, the stock out tme of a staton occurs wthn the tme wndow [a, b ]. It should be noted that the tme dfferences among l, T e, and (a +b )/ wthn [a, b ] play a sgnfcant role n evaluatng the expected total cost. There wll be a send-back cost and no shortage cost f a staton receves the order pror to runnng out of petroleum, whch s the stuaton gven by l < T e. Conversely, there wll be a shortage cost and no send-back cost f a staton runs out of product before recevng petroleum (l T e ). Thus, we consder these two cases n scenaro : Case 1: the petroleum s receved before the mdpont of the tme wndow (l (a + b )/) a. a l T e (a + b )/ b As depcted n Fgure 4(a), gas staton wll not run out of petroleum, but t wll ncur n a send-back cost. The expected daly holdng volume before recevng petroleum s r l (l d )/ and the expected daly holdng volume after the order arrval s Q (4 l ) [(4 l ) d ]/. Hence, the daly holdng cost s: [( r l l d ) ( + Q (4 l ) (4 l ) )] d h, Snce the tme to empty occurs before the mdpont of the tme wndow, the order quantty q s equal to the underground tank capacty Q. Also, the quantty that exceeds the tank capacty when recevng petroleum s r l d, so the send-back cost s (r l d )p. Therefore, the total cost for ths case denoted by C SC1a (l ) s: [( C SC1a (l ) = (k + F Q ) + r l l d ) ( + Q (4 l ) (4 l ) )] d h+ b. a T e + (r l d )p (4) l (a + b )/ b In contrast, a shortage cost [(l T e ) d ]/ and send-back cost 0 wll occur n scenaro case 1b, as shown n Fgure 4(b). For ths case, the expected daly holdng cost s [ r T e + (4 l )Q (4 l ) ] d h, 1

13 Q Q Q-(4-l)d Q-(4-l)d r r r-ld 0 a l e a + b T b 4 0 a e T l a + b b 4 (a) Case 1a (b) Case 1b Fgure 4: The nventory model of scenaro case 1. where (r T e )/ represents the expected daly holdng volume before runnng out of petroleum and Q (4 l ) [(4 l ) d ]/ s the volume after recevng ordered petroleum. The average shortage volume s [(l T e) d ]/ and therefore the total shortage cost s (l T e) d s. Let C SC1b (l ) be the total cost for scenaro case 1b. That s, C SC1b (l ) = (k + F Q ) + [ r T e + (4 l )Q (4 l ) d ] h + (l T e ) d s (5) Smlarly as for case 1, to calculate the expected cost, we take the ntegral over the nterval from a to T e for (4) and from T e to b for (5). The expected total cost of scenaro case 1, E[T C SC1 ], s gven by: T e E[C SC1 ] = C SC1a 1 b (l ) dl + C SC1b 1 (l ) dl a b a T e b a 1 = 6(a b ) [ a3 d h + 3a (4d h d p hq + hr ) + 6a (k + F Q + 4h( 1d + Q ) + pr ] + b [(178 + ( 7 + b )b )d h + 3( 48 + b )hq 6(k + F Q ) b d s) + 3( b hr pr + b d s)t e + 3d (p b s)(t e ) + d (h + s)(t e ) 3 ] (6) Case : the petroleum s receved after the mdpont of the tme wndow (l (a + b )/) a. a (a + b )/ l T e b As mentoned before, the order quantty of staton s set to the amount requred to fll the underground tank at the tme (a + b )/. In case a, snce the underground tank wll not become empty before the order expected arrval tme ((a +b )/ < T e ), 13

