 13 Technology and Investment, 2013, 4, 78-82 Published Online Febr uary 20 http://www.SciRP.org/journal/ti) Copyright © 20 SciRes. TI Heterogeneous Fleet Vehicle Routing Problem with Selec-tion of Inter-Depots ČIČKOVÁ Zuzana, PEKÁR Juraj, BREZINA Ivan, REIFF Marian Department of Operations Research and Econometrics, University of Economics, Bratislava, Slovakia Email : pekar @euba.sk Received 2012 ABSTRACT The paper presents heterogeneous fleet vehicle routing problem with selection of inter-depots. The goal of the proposed model is to schedule shipments of goods from the central depot to the inter-depots and finally to the endpoint customers, so that as inter-depots are considered the existing capacities of endpoints customers. In given type of transportation it can be expected to use two types of vehicles: large capacity vehicles ensuring the distribution of goods from the central ware ho use and small capacity vehicles ensuring the delivery of goods from inter-depots for late shipment to other cus-tomers. The principle of the model is illustrated on three demonstrative instances Keywords: mult i -depot vehicle routing problem; replenishment; mathematical programming 1. Introduction The problem that addresses the distribution of goods from a central warehouse to various delivery points using a fleet of vehicle with limited capacity is known as capa-cited vehicle routing problem (CVRP). The relevance of that problem follows from many published articles, and also fro m practical applications of distribution (e. g. , , [ 3], [ 8]). The CVRP belongs to the group of NP-hard problems, so the possibility of solution of above men-tioned problem is studied by many authors. Nowadays research attention is focused on heuristics, for example, the evolutionary approach is proposed by in , , , , , . The problem can be extended by adding in-ter-depots to the central storage facilitie s and also by considering the use of multiple types of vehicles ( , , , , , ). Delivery is then realized in two levels, the first level is to carry the goods from a central distribution warehouse into inter-depots and the second level implements the tr an s s hi p me nt of goods to the end-point customers. Generally, each facilit y tends to be ex-actly defined to belong to the group of inter-depots or to the group of final customers. However, in nowadays real life situations, companies that are dealing with this type of distribution do not open new inter-depots, which opening is associated with the construction cost and subsequently with operation cost, but they utilize endpoint customers useful storage capacities also as an intermediate storage for late shipment to other customers. In this case, as in-ter-depots stor a ge are used existing capacities of end-points customers, so that existing supply points may or may not also serve as a inter-depots. In given type of problem it can be expected to use two types of vehicles: large capacity vehic le s (L) ensuring the distribution of goods from the central warehouse and smal l capacity vehic l es (S) ensuring the delivery of goods from in-ter-depots and also from central warehouse. In this paper we formulate the considered problem as a mat h ematical programming model later used for further analysis. Due to its computational complexity, the formulated problem is solvable by classical optimization techniques (as well as other modifications of CV RP ) only for small instances. We chose the following structure of the paper: In the first part we present classical CVRP, because it forms a base of modeling for a wide variety of routing problems. The next section describes the construction of proposed heteroge-neous fleet vehicle routi ng problem with selection of inter-depots. Afterwards, the obtained experimental re-sults are given and discussed. In this section, we also analyze presented model under different vehicle cost. 2. Capacited Vehicle Routing Problem Classical capacitated vehicle routing problem is the basic routing model and it can be stated as follows: Let G = (V, E) be the full weighted graph, where V represents set of n nodes and E represents n.n set of edges. The node v1 represents the location of central warehouse and the nodes vi = 2, 3, ...n represent location of endpoint cus-tomers. Assumptions of CVRP: • Unlimited number of vehicles with a certain capacity g, which are located in a certain depot with unlimited ca-pacity. 