Operating a Fleet of Electric Taxis Essay Example for Free

Operating a Fleet of Electric Taxis Essay Abstract. The deployment of electric taxi ? eets is highly desirable from a sustainable point of view. Nevertheless, the weak autonomy of this kind of vehicles requires a careful operation. The way of managing such a ? eet and the question of locating charging terminals for the vehicles are addressed in this paper. Methods for dealing with these tasks are proposed and their e? ciency is proved through simulations. 1. Introduction 1. 1. Context. Centrale OO 1 is a pioneering project aiming to deploy in Paris a ? eet of 100 % electric taxis. The company in charge of the management of the ? eet is the Soci? t? du Taxi Electrique Parisien (STEP).ee The deployment of such ? eets ? nds is main motivation in sustainable issues: electric vehicles release almost no air pollutants at the place where they are operated and have less noise pollution than internal combustion engine vehicles. However, the main drawback of an electric vehicle is its weak autonomy – 80 km in the case of the Centrale OO project. In taxi ? eet management, two kinds of requests can be di? erentiated: booking requests and opportunistic requests. The ? rst ones can be immediate or in advance of travel and have to be processed by the taxi dispatching system which assigns the request to a taxi. The opportunistic requests correspond to the traditional taxi services picking up passengers at cab-ranks or from the side of the road. Of course, this kind of requests is not processed by the dispatching system. The constraints of the management, as expressed by the STEP, are †¢ A taxi must never break down †¢ An opportunistic demand inside Paris and its suburbs must always be satis? ed (legal environment of Paris) †¢ The number of booking demands accepted has to be maximized The charging problem of the taxis must therefore be carefully addressed. At a tactical level, a good assignment of the trips to the taxis is crucial. We propose an e? cient way to manage the electric ? eet in real-time while taking into account the charging tasks. At a strategic level, the charging problem includes the determination of the best location for the charging terminals. The signi? cant initial investment (the cost of an electrical charging terminal is about 20. 000 euros) and the restricted vehicle autonomy give a high relevancy to the charging terminal location task. Indeed, a wrong placement may in e? ect lead to a poor ? eet management with vehicles having di?culties to charge the batteries due to charging terminals saturation or even with vehicles constantly running out of charge to keep operating. Our purpose is to propose a practical way for computing the â€Å"best† locations for the charging terminals. 1. 2. Model. We describe now formally the model we deal with in this paper. We derive also some elementary relations, which gives some informations on the capacity of a given system (in terms of number of trips that can be realized by unit of time). 1. 2. 1. Input description. A complete directed graph G = (V, A) models the network. The vertices are points in the city at which trips start and ? nish. They can moreover be used to locate vehicle charging terminals. The arcs model the possible trips. The duration of a trip is a random variable Ta of expectation ? a . The Key words and phrases. charging terminal location; electric vehicles; ? eet management system; mixed integer programming; simulation; taxi dispatching. This project has been funded by R? gion Ile de France. e 1 See the website http://taxioo. com/index. html for an artistic view. 1 hal-00721875, version 2 31 Jul 2012 demand for each possible trip a ? A is assumed to follow a Poisson process of rate ? a . Actually, this demand is split between a booking demand and an opportunistic demand, see Section 5 for a more accurate description. There are n taxis. A taxi consumes ? Wh by unit of time when it is moving. It stores ? Wh by unit of time when it is charging. The number of charging terminals is denoted by r. Several terminals can be located at the same vertex. ? 1. 2. 2. Elementary relations. Let us denote by ? a the average number of demands for a trip a that are ? a ? ?a . accepted by unit of time. We have ? 