Formulations, Branch-and-Cut and a Hybrid Heuristic Algorithm for an Inventory Routing Problem with Perishable Products

Transcription

1 Formulatons, Branch-and-Cut and a Hybrd Heurstc Algorthm for an Inventory Routng Problem wth Pershable Products Aldar Alvarez Jean-Franços Cordeau Raf Jans Pedro Munar Renaldo Morabto October 2018 CIRRELT

2 Formulatons, Branch-and-Cut and a Hybrd Heurstc Algorthm for an Inventory Routng Problem wth Pershable Products Aldar Alvarez 1,2, *, Jean-Franços Cordeau 2,3, Raf Jans 2,3, Pedro Munar 1, Renaldo Morabto 1 1. Department of Producton Engneerng, Federal Unversty of São Carlos, , São Carlos- SP, Brazl 2. Interunversty Research Centre on Enterprse Networks, Logstcs and Transportaton (CIRRELT) 3. Department of Logstcs and Operatons Management, HEC Montréal, 3000 Côte-Sante- Catherne, Montréal, Canada H3T 2A7 Abstract. In ths paper, we study an nventory routng problem n whch goods are pershable. In ths problem, a sngle suppler s responsble for delverng a pershable product to a set of customers durng a gven fnte multperod plannng horzon. The product s assumed to have a fxed shelf-lfe durng whch t s usable and after whch t must be dscarded. Age-dependent revenues and nventory holdng costs are consdered. We ntroduce four mathematcal formulatons for the problem, two wth a vehcle ndex and two wthout a vehcle ndex. Branch-and-cut algorthms are proposed to solve them. In addton, we propose a hybrd heurstc soluton method for the problem. The method s based on the combnaton of an terated local search metaheurstc and two mathematcal programmng components. We present extensve computatonal experments usng nstances from the lterature as well as new larger nstances. Our results ndcate that, when compared to an arc-based formulaton from the lterature, the formulatons wthout a vehcle ndex can provde a consderably larger number of optmal and feasble solutons wthn the mposed tme lmt. Addtonally, a consderable speed-up s acheved for those nstances solved to optmalty wthn the tme lmt. The results wth the hybrd method show that t s able to provde hgh-qualty solutons n relatvely short runnng tmes for small- and medum-szed nstances whle good qualty solutons are found wthn reasonable runnng tmes for larger nstances. Keywords. Logstcs, nventory routng, pershablty, hybrd method, terated local search. Acknowledgements. Ths work was supported by the São Paulo Research Foundaton (FAPESP) under Grant numbers 2017/ and 2017/ ; the Brazlan Natonal Councl for Scentfc and Technologcal Development (CNPq); and the Coordnaton for the Improvement of Hgher Educaton Personnel (CAPES). Ths support s gratefully acknowledged. 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: Aldar.Alvarez@crrelt.ca Dépôt légal Bblothèque et Archves natonales du Québec Bblothèque et Archves Canada, 2017 Alvarez, Cordeau, Jans, Munar, Morabto and CIRRELT, 2018

3 1. Introducton Research on the ntegraton of multple actvtes throughout the supply chan has ncreased consderably n the last decades. Today, t s well known that such ntegraton can lead to sgnfcant advantages n both economc and performance terms. In partcular, the ntegraton of transportaton and nventory management actvtes has been shown to provde substantal economc benefts and to mprove the usage of the avalable resources. However, challengng problems can arse from ths ntegraton, one of whch s the nventory routng problem (IRP). The IRP conssts of defnng the optmal replenshment plan of the customers of a suppler throughout a plannng horzon as well as the routng schedule n each tme perod such that a gven objectve s optmzed. In many dfferent ndustres, raw materals, as well as ntermedate and fnal products, are often pershable. Moreover, pershablty may appear n more than one actvty throughout the supply chan and can nfluence servce levels (Amorm et al., 2013). Thus, managng pershablty becomes a relevant ssue n the supply chan, partcularly n the nventory control actvtes. Pershablty was frst studed n the context of IRPs by Federgruen et al. (1986), who addressed an nventory management and dstrbuton problem for a product that must be dscarded f t not used durng a gven fxed lfetme. The authors studed dfferent patterns and polces for the dstrbuton part of the problem. The objectve was to mnmze the sum of transportaton and expected shortage and dscardng costs. Hemmelmayr et al. (2009) studed the problem faced by a blood bank n the dstrbuton to hosptals. In ther problem, no vehcle capacty constrants were consdered (gven the small sze of the blood bags) but the maxmum length of the routes was lmted. Also, no nventory holdng costs were consdered snce t s preferable to mantan hgh nventory levels rather than to experence stockouts, gven the nature of the servce beng provded. The objectve s to mnmze travel costs over a fnte horzon. The authors proposed dfferent strateges for the dstrbuton process. Le et al. (2013) studed an IRP wth pershablty features also motvated by a healthcare applcaton. In ther problem, t was assumed that the pershable goods have a fxed shelf-lfe, and they are not usable when ths lfetme s exceeded. Upper bounds on the nventory levels of customers were determned only by the pershablty constrants snce the dscardng of products s not allowed. Thus, delveres to customers at any gven tme perod were lmted only by the shelf-lfe of the goods. The objectve was to mnmze the sum of travel and nventory holdng costs. Dabat et al. (2016) addressed the same problem as Le et al. (2013) but only mnmzng travel costs. Coelho and Laporte (2014) consdered an IRP wth a fxed shelf-lfe pershable product wth agedependent revenues and holdng costs. They presented a mathematcal formulaton and explored dfferent strateges to model the product consumpton at the customer facltes. Mrzae and Sef (2015) addressed an IRP for pershable goods n whch the objectve functon ncluded a penalty that depends on the age of the product that s used to satsfy the demands. Ths penalty was ncluded n an attempt to avod overstockng to reduce transportaton costs. The objectve was to mnmze the sum of routng, nventory and penalty costs. Soysal et al. (2015) addressed an IRP wth a fxed shelf-lfe pershable product. The authors proposed models that also consdered fuel consumpton and demand uncertanty. Splt delveres and backloggng of the demand were allowed as well. The objectve was to mnmze the sum of routng (drver wages and fuel consumpton), nventory and spolage costs. Azadeh et al. (2017) studed an IRP of a sngle pershable product wth an exponentally decayng nventory. The authors CIRRELT

