Optimal Joint Replenishment, Delivery and Inventory Management Policies for Perishable Products

Transcription

1 Optmal Jont Replenshment, Delvery and Inventory Management Polces for Pershable Products Leandro C. Coelho Glbert Laporte May 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 Optmal Jont Replenshment, Delvery and Inventory Management Polces for Pershable Products 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. In ths paper we analyze the optmal jont decsons of when, how and how much to replensh customers wth products of varyng ages. We dscuss the man features of the problem arsng n the jont replenshment and delvery of pershable products, and we model them under general assumptons. We then solve the problem by means of an exact branch-and-cut algorthm, and we test ts performance on a set of randomly generated nstances. Our algorthm s capable of computng optmal solutons for nstances wth up to 30 customers, three perods, and a maxmum age of two perods for the pershable product. For the unsolved nstances the optmalty gap s always small, less than 1.5% on average for nstances wth up to 50 customers. We also mplement and compare two suboptmal sellng prorty polces wth an optmzed polcy: always sell the oldest avalable tems frst to avod spolage, and always sell the fresher tems frst to ncrease revenue. Keywords. Pershable products, nventory control, replenshment, nventory-routng, vendor-managed nventory, branch-and-cut algorthm. 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 Calcul Québec for provdng hgh performance parallel 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 1 Introducton Inventory control consttutes an mportant logstcs operaton, especally when products have a lmted shelf lfe. Keepng the rght nventory levels guarantees that the demand s satsfed wthout ncurrng unnecessary holdng or spolage costs. Several nventory control models are avalable [3], many of whch nclude a specfc treatment of pershable products [30]. Problems related to the management of pershable products nventores arse n several areas. Applcatons of nventory control of pershable products nclude blood management and dstrbuton [5, 9, 17, 18, 20, 25, 26, 33], as well as the handlng of radoactve and chemcal materals [1, 11, 36], of food such as dary products, fruts and vegetables [4, 12, 29, 31, 34, 35], and of fashon apparel [28]. Several nventory management models have been specfcally derved for pershable tems, such as the perodc revew wth mnmum and maxmum order quantty of Hajema [15], and the perodc revew wth servce level consderatons of Mnner and Transchel [24]. Revews of the man models and algorthms n ths area can be found n Nahmas [30] and n Karaesmen et al. [19]. A unfed analytcal approach to the management of supply chan networks for tme-senstve products s provded n Nagurney et al. [27]. Effcent delvery plannng can provde further savngs n logstcs operatons. The optmzaton of vehcle routes s one of the most developed felds n operatons research [21]. The ntegraton of nventory control and vehcle routng yelds a complex optmzaton problem called nventory-routng whose am s to mnmze the overall costs related to vehcle routes and nventory control. Recent overvews of the nventory-routng problem (IRP) are those of Andersson et al. [2] and of Coelho et al. [8]. The jont nventory management and dstrbuton of pershable products, whch s the topc of ths paper, gves rse to the pershable nventory-routng problem (PIRP). Nagurney and Masoum [25] and Nagurney et al. [26] studed the dstrbuton and relocaton of human blood n a stochastc demand context, consderng the pershablty and waste of CIRRELT

4 blood related to age and to the lmted capacty of blood banks. Hemmelmayr et al. [16] studed the case of blood nventory control wth predetermned fxed routes and stochastc demand. The problem was solved heurstcally by nteger programmng and varable neghborhood search. Gumasta et al. [14] ncorporated transportaton ssues n an nventory control model restrcted to two customers only. Custódo and Olvera [10] proposed a strategcal heurstc analyss of the dstrbuton and nventory control of several frozen groceres wth stochastc demand. Mercer and Tao [23] studed the weekly food dstrbuton problem of a supermarket chan, wthout consderng product age. A theoretcal paper developng a column generaton approach was presented by Le et al. [22] to provde solutons to a PIRP. The optmalty gap was typcally below 10% for nstances wth eght customers and fve perods under the assumptons of fxed shelf lfe and flat value throughout the lfe of the product. Ths paper makes several scentfc contrbutons. We frst classfy and dscuss the man assumptons underlyng the management of pershable products. We then formulate the PIRP as a mxed nteger lnear program (MILP) for the most general case, and we also model t to handle the cases where retalers always sell older tems frst, and where they sell fresher tems frst. We devse an exact branch-and-cut algorthm for the soluton of the varous models. To the best of our knowledge, ths s the frst tme an IRP s modeled and solved exactly under general assumptons n the context of pershable products management. Our models do not requre any assumpton on the shape of the product revenue and nventory cost functons. We also establsh some relatonshps between the PIRP and the mult-product IRP recently studed by the authors [7]. The remander of the paper s organzed as follows. In Secton 2 we provde a formal descrpton of the PIRP. In Secton 3 we present our MILP model and ts two varants just descrbed, ncludng new vald nequaltes. Ths s followed by a decrpton of the branchand-cut algorthm n Secton 4. Computatonal experments are presented n Secton 5. Secton 6 concludes the paper. 2 CIRRELT

