Formulations and algorithms for the multiple depot, fuel-constrained, multiple vehicle routing problem
Kaarthik Sundar, Saravanan Venkatachalam, Sivakumar Rathinam
FFormulations and algorithms for the multiple depot, fuel-constrained,multiple vehicle routing problem
Kaarthik Sundar † , Saravanan Venkatachalam ∗ , Sivakumar Rathinam ‡ Abstract — We consider a multiple depot, multiple vehiclerouting problem with fuel constraints. We are given a set oftargets, a set of depots and a set of homogeneous vehicles, onefor each depot. The depots are also allowed to act as refuelingstations. The vehicles are allowed to refuel at any depot, andour objective is to determine a route for each vehicle with aminimum total cost such that each target is visited at leastonce by some vehicle, and the vehicles never run out fuelas it traverses its route. We refer this problem as MultipleDepot, Fuel-Constrained, Multiple Vehicle Routing Problem(FCMVRP). This paper presents four new mixed integer linearprogramming formulations to compute an optimal solution forthe problem. Extensive computational results for a large set ofinstances are also presented.
Index Terms — fuel constraints; green vehicle routing; electricvehicles; mixed-integer linear programming; unmanned vehiclerouting
I. I
NTRODUCTION
In this paper, we extend the classic multiple depot, mul-tiple vehicle routing problem (MDMVRP) to include fuelconstraints for the vehicles. We are given sets of targets, a setof depots, and a set of vehicles, with each vehicle initiallystationed at a distinct depot. The depots also perform therole of refueling stations, and it is reasonable to assumethat whenever a vehicle visits a depot, it refuels to itsfull capacity. Given this, the objective of FCMVRP is todetermine a route for each vehicle starting and ending atits corresponding depot such that (i) each target is visited atleast once by some vehicle, (ii) no vehicle runs out of fuel asit traverses its path, and (iii) the sum total cost of the routesfor the vehicles is minimum. Some of the applications for theFCMVRP are path-planning for Unmanned Aerial Vehicles(UAVs) [1], [2], [3], routing for electric vehicles based onthe locations of recharging stations [4], [5], and routing forgreen vehicles [6]. Some of these application domains areelaborated on the following sections.
A. Path-planning for UAVs
Small UAVs are being used routinely in military applica-tions such as border patrol, reconnaissance, and surveillanceexpeditions, and civilian applications like remote sensing, † Graduate Student, Dept. of Mechanical Engg., Texas A&M University,College Station, TX 77843. [email protected] ∗ Assistant Professor, Dept. of Industrial and Systems Engg., Wayne StateUniversity, Detroit, MI 48202. ‡ Assistant Professor, Dept. of Mechanical Engg., Texas A&M University,College Station, TX 77843. traffic monitoring, and weather and hurricane monitoring [7],[8], [9]. Even though there are several advantages due tosmall platforms for UAVs, there are resource constraints dueto their size and limited payload. It may not be possiblefor a small UAV to complete a surveillance mission beforerefueling at one of the depots due to the fuel constraints.For example, consider a typical surveillance mission withmultiple vehicles each starting at a depot and together arerequired to monitor a set of targets. To complete this mission,the vehicles might have to start at their respective depot,then visit a subset of targets and reach one of the depotsfor refueling before starting a new route for the rest of thetargets. This can be modeled as a FCMVRP with the depotsacting as refueling stations.
