Classification, Models and Exact Algorithms for Multi-Compartment Delivery Problems

Transcription

1 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems Leandro C. Coelho Glbert Laporte June 2013 CIRRELT Bureaux de Montréal : Bureaux de Québec : Unversté de Montréal Unversté Laval C.P. 6128, succ. Centre-vlle 2325, de la Terrasse, bureau 2642 Montréal (Québec) Québec (Québec) Canada H3C 3J7 Canada G1V 0A6 Téléphone : Téléphone : Télécope : Télécope :

2 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems Leandro C. Coelho 1,2,*, Glbert Laporte 1,3 1 Interunversty Research Centre on Enterprse Networks, Logstcs and Transportaton (CIRRELT) 2 Department of Operatons and Decson Systems, Unversté Laval, 2325, de la Terrasse, Québec, Canada G1V 0A6 3 Department of Management Scences, HEC Montréal, 3000, Côte-Sante-Catherne, Montréal, Canada H3T 2A7 Abstract. The dstrbuton of products usng compartmentalzed vehcles nvolves many decsons such as the allocaton of products to vehcle compartments, vehcle routng and nventory control. These decsons often span several perods, yeldng a dffcult optmzaton problem. In ths paper we defne and compare four man categores of the Mult-Compartment Delvery Problem (MCDP). We propose two mxed-nteger lnear programmng formulatons for each case, as well as specalzed models for partcular versons of the problem. Known and new vald nequaltes are ntroduced n all models. We then descrbe a branch-and-cut algorthm applcable to all varants of the MCDP. We have performed extensve computatonal experments on sngle-perod and mult-perod cases of the problem. The largest nstances that could be solved exactly for these two cases contan 50 and 20 customers, respectvely. Keywords. Mult-compartment delvery, vehcle-routng, nventory-routng, mult-products, mult-vehcles, branch-and-cut algorthm, fuel dstrbuton. Acknowledgements. Ths work was partly supported by the Natural Scences and Engneerng Research Councl of Canada (NSERC) under grant Ths support s gratefully acknowledged. We also thank the Réseau québécos de calcul de haute performance (RQCHP) for provdng computng facltes. Results and vews expressed n ths publcaton are the sole responsblty of the authors and do not necessarly reflect those of CIRRELT. Les résultats et opnons contenus dans cette publcaton ne reflètent pas nécessarement la poston du CIRRELT et n'engagent pas sa responsablté. * Correspondng author: Leandro.Coelho@crrelt.ca Dépôt légal Bblothèque et Archves natonales du Québec Bblothèque et Archves Canada, 2013 Copyrght Coelho, Laporte and CIRRELT, 2013

3 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems 1 Introducton Ths paper s concerned wth a mult-perod routng problem n whch several products must be delvered by compartmentalzed vehcles to customers equpped wth several tanks. No two products can be combned wthn the same compartment or wthn the same tank. The most common example arses n the dstrbuton of petroleum products by tanker trucks to underground tanks located n gas statons [8, 9, 10, 11, 27, 28]. Such problems are also encountered n the martme transportaton of bulk products by shps whose hull s dvded nto compartments [3, 14, 15, 20], n the collecton of garbage and recyclable products [24], and n lvestock transportaton [25]. We analyze the problem n the general context of nventory-routng where a centralzed agent s responsble for the dstrbuton of one or several products over several perods, and for controllng nventory levels the customer locatons. For recent surveys on the nventory-routng problem (IRP), see Andersson et al. [2] and Coelho et al. [7]. For recent branch-and-cut algorthms applcable to IRPs, see Coelho and Laporte [5, 6]. In the fuel dstrbuton problem, whch s the central applcaton of our problem, vehcles are often not equpped wth debt meters, whch mples that whenever a delvery s made, the full content of the compartment must be empted. In other words, the load of a compartment cannot be splt between dfferent tanks. Ths s the assumpton made n the fuel dstrbuton papers of Cornller et al. [8, 9, 10, 11], Popovć et al. [27] and of Vdovć et al. [28]. However, n the general case, compartments can be equpped wth debt meters and the quantty delvered to tanks then becomes a contnuous decson varable. The ablty to splt the content of a compartment between several delveres yelds a frst classfcaton of the problem. Lkewse, a customer may or may not allow dfferent vehcles to fll the same tank n a gven perod. If a tank may receve delveres from dfferent vehcles, t s sad to be splt. Ths dstncton yelds the second class of classfcaton of the problem. We show n Table 1 the four cases yelded by ths classfcaton. Note that only the upper-left case of ths table has been treated n the lterature. The other three CIRRELT

4 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems cases are new and are modeled and solved for the frst tme n ths paper. Table 1: Four cases of MCDPs yelded by the classfcaton proposed n ths paper Cases Compartments Tanks Splt Unsplt Splt Splt-Splt Splt-Unsplt Unsplt Unsplt-Splt Unsplt-Unsplt Splt compartments and splt tanks yeld an extra layer of dffculty to the problem. Even the sngle-perod verson of the problem s much more complcated than the classcal vehcle routng problem [23] because t stll contans several products, several compartments and multple tanks, whch sgnfcantly ncreases the number of bnary varables n the model. Lkewse, the sngle-product verson of the MCDP s also more complcated than the IRP [7] due to the presence of multple compartments and multple tanks. In ths paper we develop mathematcal programmng formulatons whch are adapted to handle all four combnatons of splt and unsplt compartments and tanks. We also propose an exact branch-and-cut algorthm applcable to all varants of the problem. It extends the classcal vehcle routng formulatons [23] n whch a relaxed problem s frst solved and subtour elmnaton constrants are added dynamcally as they are found to be volated. Two models are also presented for the cases where the number of customers per vehcle route s lmted to two or three. Our man goals are to formally ntroduce, model and solve four classes of the MCDP to optmalty wthn a unfed framework. A generc model s frst presented and then modfed to account for the varants of the basc case. Known and new vald nequaltes are ntroduced n all models. A byproduct of ths research s the ntroducton of a testbed of nstances whch we have used for our experments and are made avalable to the research communty. The proposed testbed s desgned to cover a large set of combnatons regardng the number of customers, products, vehcles, and compartments, as well as the 2 CIRRELT

