Modeling and solving a vehicle-sharing problem
MModeling and solving a vehicle-sharing problem
Miriam Enzi ∗ a,b , Sophie N. Parragh a , and David Pisinger ca Institute of Production and Logistics Management, Johannes Kepler University LinzAltenberger Straçe 69, 4040 Linz, Austria {miriam.enzi,sophie.parragh}@jku.at b Center for Mobility Systems, AIT Austrian Institute of TechnologyGiefinggasse 4, 1210 Vienna, Austria c Department of Management Engineering, Technical University of DenmarkAkademivej Building 358, 2800 Kgs. Lyngby, Denmark [email protected]
March 20, 2020
Abstract
Motivated by the change in mobility patterns, we present a new modeling approach for the vehicle-sharingproblem. We aim at assigning vehicles to user-trips so as to maximize savings compared to other modes oftransport. We base our formulations on the minimum-cost and the multi-commodity flow problem. These for-mulations make the problem applicable in daily operations. In the analysis we discuss an optimal compositionof a shared fleet, restricted sets of modes of transport, and variations of the objective function.
Keywords— shared mobility, vehicle-sharing, car-sharing, transportation ∗ Corresponding author. a r X i v : . [ c s . OH ] M a r ontents VShP-1T ) . . . . . . . . . . . 42.2 The vehicle-sharing problem with multiple types of shared vehicles (
VShP-xT ) . . . . . . . . . 5
VShP-1T ) . . . . . 83.3 Results for the vehicle-sharing problem with multiple types of shared vehicles (
VShP-xT ) . . . . 103.4 Comparison of
VShP-1T:car , VShP-1T:ecar , VShP-xT and the case where all trips are coveredby one car/MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Including user preferences as a restricted subset . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6 Comparing objective functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7 Managerial implications and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Introduction
Mobility – how we use it and see it – is changing. People tend to be mobile rather than owning cars. ”Mobilityas a Service” (MaaS) [33] has emerged as a widely known and used term. This change is supported by novelmobility concepts, not only in the private sector but also in the area of corporate mobility. Increasingly, companiesare trying to change their view on their corporate mobility by switching from individually assigned cars towardsMaaS for their employees, and give incentives to use (a combination of) ”greener” modes of transport to avoidpollution and congestion problems.Motivated by a project with several company partners, we study the vehicle-sharing problem in a company,having one or more offices (= depots), from which the employees have to visit various customers during officehours (e.g. for business meetings). Each visit involves one specific employee (= user). A trip covers a sequenceof meetings (= tasks) of one user, starting at a depot and terminating at the same location or in another depotof the company. Thus a trip contains several stops and it starts and ends at a predefined (but possibly different)depot. The company operates a pool of cars and provides possibilities to use other modes of transportation(MOT). Therefore, instead of having a company car for each employee, these share a pool of cars of fixed size.In addition, the users can use public transportation, bikes, taxis or walk to their meetings if they are very close.The user may specify in advance which modes she can use.We aim at assigning user-trips to the available vehicles, e.g., shared cars, in the best possible way. Thereforewe maximize the savings obtained when using a vehicle instead of any other MOT. The costs of transportationdo not only include distance cost, but also hourly wages of employees in order to properly reflect the trade-offbetween fast (but expensive) and possible slower (but cheaper) modes of transportation, such as public transportor bikes. We note that cars may not always be the fastest MOT. We also include emission cost to strengthen theuse of environmental friendly MOTs.Research addressing car-sharing systems usually tackle strategic decisions such as charging station placement[7, 11] or the relocation of cars [41, 26, 34, 9, 24, 12]. Many formulations studying car-sharing systems are basedon a time-space networks and/or flow model [16, 8, 9, 24, 26, 12, 35, 42]. Pal and Zhang [35] formulate anon-linear multi-commodity problem in a time-space network. After linearization they solve the intractablemodel using a heuristic. Zhang et al. [42] work on a vehicle to trips assignment and relay decisions in one-way electric car-sharing systems. They model the problem in a multi-commodity time-space network (withone commodity only) and develop a heuristic thereof. In Enzi et al. [18] both car-sharing and ride-sharing areconsidered simultaneously. The first step of the auxiliary graph transformation of Enzi et al. [18] is the sameas the presented graph in this work. However, they, extend the graph by duplicating trips including ride-sharingand solve the car- and ride-sharing problem by a kind of column generation algorithm, assigning cars to trips.Detailed surveys on car-sharing are provided by Jorge and Correia [23], BrandstÃd’tter et al. [10] and [31].We provide a novel modeling approach for a vehicle-sharing problem. We formulate the case where only onetype of vehicles is shared as a minimum-cost flow problem [1]. When more than one type of vehicles is shared,we base the formulation on the multi-commodity flow problem [5]. Note that, even though we will mainly baseour examples and results on cars, this problem can easily incorporate various modes of transport such as bikes orscooters.
Contribution and outline
The contributions of this paper are as follows: we introduce a novel modeling ap-proach for the vehicle-sharing problem where we use the well-known minimum-cost flow and multi-commodityflow formulation. We show that the models can be solved efficiently and thus used on a daily operational basis.2urthermore, we provide a detailed analysis with respect to the impact of using different kinds of shared vehicles,and provide insights into optimal fleet composition in a shared system. The analysis shows the cost alterationwith an increasing number of users as well as a bigger fleet of vehicles. We also analyze the number of trips percar during a day and the disadvantage (from a cost-perspective) when giving the opportunity to restrict the set ofavailable MOTs per person/trip. We compare the case where no sharing is allowed with our introduced sharingsystems. Finally, we compare the outcomes of different objective functions, whereas we once use a combinationof operational distance cost and cost of time, and then considering time only.The paper is organized as follows: We start by introducing the vehicle-sharing problem in Section 2. We firstintroduce the model with a single shared vehicle type, formulated as a minimum-cost flow problem in Section 2.1and then the multi-commodity flow problem for multiple shared vehicle types in Section 2.2. In Section 3we summarize our analysis based on an extensive computational study and give managerial implications. Weconclude this paper in Section 4.