4 ncluded the possblty of transshpments between customers (performed by an outsourced thrd-party operator) snce a sngle vehcle wth lmted capacty was consdered. Backloggng was not allowed and the objectve was to mnmze the sum of nventory and travel costs (ncludng transshpments costs) as well as spolage costs. Crama et al. (2018) addressed an IRP for a sngle pershable product wth stochastc demands (wth a known probablty dstrbuton). A maxmum tme on the duraton of the routes was mposed and no salvage value was ncluded n ther problem. Fnally, Shaaban and Kamalabad (2016) and Qu et al. (2018) studed producton-routng problems (PRP) for pershable products. PRPs add producton decsons to the IRP n an attempt to jontly optmze producton, nventory and routng decsons (Adulyasak et al., 2015). Shaaban and Kamalabad (2016) addressed the case wth multple products. In ther problem, pershablty was modeled as n Le et al. (2013),.e., upper bounds on the nventory levels are determned only by the pershablty constrants and dscardng of products s forbdden. Qu et al. (2018) addressed an PRP ncludng deteroraton rates and nventory holdng costs that are both age-dependent. The authors tested dfferent delvery and sellng prorty polces. In ths paper, we address the IRP for a sngle pershable product as proposed by Coelho and Laporte (2014), whch we wll refer to as the PIRP (pershable IRP). In ths problem, the pershablty feature s modeled by defnng a fxed shelf-lfe for an agng product as well as settng nventory holdng costs and sales revenues varyng wth the product freshness. The contrbutons of ths paper are threefold. Frst, we present and compare four mathematcal formulatons of the problem, whch are solved usng branchand-cut (B&C) algorthms. We also report an nconsstency n the mathematcal formulaton presented by Coelho and Laporte (2014) and show how we addressed t. Second, we develop a hybrd heurstc method based on the combnaton of an terated local search (ILS) metaheurstc and two mathematcal programmng components. Thrd, we report the results of extensve computatonal experments and ntroduce new large-szed problem nstances. The results show the dfferent advantages of the proposed formulatons and also reveal the effectveness of our method. We also present a further analyss of the robustness of the algorthm s behavor. The remanng sectons of ths paper are organzed as follows. In Secton 2, we descrbe the problem that we address n ths research. Secton 3 presents the mathematcal formulatons ntroduced for the problem and the B&C algorthms used to solve them. Then, the hybrd soluton method that we developed s descrbed n detal n Secton 4. Secton 5 shows the computatonal experments that we performed wth the formulatons and the hybrd method. Fnally, n Secton 6 we conclude the paper and dscuss future research. 2. Problem descrpton In the PIRP (Coelho and Laporte, 2014), a suppler s responsble for delverng a sngle pershable product to a set of customers durng a gven fnte multperod plannng horzon. The product s assumed to have a fxed shelf-lfe durng whch t s usable and after whch t must be dscarded. The problem can be defned on a complete undrected graph G = (N, E) where N = {0, 1,..., N} s the vertex set and E = {(, j):, j N, < j} s the edge set. Vertex 0 represents the suppler depot whch has a homogeneous fleet of K vehcles of capacty Q, denoted by set K = {1,..., K}. The remanng vertces of set N, denoted by C = {1,..., N}, represent the customers. Therefore, the vertex 2 CIRRELT

5 set N represents all the facltes of the dstrbuton network. The plannng horzon s denoted by a set of tme perods T = {1,..., T }. The pershable product under consderaton spols S tme perods after becomng avalable at the suppler and ts age ncreases by one unt n every tme perod. Thus, the age of the product belongs to a dscrete set S = {0, 1,..., S}. The age of the product defnes ts value, accordng to the sales revenue u s specfed for each unt of age s S consumed by customer C. A travel cost c j s assocated wth every edge (, j) E and an age-dependent nventory holdng cost h s s charged at both the suppler 0 and the customers C for each unt of product of age s S at the end of every tme perod. Each customer C has a lmted storage capacty C and each faclty N has an ntal nventory I0 0 of fresh product (of age 0) avalable at the begnnng of the plannng horzon (t = 0). Thus, the ntal nventory wll be of age 1 n the frst tme perod of the plannng horzon (t = 1). Each customer C has a known demand d t for the product n every tme perod t T, whch s the mnmum amount of product that the suppler must guarantee to be avalable at the customer at that tme perod. In addton, the suppler produces or receves a quantty r t of fresh products (of age 0) n each tme perod t T. However, ths quantty s avalable for delvery only one tme perod after becomng avalable at the suppler s faclty. To llustrate the agng process of the nventory durng the plannng horzon, Fgure 1 shows an example of the evoluton of the end-of-perod nventory for a gven customer. Assume a maxmum age of two tme perods (S = 2), no consumptons durng the plannng horzon and two delveres from the suppler of 70 and 50 unts of age 1 (s = 1) n tme perods two and three, respectvely. The ntal nventory of the customer conssts of 100 unts (s = 0 at t = 0), whch become of age 1 n the frst tme perod (s = 1 at t = 1) and then of age 2 n the second tme perod (s = 2 at t = 2) of the plannng horzon. Notce that these 100 unts of the product reached the maxmum age (s = S = 2) n tme perod 2, n whch they are stll usable to satsfy potental demand n perod 2. These unts of maxmum age wll stll be held n nventory at the end of perod 2, but they wll be dscarded n perod 3 and hence do not appear n the nventory n tme perod 3. Smlarly, the amount receved n t = 2, whch was of age 1 (s = 1), becomes of age 2 (s = 2) n tme perod 3, reachng the maxmum age but stll beng usable to satsfy potental demand n perod 3. Ths amount wll be dscarded n perod 4 and wll not be n the usable nventory from tme perod 4 onwards Inventory level Tme perod Age 0 Age 1 Age 2 Fgure 1: Agng process of the nventory at a customer n the PIRP CIRRELT