14 the order quantty wll not be equal to the full capacty of the tank, as depcted n Fgure 5(a). For ths case, the order quantty s gven by: [ ( ) ] a + b Q r d, (7) where r [(a + b )/]d represents the quantty left n the tank at the expected arrval tme. The expected daly holdng volumes before and after the order arrval are r l (l d )/ and (4 l )[Q [l (a + b )/]d [(4 l ) d ]/], respectvely. The expected daly holdng costs are: [( r l l d ) + (4 l ) ( Q ( l a ) ) + b d (4 l ) ] d h. Both the shortage cost and send-back cost are 0 snce the statons wll receve the petroleum before they run out and the ordered quantty wll not exceed the tank capacty because the actual order arrval tme occurs after the expected arrval tme (a + b )/. Let C SCa C SCa (l ) = + [ k + F (l ) denote the total cost for scenaro case a. That s, [ ( ) ]]] a + b r d [ Q [( r l l d ) + (4 l ) ( ( Q l a ) ) + b d (4 l ) ] d h (8) a + b Q-(l- )d a + b Q-(r- d) a + b Q-(l- )d-(4-l)d r a + b Q-(r- d)-(4-l)d r a + b r-( )d r-ld 0 a a + b l e T b 4 0 a a + b e T l b 4 (a) Case a (b) Case b Fgure 5: The nventory model of scenaro case. b. a (a + b )/ T e l b The orderng costs n case b are also gven by expresson (7), as T e occurs after the mdpont (a + b )/. Addtonally, the holdng cost s gven by: [ r T e [ ( ( ) )] a + b + Q r d (4 l ) (4 l ) ] d h, where (r T e )/ represents the expected daly holdng volume before runnng out of petroleum and 14

15 [ ( Q r ( a + b ) d )] (4 l ) (4 l ) d s the volume after recevng the petroleum. The average shortage volume s [(l T e) d ]/ and the total shortage cost s then gven by (l T e) d s Also, the send-back cost s 0 snce the tanks are always able to accept the full order n ths case. Thus, the total cost for scenaro case b s then: [ [ [ ( ) ]]] C SCb a + b (l ) = k + F Q r d [ r T e [ ( ( ) )] a + b + + Q r d (4 l ) (4 l ) ] d h + (l T e) d s (9) Furthermore, takng ntegral over the range a to T e for case a and over T e to b for case b after combnng (8) and (9), the expected total cost of scenaro case, ], s: E[C SC T e E[C SC 1 ] = T C SCa (l ) dl + a b a b T e 1 T C SCb (l ) dl b a 1 = + 1(a b ) [ 3a3 d h + 3a [d (F + 48h b h) + h( Q + r )] + 3a [( b )d h + 4(k + F Q + 4hQ F r )] + b [1(88d h k (F + 4h)(Q r )) 6b (d (F + 48h) + h( Q + r )) + b d (5h s)] 6((48 + b )hr b d s)t e + 6(4d h + hr b d s)(t e ) + d ( h + s)(t e ) 3 ]. (10) Scenaro 3: b T e The orderng costs n scenaro 3 are also gven by expresson (7), as T e mdpont (a + b )/. occurs after the Case 1: the petroleum s receved before the mdpont of the tme wndow (l (a +b )/) Fgure 6(a) depcts the nventory level for ths case, the shortage cost s 0 and the send-back cost s gven by: [( ) ] a + b d l d p, where [(a +b )/]d l d s the dfference between order quantty and the actual receved quantty. The expected holdng volumes before and after order arrval are r l l d 15

16 and Q (4 l ) (4 l ) d. Thus, the holdng cost s [( r l l d ) ( + Q (4 l ) (4 l ) )] d h. We denote the total cost for the scenaro 3 case 1 as C S3C1 ( ) ]]] a + b d (l ) = [ k + F [( + r l l d ) [( a + b + C S3C1 [ [ Q r ( + )] h (l ) and s thus gven by Q (4 l ) (4 l ) d ) ] d l d p. (11) Q a + b Q-(l- )d Q-(4-l)d a + b Q-(l- )d-(4-l)d r r-ld a + b r- d r a + b r- d r-ld 0 a l a + b b (a) Case 1 e T 4 0 a + b e a l b T (b) Case 4 Fgure 6: The nventory model of scenaro 3. Case : the petroleum s receved after the mdpont of the tme wndow (l (a +b )/)) Fgure 6(b) depcts the nventory level for ths case, there wll be no shortage nor sendback cost and the holdng cost s qute smlar to the holdng cost n case 1 except the expected holdng volume after order arrval s [ Q ( l ( a + b ) d )] (4 l ) (4 l ) d Thus, the total cost for the scenaro 3 case, denoted as C S3C ( ) ]]] a + b d C S3C (l ) = + [ k + F [ [ Q r [( r l l d ) [ + Q ( l ( a + b (l ), s gven by ) )] d (4 l ) (4 l ) ] d h (1) 16