B. Routing problem for green and electric vehicles
Green vehicle routing problem is a variant of the VehicleRouting Problem (VRP) and was introduced by authors in[6] to account for the challenges associated with operatinga fleet of alternate-fuel vehicles (AFVs). The US transporta-tion sector accounts for 28% of national greenhouse gasemissions [10]. Several efforts over many decades focusingtowards the introduction of cleaner fuels (e.g. ultra lowsulfur diesel) and efficient engine technologies have lead toreduced emissions and greater mileage per gallon of fuelused. Government organizations, municipalities, and privatecompanies are converting their fleet of vehicles to AFVseither voluntarily to alleviate the environmental impact offossil based fuels or to meet environmental regulations. Forinstance, FedEx, in its overseas operations, employs AFVsthat run on biodiesel, liquid natural gas, or compressednatural gas. A major challenge that hinders the increasein usage of AFVs is the number of alternate-fuel stationsavailable for refueling. The FCMVRP is a natural problemthat arises in such a scenario. An algorithm to compute anoptimal solution to the FCMVRP would generate low costroutes for the vehicles, while respecting their fuel constraints.Increasing concerns about climate changes and risinggreen house gas emissions drive the research in sustainableand energy efficient mobility. One such example is theintroduction of electrically-powered vehicles. One of themain operational challenges for electric vehicles in transportapplications is their limited range and the availability ofrecharging stations. The number of electric stations in theUS is a mere 9,571 with a total of 24,631 charging outlets a r X i v : . [ c s . D S ] A ug ig. 1. Electric station locations in Texas, USA [11] [11]. Fig. 1 shows a map with the locations of the electricstations in Texas, USA; observe that the distribution of theelectric stations is very sparse except in the four majorcities Dallas, Houston, Austin, and San Antonio. Successfuladaption of electric vehicles will strongly depend on themethods alleviating the range and recharging limitations.If we consider the range and the recharging stations forthe electric vehicles as analogues to the fuel capacity andrefueling stations of vehicles that run on fossil-based oralternate fuels respectively, then the problem of electricvehicle routing subject to the range constraints and lim-ited availability of electric stations can be modeled as anFCMVRP. Clearly, any feasible solution to the FCMVRPcan be used to implement a feasible route for an electricvehicle. II. R ELATED WORK
The FCMVRP is NP-hard because it contains the VRPas a special case. The existing literature on the FCMVRP isquite scarce. The multiple depot, single vehicle variant of theFCMVRP was first introduced by authors in [12]. When thetravel costs are symmetric and satisfy the triangle inequality,authors in [12] provide an approximation algorithm for thisvariant. They assume that the minimum fuel required totravel from any target to its nearest depot is at most equal to
F α/ units, where α is a constant in the interval [0 , and F is the fuel capacity of the vehicle. This is a reasonableassumption as, in any case, one cannot have a feasible tour ifthere is a target that cannot be visited from any of the depots.Using these assumptions, Khuller et al. [12] present a (3(1+ α )) / (2(1 − α )) approximation algorithm for the problem.Authors in [1] formulate this multiple depot single vehiclevariant as a mixed-integer linear program and present k -optbased exchange heuristics to obtain feasible solutions within of the optimal, on an average. Later, Sundar et al. [2]extend the approximation algorithm in [ ] to the asymmetriccase and also present heuristics to solve the asymmetricversion of this variant. Furthermore, variable neighborhoodsearch heuristics for FCMVRP with heterogeneous vehicles, i.e., vehicles with different fuel capacities, are presented byLevy et al. [3]. More recently, an approximation algorithmand heuristics are developed for the FCMVRP by the authors in [13].Variants of the classic VRP that are closely related to theFCMVRP include the distance constrained VRP [14], [15],[16], [17], [18], the orienteering problem [19], [20], and thecapacitated version of the arc routing problem [21], [22]. Thedistance constrained VRP is a special case of the FCMVRPwith a single vehicle and single depot that can be consideredas a fuel station. The FCMVRP is also quite different andmore general compared to orienteering problem where one isinterested in maximizing the number of targets visited by thevehicle subject to its fuel constraints. Lastly, the arc routingproblem is a single depot VRP given a set of intermediatefacilities, and the vehicle has to cover a subset of edges alongwhich targets are present. The vehicle is required to collectgoods from the targets as it traverses the given set of edgesand unloads the goods at the intermediate facilities. The goalof this problem is to find a tour of minimum length that startsand ends at the depot such that the vehicle visits the givensubset of