1 ? ? ? ?a ? a be the Let ? = ? a be the average number of trips accepted by unit of time and let ? = ? ? a? A a? A average duration of an accepted trip. ? The energy consumption of the system by unit of time is ? . The maximal rate of supply in energy is ? r. Therefore, we have the following inequality (1) ? ? ? ? r hal-00721875, version 2 31 Jul 2012 A second inequality can be derived by considerations on the time needed to perform the di? erent tasks. Let us consider a taxi over a time window of su? ciently large duration T . Denote by x the time during ? which it stores energy at a charging terminal. Over the time window, it spends in average T n unit of time with a customer on board. Therefore, we have ? T +x? T n During this duration x, it stores a quantity of energy that must cover in average the consumption over the time window. Hence ? ?T ? ? x n Combining these two inequalities leads to (2) ? (? + ? ) ? n Equations (1) and (2) can be summarized in the following inequality. (3) ? ? ? min ? r n , (? + ? )? Knowing the number of taxis, their e? ciency (encoded by ? ), the number of charging terminals, and their e? ciency (encoded by ?), then an upper bound of the number of trips that can be accepted by unit of time can be calculated. 1. 3. Plan. Section 2 is devoted to the literature review for the two problems addressed in this paper, namely ? eet management and charging terminal location. The following sections – Section 3 and Section 4 – detail the approaches proposed for each of these problems. Next, we describe a simulator that has been implemented for the evaluation of the proposed approaches (Section 5). The results of the experiments are described in Section 6. The paper ends with concluding remarks (Section 7). 2. Literature review 2. 1. Taxi dispatching. Traditional taxi dispatching systems are characterized by two principles. First, simple rules such as for example â€Å"nearest vehicle ? rst† or â€Å"least utilized ? rst† are used for dynamic vehicle assignment and second, the geographical space is usually divided into zones. In the literature, most of works on the topic basically focus on customer waiting time minimization by proposing improved methods for rule-based systems. In this context, Shrivastava et al. [SCMK97] describe a fuzzy model for rule selection and Alshamsi et al. [AAR09] propose a new technique for dynamically divide the dispatch areas. The recent apparition of transportation technologies (GPS, EDI, GIS) has widely increased the opportunities for ? eet management optimization. It is also the case for taxi dispatching. For example, Seow et 2 hal-00721875, version 2 31 Jul 2012 al. [SDL10] propose a collaborative model for taxi dispatching where a set of n taxis of the same zone are de? ned as the agents of the model and a set of n customers as the service-requests. The objective is then to maximize the total service quality solving a collaborative linear assignment problem. However, taxi dispatching is not the only aspect that can be optimized. For example, Lee et al. [LSP08] and Jia et al. [Jia08] use real-time vehicle information to propose a model for taxi relocation recommendation based on demand forecasting and a probability model for the design of taxi stops, respectively. Another approach for ? eet management optimization consists in modeling the problem as a variant of the Pick-up and Delivery Vehicle Routing Problem with Time Windows (PDVRPTW). The idea is to plan a set of routes satisfying known in advance customer requests. In the taxi management context, Wang et al. [WLC09] propose a bi-criteria two-phase method with an initial feasible assignment ? rst and a tabu search improvement later in order to minimize the number of vehicles and the sum of travel times for advanced bookings. However, the idea to block some vehicles only for advanced bookings might in some cases yield to a ? eet underutilization. Horn and al. [Hor02] and Meng et al. [MMYH10] try to ? ll the gap between simple non-optimized rule-based taxi dispatching systems and static routing approaches. The second paper describes a genetic network programming in order to ?nd the optimal balance between the waiting time and the detour time. The work of Horn [Hor02] is of particular interest in relation to the present work, proposing a taxi dispatching system architecture similar to our ? eet management system. He proposes a system for vehicle travel time minimization composed by a set of insertion algorithms to decide whether a new customer is accepted or not and a set of optimization mechanisms in order to improve the solution. However, some important di? erences exist between our work and these last two ? eet management systems. The ?rst di? erence is that in our case, the constraints related to the restricted autonomy of the vehicles have also to be taken into account by scheduling charging tasks in the routes of the vehicles. The second di? erence is that, unlike us, both articles deal with the multi-customer problem authorizing customers to share the same vehicle at the same time. 2. 