17 We then take an expectaton over the lmt for l from a to (a + b )/ for case 1 and from (a + b )/ to b for case and the expected total cost of scenaro 3, E[T C S3 ], s gven by a +b E[C S3 1 ] = T C S3C1 (l ) dl + a b a b a +b 1 T C S3C (l ) dl b a = 1 48 [ 17a d h 11b d h 48(88d h k (F + 4h)Q + F r ) + 6b (d (4F + 7h + p) + 4h( Q + r )) + a (d (4F 0( 36 + b )h 6p) + 4h( Q + r ))]. (13) The objectve of the nventory model s to select the optmal tme wndow [a, b ] for each gas staton among all possble dscrete delvery tmes specfed by transporters mnmzng the total nventory costs. Based on the relatonshps among a, b, l, and T e, we can dentfy whch scenaro wll be appled and the expected total costs can be further calculated. The optmal tme wndows and order quanttes for each gas staton can be found by solvng Equaton (14). (q, [a, b j ]) arg mn{e[c (l )] = E[C S1 (l )] + E[C SC1 (l )] + E[C SC (l )] + E[C S3 (l )]}. (14) Furthermore, notce that for a gven realzaton of l, t s possble to calculate the total cost ncurred by the gas statons by selectng the scenaro that corresponds to the value f l and evaluate the resultng value for C (l ), for all N. We now dscuss the dstrbuton part of the problem. 5 The dstrbuton problem of the transporter The proposed ntegrated routng model seeks to determne dstrbuton plan and the strategc decsons of the transporter by takng nto account the customer decsons gven by the nventory model. 5.1 Route generator As mentoned before, we attempt to generate all the possble delvery routes for the gas statons that placed orders durng each gven day, based upon the results of the nventory models. For the cases where few gas statons are served per route, we generate all possble route permutatons vstng one, two and three statons for a total of O( N 3 ) routes. As for the cases where the route vsts ether four or fve gas statons, snce we solve both the truck loadng model and the route schedulng model for all of those routes every day of the plannng horzon, we do not attempt to generate all the total possbltes, as t would requre solvng two mxed-nteger models for a total of O( N 5 ) routes per day, whch would take longer than the desred computatonal tme lmt. Notce that for each day of the plannng horzon all routes must be tested. In other words, a route that was consdered nfeasble for the frst day, may be feasble for the demands and tme wndows of the second day. Therefore, nstead of generatng all the possble permutatons of four and fve gas statons, we randomly generate a smaller subset of such routes to reduce the total computatonal load. Snce, some good routes may not be generated n the random route generaton, we test dfferent sets of randomly generated routes to see the mpact of the random generaton. 17

18 Let R be the set of routes generated. For each of the routes n R, we solve the truck loadng model that checks f the order quanttes of the gven route can be loaded nto the truck compartments. Furthermore, f the route satsfes the truck loadng constrants, we then generate the schedule for the route based on the gven tme wndows of the customers of each route. 5. Truck loadng model In addton to satsfyng the constrants gven by the tme wndows, a feasble route must also satsfy the truck compartment capacty constrants. As the trucks used to delver have multple compartments wth dfferent capactes, the demand feasblty of the routes s checked n the truck loadng model. If the demand of a route cannot be loaded nto any truck, ths route wll be elmnated from the set of canddate routes. Therefore, for every possble route r R, we solve the followng tuck loadng model to test the loadng feasblty of the route. The notaton and the decson varables are descrbed n Table 3. Notaton Descrpton Table 3: Mathematcal notaton for the truck loadng model. N r The set of gas statons served by the route r beng tested C The set of compartments for the gven truck q The delvery quantty to staton Q c The capacty of compartment c y c A bnary varable equal to 1 f demand of gas staton s assgned to compartment c, and 0 otherwse Then the truck loadng model s gven by: q Q c y c N r, (15) c C y c 1 c C, (16) p P y c {0, 1} N r, c C. (17) Constrants (15) enforce that the delvery quantty of staton cannot exceed the sum of compartment capactes of the loaded truck assgned to such a gas staton. Constrants (16) ensure only one demand can be loaded nto each compartment. Constrants (17) defne the decson varables. Notce that the loadng model s n fact a satsfablty problem, as any feasble soluton can be used for loadng the truck compartments. If route r does not satsfy the truck loadng model, t s removed from set R. Addtonally, n ths stage we compute the proft of the route, whch s calculated by the revenue perceved by delverng the petroleum mnus the travel costs of the route. Snce the gas staton vst sequence s gven by the route, and the order quanttes of each customer are calculated a prory by the nventory models, f there s a feasble loadng dstrbuton of route r gven by the soluton of model (15)-(17) we then calculate the correspondng proft of r and the transportaton cost f the route and use t as an nput for the truck routng model. 18