5 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems length of the plannng horzon, rangng from relatvely easy nstances to very challengng ones. The remander of the paper s organzed as follows. In Secton 2 we provde a formal descrpton of the problem. In Secton 3 we propose mxed-nteger lnear programmng formulatons coverng all varants of the problem. A branch-and-cut algorthm applcable to all cases s descrbed n Secton 4. Extensve computatonal experments are presented n Secton 5, followed by our conclusons n Secton 6. 2 Formal descrpton of the problem We now formally ntroduce the MCDP. The problem s defned on an undrected graph G = (V, E), where V = {0,..., n} s the vertex set and E = {(, j) :, j V, < j} s the edge set. Vertex 0 represents the suppler and the vertces of V = V \{0} represent customers. The suppler dstrbutes a set of M = {1,..., M} types of products to the customer compartments. Customers ncur unt nventory holdng costs h m per perod ( V, m M). The length of the plannng horzon s T. We assume the suppler holds enough nventory to meet all the demand durng the plannng horzon and that nventores are not allowed to be negatve,.e., backloggng s not allowed. The varables I mt defne the nventory level of product m at the end of perod t at customer V. At the begnnng of the plannng horzon the decson maker knows the current nventory level of all customers (I m0 for V, m M), and has full knowledge of the demand d mt of product m of each customer for each tme perod t. There s a set K = {1,..., K} of vehcles avalable. Each vehcle k s made up of a set L = {1,..., L} of compartments l of capacty Q lk, and tank m of customer has a capacty C m. Each vehcle can perform one route per tme perod. A routng cost c j s assocated wth edge (, j) E. The objectve of the MCDP s to mnmze the total routng and nventory holdng cost whle satsfyng the demand for every product for each customer. The replenshment plan s subject to the followng constrants: CIRRELT

6 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems the vehcle compartment capactes cannot be exceeded; the nventory level of each product at each customer can never exceed the maxmum capacty of the tanks; the suppler s vehcles can perform at most one route per tme perod, each startng and endng at the suppler. A soluton to the problem determnes whch customers to serve n each tme perod, whch vehcles and compartments to use for each product, how much of each product to delver to each vsted customer, and whch vehcle routes to use. 3 Mathematcal models We frst propose a mxed-nteger lnear programmng model for the most general case of the MCDP,.e., the verson wth splt compartments and splt tanks. We then present small modfcatons needed to account for the remanng three dfferent combnatons of these crtera. We also present two varants applcable to cases where the number of customers per vehcle route s lmted. The model works wth routng varables x kt j, (, j) E, equal to the number of tmes edge (, j) s used on the route of vehcle k n perod t. It also uses bnary varables y kt equal to one f and only f node s vsted by vehcle k n perod t, w mlkt equal to one f and only f product m s loaded n compartment l of vehcle k n perod t, and z mlkt equal to one f and only f customer receves a delvery of product m from compartment l of vehcle k n perod t. We denote by q mlkt the quantty of product m delvered to customer usng compartment l of vehcle k n perod perod t. As prevously stated, let I mt nventory level of product m at vertex V at the end of perod t T. denote the 4 CIRRELT

7 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems 3.1 Splt compartments and splt tanks In ths secton we provde two models for the MCDP wth splt compartments and splt tanks. The model presented n Secton makes an explct assgnment of products to compartments, whereas the model presented n Secton makes ths assgnment mplctly Explct compartment assgnment model for the MCDP wth splt compartments and splt tanks We frst present a formulaton for the most general verson of the problem: subject to mnmze V m M t T h m I mt + c j x kt j, (1) (,j) E k K t T I mt = I m,t 1 + l L k K q mlkt d mt V m M t T (2) 0 I mt C m V m M t T (3) q mlkt C m I m,t 1 V m M t T (4) l L k K l L q mlkt q mlkt C m y kt V m M k K t T (5) Q lk z mlkt V m M l L k K t T (6) q mlkt Q lk m M l L k K t T (7) V z mlkt y kt V m M l L k K t T (8) z mlkt w mlkt V m M l L k K t T (9) w mlkt 1 l L k K t T (10) m M CIRRELT

8 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems j V,<j Z j Z,<j x kt j + j V,j< x kt j = 2y kt V k K t T (11) x kt j y kt yn kt Z V n Z k K t T (12) Z q mlkt 0 V m M l L k K t T (13) x kt 0 {0, 1, 2} V k K t T (14) x kt j, w mlkt, z mlkt j {0, 1} V j V, < j m M l L k K t T (15) y kt {0, 1} V k K t T. (16) The objectve functon (1) mnmzes the total nventory and routng costs. Constrants (2) defne the nventory at the customers, whle constrants (3) and (4) ensure that the nventory level of product m at customer s non-negatve and does not exceed the maxmum capacty C m. Constrants (5) allow delveres to tank m of customer only f a vehcle vsts t. Lkewse, constrants (6) allow delveres of product m to customer usng compartment l of vehcle k only f the compartment s assgned to that customer. Constrants (7) ensure the vehcle compartment capactes are respected, whle constrants (8) and (9) lnk varables y kt, w mlkt and z mlkt. Specfcally they allow delveres from any compartments only f the vehcle vsts the customer. Constrants (10) lmt the use of each compartment to a sngle type of product. Constrants (11) and (12) are degree constrants and subtour elmnaton constrants, respectvely. Constrants (13) (16) enforce ntegralty and non-negatvty condtons on the varables. We propose fve classes of nequaltes to strengthen the formulaton just presented. 1. Logcal nequaltes Fschett et al. [17] and Gendreau et al. [19] have proposed the followng logcal cuts n order to lnk routng varables x wth vstng varables y n a stronger fashon: x kt 0 2y kt V k K t T (17) x kt j y kt, j V k K t T (18) 6 CIRRELT