Formally, our vehicle-sharing problem can be formulated as follows: We have a set of modes of transport K suchas walk, bikes, public transport (bus, train, metro), taxi and cars. Moreover, we have a set of users P that haveto visit meetings (= tasks). Every user p ∈ P has one or more trips π . A trip has an origin o π and destination e π whilst covering in between the set of tasks. Each task is associated with a different location and has an associatedlatest arrival time and duration. The person to task assignment is not interchangeable. With this we have a fixedsequence of tasks within a trip.Let us assume a task q and its fixed successor q (cid:48) . We have the driving time between the two tasks ( q , q (cid:48) ) using mode of transportation k , and the cost of driving between two tasks ( q , q (cid:48) ) using mode of transportation k . We consider a set of mobility types with infinitive capacity, and at least one with restricted capacity that isshared, e.g., cars. If a trip is started with one mode of transport, then it should be used for the full trip. The tripsfollow a fixed sequence of task. We calculate cost and travel time for each trip π and each MOT k and we aim atmaximizing savings obtained when using a car compared to the cheapest other mobility type.For each trip π let min k ∈ K \{ } C k π be the cost of the cheapest mobility types excluding cars k =
1. Let C π bethe cost of riding the same trip π by car k =
1. We then calculate the savings s π = C K \{ } π − C π of using a carcompared to using the cheapest possible other mobility type. Note that if traveling with a certain MOT is notpossible, we impose a penalty and set C k = ∞ .Finally, we aim at assigning user-trips to the available cars in the best possible way whilst maximizingsavings.In the following, we introduce the modeling of the two cases presented in this paper. First, we introduce themodeling approach for the case where only one type of vehicles is shared and then solved as a minimum-costflow problem. Second, we present the formulation where multiple shared vehicle types can be employed. This isthen modeled and solved as a multi-commodity flow problem.We model the problem on a directed acyclic graph (DAG). Since a MOT must be used for the full trip, wedo not model the tasks covered by a trip in the graph, and only consider nodes o π and e π for each trip π , whichrepresent starting and ending points of a trip. The savings of the edge ( o π , e π ) is s π , as explained above. In orderto connect the trips we insert additional edges ( e π , o π (cid:48) ) if trip π π has the same destination as trip π π (cid:48) has origin,and the trip π finishes before the trip π (cid:48) . The savings of such an edge is 0.3igure 1: The underlying graph of the minimum-cost flow formulation of the vehicle-sharing problem withone shared vehicle type, five trips π , and two depots d . Nodes A d , A (cid:48) d represent supra-nodes where the availablevehicles are stored at the beginning and end of the time horizon. In our example we have δ = 2 vehicles availableat A and δ = 1 car at A , the same amount of vehicles has to be returned in the evening to A (cid:48) and A (cid:48) . Nodes o π and e π give start and end points of a trip π . Finally, each edge represents a trip with a given saving s π andcapacity. The x-axis represents the time of day, and the y-axis represents the depots. VShP-1T ) For the vehicle-sharing problem with a single type of shared vehicle (
VShP-1T ) we create a node A d for eachdepot d ∈ D with a supply δ d representing the number of available vehicles. Depots represent locations wherethe shared vehicles start and end, e.g. a company’s offices. For each depot d where the vehicles must be parkedin the evening, we create a node A (cid:48) d with a demand δ (cid:48) d equal to the number of requested vehicles at the end of theplanning horizon. Every node A d is connected to all nodes o π where the trip π starts in the same node as depot d . Every node e π is connected to node A (cid:48) d where the trip π ends in the same node as depot d . We add extra edges ( A d , A (cid:48) d ) with infinite capacity and zero savings, to represent the case where a vehicle is not used and stays in thedepot. Finally, we draw the nodes in a time-space network, where the x-axis represents the time of day, and they-axis represents the depots.Figure 1 shows a simple example in which we have two depots, and five trips π . We assume that the firstdepot has two vehicles available in the morning, and two vehicles (not necessarily the same) should be returned tothe depot in the evening. Note that we show the savings and capacity for each edge in the form savings, capacity .Let V be the set of all nodes and let s i j be the savings of a trip going from node i to j (in our auxiliary graph e π , o π ). Furthermore, let δ i be the demand at the depots, being 0 for e π and o π . Parameter u i j gives the capacityof an edge, which is 1 for all arcs. Finally, the binary decision variables x i j take on value 1 if connection ( i , j ) ischosen, and 0 otherwise.With this, we show that the vehicle-sharing problem considering one single type of shared vehicles ( VShP-1T )can be modeled as the maximization equivalent to a minimum-cost flow problem, formulated in model (1)-(4).4ax ∑ ( i , j ) ∈ V s i j x i j (1)s.t ∑ j ∈ V x ji − ∑ j ∈ V x i j = δ i ∀ i ∈ V (2) x i j ≤ u i j ∀ i , j ∈ V (3) x i j ≥ ∀ i , j ∈ V (4)The objective function (1) maximizes savings. Constraint (2) restricts the out/ingoing vehicles at the begin-ning/end of the day. Further it assures flow conservation in nodes i ∈ V \ { D } . Constraint (3) makes sure that atmost one vehicle is covering a certain connection ( i , j ) .We will solve our model as a MIP as state-of-art solvers are already capable of handling these kind of prob-lems very efficiently. Nevertheless, we shortly review some of the algorithms that have been widely applied.Ford and Fulkerson [19] were first to introduce a combinatorial algorithm for the problem. Edmonds and Karp[17] proposed the scaling resulting in the first weakly polynomial-time algorithm. Tardos [39] introduced theminimum cost circulation algorithm which was the first strongly polynomial method. In the consecutive yearsmany solution approaches evolved. Scaling techniques have shown to be promising [17, 22, 21, 14]. Polynomialin time are also cycle canceling algorithms [28, 20]. Furthermore, the network simplex method was efficientlyapplied to the maximum flow problem [15, 27, 32] or adaptions of the successive shortest path algorithm [13].Kovà ˛acs [30] provide an survey of various algorithms and present an overview of the respective complexity.In what follows, we do not only consider one type of shared vehicles but multiple ones. Note that sharedvehicles can be different types of cars but also bikes or any other MOT. VShP-xT ) We start with the previously described graph. To model the vehicle-sharing problem with multiple types of sharedvehicles (
VShP-xT ), we duplicate the sources and sinks since we have different possibilities. As each commodityonly has one source, and one sink in the multi-commodity flow problem, we add a supra-source M k for each k ∈ K (cid:48) where K (cid:48) ∈ K denote the set of shared vehicles. In our example we have M for one type of cars, anda supra-source M for another type. In a similar way we add supra-terminals M (cid:48) k for the sinks. The set of allsupra-nodes, thus M k ∪ M (cid:48) k , is denoted as M . We then construct start and end depot nodes A kd , A (cid:48) kd to where weconnect the respective M k and all origins o π and end nodes e π of a trip π , respectively. We assign savings s andcapacity to each trip, i.e. edge. Drawing the nodes in a time-space network, Figure 2 shows a simple case wherewe have two shared types of vehicle.We show that the problem can then be solved as an integer multi-commodity flow problem, where edgesavings s ki j depend on the commodity transported k . In our example commodities correspond to different sharedvehicles. We now consider a demand ∆ k per MOT k and define with the capacity variable u i j how many MOTsare available at a depot d . Let x ki j be 1 if connection between ( i , j ) is covered by MOT k , 0 otherwise. Thisproblem has the formulation: 5igure 2: The underlying graph of the multi-commodity flow formulation of the vehicle-sharing problem withtwo shared types of vehicles, five trips π , and two depots d . Nodes M k , M (cid:48) k represent supra-nodes where theavailable shared vehicles are stored at the beginning and end of the time horizon and then distributed to therespective depot nodes A kd , A (cid:48) kd . We have 3 vehicles of type 1 available and 7 cars of type 2; 2 and 1 of type 1 aredistributed to depot 1 and 2, 3 and 4 type 2 vehicles to depot 1 and 2, respectively. Nodes o π and e π give start andend points of a trip π . Finally, each edge gives its respective savings and capacity. The x-axis represents the timeof day, and the y-axis represents the depots.max ∑ k ∈ K ∑ i ∈ V ∑ j ∈ V s ki j x ki j (5)s.t ∑ i ∈ V x ki j − ∑ i ∈ V x kji = ∀ k ∈ K , j ∈ V \ { M } (6) ∑ j ∈ V x ki , j − ∑ j ∈ V x kj , i = ∆ k ∀ i ∈ M k , k ∈ K (7) ∑ i ∈ V x ki , j − ∑ i ∈ V x kj , i = ∆ k ∀ j ∈ M (cid:48) k , k ∈ K (8) ∑ k ∈ K x ki j ≤ u i j ∀ i , j ∈ V (9) x ki j ≥ ∀ k ∈ K , i , j ∈ V (10)Objective function (5) maximizes the savings. Equation (6) gives the flow conservation constraints for all nodesexcept the sources and sinks. Constraints (7) and (8) restricts the number of shared MOTs. Constraint (9) givesthe capacity restriction on the arcs. Finally, constraint (10) assures positive numbers.The formulation above is polynomial in the size of the constraints, having | K | · | E | variables, where | E | is thenumber of edges, and | E | + | K | · | V | constraints. However, large-scale problems may be challenging to be solved.Therefore, efficient solution algorithm have been applied such as Langragian relaxation [36, 2], adapted branch-and-bound algorithms [4], Dantzig-Wolfe decompositions [25] and column generation algorithms [40, 3]. Withthe Dantzig-Wolfe decomposition a path-flow formulation with | E | + | K | constraints but exponential number ofvariables is generated. The column generation approach generated at most | E | + | K | paths with positive flows.However, state-of-the art commercial solvers are able to solve these kinds of problems within seconds, which we6ill show in our computational results. We provide computational results using the above presented models for the vehicle-sharing problem. The modelsare implemented in C/C++ and solved with CPLEX 12.9. Tests are carried out using one core of an Intel XeonProcessor E5-2670 v2 machine with 2.50 GHz running Linux CentOS 6.5. Tests are conducted on a number ofgenerated instances varying in size and complexity.In the following, we give a short introduction to the instance set. Afterwards we provide the results of ourcomputational study for the
VShP-1T and
VShP-xT . We further present results of varying objective functionsand restricted sets of MOTs for individual users. Lastly, we comment on the results and give some managerialinsights.
