A Branch and Price algorithm for the electric capacitated profitable tour problem with mandatory stops

9th IFAC Conference on Manufacturing Modelling, Management and Control 9th IFAC Conference on Manufacturing Modelling, Management and Control 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Available Control online at www.sciencedirect.com Berlin, Germany, August 28-30, 2019 Control 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019

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IFAC PapersOnLine 52-13 (2019) 1572–1577 A Branch and Price algorithm for the A Branch and Price algorithm for the A Branch and Price algorithm for the A Branch and Price algorithm for the electric capacitated profitable tour problem electric capacitated profitable tour problem A Branch and Price algorithm for the electric capacitated profitable tour problem electric capacitated profitable tour problem with mandatory stops with mandatory stops electric capacitated profitable tour problem with stops with mandatory mandatory stops ∗ Davidwith L. Cort´ e s-Murcia H. Murat Afsar ∗∗ mandatory stops David L. Cort´ es-Murcia ∗∗ H. Murat Afsar ∗ ∗

David L. Cort´ es-Murcia Afsar ∗ Caroline Prodhon ∗ ∗ H. Murat Caroline Prodhon David L. Cort´ es-Murcia H. Murat Afsar ∗ Caroline Prodhon ∗ ∗ es-Murcia H. Murat Afsar ∗ Caroline Prodhon ∗David L. Cort´ e de Technologie de Troyes, ∗ ∗ ICD-LOSI, Universit´ Caroline Prodhon ee de de Troyes, ∗ ICD-LOSI, Universit´ ICD-LOSI, Universit´ de Technologie Technologie deCedex Troyes, 12 Rue Marie Curie, CS 42060, 10004 Troyes France ∗ 12 Rue Marie Curie, CS 42060, 10004 Troyes Cedex France ICD-LOSI, Universit´ e de Technologie de Troyes, 12(e-mail: Rue Marie Curie, CS 42060, 10004 Troyes Cedex France david.cortes [email protected], [email protected], ∗ ICD-LOSI, Universit´ e de Technologie deCedex Troyes, david.cortes [email protected], [email protected], 12(e-mail: Rue Marie Curie, CS 42060, 10004 Troyes France (e-mail: david.cortes [email protected], [email protected], [email protected]). 12(e-mail: Rue Marie Curie, [email protected], 42060, 10004 [email protected], Troyes Cedex France [email protected]). david.cortes [email protected]). (e-mail: david.cortes [email protected], [email protected], [email protected]). [email protected]). Abstract: In this paper, a generalization of the capacitated profitable tour problem is presented. Abstract: In paper, a generalization of capacitated profitable tour problem is Abstract: In this thistime paper, generalization of the the are capacitated profitable tour is presented. presented. Since recharging andadriver’s lunch breaks commonly considered asproblem an idle time, the aim Since recharging time and driver’s lunch breaks are commonly considered as an idle time, the aim Abstract: In this paper, a generalization of the capacitated profitable tour problem is presented. Since recharging time driver’s lunch breaks are commonly considered as an the Due aim is to synchronize thoseand activities choosing restaurants where electric vehicles canidle be time, charged. Abstract: In thistime paper, adriver’s generalization of the are capacitated profitable tour problem is presented. is to synchronize those activities choosing restaurants where electric vehicles can be charged. Due Since recharging and lunch breaks commonly considered as an idle time, the aim is to synchronize those activities choosing restaurants where electric charged. with Due to visiting restaurants has an associated cost, the problem can alsovehicles be seencan as be a problem Since recharging time and driver’s lunch breaks are commonly considered as an idle time, thewith aim to visiting restaurants has an associated cost, the problem can also be seen as a problem is to synchronize those activities choosing restaurants where electric vehicles can be charged. Due to visiting restaurants has anisassociated the problem also becontext. seen asAa mathematical problem with location issues. This variant pertinent cost, especially in a citycan logistics is to synchronize thosevariant activities choosing restaurants where electric vehicles can be charged. with Due location issues. This is pertinent especially in a city logistics context. A mathematical to visiting restaurants has an associated cost, the problem can also be seen as a problem location This pertinent especially in a citywhich logistics context. A instances mathematical model is issues. proposed, asvariant well as ais Branch-and-Price algorithm is able to solve with to visiting restaurants hasas anaisassociated cost, thealgorithm problem can also becontext. seen asAa instances problem with model is proposed, as well is to location This variant pertinent Copyright especially in a citywhich logistics mathematical model is issues. proposed, asand well13 as restaurants. a Branch-and-Price Branch-and-Price algorithm is able able to solve solve instances with c 2019 which  up to 100 customers IFAC location This variant pertinent Copyright especially in a cityIFAC logistics context. A mathematical c 2019  up customers model is issues. proposed, well13 as restaurants. ais Branch-and-Price algorithm c 2019 which  IFAC is able to solve instances with up to to 100 100 customersasand and 13 restaurants. Copyright model isIFAC proposed, asand well13 as restaurants. a Branch-and-Price algorithm which able to solve with c 2019  IFAC up2019, to 100 customers Copyright © (International Federation of Automatic Control) Hosting by is Elsevier Ltd. All instances rights reserved. Keywords: Electric vehicles, Routing problems, Branch-and-price. c  2019 IFAC up to 100 customers and 13 restaurants. Copyright Keywords: Keywords: Electric Electric vehicles, vehicles, Routing Routing problems, problems, Branch-and-price. Branch-and-price. Keywords: Electric vehicles, Routing problems, Branch-and-price. 1. INTRODUCTION minutes. It would enable the use of the EV throughout Keywords: Electric vehicles, Routing problems, Branch-and-price. 1. INTRODUCTION INTRODUCTION minutes. It It would would enable enable the the use use of of the the EV EV throughout throughout 1. minutes. the day. Likewise, two operators highlight the importance the day. Likewise, two operators highlight the importance 1. INTRODUCTION minutes. It would enable the use of the EV throughout day. Likewise, two operators thetime importance of being able to synchronize the highlight recharging and the In order to reduce1.negative environmental impacts, nowa- the INTRODUCTION minutes. It would enable thethe usehighlight of the EV throughout of being able to synchronize recharging time and the the the day. Likewise, two operators the importance In order to reduce negative environmental impacts, nowaof being able to synchronize the recharging time and lunch time as a way to mitigate the impact of recharge In order to reduce negative nowadays electric vehicles (EVs)environmental are consideredimpacts, a suitable op- the day. Likewise, twoto operators highlight thetime importance lunch time as a way mitigate the impact of recharge of being able to synchronize the recharging and the days electric vehicles (EVs) are considered a suitable opIn order to reduce negative environmental impacts, nowatime a way to mitigate the impact of recharge time and theasrisk of waiting for recharge. days electric (EVs)vehicles are considered a suitable op- lunch tion to replacevehicles conventional in last mile operations of being able toa synchronize therecharge. recharging and the time and theasrisk risk of waiting waiting for lunch time way to mitigate the impacttime of recharge In order to reduce negative environmental impacts, nowation to replace conventional vehicles in last mile operations days electric vehicles (EVs) are considered a suitable optime and the of for recharge. tion to replace conventional lastrepresent mile operations (Davis and Figliozzi, 2013). vehicles EVs notinonly one of lunch time way tovariant mitigate impact of recharge In this paper, a novel