5 2 Problem Descrpton The jont replenshment and nventory problem for pershable products s concerned wth the combned optmzaton of delvery routes and nventory control for products havng a transent shelf lfe. These products typcally have an expry date, after whch they are no longer ft for consumpton. Ths s the case of some law-regulated products such as food and drugs, but also of a wde varety of unregulated products whose qualty, appearance or commercal appeal dmnshes over tme, such as flowers, cosmetcs, pant, electronc products or fashon tems. In ths secton we dscuss four man assumptons underlyng the treatment of these knds of products, and we explan how we ncorporate them n our model. Specfcally, we dscuss the types of product pershablty n Secton 2.1, the assumptons governng the nventory holdng costs of these products n Secton 2.2, ther revenue as a functon of age n Secton 2.3, and the management of tems of dfferent ages held n nventory n Secton Types of product pershablty There exst two man types of pershable products accordng to how they decay [30]. The frst type ncludes products whose value does not change untl a certan date, and then goes down to zero almost mmedately. Ths s the case of products whose utlty eventually ceases to be valued by the customers, such as calendars, year books, electroncs or maps, whch quckly become obsolescent after a gven date or when a new generaton of products enters the market. However, ths s more a case of obsolescence than pershablty. Even though these tems may stll be n perfect condton, they are smply no longer useful. Wthn the same category, we fnd products wth an expry date, such as drugs, yogurt and bottled mlk. These products can be consumed whether they are top fresh or a few days old, but after ther expry date, they are usually deemed unft for consumpton. The second type ncludes products whose qualty or perceved value decays gradually over tme. Typcal examples are fruts, vegetables and flowers. The models ntroduced n CIRRELT

6 Secton 3 can handle both types of products wthout any ad hoc modfcaton. 2.2 The mpact of tem age on nventory holdng costs As a rule, the unt nventory holdng cost changes wth respect to the age and value of a product. Ths general assumpton holds, for nstance, for nsurance costs whch are value related. All the varable costs related to the age of the product can be modeled through a sngle parameter, called the unt nventory holdng cost, whch depends on the age of the tem. In some contexts, all tems yeld the same holdng cost, regardless of ther age. Products wth a short shelf lfe usually ft n ths category. In ths case, the holdng cost, whch encompasses all other varable costs, can be captured by a unque nput parameter ndependent of the value and age of the product, whch s the case n most applcatons. 2.3 Revenue of the tem accordng to ts age A parameter that greatly affects the proft yelded by products of dfferent ages s ther perceved value by consumers. Brand new tems usually have a hgher sellng prce, whch decreases over tme accordng to some functon. In ths paper we do not make any specfc assumpton regardng the shape of ths functon. Rather, we assume that the sellng prce s known n advance for each product age. Note that the functon descrbng the relaton between prce and age can be non-lnear, non-contnuous or even non-convex, but t can stll be accommodated by our model, as wll be shown n Secton Inventory management polces The fnal assumpton we dscuss relates to the management of tems of dfferent ages held n nventory. It s up to the retaler to decde whch tems to offer to customers, whch wll nfluence the assocated revenue. In such a context, three dfferent sellng prorty polces can be envsaged. The frst one conssts of applyng a fresh frst (FF) polcy by whch 4 CIRRELT