We generate realistic benchmark instances based on available demographic, spatial and economic data of the cityof Vienna, Austria. Five different MOTs are considered: cars (combustion engine vehicles and electric vehicles),walk, bike, public transportation and taxi. In the following results we name the combustion engine car ’car-type1’, the electric vehicles ’car-type2’. For each mode of transport k ∈ K we define distances, time and costbetween all nodes for all modes of transport k ∈ K . We calculate the Aerial distance between two locations whichare then multiplied by a constant sloping factor for each MOT k in order to account for longer/shorter distances ofthe respective mode of transport. Moreover, we have emissions per distance unit, average speed, cost per distanceand cost per time as well as additional time needed for, e.g., parking the car, for each k ∈ K . The cost of time is afixed value based on the average gross salary including additional costs for employers in Austria. The objectivefunction results from these values. The values of the parameters are given in the Appendix Table A1.Each generated instance represents a distinct company operating two offices and consisting of a predefinedset of users, i.e. employees, p ∈ P . The locations of the offices (depots) are based on statistical data of officelocations in Vienna placed in the geometric centers of all 250 registration districts.Companies are defined by a fixed number of users u and depots (fixed to two in our case). Note that oneperson may have more than one trip assigned. Therefore, the number of users u does not equal the number oftrips (edges) in the graph. In Table 1 we provide an overview on the average number of trips per user. As we cansee, on average each user takes about 1.5 trips during the planning horizon.The number of meetings and their time and location, are randomly generated based on historic statistical data.We define a time horizon of one day where each user has an assigned set of meetings distributed over the day.We calculate savings based on the cheapest other MOT, whereas we always use publicly available MOTs (publictransportation, bike, taxi) to be the cheapest other possible alternative.We solve 10 instances per instance group.Table 1: Average number of trips for each instance group with u users. u 20 50 100 150 200 250 300 trips 31 76 147 218 287 358 427trips / u https://github.com/dts-ait/seamless . VShP-1T ) We start by showing the results obtained for the
VShP-1T , represented by model (1)-(4). We assume one typeof shared vehicles: in
VShP-1T:car these are combustion engine cars (car-type1), in
VShP-1T:ecar we considerelectric cars (car-type2) as our shared resource. The results are obtained for an increasing number of users u ,varying in the number of shared cars m . Walk, bike, public transportation, taxi are assumed to have no capacityrestriction. The considered cars are equally spread over the two depots.Figure 3 illustrates the average total cost as a sum of the cost of the cars and other MOTs as well as savingsfor users u = , ,
300 and cars m = , , ,
40. Note that the savings are given in the opposite direction(negative numbers) for a better distinction between savings and cost. With an increase in the number of sharedcars m we see a decrease in the overall costs, clearly visible by the declining bars in each group. For smallerinstances ( u =
20) we can observe that the cost of the cars is higher than the cost of the other MOTs. This is notsurprising as the model is able to assign the shared cars to all beneficial trips. The savings are also increasingwith the number of shared cars m and increasing number of users u considered. Figure 3(a) shows the values for VShP-1T:car , Figure 3(b) the
VShP-1T:ecar . As can be seen, the general impression as well as the total overallcost are about the same. In Figure 3(b), thus for the case where electric vehicles are shared, we have slightly lesstotal cost and less car cost. More detailed information and further results on e.g. savings, and the composition ofthe total cost regarding cars and other MOTs, can be found in Appendix Table A2 and Table A3.Tables 2 and 3 summarize the average number of trips for
VShP-1T:car and
VShP-1T:ecar and increasingnumber of cars m and users u . We observe that with an increasing number of users u the average number of tripson one route is also raising. This is because the model aims to cover as many trips by car as possible. With anincreasing number of users but the same number of cars in the system, the model will try to situate more trips onone of the few car routes. The average number of trips is higher when fewer cars are available. We observe thisfor both variants, the VShP-1T:car and
VShP-1T:ecar . Overall
VShP-1T:ecar shows a higher average of trips percar.Table 2: Average number of trips per car for an increasing number of users u and cars m for VShP-1T:car . m u=
20 50 100 150 200 250 3004 1.5 2.1 2.1 2.2 2.3 2.5 2.38 1.4 1.9 2.1 2.1 2.2 2.3 2.120 1.3 1.6 1.9 1.9 1.9 2.1 2.040 1.4 1.4 1.6 1.7 1.8 1.9 1.9Table 3: Average number of trips per car for an increasing number of users u and cars m for VShP-1T:ecar . m u=
20 50 100 150 200 250 3004 1.6 2.1 2.2 2.3 2.4 2.7 2.48 1.5 2.0 2.2 2.2 2.2 2.4 2.220 1.3 1.7 2.0 1.9 2.0 2.2 2.140 1.4 1.4 1.8 1.8 1.9 2.0 2.0In Appendix Table A5 we give an overview of the solution times as well as total times for
VShP-1T:car and8 a) VShP-1T:car (b)
VShP-1T:ecar
Figure 3: Total cost split into cost of MOTs and cost of cars, respectively for car-type1 (=combustion engine cars)and car-type2 (=electric cars) as well as savings (negative bars) for an increasing number of u = , ,
300 andcars m = , , ,
40. 9igure 4: Total cost split into cost of MOTs, cost of car-type1 (=combustion engine cars), and cost of car-type2(=electric cars) as well as savings (negative bars) comparison for u = , ,
300 and cars m = , , ,
40 for
VShP-xT . VShP-1T:ecar . For an increasing number of users u , we observe an increase in the times used to solve the models.However, we always stay below 8 seconds of solution time. VShP-xT ) In the following, we present the results obtained by solving the
VShP-xT , given in model (5)-(10). We nowassume different types of shared vehicles in one model. For our tests we use combustion engine cars (car-type1)and electric vehicles (car-type2). Note that this can be easily extended/changed in order to include, e.g., bikes ore-scooters. We are given an equal number of each car type, denoted as m k respectively. Thus if m k =
2, then twocars of each type are available. These are then again equally assigned to the depots. In our example, since weassume 2 depots, this would give us one combustion engine car (car-type1) and one electric vehicle (car-type2) ateach depot. The total number of vehicles m is simply given by M + M where k = k = u and shared cars m .Figure 4 plots the total cost as a result of the cost of the two car types, and cost of the other MOTs, as wellas savings which are given in the opposite direction as negative numbers. Note that we observe again a similarpicture as in Figure 3. We increase the cost of cars (car-type1 and car-type2) by adding more available cars m to the system whilst reducing total cost. The share of the car-type2 cost are constantly higher than the cost ofcar-type1. This means, as e-cars (= car-type2) are usually cheaper, that more electric cars are assigned. Note thatfor the smallest instances ( u =