2. Location issue. The location problem was originally de? ned by Webber when he considered how to position a single warehouse minimizing the total distance between the warehouse and a set of customers [Web29]. In 1964, Hakimi [Hak64] de? nes the P-median problem, the problem consists in determining the best location for a set of limited facilities in order to minimize the sum of the weighted distances between the clients and the facilities serving these clients. The problem increases its relevance during the last decades. High costs related to property acquisition and facility construction make facility location projects a critical aspect of strategic planning for a wide range of private and public ? rms. Indeed, the fact that facility location projects are long-term investments leads the researchers to focus on dynamic and stochastic location problems (see [OD98] for a review of this extension of the problem). Another important variant of the problem is the Capacitated Facility Location Problem (CFLP) where facilities have a constraining upper limit on the amount of demand they can satisfy. An extension of the CFLP closely to our problem is the Capacitated Facility Location Problem with Multiple facilities in the same site (CFLPM). In charging terminal location, the positions of the terminals are not the only decision variables, the number of terminals at each position have to be ? xed too. However, in some real-world applications, selecting the best location for distance minimization is not the best suitable choice. For example, in electric vehicle charging terminal location, like in other critical applications such as ambulance and ? re terminal location, the interest is to guarantee that the di? erent geographic zones are covered by a facility (closer than a previously ?xed covering distance). This class of problems are known as Covering Location Problems (see [WC74], [SVB93] and more recently [VP10] for a complete review of covering problems). In that context, the covering issue can be sometimes modelled as a problem constraint. However, if the covering distance is ? xed to a small value the problem might become unfeasible. The Maximal Covering Location Problem (MCLP) [CR74] locates the facilities in order to maximize the number of covered customers (customers with a distance to the nearest facility smaller than an initial ?xed distance). An extension of the problem is the maximal covering with mandatory closeness problem which imposes a maximal distance (less stringent than the covering distance) between the geographical zones and the nearest facility [CR74]. These covering models implicitly assume that if a geographical zone is covered by a facility then the facility will be always available to serve the demand. However, in some applications, when facilities have a ? xed capacity, being covered is not su? cient to guarantee the demand satisfaction. We ? nd 3 in the literature some models attempting to overcome this issue by maximizing the number of geographical zones covered by multiple facilities [DS81, HR86, GLS97]. 3. Fleet management We describe in this section two ways for managing the ? eet, a classical and rule-based one (Subsection 3. 1), and an improved one trying to address explicitly the charging issue (Subsection 3. 2). Let us ? rst introduce some notations. Let CRi be a booking customer request. Each customer request CRi is de? ned by a start time Si and an origin-destination pair Oi ? Di . The Si is ? xed by the customer when the customer request arrives. The completion time of a trip is Ci = Si + ? Oi Di , where ? Oi Di is the travel time between the origin and destination of the customer request CRi . Finally, let R : CTj be a taxi charging task scheduled on the charging terminal CTj . 3. 1. A classical rule-based taxi dispatching system. A taxi dispatching system based on the principles of the most common real-world systems (see for example [SCMK97], [LWCT04] or [AAR09]) is described in this section. The architecture of the current taxi dispatching systems are very similar to the system illustrated in Figure 1. The two main components of the system are (1) a customer acceptation mechanism deciding for each new customer if it is accepted (the accepted customers are inserted into a queue of customers) or rejected and (2) a rule-based mechanism assigning accepted customer requests (trips) to the free taxis. For each accepted trip i, the assigning process has to start a few minutes (? ) before the ? xed start time (Si ) in order to maximize the chances to ? nd a taxi to attend the demand. Once a trip is assigned to a taxi, the vehicle is automatically blocked and the taximeter begins