6 The PIRP conssts of determnng the tme perods n whch the customers wll be vsted; the quantty of product of each avalable age that wll be delvered n every vst; the quantty of product of each avalable age that wll be used to satsfy the demand; and the delvery routes to perform those vsts. The objectve s to maxmze the total proft, gven by the sales revenue mnus the sum of nventory holdng and routng costs. The holdng costs are charged on the nventores at the end of each tme perod at both the suppler and customers. We consder that products of dfferent ages share the same jont holdng space at all facltes. It s also assumed that the suppler holdng capacty s unbounded. In addton, accordng to the usual practce n the lterature, we assume that the customers who receve a delvery n a gven tme perod can use ths to fulfll the demand n the same tme perod. As n Coelho and Laporte (2014), we assume that the product that has reached ts maxmum age (s = S) at the end of a tme perod, wll not go nto the regular storage area, but wll be kept separately n nventory to be dscarded n the next perod. Thus, these amounts ncur the nventory holdng costs but do not lmt the quantty that the customer can receve n the next tme perod. Table 1 summarzes all the prevously ntroduced notaton as well as the one that wll be used n the followng sectons. Sets: C N E T S K Set of customers Set of vertces/facltes Set of edges Set of tme perods Set of ages of the product Set of vehcles Indces:, j, h Vertces/facltes s Ages of the product k Vehcles t, m, p Tme perods Parameters: u s h s c j d t r t C I0 0 Q Unt revenue for product of age s at customer Unt nventory holdng cost for product of age s at faclty Transportaton cost between facltes and j Demand of customer n tme perod t Amount made avalable at the suppler n tme perod t Storage capacty of customer Intal nventory at faclty Capacty of the vehcles Table 1: Sets, ndces and parameters of the problem 3. Mathematcal formulatons Ths secton presents the mathematcal formulatons we ntroduce for the PIRP. Frst, we present a corrected verson of the arc-based formulaton ntroduced by Coelho and Laporte (2014) and show why there s an nconsstency n ther formulaton. Then, n the subsequent sectons we present several reformulatons of the problem Arc-based formulaton To formulate the PIRP usng arc varables as n Coelho and Laporte (2014), consder the followng notaton. Frst, we ntroduce the set S t = {s S : 1 s t}, whch s the subset of product ages 4 CIRRELT

7 that can be avalable at all facltes n tme perod t. Ths set ndcates the ages that can be delvered by the suppler n each tme perod and also specfes the ages that can be used to satsfy the demand of the customer n the gven tme perod. Notce that ths set does not contan age 0, whch s also part of the ages set S and s avalable at the suppler n each tme perod, gven that the suppler never delvers products of age 0 to the customers because the amount made avalable at the suppler faclty n a certan perod can only be delvered n the followng perod. Also, let U = mn{q, C } be an upper bound on the amount that can be delvered to customer n tme perod t. Fnally, consder the followng decson varables: x kt j {0, 1, 2} : number of tmes vehcle k K traverses edge (, j) E n tme perod t T, y kt {0, 1} : 1 f faclty N s vsted by vehcle k K n perod t T, 0 otherwse; I t s 0 : nventory of age s S at faclty N at the end of tme perod t T ; q kt s 0 : quantty of product of age s S delvered to customer C by vehcle k K n perod t T ; d t s 0 : quantty of product of age s S used to fulfll the demand of customer C n perod t T. Gven these varables, the arc-based (AB) formulaton of the problem can be stated as: max C t T s S t u s d t s (,j) E k K t T c j x kt j N t T h s Is t h 00 r t (1) s S t t T s.t. I0s t = r t t T, s = 0, (2) I0s t = I0,s 1 t 1 qs kt t T, s S t, (3) C k K Is t = I,s 1 t 1 + qs kt d t s C, t T, s S t, (4) k K d t = s S t d t s C, t T, (5) s S t 1 \{S} I t 1 s + k K s S t q kt s C C, t T, (6) qs kt U y kt C, k K, t T, (7) s S t qs kt Qy0 kt k K, t T, (8) C s S t j N :j< x kt j + B j B:j> k K j N :j> x kt j = 2y kt N, k K, t T, (9) x kt j y kt ye kt B C, B 2, k K, t T, e B, (10) B y kt 1 C, t T, (11) I t s 0 N, t T, s S t, (12) q kt s 0 C, k K, t T, s S t, (13) CIRRELT

8 d t s 0 C, t T, s S t, (14) y kt {0, 1} N, k K, t T, (15) x kt j {0, 1} (, j) E : 0, k K, t T, (16) x kt j {0, 1, 2} (, j) E : = 0, k K, t T. (17) The objectve functon (1) conssts of maxmzng the total proft, gven by the total revenue mnus the sum of transportaton and nventory holdng costs. The last term of the objectve functon accounts for the nventory holdng cost ncurred by the amount of fresh product made avalable at the suppler n each tme perod of the plannng horzon. Ths term can be gnored as t s a constant, but for the sake of completeness, we decded to keep t n the objectve functon. Constrants (2) (3) defne the nventory conservaton at the suppler, where the frst constrant set explctly defnes the nventory of age 0 n each tme perod and the second constrant set defnes the nventory for ages n set S t. Constrants (4) defne the nventory conservaton at the customers. Constrants (5) guarantee the fulfllment of the customer demands, whch can be done wth products of dfferent ages (but only wth the ages that are avalable n the tme perod of the demand). Constrants (6) mpose that the nventory level after delvery at the customer facltes cannot exceed ther storage capacty. Notce that products of dfferent ages share the same storage space. Also, note that products of age S stll avalable at the end of tme perod t 1 wll not enter nto the storage space and hence do not lmt the amount that can be delvered n perod t. Constrants (7) permt a vehcle to perform a delvery to a specfc customer only f ths customer s vsted by the vehcle. Constrants (8) guarantee that the capacty of each vehcle s respected. Constrants (9) ensure the flow conservaton. Constrants (10) are subtour elmnaton constrants (SECs). Constrants (11) mpose that each customer can be vsted at most once n each tme perod. Fnally, the doman of the decson varables s defned n constrants (12) (17). Notce that when 0 and j >, x kt j can only take the values 0 or 1; f = 0, then xkt j can also be equal to 2, ndcatng that vehcle k makes a round trp between the depot and customer j n tme perod t. Ths formulaton has two man dfferences wth respect to the one proposed by Coelho and Laporte (2014). Frst of all, n ther formulaton, sums over varables of dfferent ages (as n constrants (5) (8) and n the objectve functon) consder the whole set of ages S, nstead of the subset S t, whch we ntroduce here. Second, the authors defne nventory conservaton constrants for products of age 0 for the customers (n the form I0 t = k K qkt 0 dt 0, C, t T ), although the suppler cannot delver these products and the customers never receve products of age 0, accordng to the assumptons of the problem as defned by Coelho and Laporte (2014) va ther suppler nventory constrants. These two dfferences can lead to an nternal nconsstency n ther formulaton. More specfcally, n the Coelho and Laporte (2014) formulaton, the varable q0 kt s defned, but only appears n the nventory conservaton constrant for products of age 0 at the customers. Furthermore, ther demand fulfllment constrants enable the satsfacton of the demand usng products of age 0. As a result, the solutons of ther formulaton can have consumptons (d t s ) and delveres (qt s ) of products of age 0 wthout subtractng these amounts from the suppler s nventory. Ths can be benefcal n a soluton because the products of age 0 have a hgh revenue n the nstances proposed by those authors. Notce that f k K qkt 0 delveres of products of age 0 were to be allowed, the term C should be subtracted from the rght-hand sde of constrants (2) and the set S t should nclude 0 as well. 6 CIRRELT