7 the suppler always sells the fresher tems frst. Ths polcy ensures a longer shelf lfe and ncreases utlty for the customers but, at the same tme, yelds a hgher spolage rate. The second polcy s the reverse. Under an old frst (OF) polcy, older tems are sold frst, whch generates less spolage, but also less revenue. The thrd polcy, whch we ntroduce n our model, s more flexble and general, and encompasses these two extremes. The optmzed prorty (OP) polcy lets the model determne whch tems to sell at any gven tme perod n order to maxmze proft. Ths means that dependng on the parameter settngs, one may prefer to spol some tems and sell fresher ones because they generate hgher revenues. In order to llustrate the FF and the OF polces, frst consder the case of bottled mlk havng a lmted shelf lfe. A retaler holds n nventory one unt of old mlk havng a remanng shelf lfe of one day, and one unt of one-day old mlk stll good for several days. The unt revenue s $2. If the retaler apples an FF polcy, he sells hs one-day old mlk today, makng $2 of revenue. Tomorrow, the remanng bottle wll be spoled and he wll make no revenue. The total revenue under the FF polcy s then $2. If, on the other hand, he apples an OF polcy, he sells hs old bottle today, and the newer one tomorrow, makng a total revenue of $4, or twce the revenue acheved under the FF polcy. Now consder the case of flowers, whose value declnes quckly from one day to the next. A one-day old bouquet of flowers generates a revenue of $10, whereas a two-day old bouquet yelds only $4. Under an FF polcy, she wll sell the one-day old flowers today, and nothng tomorrow, makng a total revenue of $10. Under an OF polcy, the retaler wll sell the older flowers today and the other ones tomorrow, achevng a smaller revenue of $8. Note that n these two examples, the OP polcy concdes wth ether the OF or the FF polcy. However, ths s not always the case, namely when the revenue functon s not monotonc wth respect to the age of the product. Consder for example the case of bananas, whch start ther shelf lfe as green products, not yet rpe for consumpton, then turn yellow when they reach ther peak value, and fnally become brown close to ther spolage date. Suppose there are two hands of bananas of each color n nventory. Let CIRRELT

8 the revenue be $1.50 for a hand of green bananas, $2 for a yellow hand, and $0.50 for a brown hand. Note how the green product s valued hgher than the brown one, because t wll mature over tme and wll eventually become yellow. For a daly demand of one hand over two perods, the FF polcy yelds a revenue of $3, the OF polcy yelds only $1, but an OP polcy consstng of sellng yellow bananas each day yelds an optmal revenue of $4. If the nventory contans only green and yellow bananas, then the OF and OP polces concde; smlarly, f only yellow and brown bananas are consdered, then the FF and OP polces concde. Thus, the choce of whch of the FF or OF polcy to apply depends on the trade-off between the nventory level and the revenue functons of the product under consderaton. The advantage of the OP polcy s that t does not mpose any constrants on the age of the tems to sell and s able to generate the most general and proftable solutons. We mplement all three polces and we analyze ther trade-offs n the context of proft maxmzaton. 3 Mathematcal Formulatons We now formally descrbe the mathematcal formulaton of PIRP under the assumptons just presented for a sngle product and under the three nventory management polces just descrbed. The case of several products s conceptually smlar, but requres an addtonal ndex [7]. We assume that the routng cost matrx s symmetrc. Thus, we defne the problem 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 remanng vertces V = V \{0} correspond to n customers. A routng cost c j s assocated wth edge (, j) E. Because of the general assumptons presented n Secton 2, we consder that both the suppler and customers are fully aware of the number of tems n nventory accordng 6 CIRRELT