counting. hal-00721875, version 2 31 Jul 2012 Customer Request Rule? based Customer Acceptation Mechanism Time? ordered queue of customers CRn CR2 CR1 Figure 1. Rule-based taxi dispatching system A rule for customer acceptation using the time windows for the trips already accepted is proposed. The idea is to limit the trips that have to be performed at the same time in order to minimize the number of not served customers and to establish a margin of k vehicles to attend opportunistic customers. For each new customer request CRnew the Algorithm 1 determines if it is accepted or not. Algorithm 1: Rule-based checking for customer acceptation for a margin of k vehicles L = {CR1 , CR2 , . . . , CRn }, list of already accepted customers CRnew , new booking customer request nC 0, number of trips performed at the same time than CRnew foreach CRi of L do if CRi is executed at the same time than CRnew ((Si ? Snew Ci ) or (Snew ? Si Cnew ) then Step 1: Increase the number of customers performed at the same time than CRnew (nC nC + 1) if condition to accept the customer (nC n ? k) then Step 2: Insert CRnew to the list of accepted customers L 4. Once the customer request CRi is accepted, it remains in the queue of customers until Si ? ? (? is usually ? xed around 20 minutes). At that moment, the system automatically starts looking for a free taxi having su? cient charge to operate the trip. If di? erent taxis are available, the system assigns the trip to the taxi minimizing the customer waiting time (a parameterizable not announced customer waiting time can be authorized). In the case of no vacant taxis are available, the system waits for a vehicle to become available. If the waiting time for any request exceeds the authorized maximal customer waiting time ?, the customer request is then canceled. Note that the number of unsatis? ed customers can be reduced by using a more restrictive rule for the customer acceptation mechanism. The main advantage of such a system where no future work is planned is the high degree of independence for taxi drivers. On the other hand, the drawbacks are the underutilization of the ? eet and the lost of e? ciency during the peak hours when most of the companies have to close their booking requests systems in order to avoid unsatis? ed customers. Indeed, some real-world systems do not integrate a customer acceptation mechanism leading, in rush hours, to unsatis?ed customers who had been initially accepted and they are ? nally served with an unannounced and, sometimes, intolerable delay or, eventually, never served at all. Furthermore, the charging tasks of the vehicles cannot be controlled leading to a poor ? eet management with vehicles having di? culties to charge the batteries due to charging terminals saturation. 3. 2. The improved electric vehicle management system. An improved ? eet management system aiming to overcome the weakness of the rule-based taxi dispatching system is proposed in this section. The main objectives of the system are to maximize the number of accepted customers and to minimize the customer waiting time. One of the major issues is how to deal with opportunistic demand. Indeed, this kind of demand is unpredictable and must always be satis? ed, so free taxis must be at any moment able to satisfy the longest trip without running out of charge. This constraint makes the problem considerably more complex forcing the system to provide a mechanism ensuring the feasibility of the already accepted trips each time an opportunistic demand is accepted. The approach proposed consists in maintaining continuously a feasible planning for the taxis and the charging terminals (see Figure 2). Each time a customer asks for a trip, a simple insertion algorithm is run, at the end of which either the trip has been successfully inserted or not. The objective is to assign the customer to the taxi minimizing the customer waiting time (a parameterizable announced customer waiting time can be authorized). If none of the tried delays on the pick-up time leads to a feasible planning, a rescheduling algorithm allowing to reallocate the already accepted customers to the taxis is run. In all these processes, a key routine is often called, namely the charging task manager, which schedules the charging tasks of a taxi, given a planning for the other taxis and the charging terminals. Feasible planning:Temporal and autonomy? related constraints are satisfied Taxi 1 hal-00721875, version 2 31 Jul 2012 Taxi 2 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111 000 111 000 CR2 111 000 111 000 CR1 R : CT1 CR4 R : CT2 CR5 VEHICLES Customer Customer Request Acceptation Mechanism Feasible Planning Taxi 000000 n 111111 111111 000000 111111 000000 CR3 R : CT1 111111 000000 111111 000000 1111111111111111111111 0000000000000000000000 Taxi n Taxi 1 CT1 CHARGING TERMINALS CT2 Taxi 2 Figure 2. Customer acceptation mechanism of the electric vehicle management system In the case of an opportunistic demand, which is necessarily accepted, we follow exactly the same scheme except that there is no degree of freedom in the insertion process: the trip is inserted at the front of the planning of the taxi stopped by the customer, and the rescheduling algorithm is also run if it is necessary. 