9 3.2. Transportaton formulaton I The frst reformulaton we propose uses decson varables that explctly ndcate the detaled use of the delveres of each age,.e., the tme perods n whch the delvery wll cover all or part of the demand, as n the faclty locaton formulaton of the sngle tem uncapactated lot szng problem, ntroduced by Krarup and Blde (1977). For ths, we ntroduce some addtonal notaton. Let Ts t = mn{t, t s + S} be the last tme perod n whch product that s of age s n tme perod t can be used to satsfy any demand. We also consder an addtonal fcttous tme perod T + 1 n order to handle nventores at the end of the plannng horzon. Consder the followng decson varables: q ktm s 0 : quantty of product of age s S delvered to customer C by vehcle k K n perod t T to cover the demand of perod m {t,..., T t s + 1}; b t 0 : amount of the ntal nventory of customer C used to fulfll ts own demand n tme perod t T. Note that n the defnton of the varable qs ktm, the ndex age (s) refers to the age of the product at the tme of the delvery. Notce that when m = T + 1 n the delvery varables (q), t ndcates that the quantty delvered wll reman n the customer nventory at the end of the plannng horzon. Also, when m = (t s + S) + 1 t means that the product wll spol and wll be dscarded at the customer faclty n perod m. Usng the ntroduced notaton and varables, the transportaton formulaton (TP-I) can be stated as follows: max (1) s.t. I t 0s = r t s C I t s = k K s 1 Is t = k K t =0 d t s = s 1 k K t =0 d t s = s 1 k K s S t 1 \{S} T t s +1 s S t m=t t =0 m=t+1 s 1 t =0 I t 1 s q ktm s s 1 k K t =0 T t s +1 T t s +1 m=t+1 T t s +1 q k,t t,m,s t t T, s S t, (18) m=t t q k,t t,m,s t C, t T, s S t : s < t, (19) q k,t t,m,s t + I 0 0 t t =1 b t C, t T : t S, s = t, (20) q k,t t,t,s t C, t T, s S t : s < t, (21) q k,t t,t,s t + b t C, t T : t S, s = t, (22) + k K T t s +1 s S t m=t q ktm s C C, t T, (23) U y kt C, k K, t T, (24) CIRRELT

10 T t s +1 C s S t m=t q ktm s Qy kt 0 k K, t T, (25) q ktm s 0 C, k K, t T, s S t, m {t,..., T t s + 1}, (26) b t 0 C, t T : t S, (27) (5), (9) (12) and (14) (17). Constrants (18) defne the nventory at the suppler for all tme perods and avalable ages (where r 0 = I 0 00 ). Constrants (19) (20) defne the nventory at the customers for all the dfferent ages of the product, wth (20) ncludng the amount from the ntal nventory that s not used to satsfy any demand. These constrants can be easly generalzed for the case when the ntal nventory s composed of products of dfferent ages. Constrants (21) (22) defne the amount of product of each dfferent age used to fulfll the demand of the customers. Notce that constrants (22) nclude the demand that can be fulflled usng the ntal nventory as well. Constrants (23) mpose that the nventory level after delvery at the customers cannot exceed ther storage capacty. Constrants (24) allow a vehcle to perform a delvery to a specfc customer only f ths customer s vsted by the vehcle. Constrants (25) guarantee that the capacty of each vehcle s respected. Fnally, constrants (26) (27) defne the doman of the new decson varables Formulaton TP-I wthout a vehcle ndex The prevous formulaton can be reformulated by droppng the vehcle ndex of the varables, as we consder a homogeneous vehcle fleet (n both capacty and travel cost terms) and assume at most one vst to each customer n each tme perod. All the varables mantan the same meanng, except for y0 t whch becomes an nteger varable ndcatng the number of vehcles used n perod t T. The formulaton, whch we wll refer to as TP-I-nk, can be stated as: max s.t. C t T s S t u s d t s I t 0s = r t s C I t s = I t s = s 1 T t s +1 t =0 m=t+1 s 1 s 1 d t s = t =0 m=t+1 t =0 s 1 d t s = t =0 s S t 1 \{S} T t s +1 s 1 t =0 (,j) E t T T t s +1 c j x t j h s Is t h 00 r t (28) N t T s S t t T m=t t q t t,m,s t t T, s S t, (29) q t t,m,s t C, t T, s S t : s < t, (30) q t t,m,s t + I 0 0 t t =1 b t C, t T : t S, s = t, (31) q t t,t,s t C, t T, s S t : s < t, (32) q t t,t,s t + b t C, t T : t S, s = t, (33) I t 1 s + s S t T t s +1 m=t q tm s C C, t T, (34) 8 CIRRELT