9 to ther age. Ths s mportant because the sales revenue and nventory holdng costs are affected by the age of the product. The suppler has the choce to delver fresh or aged product tems, and each case yelds dfferent holdng costs. Each customer has a maxmum nventory holdng capacty C, whch cannot be exceeded n any perod of the plannng horzon of length p. At each tme perod t T = {1,..., p}, the suppler receves or produces a fresh quantty r t of the pershable product. We assume the suppler has suffcent nventory to meet the demand of ts customers durng the plannng horzon, and all demand has to be satsfed. At the begnnng of the plannng horzon the decson maker knows the current nventory level of the product at each age held by the suppler and by the customers, and receves nformaton on the demand d t of each customer for each tme perod t. Note agan that, as dscussed n the prevous secton, the demand can be equally satsfed by fresh or aged products, whch wll n turn affect the revenue. As s typcally the case n the IRP lterature [8], we assume that the quantty r t made avalable at the suppler n perod t can be used for delveres to customers n the same perod, and the delvery amount receved by customer n perod t can be used to meet the demand n that perod. A set K = {1,..., K} of vehcles are avalable. We denote by Q k the capacty of vehcle k. Each vehcle can perform at most one route per tme perod, vstng a subset of customers, startng and endng at the suppler s locaton. Also, as n other IRP papers, we do not allow splt delveres,.e., customers receve at most one vehcle vst per perod. The pershable product under consderaton becomes spoled after s perods,.e., the age of the product belongs to a dscrete set S = {0,..., s}. The product s valued accordng to ts age, and the decson maker s aware of the sellng revenue u g of one unt of product of age g. Lkewse, the nventory holdng cost h g n locaton V s a functon of the age g of the product. Ths general representaton allows for flat or varable revenues, and for flat or varable holdng costs dependng on the age and value of the product, thus coverng all stuatons descrbed n Secton 2. The nventory level I t held by customer n perod t comprses tems of dfferent ages. We CIRRELT

10 break down ths varable nto I t = I gt, where I gt represents the quantty of product of g S age h n nventory at customer n perod t. Lkewse, we decompose the demand d t nto d gt. g S The am of the problem s to smultaneously construct vehcle routes for each perod and to determne delvery quanttes of products of dfferent ages for each perod and each customer, n order to maxmze the total proft, equal to the sales revenue, mnus the routng and nventory holdng costs. Ths problem s extremely dffcult to solve snce t encompasses several NP-hard problems such as the vehcle routng problem [21] and a number of varants of the classcal IRP [8]. Our MILP model works wth routng varables x kt j equal to the number of tmes edge (, j) s used on the route of vehcle k n perod t. We also use bnary varables y kt equal to one f and only f node s vsted by vehcle k n perod t. Formally, varables I t = g S denote the nventory level at vertex V at the end of perod t T, and d gt denotes the quantty of product of age g used to satsfy the demand of customer n perod t, and we denote by q gkt the quantty of product of age g delvered by vehcle k to customer n perod t. The problem can then be formulated under an OP polcy as follows: I gt (PIRP) maxmze g S t T u g dgt V g S t T h g Igt c j x kt j, (1) (,j) E k K t T subject to I gt 0 = I g 1,t 1 0 V k K q gkt g S\{0} t T (2) I 0t 0 = r t t T (3) I gt = I g 1,t 1 + k K q gkt d gt V g S\{0} t T (4) I 0t = k K q 0kt d 0t V t T (5) 8 CIRRELT

11 g S k K q gkt j V,<j S j S,<j I gt C V t T (6) g S d t = g S q gkt V g S C g S q gkt d gt V t T (7) I g,t 1 V t T (8) C y kt V g S k K t T (9) Q k y0 kt k K t T (10) x kt j + j V,j< x kt j = 2y kt V k K t T (11) x kt j y kt ym kt S V k K t T m S (12) S k K y kt 1 V t T (13) I gt, d gt, qgkt 0 V g S k K t T (14) x kt 0 {0, 1, 2} V k K t T (15) x kt j {0, 1}, j V k K t T (16) y kt {0, 1} V k K t T. (17) The objectve functon (1) maxmzes the total sales revenue, mnus nventory and routng costs. Constrants (2) defne the nventory conservaton condtons for the suppler, agng the product by one unt n each perod. Constrants (3) ensure that the suppler always produces or receves top fresh products. Constrants (4) and (5) defne nventory conservaton and agng of the tems for the customers. Constrants (6) mpose a maxmal nventory capacty at each customer. Constrants (7) state that the demand of each customer n each perod s the sum of product quanttes of dfferent ages. Note that by desgn, any product whose age g s hgher than s s spoled, e.g., t no longer appears n the nventory nor can t be used to satsfy the demand. Constrants (8) and (9) lnk the quanttes delvered to the routng varables. In partcular, they only allow a vehcle to delver products to a customer f a vehcle has been assgned to t. Constrants (10) CIRRELT