5 3. 2. 1. Insertion algorithm. This algorithm is the ?rst step in order to decide if a new trip CRnew is accepted or not. The objective is to assign the trip to the taxi minimizing the delay on the pick-up time (see Algorithm 2). The algorithm increasely tests the di? erent authorized pick-up times. Once the start time is ? xed, we sequentially try for each vehicle to insert the new request. First the scheduled charging tasks are removed. Then the new request is accepted only if it can be inserted with no constraint violation (the pick-up times of the rest of customers are respected and the current autonomy of the vehicle, without any charging task, is su? cient). In the case that the vehicle autonomy-related constraint is violated, a greedy algorithm trying to schedule a charging task between each pair of trips is proposed. After the charging tasks are inserted, if the taxi is able to perform the trips without running out of charge, then the customer request is also accepted. Algorithm 2: New request insertion algorithm for a maximal authorized delay of ? minutes V = {V1 , V2 , . . . , Vr }, list of taxis CRnew , new booking customer request accepted f alse, variable indicating if the new request is accepted st Snew , start time of the trip while st ? Snew + ?and accepted = f alse do foreach vi of V do Step 1: Delete the charging tasks of the vehicle vi if CRnew starting at st can be inserted in the route of the vehicle vi then if the vehicle autonomy-related constraint is satis? ed then Step 2: CRnew starting at st is inserted in the route of the vehicle vi (accepted true) else Step 3: Insert charging tasks for vi between each pair of trips if the vehicle autonomy-related constraint is satis? ed then Step 2: CRnew starting at st is inserted in the route of the vehicle vi (accepted true) if accepted = f alse then Step 4: Increase the pick-up time for the CRnew (st st + 1) hal-00721875, version 2 31 Jul 2012 3. 2. 2. Rescheduling algorithm. The rescheduling algorithm is proposed when the new customer is still not accepted after the insertion algorithm. As for the insertion algorithm, the goal is to ? nd a new feasible planning for the vehicles integrating the new request CRnew . The main di? erence is that the trips can be reassigned to di? erent vehicles. The problem without taking into account the autonomy-related constraints can be solved in polynomial time [NSZ02]. The idea is to convert the schedule of trips (without the charging tasks) into a graph and to verify using a max ? ow computation that all trips can be performed by the taxis. To construct the network two vertices are considered for each customer request CRi , the ? rst one vi represents the pick-up time and the second one vi the completion time of the customer request. Four dummy vertices are required: 0, 0 , a source s and a sink t. The arcs of the graph are (s, 0), (0 , t), all the (s, vi ), all the (vi , t), all the (vi , vi ), and all the (vi , vj ) such that the customer request CRj can be performed by the same taxi than the customer request CRi and after CRi , that means if Sj ?Ci + ? Di Oj . Except the arcs (s, 0) and (0 , t), they all have a capacity equal to 1. The arcs (s, 0) and (0 , t) have a capacity equal to n. A maximum ? ow computation in this directed graph determines the schedule feasibility and also proposes a new planning for the vehicles respecting the customers pick-up times. The max ? ow computation is integrated in the rescheduling algorithm in order to check the feasibility of the schedule for a given pick-up time st ? [Snew , Snew + ? ] and, if it is the case, to ? nd a reference planning (planning without charging tasks). A local search explores the neighborhood of the reference planning de? ned by the swap and the reallocation operators [Sav92]. Finally, for each explored planning respecting temporal constraints, the greedy algorithm for charging task scheduling is sequentially applied to the taxis that do not satisfy autonomy-related constraints (that is, taxis whose current charge is not enough to perform all 6 the trips assigned to them without adding charging tasks). If a feasible solution is found, the new customer is then accepted. 