9 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems y kt y0 kt V k K t T. (19) Constrants (17) and (18) are referred to as logcal nequaltes. They enforce the condton that f the suppler s the successor of a customer on the route of vehcle k n perod t,.e., x kt 0 = 1 or 2, then must be vsted by vehcle k,.e., y kt = 1. A smlar reasonng apples to customer j n nequaltes (18). Constrants (19) nclude the suppler n the route of vehcle k f any customer s vsted by that vehcle n that perod. 2. Extended logcal nequaltes We propose the followng sets of constrants, whch we call extended logcal nequaltes because they further enforce logcal relatonshps between nteger varables of the problem: y kt m M l L m M l L z mlkt y kt V k K t T (20) w mlkt y kt V k K t T (21) z mlkt V m M l L k K t T (22) w mlkt z mlkt V m M l L k K t T (23) x kt 0 k K t T (24) V V m M l L m M l L z mlkt w mlkt V x kt 0 k K t T. (25) Inequaltes (20) ensure that f a vehcle k s assgned to a customer n perod t,.e., y kt to customer. = 1, then a product from some compartment of vehcle k must be delvered Inequaltes (21) apply a smlar reasonng to the assgnment of a product to a compartment. Inequaltes (22) and (23) tghten the relatonshps between customers, products and compartments. Specfcally, f a customer s set to receve the delvery of a product from a gven compartment,.e., z mlkt = 1, then CIRRELT

10 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems vehcle k must be assgned to customer n perod t (.e., y kt = 1 n nequalty (22)) and product m must be assgned to compartment l n vehcle k n perod t (.e., w mlkt = 1 n nequalty (23)). Fnally, nequaltes (24) and (25) enforce the delvery of a product to a customer and the assgnment of a product to a compartment whenever a delvery route exsts n perod t usng vehcle k. 3. Outgong degree nequaltes at the depot We adapt the outgong degree of the depot constrants of the splt delvery VRP [12] to the MCDP: V x kt 0 V q mlkt /Q lk m M l L k K t T. (26) 4. Symmetry breakng nequaltes We also tghten the formulaton by mposng the followng vehcle symmetry breakng constrants vald for the case where the vehcle fleet s homogeneous: y kt j< y kt 0 y k 1,t 0 k K\{1} t T (27) y k 1,t j V k K\{1} t T. (28) Constrants (27) ensure that vehcle k cannot leave the depot f vehcle k 1 s not used. Ths symmetry breakng rule s then extended to the customer vertces through constrants (28) whch state that f customer s assgned to vehcle k n perod t, then vehcle k 1 must serve a customer wth an ndex lower than n the same perod. These constrants are nspred from those proposed by Fschett et al. [16] for the capactated VRP and by Albareda-Sambola et al. [1] for a plant locaton problem. They have also been used n an IRP settng by Coelho and Laporte [4, 6]. Fnally, we also propose breakng the symmetres nduced by vehcle compartments and product allocatons wth the followng constrants whch are vald f the vehcle fleet and compartments are homogeneous: 8 CIRRELT

11 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems z hlkt z hlkt j V z hlkt h h m M z m,l 1,kt j V h M l L\{1} k K t T (29) w hlkt m M w m,l 1,kt h M l L\{1} k K t T (30) j V,j m M z m,l 1,kt j j V h M l L\{1} k K t T (31) z h,l 1,kt +1 z m,l 1,kt m M V h M l L\{1} k K t T. (32) Constrants (29) and (30) allow delveres usng compartment l only f compartment l 1 s already used. Constrants (31) ensure that lower-ndex compartments are assgned to lower-ndex customers. They are smlar to constrants (27) and (28). Constrants (32) break symmetry n terms of product allocaton to compartments to the same customer, ensurng that lower-ndex products are assgned to lower-ndex compartments for the same customer. 5. Demand-based nequaltes We adapt to the MCDP addtonal cuts derved from the nstance data. These were frst proposed by Coelho and Laporte [6] for the IRP. If the total demand of customer from perod t 1 to perod t 2 s at least equal to the maxmum possble nventory held, then a lower bound on the number of vsts to ths customer n the nterval [t 1, t 2 ] s obtaned by dvdng the quantty needed to satsfy future demands by the customer nventory capacty, and roudng up, whch yelds constrants (33): t 2 y kt k K t=t 1 t 2 d mt t=t 1 C C V m M t 1, t 2 T, t 2 t 1. (33) Snce these constrants are non-lnear, one can use the weaker form (34): CIRRELT