20) the cost of the car-type1 is diminishing, meaning that almost all of the tripsare covered by car-type2. Table A4 in the Appendix shows more details on the cost as well as the breakdown ofthe total cost into cost of the respective car types and other MOTs.Table 4 shows the average number of trips on car routes for increasing number of users u and cars m . Theresults are split into values for the different car types. We can see that the average number of trips on a route ofcar-type2 (= electric car) is always greater that the number of trips for the other type. This means, that if possiblethe model aims to put more trips on the routes taken by e-cars. In the extremest case ( u = m =
40) almostno trip is covered by a conventional car (car-type1). Moreover, we can again observe an increase in the averagenumber of trips per route for a higher number of users u as well as smaller number of cars m .10able 4: Average number of trips per car for increasing number of users u and cars m for VShP-xT . Car-type1 arecombustion engine cars, car-type2 electric cars. m u=
20 50 100 150 200 250
VShP-1T:car , VShP-1T:ecar , VShP-xT and the case where all trips are cov-ered by one car/MOT
Now we compare the models
VShP-1T:car , VShP-1T:ecar , and
VShP-xT regarding cost for an increasing numberof users u and vehicles m .Figure 5 shows the cost of the different cases for u = u = u =
300 and increasing m . Therespective lines give the output for the following cases: no trip is covered by a car, every trip is covered bya combustion engine vehicle (car-type1), all trips are covered by electric vehicles (car-type2), cost for VShP-1T:car , VShP-1T:ecar and
VShP-xT . Note that the fleet restrictions only apply for
VShP-1T:car , VShP-1T:ecar and
VShP-xT . We can see that it is always most expensive if no trips are covered by any car. In all three figures,the line representing cost of using only car-type2, which are electric vehicles, lies below the line showing costwhen using car-type1, thus combustion engine vehicles, only. When considering u = VShP-1T:car , VShP-1T:ecar , and
VShP-xT are always cheaper than employing convectional cars only. For u =
100 and u =
300 thecost curves of the three models (
VShP-1T:car , VShP-1T:ecar , VShP-xT ) start above the conventional car cost,however break even between after m = u = m =
20 ( u = u =
100 (Figure 5(b)) thecost line representing
VShP-1T:ecar crosses the line where only electric cars are employed at around m =
40. For u =
20 the
VShP-xT merges at some point with
VShP-1T:ecar , as there are enough electric vehicles to cover therespective beneficial trips. Overall,
VShP-1T:ecar is always the cheapest option. The cost line of
VShP-1T:car is always above
VShP-xT and
VShP-1T:ecar . Thus, employing a shared system with combustion engine cars ismost expensive.Table 5 compares the cost of the three models
VShP-1T:car , VShP-xT , and
VShP-1T:ecar , whereas the latteris taken as the base. We show the averages over m = , , ,
40 for an increasing number of users u . As we cansees, VShP-1T:ecar is the cheapest, as we have already seen in the previously discussed figure.
VShP-xT showson average slightly higher cost. Lastly, as expected, the case where only combustion engine cars are employed ina pool of shared vehicles, is the most expensive alternative, ranging up to 1.05 times the cost to
VShP-1T:ecar .11 a) u = 20 (b) u = 100(c) u = 300
Figure 5: Total cost comparison where all trips are either covered by electric cars, combustion engine cars or notby cars at all, and the different introduced models (
VShP-1T:car , VShP-1T:ecar , VShP-xT ). The restricted fleet(given on the x-axis, m = , , ,
40) is only applicable to the cases where vehicles are shared (
VShP-1T:car , VShP-1T:ecar , VShP-xT ) as otherwise all trips are covered by the respective MOT.Table 5: Total cost comparison split for an increasing u and averaged over all m for VShP-1T:car , VShP-1T:ecar ,and
VShP-xT . Column ’cost’ gives the absolute cost of the respective model, ’comp.’ compares the cost to
VShP-1T:ecar where it is set as (cost of the model / cost of
VShP-1T:car ). VShP-1T:ecar VShP-xT VShP-1T:car u cost cost comp. cost comp.20 509 514 1.01 533 1.0550 1,541 1,558 1.01 1,599 1.04100 3,152 3,184 1.01 3,235 1.03150 5,091 5,127 1.01 5,184 1.02200 6,962 7,003 1.01 7,065 1.01250 8,909 8,954 1.00 9,022 1.01300 9,822 9,866 1.00 9,932 1.0112able 6: Categorization of the different preference variants. Percentage of the users with the respective set ofaccepted MOTs, where (1) all: no restricted set is applied, user takes all MOTs, (2) cars only: the user only wantsto drive by car, (3) no cars: no cars are given in the restricted set, only other MOTs are accepted.variant all cars only no carsprefVar0 see textprefVar1 40% 40% 20%prefVar2 10% 10% 80%prefVar3 25% 25% 50%prefVar4 0% 80% 20%prefVar5 0% 20% 80%prefVar6 0% 50% 50%
We assume that every user p has a set K p ⊆ K of possible modes of transport that can be used, reflecting herpreferences. Depending on the user that is covering a trip π , we can then define a set of modes of transportpossible to be assigned for a trip K π ⊆ K . Note that if a MOT is not in the respective set K π we impose a penaltyand set C k = ∞ . We define seven different cases aiming to represent differences in preference distribution.For the first case, prefVar0, we make use of available statistical data representing the working population ofVienna. For this we define different combinations of possible accepted MOTs in the instance generation: generic,motorised only, no public transport, no motorised, cars only, public transport only and bike only. For each ofthem we have a probability for female and male user, where we have [0.19, 0.03, 0.01, 0.04, 0.18, 0.42, 0.13]and [0.18, 0.03, 0.02, 0.03, 0.26, 0.35, 0.13], respectively. [37] We assume that 53% of the working population ismale, and 47% female. [38] Further, we incorporate the probability that 87% of them have a driving license and13% are not allowed to drive a car. [6] The combinations are then chosen randomly based on the set probabilitydistribution. We assume that if a user includes a combustion engine car in her set of MOTs, then she will also havethe electric car and vice versa. For the other cases, naming prefVar1-prefVar6, we adopt more straightforwardstrategies to represent the preferences of the users. Depending on the variant, we define a fixed percentage ofusers with a given setting. We say this may either be mixed (= accepting all MOTs), cars only or other MOTsexcept cars (=no cars). Let us assume an instance with 20 users and 40% mixed, 40% cars only and 20% otherMOTs only. Then user 1-8 accept all MOTs, users 9-16 only cars and users 17-20 anything but cars. Table 6shows the setting of each of the applied variants.Figure 6 shows the total cost divided into cost of cars and other MOTs for VShP-1T:car for increasing numberof users u = , ,
300 and cars m =
40 for the different preference settings, namely prefVar0-prefVar6. Thefirst bar in every subfigure gives the respective cost of the base case where no restricted set of preferred MOTs isgiven, which is
VShP-1T:car . For u =
20 in Figure 6(a) we can still see a substantial difference in the compositionof the cost of the different settings, yet similar total cost. It is clearly visible, that
VShP-1T:car without anyrestricted set of MOTs, outputs the least cost, however, have the highest cost for operating cars. The cost for carstake up about 60% of the total cost. The lowest share of cost for cars is used in prefVar3, only having 27% ofcar cost compared to total cost. The difference between the cheapest and most expensive variant is about 13%.We cannot observe a big difference for giving mixed preference setting and restricting to cars only (comparingprefVar1 and prefVar4, prefVar2 and prefVar5, prefVar4 and prefVar6). Similar pictures are given for u = u =