3. 2. 3. Charging task manager. As we have already seen, the insertion and the rescheduling algorithm constantly runs a greedy algorithm aiming to insert a charging task between each pair of successive trips of the same route. The algorithms proposed to determine if a new charging task can be integrated in a speci? c charging terminal planning are described in this subsection. The main feature of our problem is that the processing time of the new charging task is not ? xed, instead it is a decision variable de?ned between the interval limited by the minimal charging time for a vehicle pmin (customizable parameter) and the maximal charging time corresponding to the time necessary for a full charge. The problem to be solved by the charging terminal manager can be then formally stated as follows. A charging task Ri is de? ned by its time window [ri , di ], where ri is the earliest start time (earliest arrival time to the terminal) and di the latest end time (latest departure time from the terminal). Let pi be the decision variable corresponding to the processing time of the task Ri , then ri ? Si and Si + pi ? di , where Si is the e? ective start time of Ri . Given a feasible schedule of n charging tasks S n = {S1 , S2 , . . . , Sn , } for the charging terminals located at the same geographical position. We are given a new charging task Rn+1 with a time window [rn+1 , dn+1 ] and a processing time pn+1 inside the interval pmin ? pn+1 ? pmax . The problem consists in ? nding a new n+1 feasible schedule S n+1 = {S1 , S2 , . . . , Sn , Sn+1 } maximizing the processing time of the task Rn+1 (whence without changing the processing time of the other tasks). The mechanism tests ? rst a task insertion aiming to ? nd quickly a feasible solution. The complexity of the algorithm for task insertion maximizing the processing time of the new task is O(n) where the start times and completion times of the scheduled jobs are non-decreasing ordered. If no solution is found after the task insertion algorithm, a dichotomous algorithm allowing to reschedule the tasks is proposed in order to ? nd a solution maximizing the processing time of the new task. For each iteration of the algorithm, a satis? ability test based on constraint propagation involving energetic reasoning is ?rst triggered. The goal of the feasibility test is to detect an inconsistency indicating that it is not possible to ? nd a feasible schedule integrating the new task. Finally, if the energetic reasoning is not conclusive a local search algorithm is proposed in order to ? nd a solution. Satis? ability test: Energetic reasoning. A satis? ability test based on constraint propagation involving energetic reasoning is proposed [LE96]. A ? ctitious energy (which has nothing to do with the electricity) is produced by the charging terminals and it is consumed by the charging tasks. We determine the ? ctitious energy consumed by the tasks (Econsumed ) over a time interval ? = [t1 , t2 ] and we compare this ? ctitious energy with the available ? ctitious energy (Eproduced = m ? (t2 ? t1 )). The minimal ? ctitious energy consumed by the tasks in an interval ? = [t1 , t2 ] is: n+1 hal-00721875, version 2 31 Jul 2012 (4) Econsumed = i=1 max{0, min{pi , t2 ? t1 , ri + pi ? t1 , t2 ? di + pi }} If Econsumed Eproduced , it is then impossible to ? nd a feasible schedule S n+1 integrating the new task. The relevant intervals ? for a complete satis? ability analysis can be enumerate in O(n2 ). The test is restricted to the intervals [t1 , t2 ] speci? ed by {ri } ? {di } ? {ri + pi } ? {di ? pi } where the new task Rn+1 may consume (t1 ? dn+1 and t2 ? rn+1 ). Dichotomous algorithm. A dichotomous algorithm maximizing the processing time of the new task is described in this section (see Algorithm 3). A dichotomy is run on the processing time p as follows. For processing times p ? [pmin , pmax ], the satis? ability test based one the energetic reasoning indicates whether n+1 the necessary conditions are satis? ed or not. If it is the case, a local search mechanism tries to ? nd a feasible schedule. The parallel machine scheduling problem with time windows can be solved by a list scheduling algorithm. It means there exists a total ordering of the jobs (i. e. , a list) that, when a given machine assignment rule is applied, reaches the optimal solution. For our problem, this rule consists in allocating each task to the machine that allows it to start at the earliest (Earliest Start Time or EST rule). The local search mechanism proposed to solve the problem is based on this result. First, the tasks are