12 ensure that the vehcle capactes are respected. Constrants (11) and (12) are degree constrants and subtour elmnaton constrants, respectvely. Inequaltes (13) ensure that at most one vehcle vsts each customer n each perod, thus forbddng splt delveres. Constrants (14) (17) enforce ntegralty and non-negatvty condtons on the varables. Ths model can be strengthened through the ncluson of the followng famles of vald nequaltes [6]: y kt x kt 0 2y kt V k K t T (18) x kt j y kt, j V k K t T (19) y kt y kt 0 V k K t T (20) y kt 0 y k 1,t 0 k K\{1} t T (21) j< y k 1,t j V k K\{1} t T. (22) Constrants (18) and (19) enforce the condton that f the suppler s the mmedate successor of a customer n the route of vehcle k n perod t, then must be vsted by the same vehcle. A smlar reasonng s appled to customer j n nequaltes (19). Constrants (20) ensure that the suppler s vsted f any customer s vsted by vehcle k n perod t. When the vehcle fleet s homogeneous, one can break some of the vehcle symmetry by means of constrants (21), thus ensurng 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 by constrants (22) whch state that f customer s assgned to vehcle k n perod t, then vehcle k 1 must serve a customer wth an ndex smaller than n the same perod. We also ntroduce addtonal cuts n order to strengthen ths formulaton. If the sum of the demands over [t 1, t 2 ] s at least equal to the maxmum possble nventory held, then there must be at least one vst to ths customer n the nterval [t 1, t 2 ]. Ths constrant can be strengthened by consderng that f the quantty needed to satsfy future demands 10 CIRRELT

13 s larger than the maxmum nventory capacty, then several vsts are needed. Snce the maxmum delvery sze s the mnmum between the holdng capacty and the maxmum vehcle capacty, one can round up the rght-hand sde of (23). Makng the numerator tghter by consderng the actual nventory nstead of the maxmum possble nventory yelds nequaltes (24), whch cannot be rounded up because they would then become non-lnear due to the presence of the I t 1 varable n ther rght-hand sde: t 2 y kt k K t =t 1 t 2 d t C t =t 1 mn{max k {Q k }, C } V t 1, t 2 T, t 2 t 1 (23) t 2 y kt k K t =t 1 t 2 d t I t 1 t =t 1 mn{max k {Q k }, C } V t 1, t 2 T, t 2 t 1. (24) A dfferent verson of the same nequaltes can be wrtten as follows. It s related to whether the nventory hold at each perod s suffcent to fulfll future demands. In partcular, f the nventory held n perod t 1 by customer s not suffcent to fulfll future demands, then a vst to ths customer must take place n the nterval [t 1, t 2 ]. Ths condton can be enforced by the followng set of nequaltes: t 2 y kt k K t =t 1 t 2 d t I t 1 t =t 1 t 2 d t t =t 1 V t 1, t 2 T, t 2 t 1. (25) Even f these nequaltes are redundant for our model, they are useful n helpng CPLEX generate new cuts. It s relevant to note that ths model dstngushes tems of dfferent ages through the use of ndex g. The varables have a meanng smlar to those of the mult-product IRP [7]. In the case of a sngle pershable product, the model works as f products of dfferent ages are dfferent from each other (through ther ndex) and have dfferent profts, but contrary CIRRELT

14 to what happens n the mult-product case, any of these products can be used to satsfy the same demand. Another partcularty of ths model s that at each perod, an tem transforms tself nto another one through the process of agng. Thus, our problem shares some features of the mult-product problem [7], but t s structurally dfferent from t. 3.1 Modelng an FF polcy We now show how the formulaton just descrbed can be used to solve the problem under an FF polcy under whch the retaler sells fresher tems frst. We add extra varables and constrants to the PIRP formulaton n order to restrct the choce of products age to be sold. We mplement ths dea as follows. We frst ntroduce new bnary varables L gt equal to one f and only f products of age g can be used to satsfy the demand of customer n perod t. The frst set of new constrants restrcts the use of varables d gt,.e., the use of products of age g to satsfy the demand of customer n perod t, only to those products allowed by the respectve L gt varables, that s: d gt U L gt V g S t T. (26) We also order the new varables n ncreasng order of age ndex. The followng set of constrants allows sellng products of age g + 1 only f products of age g have been used to satsfy the demand of customer n perod t: L gt L g+1,t V g S\{s} t T. (27) We then mpose the followng constrants to dsallow the use of older products f there exsts enough nventory of fresher products. The use of products of age g + 1 s allowed f and only f the total nventory of products of ages g, g 1,..., 0 s nsuffcent to satsfy the demand of customer n perod t. Ths can be enforced through the followng constrants: 12 CIRRELT