edges, and the total amount of goods carried bythe vehicle does not exceed the capacity of the vehicle alongthe tour. One of the key differences between the arc routingproblem and the FCMVRP is that there is no requirementthat any subset of edges must be visited in the FCMVRP.The aim of this paper is to introduce and compare fourdifferent formulations for the FCMVRP and present branch-and-cut algorithms for the formulations. The first two for-mulations are arc-based, and the rest are node-based formu-lations that use the Miller-Tucker-Zemlin (MTZ) constraints[23]. The major contributions of this paper are as follow:(1) present four new formulations for the FCMVRP, (2)compare the formulations both analytically and empirically,and (3) through extensive computational experiments, showthat instances with maximum of 40 targets are within thecomputational reach of a branch-and-cut algorithm based onthe best of the four formulations.The rest of the paper is organized as follows. Sec. III statesthe formal definition of the problem and introduces notations.In Sec. IV, we develop the four mixed integer linear pro-gramming formulations. The first two formulations are arc-based and the rest are node-based formulations i.e., decisionvariables for enforcing the fuel constraints are introduced foreach edge and each target for the arc-based and the node-based formulations, respectively. The linear programmingrelaxations of the formulations are analytically compared inthis section. In Sec. V, we present the computational resultsfollowed by conclusions and possible extensions.III. P ROBLEM DEFINITION
Let T denote the set of targets { t , . . . , t n } . Let D denotethe set of depots or refueling stations { d , . . . , d k } ; eachdepot d k is equipped with a vehicle v k . The FCMVRP isdefined on a directed graph G = ( V, E ) where V = T ∪ D and E is the set of edges joining any two vertices in V .We assume that G does not contain any self-loops. Eachdge ( i, j ) ∈ E is associated with a non-negative cost c ij required to travel from vertex i to vertex j and f ij , the fuelspent by traveling from i to j . It is assumed that the cost oftraveling from vertex i to vertex j is directly proportional tothe fuel spent in traversing the edge ( i, j ) i.e. , c ij = K · f ij ( c ij and c ji may be different, but for the purpose of thispaper, we assume c ij = c ji ). It is also assumed that travelcosts satisfy the triangle inequality i.e. , for every i, j, k ∈ V , c ij + c jk ≥ c ik . Furthermore, let F denote the fuel capacityof all the vehicles. The FCMVRP consists of finding a routefor each vehicle such that the vehicle v k starts and ends itsroute at its depot d k , each target is visited at least once bysome vehicle, the fuel required by any vehicle to travel anysegment of the route which joins two consecutive depots inthe route must be at most equal to F , and the sum of thecost of all the edges present in the routes is a minimum.IV. M ATHEMATICAL FORMULATIONS
This section presents four formulations for the FCMVRP.The first two formulations are arc based, and the remainingformulations are node based. The arc based and edge basedformulations have additional decision variables for each edgeand vertex respectively, to impose the fuel constraints. Forany given formulation F , let F L denote its linear program-ming relaxation obtained by allowing the integer variablesto take continuous values within the lower and upper integerbounds, and opt( F ) denote the cost of its optimal solution. A. Arc-based formulations
We first present an arc based formulation F for theFCMVRP, inspired by the models for standard routing prob-lems [24], [17]. Each edge ( i, j ) ∈ E is associated with avariable x ij , which equals if the edge ( i, j ) is traversedby the vehicle, and otherwise. Also, associated with eachedge ( i, j ) is a flow variable z ij which denotes the total fuelconsumed by any vehicle as it starts from a depot to thevertex j , when the predecessor of j is i . Using the abovevariables, the formulation F is given as follows: ( F ) Minimize (cid:88) ( i,j ) ∈ E c ij x ij subject to: (cid:88) i ∈ V x di = (cid:88) i ∈ V x id ∀ d ∈ D, (1) (cid:88) i ∈ V x ij = 1 and (cid:88) i ∈ V x ji = 1 ∀ j ∈ T, (2) (cid:88) j ∈ V z ij − (cid:88) j ∈ V z ji = (cid:88) j ∈ V f ij x ij ∀ i ∈ T, (3) ≤ z ij ≤ F x ij ∀ ( i, j ) ∈ E, (4) z di = f di x di ∀ i ∈ T, d ∈ D, and (5) x ij ∈ { , } ∀ ( i, j ) ∈ E. (6)In the above formulation the Eqs. (1) – (2) impose the degreeconstraints on the depots and the targets. The constraints in(3) are the connectivity constraints; they eliminate sub toursof the targets. (4) and (5) together impose ≤ z ij ≤ F andthey ensure that the fuel consumed by the vehicle to travelup to a depot does not exceed the fuel capacity F . Finally,the constraints in Eqs. (6) impose the binary restrictions onthe variables.Now, we present another arc-based formulation F whichis a strengthened version of F . The following propositionis a modified version of the Prop. 1 presented in [17] for thedistance constrained vehicle routing problem; it strengthensthe bounds given by the constraints in (4). Proposition 1.
