Theorem 1 Let m≥3. The growth constant for the family of m-barrel fullerenes is equal to ρ(m) =
⌊m+13 ⌋
∏
j=1
(
2 cosπ(2j−1) m
)2
. (2)
References
[1] Afshin Behmaram , Tomislav Doslic, Shmuel Friedlsnd.Matchings in generalized Fullerene.
Ars Mathematica contemporanea ,Vo11, No2 (2016), pp 301-311.
[2] K. Kutnar, D. Marušič, On cyclic edge-connectivity of fullerenes. Discrete Appl. Math, 156 (2008) 1661–1669.
Solving the Green Vehicle Routing Problem with Capacitated Alternative Fuel Stations
Maurizio Bruglieri1, Simona Mancini2, Ornella Pisacane3
1 Dipartimento di Design, Politecnico di Milano, Milano, Italy maurizio.bruglieri@polimi.it
2 Diparimento di Matematica e Informatica, Universitá degli Studi di Cagliari, Cagliari, Italy simona.mancini@unica.it
3 Dipartimento di Ingegneria dell’Informazione, Universitá Politecnica delle Marche, Ancona, Italy
pisacane@dii.univpm.it
Abstract
The Green Vehicle Routing Problem (GVRP) aims to efficiently route a fleet of Alternative Fuel Vehicles (AFVs), in order to serve a set of customers, minimizing the total travel distance. Each AFV leaves from a common depot, serves a subset of customers and returns to the depot, without exceeding a maximum duration. Due to their limited driving range, the AFVs may need to refuel one or more times at the Alternative Fuel Stations (AFSs), along their route. In this work, we introduce the GVRP with Capacitated AFSs (GVRP-CAFS) in which only a limited number of AFVs can refuel at the same time at each AFS to account for their limited capacity.
In order to solve the GVRP-CAFS, we propose an exact approach in which a route is the composition of paths, each handling a subset of customers without intermediate stops at AFSs. Firstly, all feasible non-dominated paths are generated. Secondly, via a path-based Mixed Integer Programming model, the paths are selected and properly combined each other to generate the routes of the optimal GVRP-CAFS solution. To reduce the computational times, a relaxed version of the path-based model is solved and then, the violated constraints are iteratively added. Some preliminary results are also discussed.
Keywords: Vehicle Routing Problem, Alternative Fuel Vehicles, Mixed Integer Program- ming
1 Introduction and statement of the problem
Nowadays, the transportation companies are requested to provide more competitive services in a more sustainable way, through efficient trip planning, smart distribution systems and the use of new technologies. Among the latter, the Alternative Fuel Vehicles (AFVs), i.e., vehicles that use alternative fuel (e.g., methanol and electricity), play a key role, contributing
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to reduce both the CO2 emissions and the noise pollution. However, purchasing an AFV still remains very expensive. Moreover, the AFV driving range is still limited and, in fact, it may require several stops at theAlternative Fuel Stations(AFSs) in a trip. In addition, the AFSs are currently not widespread on the territory. Therefore, it becomes very significant to properly plan the AFV trips in order to prevent drivers remaining without enough fuel to either reach the closest AFS or return to the depot.
A relatively new operational research area is focused on the Green Vehicle Routing Prob- lem-GVRP ([1]), introduced in [2]. It aims to route a fleet ofmAFVs, based on a common depot (denoted by 0), minimizing the total travel distance. Each route starts/ends from/to 0, handling a subset of customers within the time limit Tmax and refueling (even more than once) at AFSs. The GVRP is formally represented on a complete directed graph G= (N, A), where the set of the nodesN =ISFS{0}contains the setI of customers and the setF of AFSs whileAindicates the arc set. For each (i, j)∈A, both the travel time and distance,tij anddij, are known. Moreover, for eachs∈F and for eachi∈I, the refueling time ps and the service timepi are given. For each AFV, both the maximum fuel capacity Q and the average speed v are known. The fuel consumption is linearly proportional to the travel distance through the fuel consumption rate r. Due to the limited fuel capacity, the maximum distanceDmax an AFV can travel without stopping at any AFS is given by Q/r. Each AFV is supposed to leave the depot fully refueled and to be fully refueled when it stops at an AFS. Finally, each AFSs is assumed to have an infinite capacity, i.e., there is no limit on the number of AFVs that can simultaneously refuel at s. But, really, the AFSs have a limited number of refueling docks. Neglecting this aspect, during the route planning, may yield long waiting times at the AFSs with a significant negative impact on the solution, especially when the refueling time is high (e.g., in the case of electric vehicles) and/or Tmax is very tight. In these cases, omitting AFSs capacity may produce infeasible route plans. To overcome this issue, we introduce a new variant of the GVRP, i.e., the GVRP with Capacitated AFSs (GVRP-CAFS), in which only a limited number of AFVs can refuel at the same time at the same AFS. Without loss of generality, we assume that the capacityηsof each AFSsis unitary. In fact, the caseηs>1 can be reduced to the one with ηs = 1 by properly adding clones of s with unitary capacity. We solve the GVRP-CAFS through the exact approach described in Section 2.