11 T t s +1 s S t m=t j N :j< q tm s U y t C, t T, (35) x t j + j N :j> x t j = 2y t N, t T, (36) Q x t j (Qy t T t s +1 qs tm ) B C, B 2, t T, (37) B j B:j> B s S t m=t y t 0 K t T, (38) q tm s 0 C, t T, s S t, m {t,..., T t s + 1}, (39) y t 0 Z + t T, (40) y t {0, 1} C, t T, (41) x t j {0, 1} (, j) E : 0, t T, (42) x t j {0, 1, 2} (, j) E : = 0, t T, (43) (5), (12), (14) and (27). The objectve functon (28) conssts of maxmzng the total proft. Constrants (29) and (30) (31) defne the nventory level for the dfferent ages of the product at the suppler and customers, respectvely, where r 0 = I00 0 n constrants (29). Constrants (32) (33) defne the amount of each dfferent age used to fulfll the demand of the customers. Constrants (34) mpose the maxmum storage capacty at the customers. Constrants (35) allow to perform delveres to a specfc customer only f t s vsted by a vehcle. Constrants (36) ensure the conservaton of the flow. Constrants (37) are SECs and ensure that the vehcle capactes are respected as well. Constrants (38) lmt the number of vehcles that can be used n each tme perod. Constrants (39) (43) defne the doman of the decson varables. Notce that, as ponted out by Adulyasak et al. (2014), f one dvdes the nequaltes (37) by Q, they have a form smlar to the generalzed fractonal SECs (GFSECs) for the vehcle routng problem (VRP) (Toth and Vgo, 2002). However, GFSECs n the form (37) are numercally more stable than the orgnal GFSECs, whch contan a fractonal rght-hand sde Transportaton formulaton II Ths reformulaton, smlar to TP-I, uses decson varables that explctly ndcate the detaled use of the delveres of each age. However, n ths case we consder mplctly the age of the product beng delvered. Thus, the delvery varable s defned as follows: q ktpm 0 : amount of product that was made avalable at the suppler n tme perod t {0} T and was delvered to customer C by vehcle k K n perod p T to cover the demand of perod m {p,..., T t 0 + 1}. Note that n the defnton of the varable q ktpm, the age of the product at delvery and consumpton perods s gven by the dfference between ndex t and ndces p and m, respectvely. Notce that when t = 0, the amount delvered comes from the ntal nventory of the suppler. Notce also that, analogously to formulaton TP-I, when m = T + 1 n the delvery varables (q) the quantty delvered CIRRELT

12 wll reman n the customer nventory at the end of the plannng horzon and when m = t + S + 1 the product delvered wll spol and be dscarded at the customer faclty n perod m. The formulaton, whch we wll refer to as TP-II, can be stated as: max (1) s.t. I t 0s = r t s C I t s = k K I t s = k K d t s = k K d t s = k K s S t 1 \{S} T t s +1 s S t m=t t t T t s +1 k K p=t s+1 m=p T t s +1 p=t s+1 m=t+1 t T t s +1 p=1 m=t+1 t p=t s+1 t p=1 I t 1 s C s S t m=t q k0pm q k,t s,p,m t T, s S t, (44) q k,t s,p,m C, t T, s S t : s < t, (45) + I 0 0 t t =1 b t C, t T : t S, s = t, (46) q k,t s,p,t C, t T, s S t : s < t, (47) q k0pt + b t C, t T : t S, s = t, (48) q k,t s,tm T t s +1 + k K q k,t s,tm T t s +1 s S t m=t q k,t s,tm C C, t T, (49) U y kt C, k K, t T, (50) Qy kt 0 k K, t T, (51) q ktpm 0 C, k K, t {0} T \{T }, p {t + 1,..., T t 0}, m {p,..., T t 0 + 1}, (52) (5), (9) (12), (14) (17) and (27). Constrants (44) calculate the nventory at the suppler for the avalable ages, where r 0 = I00 0. Constrants (45) (46) defne the nventory at the customers for all the dfferent ages of the product. Constrants (47) (48) state that the demand of the customers can be satsfed usng products of all the avalable ages. Notce that constrants (48) nclude the demand that can be fulflled usng the ntal nventory as well. Constrants (49) mpose the maxmum storage capacty after delvery at the customer facltes. Constrants (50) allow delveres to a customer by a specfc vehcle only f t s vsted by the same vehcle. Constrants (51) guarantee that the capacty of each vehcle s respected. Fnally, constrants (52) defne the doman of the new decson varable Formulaton TP-II wthout a vehcle ndex As n Secton 3.3, an addtonal formulaton can be obtaned by droppng the vehcle ndex of the varables for cases n whch the vehcle fleet s consdered to be homogeneous and at most a sngle vst s allowed to each customer n each tme perod. Ths formulaton (TP-II-nk) can be stated as: 10 CIRRELT

13 max (28) s.t. I t 0s = r t s C I t s = I t s = d t s = d t s = t p=t s+1 m=t+1 t T t s +1 p=1 m=t+1 t p=t s+1 t p=1 s S t 1 \{S} T t s +1 s S t m=t t T t s +1 p=t s+1 m=p T t s +1 q 0pm q t s,p,m t T, s S t, (53) q t s,p,m C, t T, s S t : s < t, (54) + I 0 0 t t =1 b t C, t T : t S, s = t, (55) q t s,p,t C, t T, s S t : s < t, (56) q 0pt + b t C, t T : t S, s = t, (57) I t 1 s + s S t T t s +1 m=t q t s,tm C C, t T, (58) q t s,tm U y t C, t T, (59) Q x t j B j B:j> B (Qy t T t s +1 s S t m=t q t s,tm ) B C, B 2, t T, (60) q tpm 0 C, t {0} T \{T }, p {t + 1,..., T t 0}, m {p,..., T t 0 + 1}, (61) (5), (12), (14), (27), (36), (38) and (40) (43). Constrants (53) and (54) (55) defne the nventory levels of the avalable ages of the product at the suppler and customers, respectvely, where r 0 = I00 0 n constrants (53). Constrants (56) (57) specfy that the demand of the customers can be satsfed usng products of all the avalable ages. Constrants (58) enforce the maxmum storage capacty at the customer facltes. Constrants (59) allow delveres to customers n a gven tme perod f they are vsted by a vehcle n the same tme perod. Constrants (60) are the GFSECs and guarantee that the capacty of each vehcle s respected as well. Fnally, constrants (61) defne the doman of the new decson varable Branch-and-cut algorthms Gven that all the presented formulatons contan an exponentally large number of SECs, we must apply a B&C algorthm to solve them. These constrants are dropped from the formulatons and added n an teratve fashon every tme they are volated at the nodes of the branch-and-bound (B&B) tree. In ths secton we provde the detals of our B&C approaches for both the formulatons wth and wthout a vehcle ndex as well as further mprovements. Addtonally, n the supplementary materal of ths paper, we show some vald nequaltes that we used to further strengthen the formulatons and provde detaled results of all the formulatons when ncludng them. CIRRELT