12 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems t 2 y kt k K t=t 1 t 2 d mt I mt 1 t=t 1 C V m M t 1, t 2 T, t 2 t 1. (34) A dfferent verson of the same nequaltes can be wrtten as follows. It s related to whether the nventory held at each perod s suffcent to fulfll future demands. In partcular, f the nventory held n perod t 1 by customer s suffcent to fulfll ts demand for perods [t 1, t 2 ], then no vst s needed for ths customer,.e., f I mt 1 t 2 d mt, then t 2 y kt 0. On the other hand, f the nventory s not t=t 1 k K t=t 1 suffcent to fulfll future demands, then at least one vst must take place n the nterval [t 1, t 2 ]. Ths can be enforced through the followng set of vald nequaltes: t 2 y kt k K t=t 1 t 2 d mt I mt 1 t=t 1 t 2 d mt t=t 1 V m M t 1, t 2 T, t 2 t 1. (35) Implct compartment assgnment model for the MCDP wth splt compartments and splt tanks When the vehcle compartments are homogeneous wthn the same vehcle, the assgnment of product types to compartments becomes rrelevant, and a more compact model can be derved. capacty s respected. Ths model gnores ths assgnment, whle ensurng that the total vehcle Ths formulaton contans no assgnment varables w mlkt. Bnary varables z mlkt are redefned as z mkt and are equal to one f and only f product m s delvered to customer n the route of vehcle k n perod t. The contnuous quantty varables q mlkt q mkt are changed to and represent the quantty of product m delvered to customer usng vehcle k n perod t. Moreover, we redefne the parameter Q k as the capacty of each compartment of vehcle k. Ths formulaton uses upper bounds q mkt on the sum of the q mkt varables, and nteger varables v mkt equal to the number of compartments needed to perform delvery 10 CIRRELT

13 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems q mkt. Lettng S k be the number of compartments n vehcle k, the mplct model can then be formulated as follows: subject to mnmze V m M t T h m I mt + c j x kt j, (36) (,j) E k K t T I mt = I m,t 1 + k K q mkt d mt m M V t T (37) q mkt j V,<j Z j Z,<j 0 I mt C m V m M t T (38) z mkt S k Q k V m M k K t T (39) q mkt q mkt m M k K t T (40) V q mkt S k Q k k K t T (41) z mkt m M q mkt = v mkt Q k m M k K t T (42) y kt V m M k K t T (43) x kt j = 2y kt V k K t T (44) x kt j + j V,j< x kt j y kt yn kt Z V n Z k K t T (45) Z q mkt 0 V m M k K t T (46) x kt 0 {0, 1, 2} V k K t T (47) x kt j, y kt, z mkt j {0, 1} V j V, < j m M k K t T. (48) The objectve functon (36) mnmzes the total nventory and routng costs. Constrants (37) defne the nventory at the customers, whle constrants (38) ensure that the nventory level of product m at customer s non-negatve and does not exceed the maxmum capacty C m. Constrants (39) allow delveres only to those customers assgned to receve CIRRELT

14 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems such delveres. Constrants (40) (42) mean that for each product m, vehcle k and perod t, an nteger number of compartments s used to perform the delveres, whle respectng the sze of the compartments and the total number of compartments avalable n the vehcle. Constrants (43) lnk the vstng varables y wth the delvery assgnment varables z. Constrants (44) and (45) are degree constrants and subtour elmnaton constrants, respectvely, whle constrants (46) (48) enforce ntegralty and non-negatvty condtons on the varables. Inequaltes (17) (19), (27) and (28) stll hold for the mplct formulaton. In addton, by makng the approprate changes n the varable defntons, nequaltes (20), (22) and (24) also reman vald. Because of the way that new varables z mkt are defned, t s possble to derve a new class of vald nequaltes for the MCDP, called the double splt delveres nequaltes, descrbed by (49). These vald nequaltes avod splttng the delvery of the same product type over two customers usng the same two vehcles (see Fgure 1). They help break symmetry when both customers have to be vsted by the two vehcles to delver dfferent products, and they avod splttng the delveres of the same products. It s possble to avod double splt delveres, as llustrated n Fgure 1. vehcle k j vehcle k Fgure 1: Example of double splt delveres for customers and j z mkt + z mkt j + z mk t + z mk t j 2, j V, j m M k, k K, k k t T. (49) 12 CIRRELT

15 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems 3.2 Splt compartments and unsplt tanks In the MCDP wth splt compartments and unsplt tanks, compartments are equpped wth debt meters so they can delver less than full compartment loads. However, tanks cannot receve two vsts n the same perod. In ths secton we develop two models for ths verson of the problem. The explct compartment assgnment for the MCDP wth splt compartments and unsplt tanks s presented n Secton and the mplct verson of ths model n Secton Explct compartment assgnment model for the MCDP wth splt compartments and unsplt tanks In order to prevent tanks from recevng vsts from more than one vehcle per perod, we need a new bnary varable u mkt equal to one f and only f customer receves product m from vehcle k n perod t, regardless of the compartment. The followng sets of constrants must then be added to the model presented n Secton 3.1: z mlkt k K u mkt 1 V m M t T (50) u mkt V m M l L k K t T (51) u mkt y kt V m M k K t T (52) u mkt {0, 1} V m M k K t T. (53) Constrants (50) lmt the number of vehcles delverng to tank m of customer n perod t, whle constrants (51) and (52) lnk the new varables to the exstng ones. Constrants (53) ensure the bnary condtons on the new varables. All vald nequaltes (17) (32) stll hold for ths verson of the problem. In addton, the followng nequaltes are vald for the MCDP wth splt compartments and unsplt tanks: CIRRELT