15 U (1 L g+1,t ) g j=0 I jt + g q jkt d t + 1 V g S\{s} t T. (28) j=0 k K 3.2 Modelng an OF polcy It s straghtforward to model the OF polcy from the constrants developed for the FF case. Ths polcy can be enforced by consderng the same L gt three sets of constrants: varables and the followng d gt U L gt V g S t T. (29) Constrants (29) only allow the use of nventory of age g to satsfy the demand f ts assocated L gt varable s set to one. Then, we also rank the L gt varables n ncreasng order of age ndex. The followng set of constrants allow sellng products of age g 1 only f products of age g are beng used to satsfy the demand of customer n perod t: L g 1,t L g,t V g S\{0} t T. (30) Fnally, we force some of the L varables to take value zero by addng the followng constrants to the model. If the total nventory avalable of ages {g, g + 1,..., s} s suffcent to satsfy the demand of customer n perod t, then the rght-hand sde of nequaltes (31) s postve, whch n turn guarantees that L g 1,t wll take value zero: U (1 L g 1,t ) s j=g I jt + s q jkt d t + 1 V g S\{0} t T. (31) j=g k K 4 Branch-and-Cut Algorthm For very small nstances szes, the model presented n Secton 3 can be fully descrbed and all constrants and varables generated. It can then be solved by feedng t drectly nto an CIRRELT

16 nteger lnear programmng solver. However, for nstances of realstc szes, the number of subtour elmnaton constrants (12) s too large to allow full enumeraton and these must be dynamcally generated throughout the search process. The exact algorthm we present s then a classcal branch-and-cut scheme n whch subtour elmnatons constrants are only generated and ncorporated nto the program whenever they are found to be volated. It works as follows. At a generc node of the search tree, a lnear program contanng a subset of the subtour elmnaton constrants s solved, a search for volated nequaltes 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 s reached, or untl there are no more cuts to be added. At ths pont, branchng on a fractonal varable occurs. We provde a sketch of the branch-and-bound-and-cut scheme n Algorthm 1. 5 Computatonal Experments In order to evaluate the proposed algorthm, we have coded t n C++ and used IBM Concert Technology and CPLEX 12.5 runnng n parallel wth two threads. All computatons were executed 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. 5.1 Instances generaton We have created randomly generated nstances to assess the performance of our algorthm on a wde range of stuatons. We have generated a total of 60 dfferent nstances whch vary n terms of the number of customers, perods, vehcles and maxmum age of the product. Our testbed s composed of nstances generated wth the followng parameters: Number of customers n: 10, 20, 30, 40, 50; Number of perods H: 3 for up to n = 50; 6 for up to n = 40; and 10 for up to n = 30; 14 CIRRELT

17 Algorthm 1 Branch-and-cut algorthm 1: At the root node of the search tree, generate and nsert all vald nequaltes nto the program. 2: Subproblem soluton. Solve the LP relaxaton of the current 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 [32]. 11: Determne whether the component contanng the suppler s weakly connected as n Gendreau et al. [13]. 12: Add all volated subtour elmnaton constrants (12). 13: Subproblem soluton. Solve the LP relaxaton of the current 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 CIRRELT

18 Number of vehcles K: 1 for n = 10; 2 for n = 20 and 30; 3 for n = 40 and 50; Maxmum age of the products s: 2 for H = 3; 3 for H = 6; 5 for H = 10; Demand d t : randomly selected from the nterval [30, 210]; Poston (x, y) of the suppler and customers: randomly selected from the nterval [0, 1000]; Customers nventory capacty C : R max t {d t }, where R s randomly selected from the set {2, 3}; Intal nventory I 0 of fresh products: equal to C d 1 ; Revenue u g : equal to R 1 (R 1 R 2 ) g/s, where R 1 and R 2 are randomly selected from the ntervals [10, 20] and [4, 7], respectvely; Inventory holdng cost h g : equal to (R 1 + gr 2 / (1 + g)) /100, where R 1 and R 2 are randomly selected from the ntervals [0, 100] and [0, 70], respectvely; Vehcle capactes Q k : equal to 1.25 d t /(HK). V t T For each combnaton of the n, s, K and H parameters we have generated fve nstances, yeldng a total of 60 nstances. In what follows we provde average statstcs over fve nstances per combnaton. Detaled results are presented n Appendx A. These results along wth the nstances are also avalable for download from coelho.com. 5.2 Solutons for an OP polcy We provde n Table 1 average computatonal results for these nstances under the OP polcy. We have allowed the algorthm to run for a maxmum of two hours. When the tme lmt s reached, we report the best avalable lower and upper bound (soluton value) and 16 CIRRELT