athe routing problem with and theasrisk of waiting forofrecharge. days electric vehicles (EVs)vehicles are considered a suitable (Davis and Figliozzi, Figliozzi, 2013). EVs notinonly only represent oneopof time tion to replace conventional lasturban mile operations In this paper, a novel novel variant ofrecharge. a routing routing problem problem with with (Davis and EVs not represent one of the cleanest means of2013). transportation in areas, but time and the risk of waiting for In this paper, a variant of a profits where it is possible to recharge the EVs during tion to replace conventional vehicles in last mile operations the cleanest means of transportation in urban areas, but (Davis and Figliozzi, EVs not only represent of In profits where ita novel is possible possible to ofrecharge recharge theproblem EVs during during this paper, variant a routing with the means transportation in urban areas, but also cleanest contribute withof2013). noise reduction, and present aone high profits where it is to the EVs the lunch time is presented. Lunch break is considered (Davis and Figliozzi, EVs not only represent one of In also cleanest contribute withof2013). noise reduction, and present high the means transportation in urban areas, but paper, a isnovel variant ofrecharge a routing problem with thethis lunch time presented. Lunch break is considered profits where it is possible to the EVs during also contribute with noise reduction, and present aatohigh tank-to-wheel efficiency (Pelletier et al., 2016). Due the the lunch time is presented. Lunch is considered a mandatory stop. Additionally, it isbreak assumed that it is the cleanest means transportation in urban areas, but tank-to-wheel efficiency (Pelletier et al., al., 2016). Due tohigh the profits also contribute withofnoise reduction, and present ato where it isis presented. possible toLunch recharge theisEVs during a mandatory stop. Additionally, it is assumed that it is is the lunch time break considered tank-to-wheel efficiency (Pelletier et 2016). Due the cutting-edge technology, the governmental subsidies and mandatory stop. it isa assumed that it possible to have anAdditionally, agreement with set of restaurants also contribute with noise reduction, and present atohigh cutting-edge technology, the governmental subsidies and tank-to-wheel efficiency (Pelletier al., 2016). Due the athe lunchtotime isanpresented. Lunch break is restaurants considered possible have agreement with a set of a mandatory stop. Additionally, it is assumed that it is cutting-edge technology, the governmental subsidies and new supportive policies aiming to et facilitate the implemento have aanRS agreement with a set for of restaurants which provides and a reservation recharging tank-to-wheel (Pelletier al., 2016). Due to and the possible new supportive supportiveefficiency policies aiming aiming to et facilitate the implemencutting-edge the subsidies apossible mandatory stop. Additionally, it isa assumed that it is which provides aanRS RS and aa reservation reservation for recharging to have agreement with set of restaurants new policies to facilitate the implementation of EVs,technology, companies like governmental Chronopost, UPS, La Poste which provides a and for recharging during lunch. The aim is to benefit from idle time caused cutting-edge the subsidies and possible tation of EVs, EVs,technology, companies like governmental Chronopost, UPS, La Poste Poste new policies to facilitate the implemento have agreement withfrom a setidle of time restaurants during provides lunch. The aim is to benefit benefit caused which aanaim RS and a reservation for recharging tation of companies like Chronopost, La and supportive TNT Express haveaiming included EVs intoUPS, their delivery during is to from this idle time caused for the lunch. lunch The break, and synchronizing with vehicle new supportive policies aiming to facilitate the implemenand TNT Express have included EVs into their delivery tation of EVs, companies like Chronopost, UPS, La Poste which a aim RSand and a reservation forwith recharging for the the provides lunch break, synchronizing this vehicle during lunch. The is to benefit from idle time caused and TNT Express have operations (Nesterova et included al., 2013).EVs into their delivery for lunch andissynchronizing thisvisits with to vehicle recharging. A break, fixed cost associated with each tation of EVs, companies like2013). Chronopost, UPS, La Poste during operations (Nesterova et included al., and TNT Express have lunch. The aim is isto benefit from idlevisits timeto caused recharging. A fixed cost associated with each for the lunch break, and synchronizing this with vehicle operations (Nesterova et al., 2013).EVs into their delivery recharging. A fixed cost isthe associated with visits each restaurant. It represents cost established for to restauand TNT limited Express have EVs intolimited their delivery However, range vehicle mileage for operations (Nesterova etofincluded al., 2013).models, the lunch andisthe synchronizing thisvisits with vehicle restaurant. It represents cost established established for to restaurecharging. A break, fixed cost associated with each However, limited rangeetof ofal., vehicle models, limited limited mileage mileage restaurant. It represents cost for restaurants to provide food andthe recharging service, so to visit operations (Nesterova 2013).models, However, range vehicle range, lowlimited vehicle speed, relatively long charging times and recharging. A fixed cost is associated with visits to each rants to provide food and recharging service, so visit restaurant. It represents the cost established for restaurange, low lowlimited vehicle range speed,of relatively long charging charging times times and rants However, vehicle models, mileage to provide food andasrecharging to visit a restaurant can be seen a location service, decision.soLikewise, range, vehicle speed, relatively long and limited payload are some challenges that limited must be faced Itcan represents the cost established restaua restaurant restaurant be seen as location decision.for Likewise, rants to provide food and recharging service, soLikewise, to visit However, limited range of vehicle models, limited mileage limitedlow payload are some challenges that musttimes be faced range, vehicle speed, relatively long charging and arestaurant. can be seen as aa location decision. an estimated profit for visiting customers is defined. The limited payload some challenges that must be faced by operators in are implementation of EVs (Nesterova and rants to provide food and recharging service, soLikewise, to visit an estimated profit for visiting customers is defined. The a restaurant can be seen as a location decision. range, low vehicle speed, relatively long charging times by operators in implementation of EVs (Nesterova and limited payload are some challenges that must be faced estimated profit for visiting customers is defined. The objective is to maximize the total operational benefit. by operators in implementation of EVs and an Quak, 2015). According to Morganti and (Nesterova Browne (2018), aan restaurant be for seenvisiting asthe a location decision. Likewise, objective is can to maximize total operational benefit. estimated profit customers is defined. The limited payload are sometochallenges that(Nesterova must be(2018), faced Quak, 2015). According Morganti and Browne by operators in implementation of transport EVs and objective to maximize thegeneralizes total operational benefit. Thus, the isproposed problem the capacitated Quak, 2015). to Morganti and Browne in France and According UK, the urban freight and (2018), service an estimated profit for visiting customers is defined. The Thus, the proposed problem generalizes the capacitated objective is to maximize the total operational benefit. by operators in implementation of EVs (Nesterova and in France France and According UK, the the urban urban freight transport