2 An exact approach for the GVRP-CAFS
Each route in a GVRP solution can be seen as the composition of paths, each handling a subset of customers without intermediate stops at AFSs. Each path can be between: 0 and AFS, AFS and 0, two AFSs and finally, 0 and itself (complete route). Moreover, for each path k, the origin (starting node)sk, the destination (arrival node) ak, the travel distance dk and the duration γk are known. In particular,γk is the sum of the travel times and the service times at the nodes of k. In the GVRP solution in Figure 1, where C1, C2, C3, C4 denote the customers, the route{0, C1, C2, C3, AF S2, C4,0}is the composition of the paths {0, C1, C2, C3, AF S2} and{AF S2, C4,0}.
The proposed solution approach is then based on two steps. Firstly, the set K of all
FIG. 1: A solution feasible for a GVRP.
feasible non-dominated paths is generated. A pathk is feasible (feasibility rules) if: dk ≤ Dmax andγk+t0sk +tak0+pak ≤Tmax. Moreover, a feasible path k1 dominates a feasible path k2 (dominance rules) if: sk1 = sk2, ak1 = ak2, they handle the same customers and dk2 ≥dk1. From the setK, the setP of all the pairs of paths is generated. Givenk1, k2 ∈K, a pair (k1, k2) exists (compatibility rules) if: ak1 =sk2,sk2 6= 0, the set of customers handled in the two paths are disjoint andt0sk1 +γk1+γk2+pak1+tak20≤Tmax. In the second step, a path-based Mixed Integer Programming (MIP) model is used to select the paths and properly combine them to generate the routes of the optimal GVRP-CAFS solution. A coverage parameter, cik, is then introduced, equal to 1 if i ∈ I is handled in k ∈ K, 0 otherwise. The refueling time pak needed at ak is equal to 0 if ak = 0. The following decision variables are introduced: zk, equal to 1 ifk∈K is selected, 0 otherwise;xkl, equal to 1 if l ∈ K is covered just after k ∈ K, 0 otherwise and finally, τk, a positive variable representing the starting refueling time atak ofk ∈K. The path-based MIP model is given in the following.
minX
k∈K
dkzk (1)
X
k∈K
cikzk = 1 ∀i∈I (2)
X
k∈K
x0k ≤m (3)
X
k1∈K:(k1,k2)∈P
xk1k2 = X
k1∈K:(k2,k1)∈P
xk2k1 ∀k2∈K (4) X
k1∈K:(k1,k2)∈P
xk1k2 =zk2 ∀k2∈K|k26= 0 (5) τk2 ≥τk1+pak
1 +γk2−Tmaxxk1k2 ∀(k1, k2)∈P (6)
|τk1−τk2| ≥pak1 ∀k1, k2 ∈K|ak1 =ak2, ak1 6= 0 (7) γk ≤τk ≤Tmax−pak +Tmax(1−zk) ∀k ∈K (8)
zk ∈ {0,1} ∀k ∈K (9)
xk1k2 ∈ {0,1} ∀(k1, k2)∈P (10) The objective function (1) concerns the minimization of the total travel distance. Each customer has to be visited exactly once (2) and the number of routes selected does not exceed the number of available AFVs (3). Route continuity is ensured by constraints (4).
A path can be inserted in a route only if it is selected (5). If xk1k2 = 1, τk2 cannot start before both the refueling operation atak1 is completed andk2 is performed (6). Two AFVs cannot simultaneously refuel at the same AFS (7). The refueling of the AFV in the path k cannot start before the time necessary to travel the path and it cannot finish afterTmax (8). Constraints (7) are linearized by (11)-(14), through auxiliary variablesξandà. In the Linear MIP (MILP) model, constraints (13) avoid overlapped refueling operations.