The inequalities in (4) can be strengthenedas follows: z ij ≤ ( F − t j ) x ij ∀ j ∈ T, ( i, j ) ∈ E (7) z id ≤ F x id ∀ i ∈ T and d ∈ D (8) z ij ≥ ( s i + f ij ) x ij ∀ i ∈ T, ( i, j ) ∈ E (9) where, t i = min d ∈ D f id and s i = min d ∈ D f di .Proof. When j is a depot, the constraints in (8) and (4)coincide. We now discuss the case when both i and j aretargets. When x ij = 1 , any vehicle that traverses this edge ( i, j ) consumes at least ( s i + f ij ) amount of fuel. As a result,the constraint in (9) strengthens the lower bound of z ij in(4). Similarly, the total fuel consumed by any vehicle thattraverses the edge ( i, j ) cannot be greater that ( F − t j ) , where t j is the minimum amount of fuel required by any vehicleto reach a depot from target j . Therefore, the constraint in(7) strengthens the upper bound of z ij in (4).Hence, the second arc-based formulation is as follows: ( F ) Minimize (cid:88) ( i,j ) ∈ E c ij x ij subject to: (1) – (3), (5) – (6), and (7) – (9). Corollary 1. opt( F L ) ≥ opt( F L ) .B. Node-based formulations In this section, we present a node-based formulation forthe FCMVRP based on the models for the distance con-strained VRP in [25], [16]. For the node based formulation,apart from the binary variable x ij for each edge ( i, j ) ∈ E ,we have an auxiliary variable u i for each vertex i , thatindicates the amount of fuel spent by a vehicle when itreaches the vertex i . We assume u d = 0 as the vehiclesare refueled to their capacity when they reach a depot. Inaddition, we will also use the following two parameters: t i = min d ∈ D f id and s i = min d ∈ D f di for every vertex ∈ V . For any d ∈ D , t d = 0 and s d = 0 . Using theabove notations, the formulation F is given as follows: ( F ) Minimize (cid:88) ( i,j ) ∈ E c ij x ij subject to: (1), (2), and (6) ,u i − u j + M x ij ≤ M − f ij ∀ i ∈ V, j ∈ T, (10) u i ≥ s i + (cid:88) d ∈ D ( f di − s i ) x di ∀ i ∈ T, and (11) u i ≤ F − t i − (cid:88) d ∈ D ( f id − t i ) x id ∀ i ∈ T. (12)The constraint in Eq. (10) serves both as sub tour eliminationand fuel constraints. It eliminates sub tours of the targets andensures any route that starts and ends at a depot consumesat most F amount of fuel. This can be easily observed byaggregating the constraints for any sub tour of the targets andfor any route starting and ending at a depot [25]. The valueof M in the constraint is given by M = max ( i,j ) ∈ E { F − s j − t i + f ij } . The constraints in Eqs. (11) and (12) specifythe upper and lower bounds on u i , for every vertex i . Thefollowing proposition strengthens the fuel constraints and thebounds on u i . Proposition 2.
The inequalities in (10) , (11) , and (12) canbe strengthened as follows: u i − u j + M x ij + ( M − f ij − f ji ) x ji ≤ M − f ij ∀ i, j ∈ T, (13) u i ≥ (cid:88) j ∈ V ( s j + f ji ) x ji ∀ i ∈ T, (14) u i ≤ F − (cid:88) j ∈ V ( t j + f ij ) x ij ∀ i ∈ T, and (15) u i ≤ F − t i − (cid:88) d ∈ D ( F − t i − f di ) x di ∀ i ∈ T. (16) where, x ii = 0 and x ij = 0 whenever s i + f ij + t j > F .Proof. The constraint in Eq. (13) can be obtained by liftingthe variable x ji in Eq. (10). A constraint is said to be “valid”if it does not remove any feasible solution to the FCMVRP.We compute the value of the coefficient α that makes thefollowing constraint valid: u i − u j + M x ij + αx ji ≤ M − f ij . The equation is valid when x ji = 0 , as it reduces to (10).When x ji = 1 , we have x ij = 0 and u j + f ji = u i . Hence,the best value of α that makes the equation valid is given by M − f ij − f ji .Similarly, Eq. (14) can be obtained by lifting every x ji variable for j ∈ T in any order. We will illustrate the lifting procedure for one of the x ji variables. This involves com-puting the coefficient α that makes the following constraintvalid: u i ≥ s i + (cid:88) d ∈ D ( f di − s i ) x di + αx ji . The above equation is valid when x ji = 0 , and when x ji = 1 ,we have x di = 0 and α ≤ u i − s i . The best value of α thatdoes not remove any feasible FCMVRP is hence given by s j + f ji − s i . Similarly, the coefficients of the other x ji variables can be computed. The resulting constraint is givenby u i ≥ s i + (cid:88) j ∈ V ( s j + f ji − s i ) x ji ∀ i ∈ V. In the above equation, s j = 0 for j ∈ D . The above equationreduces to Eq. (14) due to the degree constraints in (2). Theconstraints in Eq. (15) are similarly obtained from (12) bylifting the x ij variable for every j ∈ T . The proof is omittedas it is similar to the previous ones in the proposition. Theconstraints in Eq. (16) are valid bounding constraints for theFCMVRP when the target i is the first target that is visitedby any vehicle as it leaves the depot. In this case, the Eq.(12) reduces to u i ≤ F − t i . We further strengthen thisconstraint by lifting the variable x di for every d ∈ D . Thelifting coefficient α for x di takes the value − ( F − t i − f di ) and the resulting constraint is given by Eq. (16).Hence, the second node-based formulation is as follows: ( F ) Minimize (cid:88) ( i,j ) ∈ E c ij x ij subject to: (1), (2), (6), and (13) – (16) . Corollary 2. opt( F L ) ≥ opt( F L ) . V. C
OMPUTATIONAL RESULTS
In this section, we discuss the computational performanceof the four formulations presented in the previous section.The mixed integer linear programs were implemented inJava, using the traditional branch-and-cut framework ofCPLEX version 12.4. All the simulations were performedon a Dell Precision T5500 workstation (Intel Xeon E5630processor @2.53 GHz, 12 GB RAM). The computationtimes reported are expressed in seconds, and we imposed atime limit of 3,600 seconds for each run of the algorithm.The performance of the algorithm was tested with randomlygenerated test instances.