300 in Figures 6(b)(c). For u =
150 we see a difference in total cost between the cheapest and mostexpensive one, which is about 15%, and for u =
300 the difference between the extremes in terms of total cost is13 a) u = 20 (b) u = 150(c) u = 300
Figure 6: Total cost split into cost of MOTs and cost of cars (car-type1) for u = , ,
300 and m =
40 fordifferent variants of MOT-preference setting and
VShP-1T:car .15%. The most expensive variant in both instances is prefVar0. The lowest/highest share of car cost is 13/43%for u =
150 and 12/26% for u = VShP-1T:car . For each variant (prefVar0-prefVar6)we show the average cost of using conventional cars (car-type1), cost used for all other MOTs and in total for u =
300 and m =
40. We also have a second column for each variant, stated as ”comp.”, where we comparethe cost to the base case calculated as variant /
VShP-1T:car . We see that our base case is the most expensiveregarding car usage. In prefVar2, where most of the users prefer all MOTs except cars, we only use 0.59 timesthe cost of cars compared to the
VShP-1T:car . Conversely, regarding other MOTs, the simple
VShP-1T:car isthe cheapest variant, where prevVar0 uses 1.39 times more the cost on average. This comparably big differencein cost is mainly attributable to the more subtle differentiation of the preference settings. As in prefVar0 we alsodistinguish whether a person would, e.g., only take public transportation. In total we confirm the picture fromabove, that
VShP-1T:car without any restriction, is the cheapest setting, however prefVar1 or prefVar3 only have1.03 times the cost, which is negligible. Further results can be found in Table A6 in the Appendix.Table 7: Total cost comparison split into cost of car-type1 (=combustion engine cars) and other MOTs for u = m =
40 for different variants of MOT-preference setting and
VShP-1T:car . Column ’cost’ gives the absolutecost of the respective variant, ’comp.’ compares the cost to
VShP-1T:car where it is set as (cost of the variant /cost of
VShP-1T:car ). VShP-1T:car prefVar0 prefVar1 prefVar2 prefVar3 prefVar4 prefVar5 prefVar6cost cost comp. cost comp. cost comp. cost comp. cost comp. cost comp. cost comp.car-type1 1,227 872 0.71 1,169 0.95 722 0.59 1,030 0.84 1,188 0.97 747 0.61 1,045 0.85other MOTs 8,704 12,093 1.39 9,014 1.04 9,669 1.11 9,231 1.06 9,220 1.06 9,682 1.11 9,337 1.07total 9,932 12,964 1.31 10,184 1.03 10,391 1.05 10,261 1.03 10,408 1.05 10,429 1.05 10,382 1.05
Figure 7 shows the differences in cost of the different preference settings when solving
VShP-xT . Figure 7(a)gives the averages for u =
20, Figure 7(b) for u =
150 and Figure 7(c) represents u = a) u = 20 (b) u = 150(c) u = 300 Figure 7: Total cost split into cost of MOTs and cost of car-type1 and car-type2 (both combustion engine andelectric cars) for u = , ,
300 and m =
40 for different variants of MOT-preference setting and
VShP-xT .shared cars is always set to m =
40. We again observe structural differences between the variants, however nowalso between the similar variants (prefVar1 and prefVar4, prefVar2 and prefVar5, prefVar4 and prefVar6). InFigure 7(a) the base case and prefVar0 are covered by electric cars (car-type2) and other MOTs only. The othervariants employ a mix of conventional cars and electric cars. This is then also visible for
VShP-xT and prefVar0in Figure 7(b)(c).Table 8 summarizes the average cost of the basic setting (
VShP-xT ), and prefVar0-prefVar6 partitioned intocost for combustion engine cars (car-type1), electric cars (car-type2), other MOTs and in total for u =
300 and m =
40. For each variant we again show the cost and the comparison (comp.) to the
VShP-xT calculated asvariant /
VShP-xT . Here we have a somewhat different picture than above.
VShP-xT is still the most cost efficientin total, with prefVar0 using up to 1.31 times the cost. Again, this comparably big difference in cost is mainlyattributable to the more subtle differentiation of the preference settings. For car-type2, which is electric cars,the base case where no preferences are taken, is the most expensive one, as can be seen that all numbers of the’comp.’ columns are below 1. Comparing the cost of car-type1, this differentiates. For some cases (prefVar1,prefVar3, prefVar4, prefVar6) the cost are higher or equal to
VShP-xT . For the others the results show lower cost,e.g., prefVar0 only 0.69 of total cost. Further results can be found in Table A7 in the Appendix.
In the following, we compare two objective functions: (1) we take the objective function as presented above,consisting of operational distance cost including cost of time (OF: base), (2) only incorporating the time factor(OF: time). Again, we show the results for both
VShP-1T:car and
VShP-xT as our base cases. With this we aimto see the main driver of our outputs. Note that for the following result we solve the models with the differentobjective functions, but afterwards calculate the total cost to make them comparable.Figure 8 shows the composition of the total cost for
VShP-1T:car having a number of users u = , , u =
300 and m =
40 for different variants of MOT-preference setting and
VShP-xT .Column ’cost’ gives the absolute cost of the respective variant, ’comp.’ compares the cost to
VShP-xT where itis set as (cost of the variant / cost of
VShP-xT ). VShP-xT prefVar0 prefVar1 prefVar2 prefVar3 prefVar4 prefVar5 prefVar6cost cost comp. cost comp. cost comp. cost comp. cost comp. cost comp. cost comp.car-type1 491 339 0.69 558 1.14 348 0.71 491 1.00 593 1.21 383 0.78 515 1.05car-type2 683 505 0.74 573 0.84 371 0.54 505 0.74 595 0.87 361 0.53 531 0.78other MOTs 8,692 12,075 1.39 9,010 1.04 9,646 1.11 9,227 1.06 9,220 1.06 9,685 1.11 9,337 1.07total 9,866 12,919 1.31 10,141 1.03 10,365 1.05 10,222 1.04 10,408 1.06 10,429 1.06 10,382 1.05 (a) u = 20 (b) u = 150(c) u = 300
Figure 8: Total cost split into cost of MOTs and cost of car-type1 (=combustion engine cars) for u = , , m =
40 and different objective function for
VShP-1T:car . OF:base shows the result for previously introducedobjective function, OF:time only considers the time part. Note we solve the models with the different objectivefunctions, but afterwards calculate the total cost to make them comparable.and cars m =
40. We show the cost share of cars and other MOTs. In all three cases we observe a higher costof the cars when using OF:time as the objective. However, the total cost is only slightly higher for OF:time,resulting in less cost for other MOTs when taking time components as the objective only.In Table 9 we confirm the above figures with numbers. The table is decomposed into results for OF:base,OF:time and the comparison of the two, where we assume OF:time / OF:base. The first two are given in absolutenumbers, the latter as a ratio of the two. Each partition gives the results of the combustion engine cars (car-type1),other MOTs and in total. The numbers are given on average over all instances and all sizes of m . We can see,that using time only as an objective function gives slightly higher overall cost. The smallest difference can beobserved for u = , m ) can befound in Table A8 in the Appendix.Figure 9 plots the composition of the total cost for u = , ,
300 and m =
40 solving
VShP-xT . The cost16able 9: Total cost for OF:base and OF:time and their comparison stated as OF:time / OF:base split into cost ofMOTs and cost of car-type1 (=combustion engine cars) for increasing u and averaged over all m . OF:base showsthe result for previously introduced objective function, OF:time only considers the time part. Note we solve themodels with the different objective functions, but afterwards calculate the total cost to make them comparable. u