ξk1k2 ≥ − 1 Tmax−pak
1
(τk1 −τk2 −pak1) ∀k1, k2 ∈K|ak1 =ak2, ak1 6= 0 (11) àk1k2 ≥ − 1
Tmax−pak
1
(τk2−τk1−pak
1) ∀k1, k2 ∈K|ak1 =ak2, ak1 6= 0 (12) ξk1k2 +àk1k2 ≤3−zk1 −zk2 ∀k1, k2∈K|ak1 =ak2, ak1 6= 0 (13) ξk1k2, àk1k2 ∈ {0,1} ∀k1, k2 ∈K|ak1 =ak2, ak1 6= 0 (14) For limiting the amount of time required by our approach, a relaxation of the GVRP- CAFS (RP) generated by omitting (11)-(14) is solved. On each iteration, we solve RP to optimality and check if any of those constraints are violated in the optimal RP solution. If this is the case, ∀(k1, k2) ∈ P for which those constraints are violated, i.e., for which the related refueling operations overlap, we add to the RP the corresponding violated constraints (11)-(14) and we reiterate; otherwise, we stop because the current optimal RP solution is optimal for the GVRP-CAFS too. This approach allows us strongly limiting the number of constraints involved to address the capacity issue.
3 Results and conclusions
We introduced the GVRP-CAFS, a more realistic variant of the GVRP, in which the AFS capacity, i.e., the number of AFVs that can simultaneously refuel at the same AFS, is limited.
For the GVRP-CAFS, we formulated a MILP model and we proposed an exact method to solve it in more reasonable time. Preliminary tests were carried out on a set of challenging instances with tight AFS capacity and on average 15 customers and 3 AFSs. The proposed exact approach solved to optimality all the instances within an average computational time of 22 seconds against an average of 557 seconds of the MILP model.
References
[1] Bektaş, T. and Demir, E. and Laporte, G. (2016), “Green vehicle routing”, Green Transportation Logistics, 243-265, Springer International Publishing.
[2] S. Erdoğan and E. Miller-Hooks, “A green vehicle routing problem”, Transportation Research Part E: Logistics and Transportation Review 48(1),100-114, 2012.
The Electric Vehicle Relocation Problem in Carsharing Systems with Collaborative Operators
Maurizio Bruglieri1, Fabrizio Marinelli2, Ornella Pisacane2
1 Politecnico di Milano, Milano, Italy maurizio.bruglieri@polimi.it
2 Universitá Politecnica delle Marche, Ancona, Italy {marinelli,pisacane}@dii.univpm.it
Abstract
We address the problem of balancing the demand and the availability of vehicles be- tween stations in urban one-way electric carsharing systems through operator relocations.
Unlike the previous papers, we assume that the operators can collaborate among them through thecarpooling, i.e., giving a lift to the others when moving an EV from a pick-up request station to one of delivery. For this new problem, we propose a Mixed Integer Linear Programming formulation and a column generation based heuristic solution ap- proach.
Keywords : Mixed Integer Linear Programming, column generation, Pick-up and Delivery Problem with Time Windows, operator based relocation, one-way carsharing.
1 Introduction
The carsharing systems allow users renting cars by paying a charge that depends on the actual time of use (also a fraction of an hour) eliminating the fixed costs due to both the ownership and the maintenance of the vehicles. However, theone-way carsharing systems, in which a user can deliver the vehicle to a station different from the one of pick-up, pose the management problem of balancing the demand and the availability of vehicles between the stations [4].
Moreover, when the carsharing fleet is made up of Electric Vehicles (EVs), the relocation is more complicated due to their recharge needs.
We address the operator-based EV relocation problem in urban one-way carsharing systems assuming that: the requests are known in advance (exact predictive relocation); the operators directly drive the EVs from stations with exceeding EVs (pick-up requests) to stations that need EVs (delivery requests); they move from the latter to the former by folding bicycles as introduced in [1]. A revenue is associated with each relocation request as well as a fixed cost with each operator used. The objective is to maximize the total profit given by the difference between the total revenue due to the requests satisfied and the total cost of the operators employed, as introduced in [2, 3].
Unlike the previous papers, where the operators do not interact with each other, we assume that they can collaborate among them through thecarpooling, i.e., giving a lift to other opera- tors when moving an EV from a pick-up request station to one of delivery. We assume that the lift is given with no intermediate stop, i.e., all the passengers can get out of the EV only at the driver’s delivery station. We call this new version of the Electric Vehicle Relocation Problem (E-VReP), the E-VReP with Collaborative Operators.
For this problem, we propose a Mixed Integer Linear Programming (MILP) formulation and a column generation based solution method.
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2 Statement of the problem and MILP formulation
Let L be the maximum distance a fully recharged EV can cover. When the EV is not fully recharged, such distance is supposed linearly proportional to its residual battery charge. Al- though the charging time function depends on the battery technology used, it is assumed to be linear andΓis the time necessary for a full recharge. We assume that each parking station has a charger to which an EV is always connected when it is unused.