14 Branch-and-cut for the vehcle ndex formulatons To solve the formulatons wth a vehcle ndex, we use an exact separaton algorthm that solves a seres of mnmum s t cut problems to detect volated SECs for each vehcle n each tme perod of the plannng horzon. At a gven node of the B&B tree, let ȳ kt and x kt j denote the values of the vst (y) and flow varables (x) of the soluton, respectvely. A graph for vehcle k n tme perod t s constructed from the set of nodes where ȳ kt > 0, settng the weghts of the graph edges to x kt j, (, j) E. Then, for each customer node of the constructed graph, we solve a mnmum s t cut problem, settng the suppler node as the source node (s) and the customer node as the snk node (t). A volated SEC s dentfed f the capacty of the mnmum cut s less than 2ȳ kt (Adulyasak et al., 2014). If a subtour on a set of nodes B C s found for vehcle k n perod t, we add constrants (10) wth e = arg max B {ȳ kt} to the formulaton, for all vehcles and tme perods of the plannng horzon. To solve the mnmum s t cut problem, we used the Concorde solver (Applegate et al., 2018). These SECs are separated only at the root node and then every tme an nteger soluton s found at a node of the B&B tree, to avod generatng too many cuts n the tree. Notce that constrants (10) can be added to the formulaton n many dfferent ways, among whch we tested: addng the cut only for the specfc vehcle and tme perod for whch t was volated; addng the cut for all vehcles n the same tme perod n whch the volated cut was dentfed; and, fnally, addng the cut for all vehcles and tme perods. The latter strategy resulted n a slghtly better performance. Regardng the selecton of the customer e B for whch the cut would be set, we tred ncludng the cut only for the customer e such that e = arg max B {ȳ kt } and, for every customer n the dentfed subset of customers B. In ths case, the former strategy resulted n a better performance of the B&C algorthms Branch-and-cut for the formulatons wthout a vehcle ndex To separate GFSECs (37) and (60) of formulatons TP-I-nk and TP-II-nk, respectvely, we use the separaton package developed by Lysgaard et al. (2004) for the VRP, as n Adulyasak et al. (2014). Ths algorthm conssts of four heurstc algorthms, whch are appled sequentally. One of these heurstcs s an exact separaton algorthm when all the flow varables (x) take nteger values. For a gven soluton, we call the algorthm for each tme perod t T. At a gven node of the B&B tree, let ȳ kt, xkt j and qtm s ( q tpm ) denote the values of varables y kt, xkt j and qtm s (q tpm ) for the formulaton TP-I-nk (TP-II-nk). The nput requred by the separaton package (a VRP soluton) n perod t s constructed consderng customer nodes wth ȳ kt > 0, settng the weght of each edge (, j) to x kt j and settng the delvery quantty for customer to T t s +1 s S t m=t qtm s and to T t s +1 s S t m=t qt s,tm for the formulatons TP-I-nk and TP-II-nk, respectvely. Smlar to the formulatons wth a vehcle ndex, we separate GFSECs only at the root node and then whenever an nteger soluton s found at a gven node of the B&B tree to avod generatng too many cuts. Every tme we dentfy a volated cut, we add the correspondng constrant for all tme perods. In addton, to further strengthen formulatons TP-I-nk and TP-II-nk, we ncluded the followng SECs, as used n the formulatons wth a vehcle ndex (AB, TP-I, TP-II): B j B:j> x t j B y k y k e B C, B 2, k K, e B. (62) Usng these constrants together wth GFSECs resulted n an mproved performance by the formu- 12 CIRRELT

15 latons wthout a vehcle ndex. These last SECs are separated as descrbed n Secton Optmzaton-based terated local search In ths secton, we present the hybrd algorthm that we propose to solve the PIRP. Ths algorthm s based on the ILS metaheurstc for the basc varant of the IRP presented by Alvarez et al. (2018). Addtonally, we nclude two mathematcal programmng components wthn ts structure. The basc dea of ILS s to teratvely apply a local search algorthm to solutons resultng from the perturbaton of the prevously vsted local optma, whch leads to a randomzed walk n the space of local optmal solutons (Lourenço et al., 2003). In the proposed method, the varous decsons of the problem are handled by dfferent components. On the one hand, routng decsons (x) are managed by the local search phase of the method whle the vst varables (y) are mostly handled n the perturbaton phase. On the other hand, a multcommodty flow (MCF) problem formulaton s used to determne the optmal values of the contnuous varables (q, d and I) for a gven set of vst varables (y). Addtonally, a mxed-nteger programmng (MIP) formulaton that can remove and nsert customers from a soluton gven as nput s used as a soluton mprovement (SI) step n the fnal phase of the method. An overvew of the proposed method s shown n Algorthm 1. Algorthm 1: Optmzaton-based terated local search 1 begn 2 O 0 constructon heurstc(); 3 f O 0 then 4 O rvnd heurstc(o 0 ); 5 whle stop crteron s not met do 6 O perturbaton(o ); 7 O rvnd heurstc(o ); 8 O optmze amounts(o ); 9 f f(o ) > f(o ) then O O ; 10 end 11 O SI formulaton(o ); 12 end 13 end The algorthm starts wth an ntal feasble soluton (lne 2), whch s generated usng the constructon heurstc that wll be descrbed n Secton 4.1. If the constructon heurstc cannot fnd a feasble soluton, the algorthm stops; otherwse, the search process contnues. A randomzed varable neghborhood descent (RVND) heurstc s used as local search algorthm (lnes 4 and 7), and a multoperator algorthm s used as a perturbaton mechansm (lne 6). The contnuous varables (delveres, consumptons and nventores) of the soluton are then optmzed by solvng a MCF formulaton (lne 8). The acceptance crteron admts the resultng soluton only f t s better than the current best soluton (lne 9). Fnally, after reachng a stoppng crteron, the method apples the SI formulaton (lne 11). All components of the hybrd method are descrbed n detal n the followng sectons. In the descrpton of the hybrd method, we use the subsequent notaton. Gven a soluton O, we denote by Īt s, qkt s, dt s and ȳ kt respectvely. In addton, R(O) s the set of all vehcle routes of the soluton; the values of ts nventory, delvery, consumpton and vst varables, CIRRELT