transport and (2018), service Thus, Quak, 2015). to Morganti Browne the tour proposed problem generalizes the capacitated profitable problem (CPTP), where decision related in and UK, freight and service operators report the limited range andand the risk of queueing objective is to problem maximize thegeneralizes total operational benefit. profitable tour (CPTP), where decision related Thus, the proposed problem the capacitated Quak, 2015). According to Morganti and Browne (2018), operators report the limited range and the risk of queueing in France and UK, the urban freight transport and service problem (CPTP), where must decision related to visit or tour not customers and restaurants be made. operators report the limited and the and risk operational of queueing profitable at recharging stations (RSs)range as technical Thus, proposed problem generalizes the capacitated to visitthe or tour not customers and restaurants restaurants must be made. made. profitable problem (CPTP), where must decision related in and UK, urbanrange freight transport and service to at France recharging stations (RSs) as technical and operational operators report thethe limited and risk of queueing visit or not customers and be at recharging stations (RSs) as technical and operational obstacles after having adopted EVs in the their operations. problem as (CPTP), where decision related The paper is structured follows: in Section 2,bea made. review to visit or tour not customers and restaurants must operators report the limited range and the risk of queueing profitable obstacles after having adopted EVs in their operations. at recharging stations (RSs) as technical and operational The paper is structured structured as follows: in Section Section 2,beaa made. review obstacles after having adopted EVs in their operations. to visit or not customers and restaurants must The paper is as follows: in 2, review of related literature is discussed. In Section 3, the problem at recharging stations (RSs) as technical and operational Moving towards the use of electric requires to of obstacles after having adopted EVs invehicles their operations. related literature is discussed. discussed. In Section Section 3, the the problem paper is structured as follows: in Section 2,presented. aproblem review Moving towards the use use of of electric electric vehicles requires to to The related literature is In 3, and corresponding mathematical model are obstacles after EVs decisions invehicles their operations. Moving requires adapt thetowards tools having tothe makeadopted operational by including of The paper is structured as follows: in Section 2, a review and corresponding mathematical model are presented. of related literature is discussed. In Section 3, the problem adapt the tools to make operational decisions by including Moving towards the use of electric vehicles requires to and corresponding mathematical model are presented. The objective is to solve the problem to optimality, so adapt the tools to make operational by including the characteristics mentioned above. decisions In consequence, stud- of related literature is discussed. In Section 3, the problem The objective is to solve the problem to optimality, so and corresponding mathematical model are presented. Moving towards the use of electric vehicles requires to the characteristics mentioned above. In consequence, studadapt the tools to make operational decisions by including The objective is to solve the problem to optimality, so a Branch-and-Price algorithm is presented as solution the characteristics mentioned above.(VRP) In consequence, stud- and ies on the Vehicle Routing Problems which consider corresponding modelto are a Branch-and-Price Branch-and-Price algorithm is presented presented aspresented. solution objective is to4.mathematical solve the Computational problem optimality, so adapt the Vehicle tools toRouting make operational decisions by including ies on on the Problems (VRP) which consider the characteristics mentioned above. In consequence, stud- aThe algorithm is as solution method in Section Finally, results are ies the Vehicle Routing Problems which consider usage of EVs increase in recent years (VRP) (Afroditi et al., 2014). The objective is to solve the problem to optimality, so method in Section 4. Finally, Computational results are a Branch-and-Price algorithm is presented as solution the characteristics mentioned above. In consequence, stud- method usage of EVs increase in recent years (Afroditi et al., 2014). ies on the Vehicle Routing Problems (VRP) which consider in Section 4. Finally, Computational results are shown in Section 5 and conclusions in Section 6. usage of EVs in recentthe years (Afroditi et al., 2014). In general in increase those variants, charging time turns out amethod Branch-and-Price algorithm is presented as solution shown in Section 5 and conclusions in Section 6. in Section 4. Finally, Computational results are ies on of theEVs Vehicle Routing Problems (VRP) which consider In general general in increase those variants, the charging time turns out shown in Section 5 and conclusions in Section 6. usage indriver’s recent years (Afroditi et al., 2014). In in those variants, the charging time turns out to be an idle time and mandatory break times are shown ininSection Section 4. Finally, Computational results are 5 and conclusions in Section 6. usage of EVs increase indriver’s recentthe years (Afroditi et al., 2014). to be an idle time and mandatory break times are method In general in those variants, charging time turns 2. RELATED LITERATURE to an idle time and driver’s mandatory break times out are notbeexplicitly considered. 2. RELATED RELATED LITERATURE shown in Section 5 and conclusions in Section 6. In general in time those variants, the charging break time turns out 2. LITERATURE not explicitly considered. to be an idle and driver’s mandatory times are not explicitly considered. RELATED LITERATURE to anwork idle time and driver’s break operators times are We present a2.brief summary of the most recent works In be the of Morganti andmandatory Browne (2018) not explicitly considered. 2. RELATED LITERATURE We present a brief summary of the the most most recent recent works works In the work of Morganti and Browne (2018) operators not explicitly considered. present brief summary of In the work of eventual Morgantiboost and charging, Browne in (2018) operators We related to theaelectric vehicle routing problems and vehicle mention that an a fast-charging related to the electric vehicle routing problems and vehicle mention that an eventual boost charging, in a fast-charging We present a brief summary of the most recent works In the work of Morganti and Browne (2018) operators the electric mention that antoeventual boost charging, in a fast-charging routing to problems withvehicle profits.routing problems and vehicle station, allows recharge up to 80% battery in about 30 related We present aelectric brief summary of theproblems most recent works In the work oftoeventual Morganti andto Browne (2018) operators routing problems with profits. station, allows recharge up 80% battery in about 30 related to the vehicle routing and vehicle mention that an boost charging, in a fast-charging station, allows to recharge up to 80% battery in about 30 routing problems with profits. the electric mention that antoeventual boost charging, in a fast-charging routing to problems withvehicle profits.routing problems and vehicle station, allows recharge up to 80% battery in about 30 related Copyright © 2019 to IFAC routingbyproblems with profits. station, recharge up to Federation 80% battery in about 301592 2405-8963 allows © IFAC (International of Automatic Control) Elsevier Ltd. All rights reserved. Copyright © 2019, 2019 IFAC 1592Hosting Copyright 2019 responsibility IFAC 1592Control. Peer review© of International Federation of Automatic Copyright ©under 2019 IFAC 1592 10.1016/j.ifacol.2019.11.424 Copyright © 2019 IFAC 1592