16 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems y kt M V t T. (54) k K Constrants (54) lmt the number of maxmum vsts to customer n perod t to the maxmum number M of exstng tanks n each customer Implct compartment assgnment model for the MCDP wth splt compartments and unsplt tanks The mplct formulaton presented n Secton uses varable z mkt ndcatng whether product m s delvered to customer by vehcle k n perod t or not. Smply addng the followng constrants to t allows for the resoluton of ths varant of the problem: k K z mkt 1 V m M t T. (55) The nterpretaton of constrants (55) s obvously the same as that of constrants (50). 3.3 Unsplt compartments and splt tanks In the MCDP wth unsplt compartments and splt tanks, compartments are not equpped wth debt meters so that full compartment loads must be delvered to each vsted tank. However, tanks are allowed to receve the vst of more than one vehcle per perod. The explct compartment assgnment model for the case wth unsplt compartments and splt tanks s presented n Secton 3.3.1, whle the mplct assgnment model s developed n Secton CIRRELT

17 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems Explct compartment assgnment model for the MCDP wth unsplt compartments and splt tanks In order to prevent the loads of compartments from beng splt, we add the followng constrants to the formulaton presented n Secton 3.1: q mlkt Q lk z mlkt V m M l L k K t T. (56) Together wth constrants (6), constrants (56) ensure that f a compartment s assgned to a tank, the full load of the compartment s delvered. All vald nequaltes (17) (32) stll hold for ths verson of the problem. In addton, the followng nequaltes are vald for the MCDP wth unsplt compartments and splt tanks: V m M V y kt L k K t T (57) z mlkt 1 l L k K t T. (58) Constrants (57) lmt the number of maxmum vsts performed by vehcle k n perod t to the number L of exstng compartments n each vehcle. Constrants (58) ensure that each compartment s assgned to at most one customer and one type of product for each vehcle and each perod Implct compartment assgnment model for the MCDP wth unsplt compartments and splt tanks In order to adapt the mplct model presented n Secton 3.1 to the MCDP wth unsplt compartments and splt tanks, one needs to control the specfc amounts of product delvered to each customer by each vehcle. To ths end, we defne a new nteger varable v mkt equal to the number of compartments used to delver product m to customer. Then, the followng constrants must be mposed: CIRRELT

18 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems q mkt = v mkt Q k V m M k K t T. (59) Constrants (59), whch are smlar to (42), apply to each customer and ensure that full compartment loads are delvered. 3.4 Unsplt compartments and unsplt tanks In ths most restrctve verson of the MCDP, no splttng s allowed. Thus, one must ensure that the full content of a compartment s delvered to a tank and that tanks only receve the vst of at most one vehcle per tme perod. The explct compartment assgnment model s presented n Secton 3.4.1, and the mplct assgnment model n Secton Varants of ths verson of the problem are presented n Secton Explct compartment assgnment model for the MCDP wth unsplt compartments and unsplt tanks Constrants (50) (53) and (56) must be added to the model descrbed n Secton 3.1 n order to account for these changes. All vald nequaltes (17) (32), (54), (57) and (58) are stll vald for ths verson of the problem Implct compartment assgnment model for the MCDP wth unsplt compartments and unsplt tanks The mplct model presented n Secton must be consdered wth constrants (55) and (59) n order to formulate ths verson of the problem. 16 CIRRELT

19 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems Varants of the MCDP wth unsplt compartments and unsplt tanks When the number of statons vsted by a vehcle s very small (two or three) and no splt decsons need to be made, as n Cornller et al. [8] and Popovć et al. [27], one can easly enumerate all possble routng combnatons and avod usng the related x kt j varables and constrants. Usng ths approach, these researchers have obtaned excellent results on the unsplt-unsplt case of the fuel dstrbuton problem. We extend ther formulaton to the remanng three cases of the MCDP. To ths end, we now ntroduce two varants of the routng decsons explotng the reduced number of statons per route. Note that ths new approach s not effcent f more than three statons are allowed to be vsted per route Model for up to two statons per route For the specal case n whch the number of statons per route s restrcted to be at most two, one can smplfy the search by enumeratng all possble combnatons of statons consstng of one or two statons. To ths end, we ntroduce a new bnary varable R kt j equal to one f and only f statons V and j V, j > are vsted by vehcle k n perod t. If the route conssts of only one staton, then = 0 and j represents the staton. If two statons are vsted, they are ordered n such a way that < j, thus avodng symmetry n ths new varable. The cost ĉ j of vstng statons and j, or vstng only staton j can be precomputed as 2c 0j f = 0, ĉ j = c 0 + c j + c 0j otherwse. (60) The objectve functons of the prevous models have to be changed to reflect ths new varable. The term must be replaced wth c j x kt j (61) (,j) E k K t T CIRRELT

20 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems V j V,j> k K t T ĉ j Rj kt. (62) Fnally, two sets of constrants must be added to the models, replacng the routng varables: <m V j V,j> R kt j 1 k K t T (63) Rm kt + Rmj kt = ym kt m V k K t T. (64) j>m Constrants (63) ensure that only one combnaton of customers s assgned to any gven vehcle and perod, effectvely allowng only one route per vehcle per perod. Constrants (64) lnk the new varable to the exstng varables of the model. Specfcally, they set varable y kt m to one f staton m s selected to be vsted by vehcle k n perod t Model for up to three statons per route When the number of statons per route s lmted to three, we ntroduce a new bnary varable R kt jl equal to one f and only f statons V, j V, j > {0} and l V, l > j are vsted by vehcle k n perod t. Note that f only one staton s vsted n a gven route, t s represented by the last ndex l, whle and j are equal to zero. If two statons are vsted, then they are represented by j and l, where s equal to 0. Moreover, the defnton of the varable ensures that l > j. Fnally, f three statons are vsted, they are represented n ncreasng order of ther labels. Ths s done n order to avod symmetry n the representaton of the routes. The cost ĉ jl of vstng all combnatons of one, two or three statons per route can be easly precomputed as follows: 18 CIRRELT