Let K be the number of operators available and C the cost associated with their employ- ment. LetDandP be the set of delivery requests (i.e. EVs delivery to try to prevent a station from running out of EVs) and of pick-up requests (i.e, to try to prevent a station from being full of EVs), respectively. For each relocation request r∈P ∪D, the parking locationvr, the residual battery chargeρr, the earliest and the latest time allowed to carry outr, [τrmin, τrmax], are known. We assume that the requests are not mandatory and a revenueπris obtained if the requestr is satisfied. Since the carsharing fleet is supposed homogeneous, each requestr ∈D can be satisfied bringing to vr an EV of a pick-up request, compatible for both time window and the battery charge level.
We want both to route and to schedule the operators, leaving from a common depot (0), at two different times,t′0 andt′′0, e.g., corresponding to the start of the Morning Shift (MS) and of the Afternoon Shift (AS), in order to maximize the total profit. In each route, a request of pick-up is always alternated to a one of delivery. Moreover, since we assume that the operators can collaborate through thecarpooling with no intermediate stop, their routes can share some ordered pairs of pick-up and delivery requests. The same pair can be shared in at most C˜ routes, beingC˜ the capacity of an EV.
The problem is represented on a directed graph G= (N, A) where N =P ∪D∪ {0} and arc set A is the union of the arcs AEV traveled by EV, and those of AB traveled by bike.
Arcs (i, j) ∈ AEV, with i ∈ P and j ∈D, model the action of an operator that goes from a station of pick-up to one of delivery by EV, also possibly giving a lift to other operators. Arcs (j, i)∈AB, withj∈D andi∈P, model the action of an operator that moves from a station of delivery to one of pick-up by bike. For each(i, j) ∈AEV, dij is the length of the shortest path from vi to vj by EV, while cij is the corresponding operational time taking into account the time to load the bike in the EV trunk, to go fromitojby EV, to park the EV and to take the bike from the EV trunk. Instead, ∀(i, j)∈AB, cij is the time to go from i toj by bike.
The problem is mathematically modeled by introducing the following decision variables: xij, number of operators traversing (i, j) ∈ A; yij, equal to 1 if (i, j) ∈ A is traveled by at least one operator, 0otherwise;ti, the latest arrival time ati∈N and ξrequal to1 ifr∈P ∪D is handled in the MS, 0if it is served in the AS.
max X
(i,j)∈AEV
(πi+πj)yij− X
j∈δ+(0)
Cx0j (1)
X
j∈δ+(0)
x0j ≤K (2)
X
j∈δ+(i)
yij ≤1 ∀i∈P (3)
X
i∈δ−(j)
yij ≤1 ∀j∈D (4)
X
j∈δ+(i)
xij − X
j∈δ−(i)
xji= 0 ∀i∈N (5)
xij ≤Cy˜ ij ∀(i, j)∈A (6)
yij ≤xij ∀(i, j)∈A (7)
t′0ξj+t′′0(1−ξj) +c0jy0j ≤tj ∀j∈δ+(0) (8) ti+cijyij−T(1−yij)≤tj ∀(i, j)∈A:i6= 0, j 6= 0 (9) ti+ci0yi0−t′0ξi−t′′0(1−ξi)≤T ∀i∈δ−(0) (10) ξi−ξj ≤1−yij ∀(i, j)∈A:i6= 0, j6= 0 (11) ξj−ξi ≤1−yij ∀(i, j)∈A:i6= 0, j6= 0 (12) τimin ≤ti≤τimax ∀i∈P ∪D (13) L(ρi+ti−τimin
Γ )≥dijyij ∀(i, j)∈AEV (14)
ρi+ti−τimin Γ −dij
L yij ≥ρj−τjmax−tj
Γ −(ρj+ 1)(1−yij) ∀(i, j)∈AEV (15) 1−dij
L yij ≥ρj−τjmax−tj
Γ −(ρj+ 1)(1−yij) ∀(i, j)∈AEV (16) xi,j ≥0, integer, yij ∈ {0,1}, ∀(i, j)∈A, ti≥0 ∀i∈N (17) where δ−(i)and δ+(i)denote the ingoing and outgoing arcs in/from i∈N, respectively. The objective function (1) represents the total profit to be maximized. Constraint (2) ensures that no more than K operators are employed. Constraints (3) avoid that the same picked up EV is used to satisfy more than one delivery request. Vice versa, constraints (4) avoid that more than one pick-up request are used to satisfy the same delivery request. Conditions (5) ensure the flow conservation on x.