2019 IFAC MIM Berlin, Germany, August 28-30, 2019 David L. Cortés-Murcia et al. / IFAC PapersOnLine 52-13 (2019) 1572–1577

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EVRP is an extension of the vehicle routing problems where fleet is composed of EVs, a literature review of this field is presented by Afroditi et al. (2014). Schneider et al. (2014) introduce the Electric Vehicle Routing Problem with Time Windows which is devoted to manage EVs, visits to recharging stations and customer time windows. From that moment on, multiple variants of the problem have been studied. i.e. Goeke and Schneider (2015) present the Electric Vehicle Routing Problem with Time Windows and Mixed Fleet where the fleet is composed by EVs and conventional vehicles. Keskin and C ¸ atay (2016) relax the full charge policy and present the EVRP with Partial Recharching. Hiermann et al. (2016) include a fleet size problem with heterogeneous fleet, while Montoya et al. (2017) present an extension of the EVRP which considers nonlinear recharging times. Among the previous works, some heuristic and metaheuristic methods to solve the EVRP and its variants are presented. Desaulniers et al. (2016) come up with an exact algorithm to solve four variants of the EVRPTW. They present exact branchprice-and-cut algorithms which use monodirectional and bidirectional labeling algorithms for generating feasible routes.

due to Bianchessi et al. (2018) who present a new twoindex formulation with a polynomial number of variables reinforced by connectivity constraints. A branch-and-cut algorithm is presented and 24 previously unsolved instances were closed to optimality. A B&P algorithm for the CPTP can be found in the work of Archetti et al. (2009). The algorithm is adapted from the one defined by Boussier et al. (2007). Later, an improved B&P algorithm for CPTP was developed by Archetti et al. (2013). It implements a bi-directional dynamic programing and decremental space relaxation. Jepsen et al. (2014) presents a Branch-andCut algorithm based on a formulation for the undirected CPTP. Valid inequalities are implemented and a new family of inequalities for the CPTP is introduced.

The vehicle routing problems with profits are a combination of node selection and routing problems. The objective is to maximize the total score collected from visited (selected) nodes. For the multiple-vehicle case two problems can be found in the literature: the Team Orienteering Problem (TOP) and the capacitated profitable tour problem (CPTP). In those problems, not all available nodes can be visited due to the limited time budget. The difference between them are the objective functions. While the TOP aims to max prof it, the CTOP aims to max (prof it − cost). Gunawan et al. (2016) and Gavalas et al. (2014) present literature reviews of vehicle routing problems with profits. Due to the goal of this paper is to propose an exact method, in the following some works related to exact methods for TOP and CPTP are presented. Boussier et al. (2007) present a Branch-and-Price (B&P) algorithm based on a set-packing formulation. The authors tested the algorithm on the benchmark instances proposed by Chao et al. (1996). Dang et al. (2013) presents a Branch-and-Cut algorithm based on a linear formulation with a polynomial number of variables. The algorithm includes some dominance properties and valid inequalities such as symmetric breaking inequalities, boundaries on profits, generalized subtour eliminations and clique cuts from graphs of incompatibilities. They were able to solve 29 previously unsolved benchmark instances. Later, Keshtkaran et al. (2016) having as reference the work of Boussier et al. (2007), presents a new B&P algorithm where the pricing problem is solved by a bounded bidirectional dynamic programming algorithm with decremental state space relaxation featuring a two-phase dominance rule relaxation. Additionally, they include subset-row inequalities and presents a Branchand-Cut-and-Price algorithm as a second solution method. With these algorithms authors are able to solve 17 previously unsolved benchmark instances. El-Hajj et al. (2016) design a Cutting Plane algorithm. Several types of cuts are proposed in order to strengthen the classical linear formulation, including those presented in Dang et al. (2013). With this method 24 previously unsolved instances were solved to optimality. The most recent exact approach is

3. MATHEMATICAL MODEL

In relation to the literature review, none of the works previously presented introduces an exact method to solve a problem that considers: usage of EVs, driver’s mandatory break times, and routing with profits at the same time. Thus, we present the electric capacitated profitable tour problem with mandatory stops (ECPTPMS) and a branch-and-price algorithm to solve it to optimality.