19 the optmalty gap. We report the nstance szes as (n-s-k-h), where n s the number of customers, s s the maxmum age of the product, K s the number of vehcles, and H s the length of the plannng horzon. The next columns report the average best soluton value obtaned, the average best bound, the average optmalty gap, the number of nstances out of the fve that were solved to optmalty, and the average runnng tme n seconds. Table 1: Summary of the computatonal results for the PIRP under the OP polcy Instance sze Best known Best known (n-s-k-h ) soluton value upper bound Gap (%) # solved Tme (s) PIRP /5 0.4 PIRP /5 2.4 PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / These results clearly ndcate that the performance of the algorthm s drectly related to the number n of customers and to the length H of the plannng horzon. For the nstances wth shorter plannng horzons (H = 3), the algorthm s always able to fnd optmal solutons wthn a few seconds of computatonal tme. Ths remans true even when the number of customers and vehcles ncreases, e.g., all fve nstances wth 30 customers and three perods were solved to optmalty, takng on average fve mnutes. Larger nstances wth up to 40 and 50 customers also wth three perods were solved wth a gap of less than 1.50% on average. CIRRELT

20 5.3 Solutons for an FF and an OF polcy We also compare the soluton cost of the optmzed polcy wth respect to the age of the products sold wth the cost of the alternatve FF and OF polces. We frst consder the FF polcy whch maxmzes the revenue by always sellng fresher tems. Ths polcy, on the other hand, leads to more spolage, whch n turn ncreases the need for more delveres, thus ncreasng dstrbutons costs. The results are shown n Table 2 as percentages representng the proft decrease of the FF polcy wth respect to the OP polcy. We also report the optmalty gap, the number of nstances solved optmally, and the runnng tme n seconds. We note that the dffculty of solvng the PIRP under an FF polcy s smlar to that observed for the OP polcy, and the proft s only slghtly lower. Fnally, we provde the same comparson wth respect to the OF polcy. The summary of the results are shown n Table 3. Table 2: Summary of the computatonal results for the PIRP under an FF polcy Instance sze (n-s-k-h ) % decrease Opt gap (%) # solved Tme (s) PIRP /5 0.6 PIRP /5 3.2 PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / CIRRELT

21 Table 3: Summary of the computatonal results for the PIRP under an OF polcy Instance sze (n-s-k-h ) % decrease Opt gap (%) # solved Tme (s) PIRP /5 0.2 PIRP / PIRP / PIRP /5 6.0 PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / PIRP / CIRRELT

22 As was the case of the FF polcy, the dffculty of obtanng optmal and quas-optmal solutons s not affected by the ncluson of the new bnary varables and the new constrants. However, unlke the prevous polcy, the effect on cost of sellng older tems frst, thus dervng lower revenues, has a major effect on the total proft observed, whch decreases substantally over all nstances. 5.4 Solutons for alternatve revenue functons In order to assess the trade-off between the OP, FF and OF polces, we have changed how the product revenue vares lnearly as a functon of age. We have generated three varatons. In the frst mld scenaro, the dfference n cost between fresh and old products s reduced. In the second steep scenaro, the dfference s ncreased. Fnally, we have also created a flat scenaro case n whch the revenue of the product s constant as a functon of age. These three scenaros and the base case are depcted n Fgure 1. The slopes of the lnear functons n ncreasng order are equal to 2.4, 1.8 and " Alterna2ve&revenue&func2ons& 15" Value&of&the&product& 12" 9" 6" 3" Flat" Mld" Base"case" Steep" 0" 0" 1" 2" 3" 4" 5" Age&of&the&product& Fgure 1: Four alternatve revenue functons We have desgned the followng experments n order to evaluate the mpact of these 20 CIRRELT