The x variables are linked to the y ones in (6) and (7): these constraints ensure that if yij = 1, the arc (i, j) ∈ A can be traversed by at most C˜ operators; otherwise, it cannot be traveled. In each route, the arrival times at both the first node visited and the next ones are ruled by constraints (8) and (9), respectively.
The total duration of a route cannot exceed T thanks to (10). Constraints (11) and (12) ensure that, if two requests are served in the same route, they are served in the same time shift too. The time window of each request i ∈P ∪D is imposed in conditions (13). The distance traveled by each EV is proportional to its residual battery level (14) and each EV is delivered satisfying the required battery level (15)-(16).
3 A Column Generation based heuristic
Since the model described in the previous section can solve in reasonable time only instances of few tens of requests through a state of the art MILP solver (CPLEX), we propose a different solution method based on column generation.
For this purpose, let Ω the set of all feasible routes for the E-VReP. The following route- based formulation models a relaxation of the original problem (1)-(17) since no synchronization constraint is imposed on the routes that share the same arcs.
It is based on the binary variablesθω= 1if the routeω is chosen, 0 otherwise and on binary variables yij = 1if arc (i, j)∈Ais chosen in at least one route, 0 otherwise:
max X
(i,j)∈AEV
(πi+πj)yij−CX
ω∈Ω
θω (18)
X
ω∈Ω
θω≤K (19)
X
ω∈Ω
aijωθω≤Cy˜ ij ∀(i, j)∈A (20) yij ≤ X
ω∈Ω
aijωθω ∀(i, j)∈AEV (21)
X
j∈δ+(i)
yij ≤1 ∀i∈P (22)
X
i∈δ−(j)
yij ≤1 ∀j∈D (23)
θω∈ {0,1} ∀ω∈Ω, yij ∈ {0,1} ∀(i, j)∈A (24) where the parameter aijω is equal to 1 if the pair of requests (i, j) ∈ A is served in route ω, 0 otherwise. Indeed constraints (19) guarantees that no more thanK operators are used;
constraints (20) ensure that no more than ˜C operators are carpooling along each arc, at the same time; while (21) guarantee the coherency between variables θω and yij, i.e., if yij = 1 then at least one routeω containing the arc (i, j) must be selected; (22) ensure that with the vehicle picked up fromi only one delivery request can be satisfied; vice versa, (23) guarantees that a delivery request j can only one be satisfied by one pick-up request; finally, (24) model the variables nature.
The continuous relaxation of the route-based formulation (18)-(24) is solved through a col- umn generation approach. Then, an integer solution is heuristically detected by solving (18)- (24) restricted to the only ”good” routes found (i.e., those selected along the column generation) and possibly adding other routes. However, such integer solution could not satisfy the synchro- nization constraints among the operators, relaxed in the formulation (18)-(24). This solution is then heuristically repaired according to the synchronization infeasibilities, through proper forward and/orbackward shifts of the relocation request execution times.
We notice that in this procedure we have to guarantee not only that each route satisfies the time windows of the requests handled after the shifts, but we also have to carefully consider the battery charge levels. Indeed, if an EV is picked up too early then it may have not enough battery recharge to reach the next delivery request. Vice versa, if it is picked up too late then it may arrive to the delivery request just before the maximum allowed time window without the required battery level since there is not enough time to recharge it at the delivery station.
4 Results and conclusions
In this work, we extended the E-VReP problem concerning the relocation of electric vehicles in a carsharing system, allowing the operators to collaborate among them through thecarpooling, i.e., giving a lift to other operators when moving an EV from a pick-up request station to one of delivery. The problem was formulated by MILP and solved in more efficient way through a column generation based heuristic. Preliminary results show that thanks to the collaboration among the operators not only it is possible to decrease the distance covered via bike by the operators, but sometimes also to increase the total profit.
References
[1] M. Bruglieri, A. Colorni, A. Luè, “The vehicle relocation problem for the one-way electric vehicle sharing”,Networks, 64 (4), 292–305, 2014.
[2] M. Bruglieri, F. Pezzella, O. Pisacane, “Heuristic algorithms for the operator-based relo- cation problem in one-way electric carsharing systems”,Discrete Optimization, 23, 56–80, 2017.
[3] M. Bruglieri, F. Pezzella, O. Pisacane, “An Adaptive Large Neighborhood Search for Relocating Vehicles in Electric Carsharing Services”,Discrete Applied Mathematics, DOI:
10.1016/j.dam.2018.03.067, 2018.
[4] G. Laporte, F. Meunier, R., Wolfler Calvo, “Shared mobility systems”, 4OR-Q J Opera- tional Research, 13:341–360, 2015.