16 C t (O) = { C : k K ȳkt = 1} s the set of customers vsted by routes of the soluton n tme perod t; T (O) = {t T : k K ȳkt of the soluton. Also, gven a route r R(O) of the soluton, t(r) s the tme perod of the route; and V(r) s the set of customers vsted by the route A constructon heurstc for the PIRP = 1} s the set of tme perods n whch customer s vsted by routes To obtan feasble solutons, we devsed a decomposton constructon heurstc whch teratvely separates the decsons of the problem nto two phases. In the frst phase, the heurstc defnes the sze of the potental delvery to each customer and assgns a prorty to each one of them. Then, n the second phase, feasble delvery routes are desgned to delver the amounts set n the frst phase. The heurstc starts by usng the ntal nventory of each customer to satsfy the maxmum number of demands. The values for the respectve consumpton varables (d t s ) are set as follows: d t s = mn{īt 1, d t } f t S, s = t, 0 otherwse, C, t T, s S t. (63) Then, the heurstc computes the aggregated nventory levels I t for each customer C at the end of each tme perod t T gven the ntal consumptons, as follows: I t I0 0 = t d p p=1 p f t < S, 0 otherwse, C, t T. (64) These nventory levels wll be updated at the end of each teraton based on the delveres and consumptons. They are used to determne the delvery szes n each perod gven that the amount a customer can receve s bounded by the holdng capacty and the nventory level at the end of the prevous tme perod. Notce that, ntally, I t = 0 for t = S snce the ntal nventory wll spol at the end of ths tme perod and, as stated n Secton 2, the spoled nventory does not lmt the amount that the customer can receve n the next perod. Usng these values, the heurstc performs one teraton for each tme perod t T, startng from t = 1. In the frst phase of teraton t, the heurstc sets a potental delvery quantty to each customer by computng the dfference between ts capacty and aggregated nventory level n the prevous tme perod, also respectng the vehcle capacty. To smplfy the heurstc, we apply a greedy approach n whch all the delveres are of the freshest possble product, whch n our case s product of age s = 1. Therefore, the potental delvery ( q) to each customer s set as: q t 1 = mn{rato demand (C I t 1 ), Q}, (65) 14 CIRRELT

17 where rato demand (0, 1] s a parameter that defnes the proporton of the maxmum possble quantty that wll be actually delvered. Next, the prorty π of customer s set as the number of upcomng look ahead perods (ncludng t) n whch ts demand s not fully covered yet,.e., π s the number of tmes n whch d p s S p s < dp, for p = t,..., mn{t, t + look ahead}, C. The value of look ahead determnes how much to look forward n the plannng horzon, tryng to antcpate forthcomng stockouts. After defnng these delveres and prortes, the second phase of teraton t starts. It conssts of determnng one or more vehcle routes usng a nearest-neghbor nserton heurstc that frst routes customers wth hgher prorty as long as the nserton satsfes the vehcle capacty. At most K routes can be defned n ths phase. Then, gven the delveres actually performed, we set the values of q kt 1 and update the values of the consumpton (usng the frst-n frst-consumed (FIFC) rule) and nventory varables. Fnally, a new teraton s started for the next perod (t + 1), untl reachng tme perod T. A pseudo-code of the heurstc s gven n Algorthm 2. Snce the heurstc runs n a short tme, t was defned nsde two outer loops, explorng dfferent values for rato demand and look ahead, amng at fndng a reasonably good feasble soluton, as n Alvarez et al. (2018). Furthermore, at the end of the executon of the heurstc, the contnuous varables of the best feasble soluton found (f any) are optmzed usng the MCF formulaton of Secton 4.4. Fnally, as wll be shown n Secton 5, ths heurstc was able to fnd feasble solutons for all the benchmark nstances used n ths paper. Algorthm 2: Constructon heurstc for the PIRP 1 begn 2 O ; 3 Use ntal nventory to set as many consumptons ( d) as possble, usng (63); 4 Compute aggregated nventory levels I t as n (64), for all C and t T ; 5 rato demand 1.0; 6 whle rato demand > 0 do 7 look ahead 0; 8 whle look ahead S do 9 for t T do 10 for C do 11 q 1 t mn{rato demand (C I t 1 ), Q}; 12 π 0; 13 for p = t,..., mn{t, t + look ahead} do 14 f d p s S p s < dp then π π + 1 ; 15 end 16 end 17 Apply a nearest-neghbor nserton heurstc, routng customers wth hgher π frst; 18 For all routed customers, set the correspondng q 1 kt values and compute the correspondng d t s values usng the FIFC rule and update Īt s and It ; 19 end 20 Update best feasble soluton O ; 21 look ahead look ahead + 1; 22 end 23 rato demand rato demand 0.1; 24 end 25 f O then O optmze amounts(o ) ; 26 end CIRRELT

18 4.2. Randomzed varable neghborhood descent heurstc For the local search procedure of the proposed method, we use a varable neghborhood descent heurstc (Mladenovć and Hansen, 1997) wth random neghborhood orderng. In ths algorthm, local search operators are selected randomly from a predefned set and appled to the ncumbent soluton untl none of them can mprove t (Subramanan et al., 2012; Alvarez and Munar, 2017; Alvarez et al., 2018). The randomzed behavor of RVND enhances the dversfcaton of the method because dfferent local search operators can provde dstnct local optmal solutons. Therefore, a dfferent fnal soluton can be obtaned every tme RVND s appled over the same startng soluton. Moreover, the randomzed order leads to a more balanced exploraton of the neghborhoods, gven that when a fxed sequental order s adopted most of the effort s spent on the frst operators (Deng and Bard, 2011). In our mplementaton, a set contanng several local search operators s ntalzed at the begnnng of the procedure. Then, whle the set s not empty, an operator s chosen at random and appled to the ncumbent soluton. If the operator mproves the soluton, the set s re-establshed to ts ntal confguraton (contanng all the local search operators). Otherwse, the operator s removed from the set and the process contnues wth the remanng operators. A pseudo-code of the RVND heurstc s shown n Algorthm 3. Algorthm 3: Randomzed varable neghborhood descent heurstc 1 begn 2 O O 0 (save ntal soluton); 3 L ntalze the set of local search operators; 4 whle L > 0 do 5 l select at random a local search operator from L; 6 Apply l to O ; 7 f l mproved O then 8 L rentalze the set of local search operators; 9 else 10 remove l from L; 11 end 12 end 13 end In our method, we use the local search phase to handle the routng decsons of the soluton. For ths, the RVND heurstc uses the followng classcal VRP operators: Or-opt-k, k {1, 2, 3}; Shft(k), k {1, 2, 3}; Swap(k 1, k 2 ), k 1, k 2 {1, 2}, k 1 k 2 ; and k-opt, k {2, 3}. All operators explore the search space usng the frst mprovement strategy, allowng only feasble solutons n the search process Perturbaton mechansm Snce n the PIRP there are dfferent decsons that must be made smultaneously, we desgned a perturbaton algorthm that can change multple attrbutes of a soluton n a sngle call. The algorthm uses the followng operators to modfy the vst and delvery decsons of an nput soluton O. 1. Insert vsts: choose randomly a route r R(O) and customer such that / C t(r) (O). The customer s nserted nto the cheapest nserton poston n the route. Then the values of q, d and Ī are re-optmzed usng the MCF formulaton; 2. Remove vsts: choose a random route r R(O) and a customer V(r) and then remove from r. After that, the values of q, d and Ī are re-optmzed usng the MCF formulaton; 16 CIRRELT