23 changes n the trade-off between the dfferent polces. We have selected all 30 nstances contanng 10 and 20 customers. Each nstance was solved under the three polces and under the three alternatve revenue functons. In Table 4 we report the percentage decrease n proft wth respect to the optmzed polcy for each of the revenue functons consdered. Table 4: Percentage decrease n proft when usng alternatve revenue functons Instance sze FF polcy OF polcy (n-s-k-h ) Base case Mld Steep Flat Base case Mld Steep Flat PIRP PIRP PIRP PIRP PIRP PIRP Fnally, to better understand how dfferent revenues for products of dfferent ages affect the trade-off between each of the three polces, we have conducted the followng experments. We have selected one nstance (PIRP ) and we have solved t usng the three polces for several slopes of the revenue functons. Specfcally, we have set the revenue of a fresh product to 20 and we have set the revenue of the oldest tem rangng from zero to 20, n steps of one unt. We have then plotted the values of the objectve functons of each one n the graph of Fgure 2. These new sets of experments confrm that on our data set the FF polcy provdes soluton values that are almost dentcal to the OP polcy. Note how the thn contnuous lne of the OP polcy s only slghtly hgher than the dotted lne of the FF polcy, but vsually ndstngushable from t. Ths mples that here the optmal polcy tends to favor the sale of fresher products. The OF polcy, on the other hand, provdes solutons whose cost s greatly affected by the revenue value of older products. The dfference between the polces s largest when these products are valued very low. When the revenue value for CIRRELT

24 160000# Varable'revenue'func%ons' # Solu%on'value' # # # OP#polcy# OF#polcy# FF#polcy# # 0# 0.1# 0.2# 0.3# 0.4# 0.5# 0.6# 0.7# 0.8# 0.9# 1# Revenue'for'one'unt'of'the'oldest'tem'as'a'frac%on'of'the'revenue'of'a'fresh'tem' (low'='steep'revenue'func%on,'hgh'='mld'revenue'func%on)' Fgure 2: Varable revenue functons. The horzontal axs ndcate the revenue for one unt of the oldest tem as a fracton of the revenue of a fresh tem. Low values on the horzontal axs ndcate a steep revenue functon wth respect to the age of the products. Hgh values on the horzontal axs ndcate a mld revenue functon wth respect to the age of the products. 22 CIRRELT

25 older products ncreases, so does ncrease the proft of applyng an OF polcy, and the dfference between ths polcy and the other two tends to vansh. 6 Conclusons We have ntroduced the jont replenshment and nventory control of pershable products. We have modeled the problem under general assumptons as a MILP, and we have solved t exactly by branch-and-cut. We have also ntroduced, modeled and solved exactly two varants of the problem defned by applyng the OF and the FF sellng prorty polces, n whch the retaler sells wth hgher prorty older and fresher tems, respectvely. Our model remans lnear even when the product revenue decreases n a non-lnear or even n a non-convex fashon over tme. It keeps track of the number of tems of each age, and consders dfferent holdng costs for products of dfferent ages. The model optmally determnes whch tems to sell at each perod based on the trade-off between cost and revenue. The algorthm can effectvely compute optmal jont replenshment and delvery decsons for pershable products n an nventory-routng context for medum sze nstances. We have also shown that on our testbed, the proft s drastcally reduced when an OF polcy s appled, but the decrease s only margnal under an FF polcy. Extensve computatonal experments carred out on randomly generated nstances support these conclusons. Acknowledgments Ths work was partly supported by the Canadan Natural Scences and Engneerng Research Councl under grant Ths support s gratefully acknowledged. We also thank Calcul Québec for provdng hgh performance parallel computng facltes. CIRRELT

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27 Table 5: Detaled results of the computatonal experments for the PIRP OP polcy FF polcy OF polcy Instance sze Best known Best known Best known Best known Best known Best known Gap (%) Tme (s) Gap (%) Tme (s) (n-s-k-h ) soluton value upper bound soluton value upper bound soluton value upper bound Gap (%) Tme (s) PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP PIRP CIRRELT

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