21 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems 2c 0l f = j = 0, c 0j + c jl + c 0l f = 0, ĉ jl = c 0 + c j + c jl + c 0l mn c 0j + c j + c l + c 0l otherwse. c 0 + c l + c lj + c 0j (65) As n the prevous case, the objectve functons of the prevous models can be modfed by replacng the term (61) wth V j V,j> {0} l V,l>j k K t T ĉ jl Rjl. kt (66) Then two sets of constrants must be added to the models, replacng all routng constrants: <m l>m V j V,j> {0} l V,l>j R kt ml + <j {0} j<m R kt jl 1 k K t T (67) R kt jm = y kt m m V k K t T. (68) Constrants (67) and (68) have the same nterpretaton as (63) and (64), respectvely. 4 Branch-and-bound and branch-and-cut algorthms The MCDP s N P-hard snce t contans the IRP and thus the capactated VRP as specal cases. If the nstance sze s small or f the number of customers per route s lmted as descrbed n Sectons and , one can take advantage of the small number of subtour elmnaton constrants needed n the models and generate all of them, whch are added to the root node of the branch-and-bound tree. An alternatve s to precompute routng decsons as explaned n these two sectons, whch yelds a problem wthout any routng varable. Agan, the problem s solved by branch-and-bound. CIRRELT

22 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems If the nstance sze s not excessve, all proposed undrected formulatons can be solved exactly by branch-and-cut as follows. At a generc node of the search tree, a lnear program wth relaxed subtour elmnaton constrants s solved, a search for volated constrants s performed, and some of these are added to the current program whch s then reoptmzed. Ths process s reterated untl a feasble or domnated soluton has been reached, or untl no more cuts can be added. At ths pont branchng on a fractonal varable occurs. In Algorthm 1 we provde a sketch of the branch-and-cut scheme for the most general verson of the problem. Algorthm 1 Pseudocode of the proposed branch-and-cut algorthm 1: At the root node of the search tree, generate and nsert all vald nequaltes (17) (32). 2: Subproblem soluton. Solve the LP relaxaton of the node. 3: Termnaton check: 4: f there are no more nodes to evaluate then 5: Stop. 6: else 7: Select one node from the branch-and-cut tree. 8: end f 9: whle the soluton of the current LP relaxaton contans subtours do 10: Identfy connected components as n Padberg and Rnald [26]. 11: Determne whether the component contanng the suppler s weakly connected as n Gendreau et al. [18]. 12: Add volated subtour elmnaton constrants (12). 13: Subproblem soluton. Solve the LP relaxaton of the node. 14: end whle 15: f the soluton of the current LP relaxaton s nteger then 16: Go to the termnaton check. 17: else 18: Branchng: branch on one of the fractonal varables. 19: Go to the termnaton check. 20: end f 20 CIRRELT

23 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems 5 Computatonal experments We now descrbe the computatonal experments we have executed to evaluate our algorthms. All computatons were carred out on a grd of Intel Xeon processors runnng at 2.66 GHz wth up to 24 GB of RAM nstalled per node, wth the Scentfc Lnux 6.1 operatng system wth a sngle thread used. The algorthms were coded n C++ and we use IBM Concert Technology and CPLEX 12.5 as the MIP solver. The nstance generaton s descrbed n Sectons 5.1 and 5.2, whle detaled computatonal results are provded n Secton Instances detals Snce no exstng study deals wth all the cases we have consdered, we have created our own set of randomly generated nstances whch can cover all four categores of problems descrbed n the prevous sectons. Our testbed s made up of nstances contanng up to 50 customers, three products, fve compartments, 18 vehcles, and spannng fve perods. We have generated fve nstances of each sze. In Secton 5.3 we provde averages over these fve nstances per combnaton. All nstances as well as detaled results are publshed n the webste Checkng the feasblty of an nstance Not all nstances are feasble, whch means that before ncludng an nstance n the testbed, we must frst ensure that t s feasble; otherwse, t s dscarded. In order to guarantee the feasblty of an nstance of the MCDP, t suffces to prove t s feasble for the case wth unsplt compartments and unsplt tanks. Snce the other three cases are relaxatons of ths one, f an nstance s feasble for the unsplt-unsplt case, t wll be feasble for all other three cases. In order to prove feasblty, an NP-hard problem must be solved, snce the feasblty problem can be reduced to a mult-commodty crculaton problem, whch CIRRELT

24 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems s known to be NP-complete for nteger flows [13]. In order to obtan a crculaton problem from the MCDP to prove ts feasblty, t suffces to remove from ts formulaton all routng varables x kt j them. Constrants (69) (80) defne the feasblty problem: and all constrants contanng I mt = I m,t 1 l L k K + l L k K q mlkt d mt V m M t T (69) 0 I mt C m V m M t T (70) q mlkt C m I m,t 1 V m M t T (71) l L q mlkt q mlkt C m y kt V m M k K t T (72) Q lk z mlkt V m M l L k K t T (73) q mlkt Q lk m M l L k K t T (74) V z mlkt y kt V m M l L k K t T (75) z mlkt w mlkt V m M l L k K t T (76) m M w mlkt 1 l L k K t T (77) q mlkt 0 V m M l L k K t T (78) w mlkt, z mlkt {0, 1} V m M l L k K t T (79) y kt {0, 1} V k K t T. (80) The nterpretaton of these constrants s the same as n Secton Note that what remans here are assgnment varables, constrants that mpose ntegralty on the delveres quanttes n terms of number of compartments, and bounds on the flows,.e., mnmum and maxmum nventory levels at the customers, and maxmum capacty of each compartment. 22 CIRRELT