19 5.3 Route schedulng model The route schedulng model ams to fnd the truck schedule of each canddate route r R, as well as a set of penaltes for havng late delveres. Gven the tme wndows obtaned by the nventory models and the startng hour of the delvery shft, we check the feasblty of the tme wndow constrants for each gas staton of canddate route r R. Snce the number of trucks s lmted and the tme wndow requrements for the gas statons of route r could be too close or potentally overlap, the delvery tme wndows are not always satsfed for all customers. For that reason, n the route schedulng model we assgn penaltes to the routes for whch the statons receve petroleum at undesred tmes. The penaltes consst of addtonal costs for truck arrval tmes ether before a or after b, and are proportonal to the truck arrval tme at statons outsde the specfed tme wndows, as shown n (18). Notce that we also add addtonal penaltes for devatng from the mdpont of the tme wndows. Ths s ntended to ensure that, f possble, the tucks should try to arrve at the mdpont of the tme wndows to avod the possblty for the customers to ncur n shortage or send-back costs. Although, when consderng the fnal value for the penaltes of the route, only the penaltes for devatng from the tme wndow [a, b ] are consdered. The routes wth no devatons (postve or negatve) are routes whose schedule guarantees that the truck arrves at every staton at the desred tme (the mdpont of tme wndows). The objectve of ths model s to fnd the optmal truck arrval tme at each staton n the route so that the penaltes are mnmzed. The defnton of the notaton used for ths model s gven n Table 4. The route schedulng model s formulated as follows: mn (u + v + γm + δn ) (18) N r s.t. x (a + b )/ = v u N r, (19) x 1 h + t 0, (0) x x 1 + s t 1 + t 1 N r \ {1}, (1) x + m a N r, () x n b N r, (3) a x m N r, (4) x R +, u R +, v R +, m R +, n R + N r. (5) 19

20 Table 4: Mathematcal notaton and decson varables for the route schedulng model. Notaton Descrpton N r Subset of statons n gven route r h The startng hour of the shft m The maxmum delvery tme n one day s The servce tme of staton t The trucks travel tme from staton to staton + 1, N r a The earlest tme wndow of staton b The latest tme wndow of staton γ The penalty that truck arrval before a δ The penalty that truck arrval after b x Truck arrval tme at staton u Negatve truck arrval tme devaton from the mdpont (a + b )/ for staton v Postve truck arrval tme devaton from the mdpont (a + b )/ for staton m Amount of tme the truck arrves before a for staton Amount of tme the truck arrves after b for staton n The objectve functon (18) mnmzes penaltes assocated wth the tme devatons from the expected arrval tmes of route r. Constrants (19) defne the postve and negatve devatons of the delvery tme from the mdpont of the tme wndows for each gas staton. Constrants (0) ensure that the truck arrval tme at the frst staton of each route occurs after the startng hour of the shft plus the travel tme from termnal to the frst staton. Constrants (1) enforce that the truck arrval tme at all the gas statons n the route (except for the frst staton) s greater than the arrval tme plus the servce tme of the precedng staton, and travel tme between the statons. Constrants ()-(3) defne the truck arrval tme between the lmts of the tme wndow [a, b ]. Constrants (4) requre that the truck arrval tme at staton les wthn the earlest tme wndow a and the maxmum delvery tme n one day m. Fnally, constrants (5) defne the decson varables. Once the truck arrval tme at each staton of the route s known, one can compute the penalty for delverng petroleum to the gas statons n route r as: N r (γm + δn ). (6) In addton to the penaltes, the truck schedule can be recovered from the value of the x s varables. 5.4 ɛ-constrant method for the route schedulng problem Typcally, n the presence of tme wndows that overlap, or when some tme wndows are too narrow the tme wndow requrements are dffcult to satsfy. One of the possble stuatons that can occur s that a small set of the gas statons gets contnuously penalzed wth late delveres several days ov the tme horzon, whch may result n the potental loss of such customers. To avod ths, we attempt to balance the late delveres over the tme horzon by applyng the ɛ-constrant method for the multple tme perods. For tme perod t, we frst mnmze the maxmum tme wndow volaton of any gas staton by transformng the model as follows: 0