20 50 100 150 200 250 300 O F :ti m e car-type1 333 747 1,004 1,089 1,185 1,283 1,277other MOTs 208 873 2,267 4,270 6,100 8,130 8,871total 541 1,620 3,271 5,359 7,285 9,413 10,147 O F : b a s e car-type1 258 656 955 1,052 1,148 1,255 1,227other MOTs 275 942 2,281 5,917 4,132 7,767 8,704total 533 1,599 3,235 5,184 7,065 9,022 9,932 O F :ti m e / O F : b a s e car-type1 1.29 1.14 1.05 1.04 1.03 1.02 1.04other MOTs 0.76 0.93 0.99 1.03 1.03 1.05 1.02total 1.02 1.01 1.01 1.03 1.03 1.04 1.02 are divided into cost for combustion engine cars (car-type1), electric cars (car-type2) and other MOTs. Again,having the actual objective function, leads to lower overall cost and lower cost of cars. Note that for the timefunction there is no difference between the two car types, as the differences are only in the operational cost. Aspreviously, the VShP-xT prefers electric vehicles, as they have lower cost not related to time.Table 10 summarizes the cost for OF:base and OF:time, and gives the respective ratio of them, calculated asOF:time / OF:base. We again give the respective values for the car types, other MOTs and in total. We observe,that if only optimizing towards savings in time, we end up with higher overall cost. We have between 1.01 and1.03 times the cost compared to our basic objective function. Results separated for m = , , ,
40 can be foundin Table A9 in the Appendix.Table 10: Total cost for OF:base and OF:time and their comparison stated as OF:time / OF:base split into cost ofMOTs and cost of car-type1 and car-type2 (both combustion engine and electric cars) for increasing u and aver-aged over all m . OF:base shows the result for previously introduced objective function, OF:time only considersthe time part. Note we solve the models with the different objective functions, but afterwards calculate the totalcost to make them comparable. u
20 50 100 150 200 250 300 O F :ti m e car-type1 177 390 507 539 603 650 636car-type2 141 326 455 503 533 577 585other MOTs 211 874 2,267 4,270 6,099 8,130 8,871total 528 1,590 3,229 5,312 7,234 9,358 10,092 O F : b a s e car-type1 50 193 365 417 463 496 491car-type2 225 446 545 589 638 704 683other MOTs 239 919 2,273 4,121 7,003 7,754 8,692total 514 1,558 3,184 5,127 7,003 8,954 9,866 O F :ti m e / O F : b a s e car-type1 3.51 2.02 1.39 1.29 1.30 1.31 1.30car-type2 0.63 0.73 0.83 0.85 0.83 0.82 0.86other MOTs 0.88 0.95 1.00 1.04 0.87 1.05 1.02total 1.03 1.02 1.01 1.04 1.03 1.05 1.02 Finally, we compare the average number of trips per car in Tables 11 and 12 for
VShP-1T:car and
VShP-xT ,solving each with the different objective functions. For
VShP-1T:car we see an increase in trips per route, wherewe have 1.1 more trips on average for all user u groups. For VShP-xT we observe a different picture. We can17 a) u = 20 (b) u = 150(c) u = 300
Figure 9: Total cost split into cost of MOTs and cost of car-type1 and car-type2 (both combustion engine andelectric cars) for u = , ,
300 and m =
40 and different objective function for
VShP-xT . OF:base shows theresult for previously introduced objective function, OF:time only considers the time part. Note we solve themodels with the different objective functions, but afterwards calculate the total cost to make them comparable.see a bigger increase in average trips per combustion engine car, and decrease of trips on electric vehicles. Thisis because for OF:time the two vehicle types do not make any difference. The difference of the both vehicles isonly visible in the operational cost then.Table 11: Average number of trips per car when solving OF:base and OF:time and their comparison stated asOF:time / OF:base for increasing u and averaged over all m for VShP-1T:car . u =
20 50 100 150 200 250 300OF:time 1.6 1.9 2.1 2.2 2.3 2.4 2.3OF:base 1.4 1.8 2.0 2.0 2.0 2.2 2.1OF:time / OF:base 1.1 1.1 1.1 1.1 1.1 1.1 1.1
We have seen in all our results, that with a higher number of cars (combustion engine or electric car), we enforcelower total cost. This is true, even though cars are the most expensive MOT from an operational cost pointof view. However, they are in many cases fast MOTs. Therefore, if possible, the trips are covered by a car.Also, whenever possible, electric vehicles are preferred as they have even lower cost but the same speed as theconventional ones. This holds for the case where only one kind of car is shared. A mixed fleet is slightly moreexpensive compared to when only electric vehicles are employed. However, if a fleet of combustion engine carsis available, one can stepwise expand the fleet or replace the conventional cars with the electric vehicles. Havinga shared pool of combustion engine vehicles only is the most expensive case of the sharing concepts and leastenvironmental friendly, thus not recommendable.Employing no cars at all, is the most expensive in any case. Rather than having a small fleet size and a big18able 12: Average number of trips per car when solving OF:base and OF:time and their comparison stated asOF:time / OF:base for increasing u and averaged over all m for VShP-xT . u =
20 50 100 150 200 250 300OF:time car-type1 1.5 1.9 2.1 2.2 2.3 2.5 2.3car-type2 1.5 1.9 2.1 2.2 2.3 2.4 2.3OF:base car-type1 0.8 1.3 1.5 1.5 1.6 1.7 1.6car-type2 1.5 2.1 2.4 2.4 2.5 2.7 2.6OF:time / OF:base car-type1 1.8 1.5 1.4 1.4 1.4 1.4 1.4car-type2 1.0 0.9 0.9 0.9 0.9 0.9 0.9group of users, one is recommended to cover all trips with electric vehicles and thus adopt no sharing concept.If one decides to go with any (of the presented) sharing concepts, it is advisable that the number of cars in thefleet should be at least 20%-25% of the number of users. E.g. for 20 users this would be 4-5 cars. From thereit starts to be cost efficient to have shared vehicles, and additionally cover trips with other MOTs such as publictransport or bike. The use of electric vehicles - either exclusively or in combination with conventional cars - ishighly recommendable due to their lower operational cost and same time needed for trips.Using operational cost information and time in the objective function is crucial. As the cost of time dependson the distance too (we assume different distances for different MOTs), not all of the trips are covered by thefastest MOT, which would be a car. So the shortcuts that can be taken by different MOTs sometimes outperformsthe benefits given by fast cars. As our instance companies are based on a city, this makes sense. For longer trips,the results would lead to different trade-offs.If enough cars are available to cover the beneficial trips without handing over the cars at the depots, then thiswill be done so and is also recommendable. A profound sharing concept is only advisable if the car is a restrictedresource (however, not too much as discussed above). Yet, we saw in our results, that with a constant number ofusers but a smaller fleet, the trips per car are rising above the average number of trips a user is taking. Also, theaverage number of trips per car is higher for electric vehicles.Finally, we introduced a set of restricted MOTs based on individual preferences of a user. As expected, thecase where all MOTs are always available for all trips, and thus for all users, is the one with the least cost as itis the least restricted case. However, for some of the cases we observed only a modest increase in cost. Yet, bygiving the users a restricted set of MOTs we might achieve a higher satisfaction and acceptance of the systemand therefore it can be beneficial in a non-monetary way.