 The problem is defined on a graph G = (V0,N +1 , A) with   a set of vertices V0,N +1 = {V ∪ F ∪ {0, N + 1}} and a  set of arcs given by A = {(i, j)|i, j ∈ V0,N +1 , i = j}. Let be V = {1, ..., N } the set of customers, F = {0, ..., M } the set of restaurants which provides recharging stations, which represents all and Gi is the set of dummy vertex  the visits to restaurant i ∈ F . F  = i∈F Gi is the set of restaurants and its dummy vertices. The depot is denoted as 0 when it is a departing node, and denoted as N +1 when it is the end of a route. With these notations, the subsets which includes one or both depot vertices are giving by subscripting the respective set. For example, VN +1 contains all customer vertices, all restaurants vertices and the arrival depot vertex.

For each arc (i, j) a distance dij between vertex i and j is given. The driving time through an arc (i, j) depends on d the average EVs speed v and it is computed as tij = vij .  Each vertex i ∈ V0,N +1 has: a non-negative demand / V ), a non-negative service time si (si = qi (qi = 0, i ∈ 0, i ∈ / V ) and a profit per visit pi (pi = 0, i ∈ / V ). Each restaurant i ∈ F  has a hard time window [ei , li ]. It represents the lunch time established by the restaurant or by the company policies. Likewise, selecting a restaurant has a fix cost Ci related to the food and recharge service. The time window [0, TM ax ] at the depot express the maximum tour duration. A homogeneous electric vehicle fleet of size U is available at the depot. Q is the maximum cargo loading per vehicle. On each visit to the restaurant, the lunch time is denoted as S and it is considered constant. Also, we assume that during that time the vehicle is charged to the maximum battery level B. Energy consumption is described as a linear relation between dij and the consumption rate cr. Thus, the problem can be formulated as the following mixed-integer linear program (MILP). For every arc (i, j) ∈ A the boolean decision variables xij are equal to

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1 if the arc (i, j) is traversed, 0 otherwise. The boolean decision variable zi is is equal to 1 if the vertex restaurant i is visited, 0 otherwise. rsi is used to determine if a restaurant is already visited when the vehicle arrives at vertex i. Positive variables: ui define the remaining cargo, τi define the arrival time and yi the remaining charge level  of the vehicle on arrival at vertex i ∈ V0,N +1 . max



pi

i∈V 



 j∈VN +1

xij −





 i∈V0 j∈VN +1

dij xij −



ci zi (1)

i∈F

Subject to: 

 j∈VN ,i=j +1



 j∈VN ,i=j +1



 j∈VN ,i=j +1

xij ≤ 1

xij ≤ zk

xij −



∀i ∈ V

(2)

∀i ∈ Gk , ∀k ∈ F

(3)

xji = 0

j∈V0 ,i=j



i∈V 

∀i ∈ V 

x0i ≤ U

(5)

ui − qi xij + Q(1 − xij ) ≥ uj ∀i ∈ V0 , ∀j ∈ VN +1 , i = j ui ≤ Q

∀i ∈ V

(4)



(7)

(8)

τi + (S + tij )xij − (lN +1 + S)(1 − xij ) ≤ τj ∀i ∈ F  , ∀j ∈ VN +1 , i = j

(9)

∀i ∈ FN +1 ∀i ∈ FN +1

(10) (11)

yi − cdij xij + B(1 − xij ) ≥ yj ∀i ∈ V0 , ∀j ∈ VN +1 , i = j

(12)

B − cdij xij + B(1 − xij ) ≥ yj ∀i ∈ F  , ∀j ∈ VN +1 , i = j

(13)

 yi ≤ B ∀i ∈ V0,N +1   zi = x0i i∈F

Battery level at a vertex succeeding a customer is set by constraints (12) while constraints (13) set battery level at a vertex succeeding a restaurant. Battery capacity is guarantee by constraints (14). Constraints (15) establish the number of restaurants that must be visited according to the number of routes while (16) and (17) guarantees at most one visit per route . Finally, domain variable definitions are defined in constraints (18) - (21). 4. BRANCH-AND-PRICE METHODOLOGY

(6)

τi + (si + tij )xij − lN +1 (1 − xij ) ≤ τj ∀i ∈ V0 , ∀j ∈ VN +1 , i = j

ei ≤ τi τi ≤ li

minus the operational cost associated with the total distance and minus total cost associated with the restaurant service. Constraints (2) state that all customers must be visited at most once. Relation between routing variables x and restaurant selection variables z is represented in constraints (3). Flow conservation constraints (4) guarantee for each vertex that the number of incoming arcs is equal to the number of outgoing arcs. Fleet size is guarantee by constraints (5). Load flow and fulfillment of demand are represented by constraints (6). Vehicle capacity is restricted by constraints (7). For the arrival times at each vertex two scenarios have to be considered, constraints (8) link arrival time of a vertex i preceded by a customer while constraints (9) link arrival time of a vertex j preceded by a restaurant i. Restaurant time windows and route duration are represented by constraints (10) and (11).