21 mn z (7) s.t. z u + v + γm + δn + p t cum N r, (8) (x, u, v, m, n) Ω. (9) Here p t cum represents the cumulatve penalty of staton n route r for day t whch s gven by t 1 p t cum = (γm τ + δn τ ) (30) τ=0 and Ω represents the constrant set gven by (19)-(5). Also, varables m τ and n τ are assumed to be the optmal devatons of the schedulng problems of day τ. The soluton found n the formulaton above s denoted as z. We then mnmze the summaton of volatons by replacng z by the optmal soluton z n the formulaton, whch results n the followng model. mn (u + v + γm + δn ) (31) N r 5.5 Truck routng model s.t. z u + v + γm + δn + p t cum N r, (3) (x, u, v, m, n) Ω. (33) Fnally, the objectve of the truck routng model s to assgn the optmal routes to the truck that wll be used to fulfll the dstrbuton plan. The trucks are allowed to make multple delveres provded that the delvery schedules of the routes do not overlap and the trucks return at the termnal by the allowable tme lmt. The notaton and decson varable are defned n Table 5. Table 5: Mathematcal notaton and decson varable for the truck routng model. Notaton K R k N r a r ρ r q r x rk Descrpton The truck set The possble route set of that can be served by truck k, k K The statons served by route r, r R k A bnary parameter equal to 1 f staton s served by route r, r R k, k K The proft of route r, r R The penalty of route r f route r s a late route, r R A bnary varable that takes the value of 1 f route r s operated by truck k, r R k, k K Then the routng model s gven by: max s.t. (ρ r q r )x rk (34) r R k k K r R k k K a r x rk = 1 N, (35) 1

22 r N k x rk 1 t T, k K, (36) x rk {0, 1} r R k, k K. (37) For ths formulaton, the objectve functon (34) maxmzes the total proft, whch s dfference between the proft and penalty of the routes. Constrants (35) state that each staton s vsted exactly once. Constrants (36) ensure that the delvery tmes of the selected routes cannot overlap for each dscrete tme perod t. We consder T as 4 hours here. For example, the tme slot between and 3 can only be occuped by one route for each truck assurng that the selected routes wll not overlap n the tme slot from to 3. Constrants (37) defne the decson varables. 6 Computatonal results The proposed soluton framework was coded n Java 8 wth the API of CPLEX 1.6. All the experments were performed on an computer wth an Intel(R) Xeon(R) E5645 processor and 3.0 GB of RAM. We frst present the procedure for generatng the test nstances n subsecton 6.1; The performance measures of the soluton framework are presented n subsecton 6.; and then, the mpact of several parameters on the generated dstrbuton plans s studed thereafter. 6.1 Test problems In order to evaluate the proposed petroleum dstrbuton framework, we use a test bed of 15 randomly generated nstances wth 50 customers adapted from [6], after ncludng the addtonal components ntroduced n ths study. Among all the possble tank confguratons provded n [6], we use as a base model a medum sze tank confguraton that we latter vary to analyze ts mpact on the soluton qualty (see Secton 6.3). We converted the petroleum quantty unts from lters to gallons and the dstance unts from klometers to mles as per use n the typcal context of U.S. scenaros. Furthermore, we ntroduced addtonal nformaton requred for the nstances to be used n the context of our approach. The parameters used regardng the nventory costs and penaltes can be found n Table 6. We consder a fleet of 4 trucks whose compartment compostons are gven n Table 7. As for the demands, we generate demand rates for the test nstances so that each staton orders every one or two days. All ths to see the patterns that emerge regardng the tme wndows and the mpact of balancng late delveres. We further study the mpact that the tank capacty of the gas statons and the number of randomly generated routes servng more than three gas statons have on the solutons; see Sectons 6.3 and 6.4, respectvely. We ntroduced a correlaton between the demand rates and the tank capactes of the gas staton to reflect the fact that gas statons wth larger tanks are expected to have hgher demand rates. The tank capactes we used are gven n Table 8, labeled C1 to C5. Furthermore, we also vary the total number of randomly generated routes wth more than three customers. We solve the replenshment problem for fve scenaros labeled R1 to R5 n whch we generate 100, 1000, 10000, 0000, and of such routes, respectvely. 6. Performance of the soluton framework We tested the performance of the proposed approach over the 15 nstances descrbed n Secton 6.1. We performed 50 dfferent runs for each nstance, varyng the tank capactes, the demand rates,