Inspired by the change of mobility and vehicle-sharing systems we introduced two novel modeling approachesfor the vehicle-sharing problem. In our problem we assume a set of users that have to cover certain trips on afixed time schedule. These trips are then covered by a certain mode of transport. We assume a restricted availablepool of cars which the users may use. Other modes of transport are incorporated without any capacity limits. Weaim to assign the restricted resources in the best possible way such that savings (using e.g. a car instead of anyother mobility type) are maximized.We used two well-known formulations from the literature, namely the maximization equivalent of the minimum-cost flow problem and the multi-commodity flow problem. If we assume only one shared MOT, e.g. cars, webase our formulation on the minimum-cost flow problem. We extend the problem by introducing another type of19hared vehicle, and we formulate it as a multi-commodity flow problem where the commodities are the sharedvehicles. Note that a shared resource may also be a bike or another MOT.We further provide an analysis of the different models incorporating combustion engine vehicles and electriccars as our shared vehicles. We show that a shared fleet of electric vehicles only contributes most to our objectivefunction. Instances with up to 300 users are solved in less than 20 seconds of computing time. With this we canshow that our models can be used on a daily operational basis.Besides the analysis, the present paper aims to give a theoretical foundation to future car/vehicle-sharingproblems. As the models are well studied in the literature, many efficient algorithms exist and even biggerinstances can be solved to optimality within seconds. Future work might look into adapting the structure of thetrips. Now we assume a fixed sequence, however optimizing the trips as a small-sized traveling salesman problemmay achieve even better results.
Acknowledgements
This work was supported by the Climate and Energy Funds (KliEn) [grant number 853767]; and the AustrianScience Fund (FWF): P 31366.
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Appendix
Table A1: Parameter value setting for the instances. The total cost are calculated as ((sloping factor * cost perkm) + (sloped distance * (1 / average speed) + setup time) * cost per time + (cost of emissions * emissions perkm)). sloping factor: foot: 1.1bike: 1.3car: 1.3public transport: 1.5CO2 emissions per km in gramm: foot: 0bike: 0combustion engine car: 200.9electric car: 42.7public transport: 0cost of CO2 emissions: 5 euro/taverage speed (km/h): foot: 5bike: 16car: 30public transport: 20cost per km: foot: 0bike: 0combustion engine car: 0.188electric car: 0.094public transport: 0taxi: 1.2cost per time: 19.42 euro per hoursetup time (in minutes): foot: 0bike: 2car: 10public transport: 5taxi: 5Iable A2: Comparison of total cost split into combustion engine cars (car-type1) and other MOTs, and savingsfor increasing number of u and m for VShP-1T:car . Share of total cost of the respective car and MOT costs givenin ’car-type1 / total’ and ’other MOTs / total’. u m car-type1 other MOTs total savings car-type1 / total other MOTs / total20 4 155 401 555 - 49 0.28 0.728 243 294 537 - 68 0.45 0.5520 316 204 520 - 85 0.61 0.3940 318 202 520 - 85 0.61 0.3950 4 264 1,423 1,686 - 122 0.16 0.848 467 1,160 1,626 - 182 0.29 0.7120 846 704 1,550 - 258 0.55 0.4540 1,050 482 1,532 - 276 0.69 0.31100 4 291 3,090 3,381 - 178 0.09 0.918 553 2,735 3,288 - 270 0.17 0.8320 1,155 2,008 3,163 - 395 0.37 0.6340 1,820 1,289 3,109 - 449 0.59 0.41150 4 320 5,146 5,466 - 221 0.06 0.948 575 4,752 5,327 - 360 0.11 0.8920 1,235 3,832 5,067 - 619 0.24 0.7640 2,079 2,797 4,875 - 811 0.43 0.57200 4 338 7,087 7,424 - 241 0.05 0.958 624 6,635 7,260 - 406 0.09 0.9120 1,320 5,608 6,928 - 738 0.19 0.8140 2,310 4,339 6,648 - 1,017 0.35 0.65250 4 373 9,063 9,436 - 289 0.04 0.968 670 8,567 9,238 - 487 0.07 0.9320 1,470 7,391 8,861 - 864 0.17 0.8340 2,506 6,048 8,555 - 1,170 0.29 0.71300 4 350 9,966 10,316 - 247 0.03 0.978 638 9,486 10,124 - 439 0.06 0.9420 1,427 8,348 9,775 - 789 0.15 0.8540 2,495 7,017 9,512 - 1,051 0.26 0.74IIable A3: Comparison of total cost split into combustion engine cars (car-type1) and other MOTs, and savingsfor increasing number of u and m for VShP-1T:ecar . Share of total cost of the respective car and MOT costsgiven in ’car-type1 / total’ and ’other MOTs / total’. u m car-type2 other MOTs total savings car-type2 / total other MOTs / total20 4 162 379 541 - 64 0.30 0.708 271 243 514 - 91 0.53 0.4720 348 143 491 - 114 0.71 0.2940 352 138 491 - 114 0.72 0.2850 4 243 1,420 1,663 - 145 0.15 0.858 430 1,155 1,588 - 220 0.27 0.7320 850 625 1,475 - 333 0.58 0.4240 1,098 340 1,438 - 370 0.76 0.24100 4 266 3,089 3,355 - 204 0.08 0.928 509 2,731 3,239 - 319 0.16 0.8420 1,084 1,977 3,062 - 497 0.35 0.6540 1,776 1,175 2,951 - 608 0.60 0.40150 4 298 5,139 5,437 - 250 0.05 0.958 539 4,736 5,275 - 412 0.10 0.9020 1,150 3,808 4,957 - 729 0.23 0.7740 1,960 2,733 4,693 - 993 0.42 0.58200 4 310 7,083 7,393 - 272 0.04 0.968 577 6,625 7,202 - 463 0.08 0.9220 1,237 5,572 6,809 - 857 0.18 0.8240 2,165 4,279 6,444 - 1,222 0.34 0.66250 4 351 9,050 9,401 - 324 0.04 0.968 625 8,551 9,176 - 549 0.07 0.9320 1,371 7,357 8,728 - 997 0.16 0.8440 2,338 5,994 8,332 - 1,393 0.28 0.72300 4 322 9,962 10,284 - 280 0.03 0.978 596 9,470 10,066 - 497 0.06 0.9420 1,332 8,315 9,646 - 917 0.14 0.8640 2,339 6,952 9,291 - 1,272 0.25 0.75IIIable A4: Comparison of total cost split into combustion engine cars (car-type1), electric cars (car-type2) andother MOTs, and savings for increasing number of u and m for VShP-xT . Share of total cost of the respective carand MOT costs given in ’car-type1 / total’ and ’other MOTs / total’. u = m car-type1 car-type2 other MOTs total savings car-type1 / total car-type2 / total other MOTs / total20 4 64 91 392 547 - 58 0.12 0.17 0.728 88 163 272 522 - 82 0.17 0.31 0.5220 48 298 149 495 - 110 0.10 0.60 0.3040 2 347 142 491 - 114 0.00 0.71 0.2950 4 112 139 1,421 1,673 - 135 0.07 0.08 0.858 194 249 1,159 1,603 - 205 0.12 0.16 0.7220 270 536 694 1,500 - 308 0.18 0.36 0.4640 196 859 401 