A B&P algorithm is proposed as solution method. The previous formulation can be rewrite as a route-based formulation as follows. Let Ω be the set of feasible vehicle routes. In other words, the set of arcs in A issued from the depot, visiting one restaurant, going back to the depot, satisfying capacity, battery, restaurant time windows, and visiting at most once each customer. Let ρk be the income of the route rk ∈ Ω. Let aik be a boolean variable equal to 1 if route rk visits node i, 0 otherwise. Let bijk be a boolean variable equal to 1 if route rk pass over arc (i, j) ∈ A, 0 otherwise. Let γf k be a boolean variable equal to 1 if route r k visits restaurant f ∈ F , 0 otherwise.  Note that ρk = bijk (pj − dij ), aik =  i∈V0 j∈VN j∈VN +1 bijk +1  and γf k = j∈VN +1 bf jk . Boolean variable θk is equal to 1 if route rk is part of the solution, 0 otherwise. Boolean variable µf indicates if the restaurant f is visited or not. Each restaurant f ∈ F has a fix food and recharge service cost Ci when it is selected in the solution. Thus, the model can be described with the following setpacking model (SPM):

(14) (15)

(SP M )

max

 i∈VN +1

rsi − (1 − xij ) ≤ rsj ∀i ∈ V0 , ∀j ∈ VN +1 , i = j rsi + 1 − (1 − xij ) ≤ rsj ∀i ∈ V0 , ∀j ∈ F  , i = j  xij ∈ {0, 1} ∀i, j ∈ V0,N +1 , i = j zi ∈ {0, 1} ∀i ∈ F rsi ∈ {0, 1} ∀i ∈ F  ui , τi , yi ≥ 0 ∀i ∈ V 

(16) (17) (18) (19) (20) (21)

The objective function (1) aims to maximize the total operational income computed as the sum of the profits 1594



rk ∈Ω



rk ∈Ω





rk ∈Ω

rk ∈Ω

ρk θ k −



θk ≤ U

aik θk ≤ 1

γf k θk ≤ U µf

θk ∈ {0, 1}

µf ∈ {0, 1}

cf µf

(22)

f ∈F

(23) ∀i ∈ V ∀f ∈ F

∀rk ∈ Ω ∀f ∈ F

(24) (25) (26) (27)

2019 IFAC MIM Berlin, Germany, August 28-30, 2019 David L. Cortés-Murcia et al. / IFAC PapersOnLine 52-13 (2019) 1572–1577

The objective function (22) remains the same as (1). Constraints (23) limits the number of vehicles that are used. Constraints (24) guarantee that customers can be served at most once. Constraints (25) indicates if a restaurant is visited by the solution. Finally, domain variable definitions are defined in constraints (26) - (27).

4.3 Pricing Problem Once the LRPM(Ω1 ) is solved, the dual variables λ∗0 , λ∗i and αf∗ are obtained. The purpose of the pricing problem is twofold: to find routes rk ∈ Ω \ Ω1 with positive reduce cost, namely, routes such that:

4.1 Column Generation Since the number of routes in Ω is exponential in the number of customers and the number of restaurants, they are generated dynamically by a column generation procedure. The procedure is composed of two components: the linear restricted master problem (LRMP) and an auxiliary problem, called the pricing problem. Iteratively the LRMP is solved, and the pricing problem look for routes with positive reduced cost which are added to LRMP. When no route with positive cost exists, the solution is optimal for the linear relaxation of SPM. For this problem the structure of the algorithm is based on that introduced by Boussier et al. (2007) and adapted to manage restaurant visits and some new features. 4.2 Restricted Master Problem The restricted master problem RMP(Ω1 ) is obtained from the SPM considering only a subset Ω1 ⊂ Ω of variables. In the following the linear relaxation of RMP(Ω1 ) (LRM P (Ω1 )) is presented. (LRM P (Ω1 ))

max



rk ∈Ω1

Subject to: 



rk ∈Ω1

rk ∈Ω1



rk ∈Ω1

ρk θk −

cf µf

(28)

f ∈F

θk ≤ U

aik θk ≤ 1

(29) ∀i ∈ V

γ f k θk − U µf ≤ 0 θk ≥ 0 µf ≥ 0



∀f ∈ F

∀rk ∈ Ω1 ∀f ∈ F

(30) (31)

i∈V

Subject to:   λ0 + aik λi + γf k αf ≥ ρk i∈V

f ∈F

U αf ≥ c f ∀f ∈ F λ0 ≥ 0 λi ≥ 0 ∀i ∈ V ∀f ∈ F αf ≥ 0

ρk − λ∗0 −

∀rk ∈ Ω1

(35) (36) (37) (38) (39)



i∈V

aik λ∗i −



γf k αf∗ > 0

(40)

f ∈F

or to prove that none exists. Condition (40) can be expressed as: 



 i∈V0 j∈VN +1

bijk (pi −dij −λ∗i )−

 

 f ∈F j∈VN +1

bf jk (df j +αf∗ ) > 0 (41)

Expression (41) shows that the pricing problem can be reduced to an elementary longest path problem with resource constraints. In this paper two versions of the pricing problem are used. The first seeks to find elementary paths from 0 to N + 1 maximizing the reduce cost and satisfying a set of resource constraints. The second version seeks to find ng-paths still departing from 0 to N + 1 maximizing the reduce cost and satisfying a set of resource constraints. An elementary path is a path visiting a subset of nodes where each node is visited just once (Irnich and Desaulniers, 2005). A ngpath is a path visiting a subset of nodes (even more than once) (Baldacci et al., 2011). Each customer i ∈ V  has a neighborhood Ni ⊆ V  including i itself. The cardinality of N G(i) is a parameter. A ng-path allows multiple visits to a customer i if at least one costumer j such that i ∈ / N G(j) is visited between the last |N G(i)| successive visits (Pecin et al., 2017). In the version of labeling algorithm with elementary paths, a partial path p from 0 to a vertex i ∈ VN +1 is represented by a label Li = (costi , RSi , timei , loadi , bati , Ti ), where the label components are as follows: costi : RSi : timei : loadi : bati : Ti :

(32) (33)

And let D(Ω1 ) be the dual program of LRM P (Ω1 ):  min U λ0 + λi (34) (D(Ω1 ))

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reduced cost of path p. number of visits to restaurants along path p. earliest service start time at vertex i. load delivered along path p. energy consumption since last recharge along path p. set saving sequence of nodes visited along path p.