25 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems 5.3 Computatonal results We now present computatonal results for the nstances just descrbed. We start by studyng the performance of the two models proposed for each varant of the problem. To ths end, we have run a subset of the nstances wth a lmt on the runnng tme of 1800 seconds. Table 2 shows the percentage of nstances for whch the mplct compartment assgnment model s superor to the explct compartment assgnment model. Table 3 shows that after the 1800 seconds the mplct models usually yeld smaller percentage gaps between the upper and lower bounds. Table 2: Percentage of best (or equal) results favorable to the Implct Compartment Assgnment models Crtera Problem Upper Bound Lower Bound Tme (s) Splt-Splt Splt-Unsplt Unsplt-Splt Unsplt-Unsplt Table 3: Average gaps for the Explct vs Implct Compartment Assgnment models after 1800 seconds of runnng tme Average gap (%) Problem Explct Implct Splt-Splt Splt-Unsplt Unsplt-Splt Unsplt-Unsplt Wth these results we can now put the relatve performance of each of the models nto perspectve, and we can proceed to obtan solutons for larger and more challengng n- CIRRELT

26 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems stances. To ths end, we have evaluated the mplct compartment assgnment models wth a two-hour tme lmt. We have also appled the fndngs of Coelho and Laporte [6] to order the nput data wth respect to the demand of the customers, n such a way that customers are relabeled consecutvely n non-ncreasng order of ther demand. Input orderng was frst proposed by Jans and Desrosers [21, 22] and was shown to mprove the lower bound of the problem. Labelng the customers accordng to ther demand has proved to be the most successful choce among those mplemented and tested by Coelho and Laporte [6]. We start ths detaled analyss wth nstances contanng a sngle product, n order to evaluate the effect of havng several compartments on the performance of the algorthm. As n Cornller et al. [8], sngle perod nstances were used. The results for all four categores of the problem are summarzed n Table 4. Several nstances wth up to 50 customers were solved to optmalty. Even some of the largest nstances consdered, whch contan up to 50 customers, four compartments, and 14 vehcles were solved to optmalty. Lkewse, nstances wth up to 50 customers, three compartments and 18 vehcles were also solved for some varants of the problem to optmalty n relatvely small runnng tmes. One can observe that the problem becomes more dffcult when t s more constraned, to the pont where our algorthm could not fnd any feasble soluton wthn two hours of computng tme for some nstances of the unsplt-unsplt case. Ths s also reflected by the sze of the gaps and by the ncreased runnng tme towards the rght of the table. We have then consdered nstances defned over several perods. Average results over all four varants of the problem are presented n Table 5. These nstances are consderably more dffcult to solve than ther sngle-perod counterpart, as reflected by the gaps and runnng tmes. The algorthm often could not dentfy any feasble soluton for nstances wth more than 20 customers. For the 20 nstances contaned n Table 5, we observe that, as s the case for the results of Table 4, the solutons of the frst two varants of the problem are qute smlar, and so are those of the last two. Moreover, for each par, the more restrctve case seems to be easer to solve, whch s reflected by smaller average 24 CIRRELT

27 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems Table 4: Summary of the results on sngle-perod sngle-product nstances Instance Splt-Splt Splt-Unsplt Unsplt-Splt Unsplt-Unsplt Soluton Gap (%) Tme (s) Soluton Gap (%) Tme (s) Soluton Gap (%) Tme (s) Soluton Gap (%) Tme (s) MCD MCD MCD MCD MCD MCD MCD MCD MCD MCD gaps and runnng tmes. These smlar results are, however, an effect of the structure of the problem obtaned by mposng unsplt compartments rather than a consequence of the model. Table 5: Summary of the results on mult-perod sngle-product nstances Instance Splt-Splt Splt-Unsplt Unsplt-Splt Unsplt-Unsplt Soluton Gap (%) Tme (s) Soluton Gap (%) Tme (s) Soluton Gap (%) Tme (s) Soluton Gap (%) Tme (s) MCD MCD MCD MCD Fnally, we have solved the most general nstances contanng several products, perods, vehcles and compartments. These nstances contan up to 20 customers, three products, fve compartments, eght vehcles, and fve perods. As expected, the sze of the nstances for whch optmal soluton can be obtaned decreases, as s shown n Table 6. A transversal analyss over the last three tables allows us to derve some comments on the relatve dffculty of each varant of the problem. We observe that the average ncrease on the soluton cost (or on the upper bound when optmalty s not acheved) of each CIRRELT

28 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems Table 6: Summary of the results on mult-perod mult-product nstances Instance Splt-Splt Splt-Unsplt Unsplt-Splt Unsplt-Unsplt Soluton Gap (%) Tme (s) Soluton Gap (%) Tme (s) Soluton Gap (%) Tme (s) Soluton Gap (%) Tme (s) MCD MCD MCD MCD MCD MCD MCD MCD MCD MCD MCD MCD MCD MCD MCD varant of the problem wth respect to the most general scenaro,.e., the splt-splt case, s qute stable. The splt-unsplt case often yelds solutons that are margnally more expensve than the splt-splt varant, whle solvng ether the unsplt-splt or the unspltunsplt case approxmately doubles the cost. Moreover, movng from the unsplt-splt to the unsplt-unsplt varant yelds no sgnfcant dfference n costs over the nstances we have tested. 6 Concluson We have ntroduced, classfed, modeled and solved a wde range of routng problems wth several compartments used for the delvery of several products spannng several perods, wth the am of mnmzng routng and nventory costs over the network. Ths problem typcally arses n the dstrbuton of petroleum products to gas statons. We have developed two models for each varant of the problem, and we have shown how to 26 CIRRELT