19 3. Move vst: choose a random route r R(O) and a customer V(r) such that T (O) < T,.e., a customer that s not vsted n every tme perod of the plannng horzon. Then, the vst to s removed from r and nserted nto the cheapest poston of a route of a perod p T \T (O), choosng both, p and the route, at random. Fnally, the values of q, d and Ī are re-optmzed usng the MCF formulaton; 4. Reduce delveres: choose a random route r R(O) and a delvery (of a certan age s) to a customer V(r) such that the amount delvered s not completely consumed by the customer. Ths can happen, for nstance, when t s proftable to accumulate nventory at the customer to save holdng costs at the suppler. Then, the delvery s reduced by the amount not consumed by the customer. Both the customer and the delvery to be reduced are chosen at random. After applyng each operator, the objectve functon value of the soluton s recomputed. The am of these operators s twofold. Frst, helpng to determne the perods n whch each customer must be vsted and, second, creatng slack n the routes for the local search heurstc. In the Remove and Move operators, nfeasble solutons are rejected. In such a case, the operator chooses another customer of the same route. If all customers of the chosen route are unsuccessfully explored (resultng n nfeasble solutons), the operator chooses another route and the process s appled n the same fashon. Note that the performance of an ILS-based algorthm s strongly related to the strength of ts perturbaton mechansm gven that t defnes much of the behavor of the method. Ths mechansm must be able to dversfy the search process wthout turnng t nto a randomzed restart search. For ths purpose, we use the parameter max perturb, whch defnes the maxmum number of elements of the soluton that can be changed each tme the perturbaton mechansm s called. Thus, smlar to the RVND heurstc, our perturbaton algorthm can use multple operators n a sngle call, applyng one operator at a tme (changng at most one element of the soluton) untl ether the number of changes performed to the soluton reaches max perturb or none of the operators can change the soluton Multcommodty flow (MCF) formulaton Gven the values of ȳ from a soluton O, one can determne the optmal values for the delvery, consumpton and nventory varables that maxmze the total proft by solvng the followng MCF problem formulaton: max s.t. u s d t s h s Is t (66) C t T s S t N t T s S t qs kt U ȳ kt C, k K, t T, (67) s S t qs kt Qȳ0 kt k K, t T, (68) C s S t (2) (6) and (12) (14), where ȳ0 kt = 1 ndcates that vehcle k s used n tme perod t and ȳ kt = 1 ndcates that vehcle k vsts customer n tme perod t. The objectve functon (66) conssts of maxmzng the total proft, gven by the total revenue mnus the total nventory cost. Constrants (67) allow a vehcle to perform a delvery to a specfc customer n a gven tme perod only f the customer s vsted by the vehcle CIRRELT

20 n that tme perod n the soluton O. Fnally, constrants (68) mpose the vehcle capacty. Ths s a lnear program (LP) that can be solved usng a general-purpose solver. Notce that empty vsts can result from ths phase,.e., cases wth s S t qkt s = 0 and ȳkt = 1 for a gven C, k K, t T. In such a case, the customer s removed from the route and the objectve functon value of the soluton s updated Soluton mprovement (SI) formulaton As an mprovement step, we use a MIP formulaton for a customer assgnment problem, as n Archett et al. (2012). Ths model can be used to remove and nsert customers nto a gven soluton O. Let kt be the savngs n the travel cost when customer s removed from the route of vehcle k n tme perod t. Ths value s computed as c h + c j c hj, where h and j are the predecessor and successor of the customer n the route, respectvely. We set kt n tme perod t (.e., kt = 0 f ȳ kt = 0). Smlarly, let Γ kt as 0 when the customer s not vsted by vehcle k be the cost of nsertng customer nto ts cheapest poston n the route of vehcle k n tme perod t. Γ kt equals 0 for those customers that are already vsted by vehcle k n tme perod t. The formulaton uses two bnary decson varables. Let δ kt be a bnary varable equal to 1 f and only f customer s removed from the route of vehcle k n perod t, and let γ kt be a bnary varable equal to 1 f and only f customer s nserted nto the route of vehcle k n tme perod t. Then, the soluton mprovement formulaton can be stated as follows: max s.t. u s d t s h s Is t C t T s S t N t T s S t + kt δ kt Γ kt γ kt (69) C t T C t T T t s +1 s S t m=t k K q ktm s C s S t m=t γ kt δ kt γ kt C T t s +1 U (ȳ kt q ktm s δ kt k K + γ kt ) C, k K, t T, (70) Qȳ kt 0 k K, t T, (71) 1 ȳ kt C, k K, t T, (72) ȳ kt C, k K, t T, (73) ȳ kt 0 C, k K, t T, (74) (δ kt + γ kt ) β k K, t T, (75) δ kt {0, 1} C, k K, t T, (76) γ kt {0, 1} C, k K, t T, (77) (5), (12), (14), (18) (23) and (26) (27). The objectve functon (69) conssts of maxmzng the total proft, gven by the total revenue mnus the total nventory cost plus the dfference between the savngs and addtonal travel cost gven by the removal and nserton operatons, respectvely. Constrants (70) allow vehcle k to perform delveres to customer n perod t only f ether ths customer s already vsted by the vehcle n the soluton and t was not removed from the route or f the customer was nserted nto the route of the vehcle n the gven 18 CIRRELT