An extension of Li to a node j is feasible if holds (RSj ≤ 1) ∧ (timej ≤ lj ) ∧ (timej ≤ l0 ) ∧ (loadj ≤ Q) ∧ (batteryj ≤ B). A label L1i dominates a label L2i if every feasible extension of L2i yields to a path with reduce cost smaller than the feasible route obtained by applying the same extension from L1i . Thus, the following domination rules are established. L1i dominates L2i if:

where λ0 is the nonnegative dual variable associated to fleet size constraint (29), λi is the nonnegative dual variable associated with constraint (30) and αf is the nonnegative dual variable associated with the use of restaurant f (constraint (31)). 1595

(a) (b) (c) (d) (e) (f)

cost1i ≤ cost2i RSi1 = RSi2 time1i ≤ time2i load1i ≤ load2i batteryi1 ≤ batteryi2 Ti1 ⊆ Ti2

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In the version with ng-paths the last element of the label is replaced by Πi which represents the set of vertices forbidden as immediate extensions due to ng-sets. Thus, the last domination condition is replaced by Π1i ⊆ Π2i . 4.4 Branching Strategy If the optimal solution of LRMP is not integer, a branching decision must be made. Below, the branching strategies are presented. The first type of branching focus on the variable µf of the MP, which indicates if a restaurant f ∈ F is visited or not. If there are µ variables with fractional values, the one with the largest opening cost cf is selected, and two branches are created. The first enforcing to include the full opening cost cf , and allowing to visit the restaurant F . The second forbidding the visit of restaurant f . At the master problem level, new constraints are added to LRMP(Ω1 ) as follows for restaurant f :  1 in the branch where cf must be included µf = 0 in the branch where the visit to f is forbidden When all µ variables are integer, the branching rules proposed by Boussier et al. (2007) are applied. Flow over each node is checked and then flow over arcs is examined. If there are vertices with fractional flow, the one with the largest cost pi − λ∗i is selected. Two new branches are created. In the first is mandatory to visit customer i, and in the second, it is forbidden. Enforcing or forbidding a visit to a customer is done in the LRMP(Ω1 ) by updating constraint (29) as follows:  + yi = 1 in the branch where customer i     have to be visited aik θk  =0 in the branch where the visit to  rk ∈Ω1  customer i is forbidden

Virtual variables yi are created for every node i ∈ V to handle some infeasible situations with the LRMP(Ω1 ) when a new branch is created. This variable is added with a large negative value in the objective function.

Finally, if all µ variables are integer and the flow is integer for every node, the third branching strategy looks for arcs with fractional flow. The arc (i, j) with non integer flow and with the largest cost pi − λ∗i /2 − λ∗j /2 is selected. According with previous branching decisions two or three branches are created. If one of the nodes i or j is already constrained to be visited, two new branches are created. One forcing the arc (i, j) and other forbidding it. If neither i nor j is constrained to be visited, three branches are created, two where the node i is enforced to be visited and the arc (i, j) is enforced or forbidden respectively, and a third branch forbidding the visit to node i, which also results in removing arc (i, j) from the solution.

the algorithm where dominance condition (f) is relaxed. Once the the heuristic version does not find a new route with positive reduced cost, the exact version is executed. At each iteration of the pricing problem, the algorithm is stopped once it reach 75 routes with a positive reduced cost. The labeling algorithm is improved by doing forward extensions until each path reach a restaurant. Later, taking advantage of the symmetries of the routes, a concatenation procedure evaluating the merge between every path finishing at the same restaurant is done. This allow to have the same advantages of the bi-directional labeling strategies. 5. COMPUTATIONAL RESULTS A set of instances of this problem is created based on the instances proposed by Chao et al. (1996). It is composed of three types of instances, each one with a different type of battery capacity: small, medium and large. The B&P is implemented in C++ and linear relaxations are solved by CPLEX 12.8. The algorithm is executed on a machine with Intel(R) Core(TM) i5-5300U CPU @2.30GHz 2.30GHz and 8 GB RAM. The execution time is limited up to 2 hours. In Table 1, we present the resume of the results. n is the number of available customers, rs is the number of available restaurants, Op/Inst is number of optimal solutions reached by the algorithm over the set size, and CPU is the average computational time (in seconds) for those instances solved to optimality under the time limit. BP1 are the results for the version of the B&P where the pricing problem is solved using just elementary paths, while BPN G are the results of the algorithm solving the pricing allowing ng-paths. Results show that the proposed algorithm is capable to solve instances with up to 100 customers and 13 restaurants in less than 7 minutes, on average. Additionally, the usage of ng-paths while solving the pricing problem improves the performance of the algorithm. It allows to solve to optimality 120 instances more than the version with elementary paths. 6. CONCLUSION To sum up, the electric capacitated profitable tour problem with mandatory stops is presented as a novel variant of routing problem with profits, where decisions about where to recharge are made. Also, synchronization between recharging stops and lunch stops allows to take advantage of lunch breaks. A branch-and-price algorithm is presented and the benefit of include ng-path relaxation is proved, as well as their performance to manage problems with large sizes. Finally, as an insight, to group the restaurants in clusters and to have a cost per cluster can be interesting to represent services offered by restaurant chains.

4.5 Acceleration techniques

REFERENCES

In order to improve the performance of the algorithm the following strategies are applied during the execution. Both the pricing problem with elementary paths and the pricing problem with ng-paths has an heuristic version of

Afroditi, A., Boile, M., Theofanis, S., Sdoukopoulos, E., and Margaritis, D. (2014). Electric vehicle routing problem with industry constraints: trends and insights for future research. Transportation Research Procedia, 3, 452–459.