ordered in a non-decreasing order of their due dates (Earliest Due Date or EDD rule), then the local search consists in 7 exploring di?erent permutations of the list de? ned by the insertion neighborhood (O(n2 )). For each list of task, the machines are assigned according to the EST rule in order to reach a feasible solution. If no feasible schedule is eventually found, the request is rejected. Algorithm 3: Dichotomous algorithm for processing time maximization min pmin max pmax n+1 n+1 Sbest ? while min ? max do Step 1: Fix the processing time p of the new task Rn+1 (p min+max ) 2 if Satisf iabilityT est() then Step 2: Sort the tasks according to the EDD rule Step 3: Local search using the insertion operator if a feasible schedule S n+1 = {S1 , S2 , .. . , Sn , Sn+1 } is found then Step 4: Update the lower limit (min p + 1) n+1 Step 5: Update the best solution (Sbest S n+1 ) else Step 6: No solution exists, update the upper limit (max p ? 1) else Step 7: No solution exists, update the upper limit (max p ? 1) n+1 if Sbest = ? then Step 8: No solution is found (return ? ) else n+1 Step 9: A feasible solution is found (return Sbest ) hal-00721875, version 2 31 Jul 2012 4. Electric vehicles charging terminal location The EV charging terminal location problem consists in determining the best locations of the charging terminals. The linear programming model has to take into account two important aspects. First, the charging terminals have to be conveniently spread over the geographical area in order to avoid remote geographical zones which di? cult taxi operability and ? eet management. The second aspect is that the model has to determine the number of charging points facilitating the charging process of the taxis by minimizing the risks of terminals saturation. For these purposes, we propose two models, one called the P -median model, the other the Demand-based model. V is the set of geographical points of the problem and J ? V is the set of potential locations where the charging terminals can be located. The number of terminals is limited to r. 4. 1. P -median model. Following Hakimi [Hak64], we de? ne xj to be the decision variables indicating if a facility is located to the point j and yij to be the variables indicating that the geographical point i is assigned to the facility located in j. The linear program minimizing the sum of the distances between clients and facilities can be written as follows. 8 (5) min i? V j? J distij yij s. t. (6) j? J yij yij xj j? J = 1 for all i ? V ? xj for all i ? V, j ? J ? r ? {0, 1} for all j ? J ? {0, 1} for all i ? V, j ? J (7) (8) (9) (10) xj yij hal-00721875, version 2 31 Jul 2012 4. 2. Demand-based model. Another approach consists in de? ning a model with two distances ? f ar and ? close as proposed by Church and ReVelle [CR74]. The idea is then to spread the terminals by ? xing a maximal distance (? f ar ) between the di? erent geographical zones and the nearest charging terminal and, at the same time, trying to maximize the demand that will be covered by a nearby charging terminal (? close ). We can then de? ne Jif ar (resp. Jiclose ) as the subset of points in J at distance less than ? f ar (resp. ?close ) from i ? V . Conversely, Vjclose is the set of points at distance less than ? close from the point j ? J. Let xj be the decision variable indicating the number of terminals located at point j ? J and yij to be the fraction of the demand di for i ? V covered by a charging terminal located in j at distance less than ? close from i. The linear programming model proposed to solve the problem called Demand-based model is the following. (11) max j? J i? Vjclose di yij s. t. (12) f j? Ji ar xj yij close j? Ji ? 1 for all i ? V ? 1 for all i ? V ? xj for all j ? J ? r ? Z+ for all j ? J ? R+ for all i ? V, j ? Jiclose (13) (14) i? Vjclose di yij xj j? J (15) (16) (17) xj yij The objective function (Eq. (11)) consists in maximizing the pointwise demand covered by a charging terminal considering the distance ? close . Eq. (12) imposes that a geographical zone i ? V must be covered at least for one charging terminal considering the distance ? f ar . Here the mandatory closeness is only required for the geographical zones closer than ? f ar from a potential charging terminal location in order to ? nd a solution even if this constraint is violated for some geographical zones. We stress that an adequately ? f ar make possible to spread the charging terminals over the geographical area. Eq. (13) speci? es that for each geographical zone i ? V the sum of the fractions of demand covered by a charging terminal considering the distance ? close has to be less or equal to the unit.