Instance generation
The problem instances were randomly generated in asquare grid of size [100,100] with 5 fixed depot locations.The number of targets varies from to in steps offive, while their locations were uniformly distributed inthe square grid; for each | T | ∈ { , , , , , , } ,we generate five random instances. Each depot contains avehicle. The travel costs and the fuel consumed to traveletween any pair of vertices are assumed to be equal tothe euclidean distances between the pair. For each of theseproblems, we generate four possible fuel capacities F asa function of the the distance to the farthest target fromany depot λ . The fuel capacity F of the vehicles gets thevalues . λ , . λ , . λ and λ . In total, we generate instances and run the branch-and-cut algorithm for all theformulations.Tables I and II, and Fig. 2–3 summarize the computationalbehavior of the algorithms for all the instances. Thefollowing nomenclature is used throughout the rest of thepaper: : instance number; opt( F Li ) : linear programming relaxation solution for formu-lation i ; n : instance size i.e. , number of targets in the instance;%-LB: percentage LB/opt, where LB is the objective valueof the linear programming relaxation computed at the rootnode of the branch and bound tree and opt is the cost of theoptimal solution to the instance;total: total number of test instances of a given size;succ: number of instances for which optimal solutions werecomputed within a time limit of 3,600 seconds.Table I compares the cost of the linear programming (LP)relaxations of the four formulations presented in Sec. IV forthe 40 target instances. The results in table I provide anempirical comparison of the formulations presented in IV;the observed behavior is expected because the formulations F and F are strengthened versions of F and F , respec-tively (see corollaries 1 and 2). As for the LP relaxations offormulations F and F , it is difficult to conclude that oneis better than the other since F produces better relaxationvalues than F only for 60% of the instances. Hence, therest of the computational results compares the formulations F and F .Table II shows the number of instances of different sizessolved to optimality by the formulations F and F withinthe time limit of 3600 seconds. The plot in Fig. 2 shows theaverage time taken by the two formulations to compute theoptimal solution to the FCMVRP. The table II and Fig. 2indicate that the arc-based formulation F outperforms thenode-based formulation F for the larger instances. For thesmaller sized instances, it is difficult to differentiate betweenthe two formulations. The plot in Fig. 3 shows the percentageLB/opt for both the formulations (LB is the objective valueof the linear programming relaxation computed at the rootnode of the branch and bound tree and opt is the cost ofthe optimal solution to the instance; for the instances notsolved to optimality, opt represents the cost of the bestfeasible solution obtained at the end of 3,600 seconds). Weobserve that the %LB is consistently better for formulation F . This plot also provides empirical evidence to the claim TABLE IC
OST OF THE LP RELAXATION FOR THE TARGET INSTANCES . F L ) opt( F L ) opt( F L ) opt( F L ) OMPARISON OF FORMULATIONS F AND F . F F n total succ succ10 20 20 2015 20 20 2020 20 20 2025 20 20 1430 20 20 535 20 20 1540 20 19 1 that the arc based formulation F outperforms the node basedformulation F .VI. C ONCLUSIONS AND FUTURE WORK
In this paper, we have presented four different mixedinteger linear programming formulations for the multipledepot fuel constrained multiple vehicle routing problem. Theproblem arises frequently in the context of path planning forUAVs, green vehicle routing and routing electric vehicles.The formulations have been compared both analytically andempirically, and it is observed that a strengthened arc basedformulation ( F ) performs the best in terms of computingoptimal solutions to the problem. Computational experimentson a large number of test instances corroborates this observa-tion. Future work can be directed towards developing similarmixed integer linear programming formulations and branch-and-cut algorithms to solve a heterogeneous variant of theproblem i.e., with vehicles having different fuel capacities.R EFERENCES[1] K. Sundar and S. Rathinam, “Route planning algorithms for unmannedaerial vehicles with refueling constraints,” in
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