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Table 1. Computational results with different B values

Set-1 Set-2 Set-3 Set-4 Set-5 Set-6 Set-7 Total

n 30 19 31 98 64 62 100

rs 4 3 4 12 8 8 13

Bsmall BP1 BPN G Op/Inst CPU Op/Inst CPU 54/54 0,04 54/54 0,03 33/33 0,03 33/33 0,04 60/60 14,22 60/60 3,96 36/60 284,20 43/60 405,85 65/78 141,93 76/78 205,86 39/42 121,07 42/42 159,38 38/60 110,87 42/60 141,79 325/387

350/387

Bmedium BP1 BPN G Op/Inst CPU Op/Inst CPU 54/54 3,55 54/54 0,34 33/33 0,03 33/33 0,04 55/60 12,64 60/60 84,18 24/60 490,57 33/60 331,72 51/78 166,01 65/78 123,89 34/42 178,62 38/42 16,77 31/60 141,38 47/60 375,32 282/387

Archetti, C., Bianchessi, N., and Speranza, M.G. (2013). Optimal solutions for routing problems with profits. Discrete Applied Mathematics, 161(4-5), 547–557. Archetti, C., Feillet, D., Hertz, A., and Speranza, M.G. (2009). The capacitated team orienteering and profitable tour problems. Journal of the Operational Research Society, 60(6), 831–842. Baldacci, R., Mingozzi, A., and Roberti, R. (2011). New route relaxation and pricing strategies for the vehicle routing problem. Operations research, 59(5), 1269–1283. Bianchessi, N., Mansini, R., and Speranza, M.G. (2018). A branch-and-cut algorithm for the team orienteering problem. International Transactions in Operational Research, 25(2), 627–635. Boussier, S., Feillet, D., and Gendreau, M. (2007). An exact algorithm for team orienteering problems. 4or, 5(3), 211–230. Chao, I.M., Golden, B.L., and Wasil, E.A. (1996). A fast and effective heuristic for the orienteering problem. European journal of operational research, 88(3), 475– 489. Dang, D.C., El-Hajj, R., and Moukrim, A. (2013). A branch-and-cut algorithm for solving the team orienteering problem. In International Conference on AI and OR Techniques in Constriant Programming for Combinatorial Optimization Problems, 332–339. Springer. Davis, B.A. and Figliozzi, M.A. (2013). A methodology to evaluate the competitiveness of electric delivery trucks. Transportation Research Part E: Logistics and Transportation Review, 49(1), 8–23. Desaulniers, G., Errico, F., Irnich, S., and Schneider, M. (2016). Exact algorithms for electric vehicle-routing problems with time windows. Operations Research, 64(6), 1388–1405. El-Hajj, R., Dang, D.C., and Moukrim, A. (2016). Solving the team orienteering problem with cutting planes. Computers & Operations Research, 74, 21–30. Gavalas, D., Konstantopoulos, C., Mastakas, K., and Pantziou, G. (2014). A survey on algorithmic approaches for solving tourist trip design problems. Journal of Heuristics, 20(3), 291–328. Goeke, D. and Schneider, M. (2015). Routing a mixed fleet of electric and conventional vehicles. European Journal of Operational Research, 245(1), 81–99. Gunawan, A., Lau, H.C., and Vansteenwegen, P. (2016). Orienteering problem: A survey of recent variants, solution approaches and applications. European Journal of Operational Research, 255(2), 315–332. Hiermann, G., Puchinger, J., Ropke, S., and Hartl, R.F. (2016). The electric fleet size and mix vehicle routing problem with time windows and recharging stations.

330/387

Blarge BP1 BPN G Op/Inst CPU Op/Inst CPU 54/54 21,65 54/54 0,80 33/33 0,03 33/33 0,03 55/60 46,58 60/60 107,61 24/60 563,01 33/60 411,56 51/78 230,90 65/78 144,24 34/42 227,75 38/42 20,14 31/60 169,22 46/60 372,25 282/387

329/387

European Journal of Operational Research, 252(3), 995– 1018. Irnich, S. and Desaulniers, G. (2005). Shortest path problems with resource constraints. In Column generation, 33–65. Springer. Jepsen, M.K., Petersen, B., Spoorendonk, S., and Pisinger, D. (2014). A branch-and-cut algorithm for the capacitated profitable tour problem. Discrete Optimization, 14, 78–96. Keshtkaran, M., Ziarati, K., Bettinelli, A., and Vigo, D. (2016). Enhanced exact solution methods for the team orienteering problem. International Journal of Production Research, 54(2), 591–601. Keskin, M. and C ¸ atay, B. (2016). Partial recharge strategies for the electric vehicle routing problem with time windows. Transportation Research Part C: Emerging Technologies, 65, 111–127. Montoya, A., Gu´eret, C., Mendoza, J.E., and Villegas, J.G. (2017). The electric vehicle routing problem with nonlinear charging function. Transportation Research Part B: Methodological, 103, 87–110. Morganti, E. and Browne, M. (2018). Technical and operational obstacles to the adoption of electric vans in france and the uk: An operator perspective. Transport Policy, 63, 90–97. Nesterova, N. and Quak, H. (2015). State of the art of the electric freight vehicles implementation in city logistics - update 2015. Technical report, TNO. Nesterova, N., Quak, H., Balm, S., Roche-Cerasi, I., and Tretvik, T. (2013). State of the art of the electric freight vehicles implementation in city logistics. Technical report, TNO and SINTEF. Pecin, D., Pessoa, A., Poggi, M., and Uchoa, E. (2017). Improved branch-cut-and-price for capacitated vehicle routing. Mathematical Programming Computation, 9(1), 61–100. Pelletier, S., Jabali, O., and Laporte, G. (2016). 50th anniversary invited articlegoods distribution with electric vehicles: review and research perspectives. Transportation Science, 50(1), 3–22. Schneider, M., Stenger, A., and Goeke, D. (2014). The electric vehicle-routing problem with time windows and recharging stations. Transportation Science, 48(4), 500– 520.

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