1,456 - 352 0.13 0.59 0.28100 4 124 152 3,090 3,366 - 193 0.04 0.05 0.928 235 291 2,734 3,260 - 299 0.07 0.09 0.8420 468 637 1,998 3,102 - 456 0.15 0.21 0.6440 634 1,102 1,270 3,006 - 552 0.21 0.37 0.42150 4 142 161 5,146 5,450 - 237 0.03 0.03 0.948 234 314 4,748 5,296 - 391 0.04 0.06 0.9020 490 691 3,820 5,000 - 686 0.10 0.14 0.7640 800 1,191 2,771 4,762 - 925 0.17 0.25 0.58200 4 155 168 7,085 7,408 - 258 0.02 0.02 0.968 267 331 6,628 7,226 - 439 0.04 0.05 0.9220 530 740 5,586 6,856 - 810 0.08 0.11 0.8140 902 1,314 4,306 6,522 - 1,144 0.14 0.20 0.66250 4 161 193 9,063 9,417 - 308 0.02 0.02 0.968 281 363 8,558 9,202 - 523 0.03 0.04 0.9320 587 818 7,376 8,781 - 944 0.07 0.09 0.8440 956 1,442 6,018 8,416 - 1,309 0.11 0.17 0.72300 4 154 179 9,964 10,298 - 265 0.01 0.02 0.978 271 337 9,482 10,091 - 473 0.03 0.03 0.9420 570 790 8,338 9,698 - 866 0.06 0.08 0.8640 969 1,424 6,983 9,376 - 1,187 0.10 0.15 0.74IVable A5: Solving time in seconds for
VShP-1T:car , VShP-1T:ecar , VShP-xT for an increasing number of u and m . u m VShP-1T:ecar VShP-1T:car VShP-xT
20 4 0.0 0.0 0.18 0.0 0.0 0.120 0.0 0.0 0.040 0.0 0.0 0.150 4 0.1 0.2 0.48 0.1 0.2 0.420 0.2 0.2 0.440 0.1 0.2 0.4100 4 0.7 0.7 1.68 0.7 0.7 1.620 0.7 0.7 1.640 0.7 0.7 1.6150 4 1.6 1.6 3.78 1.6 1.6 3.620 1.6 1.6 3.740 1.6 1.6 3.6200 4 3.0 3.1 7.08 3.1 3.1 6.820 3.1 3.1 6.740 3.0 3.0 6.9250 4 5.0 4.9 11.08 4.8 5.0 10.820 4.9 4.9 11.040 4.8 5.0 10.9300 4 7.3 7.4 16.38 7.2 7.6 17.120 7.3 7.4 16.840 7.4 7.4 16.8Vable A6: Average cost for one car-type1 (combustion engine cars) and other MOTs, in total and average savingsfor
VShP-1T:car and the different preference variants (prefVar0-prefVar6). The values are given for an increasingnumber of u and averages over all m .u = 20 50 100 150 200 250 300 VShP-1T:car car-type1 258 656 955 1,052 1,148 1,255 1,227other MOTs 275 942 2,281 4,132 5,917 7,767 8,704total 533 1,599 3,235 5,184 7,065 9,022 9,932savings - 72 - 209 - 323 - 503 - 601 - 703 - 631prefVar0car-type1 180 429 624 783 825 902 872other MOTs 424 1,495 3,361 5,930 8,302 10,854 12,093total 604 1,925 3,985 6,713 9,127 11,757 12,964savings - 215 - 595 - 1,119 - 1,682 - 1,939 - 2,109 - 2,264prefVar1car-type1 237 580 873 989 1,080 1,162 1,169other MOTs 323 1,084 2,475 4,358 6,198 8,154 9,014total 560 1,664 3,348 5,347 7,279 9,316 10,184savings - 76 - 209 - 343 - 590 - 698 - 818 - 787prefVar2car-type1 42 200 359 510 618 726 722other MOTs 558 1,570 3,150 5,047 6,893 8,783 9,669total 600 1,770 3,509 5,558 7,511 9,509 10,391savings - 11 - 56 - 74 - 196 - 213 - 310 - 252prefVar3car-type1 144 425 707 848 943 1,036 1,030other MOTs 437 1,281 2,698 4,569 6,415 8,331 9,231total 581 1,706 3,405 5,417 7,358 9,367 10,261savings - 39 - 149 - 245 - 444 - 471 - 642 - 522prefVar4car-type1 245 605 901 1,003 1,101 1,187 1,188other MOTs 325 1,084 2,496 4,475 6,327 8,375 9,220total 571 1,689 3,397 5,478 7,429 9,562 10,408savings - 84 - 251 - 409 - 713 - 811 - 981 - 910prefVar5car-type1 42 209 376 522 650 754 747other MOTs 559 1,565 3,142 5,053 6,882 8,788 9,682total 601 1,774 3,518 5,575 7,532 9,542 10,429savings - 11 - 63 - 84 - 233 - 255 - 376 - 306prefVar6car-type1 148 442 732 868 963 1,059 1,045other MOTs 439 1,278 2,700 4,622 6,474 8,445 9,337total 587 1,720 3,432 5,489 7,437 9,504 10,382savings - 45 - 170 - 278 - 533 - 551 - 755 - 615VIable A7: Average cost for one car-type1 (combustion engine cars), car-type2 and other MOTs, in total andaverage savings for
VShP-xT and the different preference variants (prefVar0-prefVar6). The values are given foran increasing number of u and averages over all m .u = 20 50 100 150 200 250 300 VShP-xT car-type1 50 193 365 417 463 496 491car-type2 225 446 545 589 638 704 683other MOTs 239 919 2273 4121 5901 7754 8692total 514 1558 3184 5127 7003 8954 9866savings -91 -250 -375 -560 -663 -771 -698preVar0car-type1 27 86 163 303 304 343 339car-type2 147 295 387 455 477 518 505other MOTs 418 1520 3401 5916 8303 10848 12075total 593 1900 3951 6673 9084 11709 12919savings -226 -620 -1153 -1722 -1982 -2156 -2309preVar1car-type1 110 253 413 471 510 534 558car-type2 130 330 440 486 535 588 573other MOTs 313 1058 2461 4354 6193 8150 9010total 553 1641 3314 5311 7238 9272 10141savings -83 -232 -377 -626 -738 -862 -829prefVar2car-type1 31 96 172 254 289 336 348car-type2 13 114 191 246 324 376 371other MOTs 555 1553 3133 5040 6874 8768 9646total 599 1763 3496 5539 7488 9481 10365savings -12 -63 -88 -214 -236 -338 -278prefVar3car-type1 64 180 330 398 451 483 491car-type2 80 257 374 432 466 524 505other MOTs 432 1252 2675 4554 6404 8323 9227total 576 1689 3379 5384 7321 9329 10222savings -44 -165 -271 -477 -508 -680 -560prefVar4car-type1 139 332 454 516 543 596 593car-type2 106 270 447 487 558 592 595other MOTs 325 1087 2496 4475 6327 8374 9220total 571 1689 3397 5478 7429 9562 10408savings -84 -251 -409 -713 -811 -981 -910prefVar5car-type1 41 139 260 323 363 399 383car-type2 1 75 116 194 277 348 361other MOTs 559 1560 3142 5057 6892 8795 9685total 601 1774 3518 5575 7532 9542 10429savings -11 -63 -84 -233 -255 -376 -306prefVar6car-type1 88 253 382 439 482 527 515car-type2 59 192 353 428 481 532 531other MOTs 439 1274 2698 4623 6473 8445 9337total 587 1720 3432 5489 7437 9504 10382savings -45 -170 -278 -533 -551 -755 -615VIIable A8: Comparison of total cost for OF:time split into combustion engine cars (car-type1) and other MOTsfor increasing number of u and m for VShP-1T:car . u =
20 50 100 150 200 250 3004 car-type1 186 275 293 323 341 374 355other MOTs 383 1,455 3,176 5,473 7,479 9,691 10,380total 569 1,730 3,469 5,796 7,820 10,065 10,7358 car-type1 310 482 560 593 640 690 665other MOTs 235 1,166 2,774 4,980 6,922 9,047 9,751total 545 1,648 3,334 5,573 7,562 9,738 10,41620 car-type1 418 954 1,192 1,279 1,370 1,482 1,482other MOTs 108 602 1,973 3,891 5,702 7,656 8,405total 526 1,556 3,165 5,169 7,071 9,138 9,88740 car-type1 418 1,277 1,971 2,161 2,389 2,585 2,605other MOTs 107 268 1,145 2,738 4,298 6,127 6,947total 526 1,545 3,116 4,899 6,687 8,712 9,552Table A9: Comparison of total cost for OF:time split into car-type1, car-type2 (combustion engine and electriccars) and other MOT for increasing number of u and m for VShP-xT . u =u =