29 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems adapt these models to handle specfc versons of the problem descrbed n the operatonal research lterature. Extensve computatonal results on a set of benchmark nstances show that optmal solutons can be proved for nstances contanng up to 50 customers, four compartments and 14 vehcles. However, mult-perod nstances are clearly more challengng. When several perods are consdered, nstances wth up to 20 customers, and several products, compartments, and vehcles can be solved optmally. References [1] M. Albareda-Sambola, E. Fernández, and G. Laporte. A computatonal comparson of several models for the exact soluton of the capacty and dstance constraned plant locaton problem. Computers & Operatons Research, 38(8): , [2] H. Andersson, A. Hoff, M. Chrstansen, G. Hasle, and A. Løkketangen. Industral aspects and lterature survey: Combned nventory management and routng. Computers & Operatons Research, 37(9): , [3] D. O. Bausch, G. G. Brown, and D. Ronen. Schedulng short-term marne transport of bulk products. Martme Polcy & Management, 25(4): , [4] L. C. Coelho and G. Laporte. The exact soluton of several classes of nventoryroutng problems. Computers & Operatons Research, 40(2): , [5] L. C. Coelho and G. Laporte. A branch-and-cut algorthm for the mult-product mult-vehcle nventory-routng problem. Internatonal Journal of Producton Research, forthcomng, do: / [6] L. C. Coelho and G. Laporte. Improved solutons for nventory-routng problems through vald nequaltes and nput orderng. Techncal Report CIRRELT , Montreal, Canada, CIRRELT

30 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems [7] L. C. Coelho, J.-F. Cordeau, and G. Laporte. Thrty years of nventory-routng. Transportaton Scence, forthcomng, [8] F. Cornller, F. F. Boctor, G. Laporte, and J. Renaud. An exact algorthm for the petrol staton replenshment problem. Journal of the Operatonal Research Socety, 59(5): , [9] F. Cornller, F. F. Boctor, G. Laporte, and J. Renaud. A heurstc for the multperod petrol staton replenshment problem. European Journal of Operatonal Research, 191(2): , [10] F. Cornller, G. Laporte, F. F. Boctor, and J. Renaud. The petrol staton replenshment problem wth tme wndows. Computers & Operatons Research, 36(3): , [11] F. Cornller, F. F. Boctor, and J. Renaud. Heurstcs for the mult-depot petrol staton replenshment problem wth tme wndows. European Journal of Operatonal Research, 220(2): , [12] M. Dror, G. Laporte, and P. Trudeau. Vehcle routng wth splt delveres. Dscrete Appled Mathematcs, 50(3): , [13] S. Even, A. Ita, and A. Shamr. On the complexty of tmetable and multcommodty flow problems. SIAM Journal on Computng, 5(4): , [14] K. Fagerholt and M. Chrstansen. A combned shp schedulng and allocaton problem. Journal of the Operatonal Research Socety, 51(7): , [15] K. Fagerholt and M. Chrstansen. A travellng salesman problem wth allocaton, tme wndow and precedence constrants an applcaton to shp schedulng. Internatonal Transactons n Operatonal Research, 7(3): , [16] M. Fschett, J. J. Salazar-González, and P. Toth. Experments wth a multcommodty formulaton for the symmetrc capactated vehcle routng problem. In 28 CIRRELT

31 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems Proceedngs of the 3rd Meetng of the EURO Workng Group on Transportaton, pages , Barcelona, Span, [17] M. Fschett, J. J. Salazar-González, and P. Toth. Solvng the orenteerng problem through branch-and-cut. INFORMS Journal on Computng, 10(2): , [18] M. Gendreau, G. Laporte, and F. Semet. The coverng tour problem. Operatons Research, 45(4): , [19] M. Gendreau, G. Laporte, and F. Semet. A tabu search heurstc for the undrected selectve travellng salesman problem. European Journal of Operatonal Research, 106(2-3): , [20] L. M. Hvattum, K. Fagerholt, and V. A. Armentano. Tank allocaton problems n martme bulk shppng. Computers & Operatons Research, 36(11): , [21] R. Jans and J. Desrosers. Bnary clusterng problems: Symmetrc, asymmetrc and decomposton formulatons. Techncal Report G , GERAD, Montreal, Canada, [22] R. Jans and J. Desrosers. Effcent symmetry breakng formulatons for the job groupng problem. Computers & Operatons Research, 40(4): , [23] G. Laporte. Ffty years of vehcle routng. Transportaton Scence, 43(4): , [24] L. Muyldermans and G. Pang. On the benefts of co-collecton: Experments wth a mult-compartment vehcle routng algorthm. European Journal of Operatonal Research, 206(1):93 103, [25] J. Oppen and A. Løkketangen. A tabu search approach for the lvestock collecton problem. Computers & Operatons Research, 35(10): , CIRRELT

32 Classfcaton, Models and Exact Algorthms for Mult-Compartment Delvery Problems [26] M. W. Padberg and G. Rnald. A branch-and-cut algorthm for the resoluton of large-scale symmetrc travelng salesman problems. SIAM Revew, 33(1):60 100, [27] D. Popovć, M. Vdovć, and G. Radvojevć. Varable neghborhood search heurstc for the nventory routng problem n fuel delvery. Expert Systems wth Applcatons, 39(18): , [28] M. Vdovć, D. Popovć, and B. Ratkovć. Mxed nteger and heurstcs model for the nventory routng problem n fuel delvery. Internatonal Journal of Producton Economcs, Forthcomng, CIRRELT