13 ( Č. Zuzana ET AL. TI • Set of delivery points (endpoi nts) with a certain de-man d bi, i = 2, 3, ...n (the orders have to be delivered in full). • Minimal distances between all the pairs of endp oint s customers and also between the endpoint customers and the depot are known (matrix D). • The goal is to fi nd optimal vehicle routes (so that the total distance traveled is minimized) in such a way that each customer is visited only once by exactly one ve-hicle, all routes start and end at the depot, and the total demands of all customers on one particular route must not exceed the capacity of the vehicle. • In the model is implicitly assumed bi ≤ g for each i = 2, 3, ...n. Parameters D – matrix of minimal distances between all the pairs of nodes c – transportation cost per unit b – vector of demands g – capacity of the vehic le . Variables { }0, 1ijx∈, , 1, 2, ...ij n= - binary variables, that represent the use of corresponding edge: If the edge (i,j) is on the route of the some vehicle 1ijx=, otherwise 0ijx=, 0iu≥,1, 2, ...in= - non-negative variables, that represent the cumulative demand of the individual nodes on the route of any vehicl e. Starting value of variable 10u= CVRP formulation can be interpreted as follows: ( )11minnnij ijijfc dx= == →∑∑X, u Objective function ensures minimization of total tran s-portation costs of vehicles. 11=∑=niijx, 2, 3, ...jn=, ij≠ The equations represent that the vehicle comes to each node (except the centre) exactly once. 11=∑=njijx, 2, 3, ...in=, ij≠ The equations represent that the vehicle leaves each node (except the centre) exactly once. ( )0ijj iju bux+− =, 1, 2,...in=, 2, 3,...jn=, ij≠ iibu g≤≤, 2, 3, ...in= The equations are anti-cyclic constraints which ensure also the accumulation of the load of the vehicle in such way the load must not exceed the vehicle capacity. 3. Heterogeneous Fleet Vehicle Routing Problem with Selection of Inter-Depots – Problem Formulation The model was in general introduced by authors in . Problem is useful for company dealing with one central warehouse (edge v1 denotes localization of the central ware ho use ) and n – 1 delivery points. The large capacity vehicles L are available in the central warehouse. The small capacity vehicles S are available at the endpoints and also at the central warehouse. Transportation costs per unit are known for both types of vehicles . It is as-sumed the known demand of all delivery points. All the orders have to be delivered in full. Each of the n – 1 delivery point can be used as a inter-depot, so demand of other d elivery point can be can be met from the in-ter-depot by vehicle S. The route of vehicle S can be fi-nished also in other than starting inter-depot, (either cen-tral warehouse can be assumed to serve as inter-depot), but the number of incoming and outc omin g vehicles of type S must be identical. Parameters D – matrix of minimal distances between all the pairs of nodes b – vector of demands gL – capacity of the vehicle L gS – capacity of the vehicle S cL – cost per unit of the vehicle L cS – cost per unit of the vehicle S . 4. Heterogeneous Fleet Vehicle Routing Problem with Selection of Inter-Depots – Model Variables { }0, 1ijx∈, , 1, 2, ...ij n= - binary variables, that represent the use of corresponding edge by vehicle L: If the edge (i,j) is on the route of the vehicle L 1ijx=, oth-erwis e 0ijx=. { }0, 1ijy∈, , 1, 2, ...ij n= - binary variables, that represent the use of corresponding edge by vehicle S: If the edge (i,j) is on the route of the vehicle S 1ijy=, otherwi s e 0ijy=. 0ijyy ≥, , 1, 2, ...ij n= - non-negative variables, that represent the cumulat i ve amount of unloaded goods transported on the edge (i,j). 7913 Copyright © 20 SciRes. Č. Zuzana ET AL. TI 0iu≥,1, 2, ...in= - non-negative varia bles, that represent the cumulative demand of the individual nodes on the route of vehicle L (including eventual distrib ution with a vehicles S from corresponding node). 0iz≥,1, 2, ...in= - non-negative variables, that represent t he cumulative demand of the individual nodes on the route of vehicle S. 0iw≥,2, 3, ...in= - non-negative variables, that represent the quantity of goods that is distributed from corresponding node by vehicl e S. 0iq≥,1, 2, ...in= - non-negative variables, that represent the quantity of imported goods to the corres-ponding node by a vehicle L. Starting values of variables 111x= 0iix=, 2, 3, ...in= 10u= 10z= The model can be interpreted as follows: ( )11 11, minnn nnLij ijSijijij ijfcdx cdy= == ==+→∑∑ ∑∑X,Y,YY,u,z w,q Objective function ensures minimization of total tran s-portation costs for vehicles L and vehicles S. 11nijix=≤∑, 2, 3, ...jn=,ij≠ The equations represent that the vehicle L comes to each node (except the centre) no more than once. 11nijjx=≤∑, 2, 3, ...in=,ij≠ The equations represent that the vehicle S leaves to each node (except the centre) no more than once. 11nnij jijjxx= ==∑∑, 1, 2, ...in=,ij≠ The equations represents that the number of a rriva ls and exits of ve hic le L into and from the node is identical. ( ).0ijjijuqux+− =,1, 2, ...in=, 2, 3, ...jn=, ij≠ The equations provide the accumulation of the load of the vehicle L. iLug≤, 2, 3, ...in= The equations represent that the capacity of vehicle L must not be exceeded. 11nniiiiqb= ==∑∑ The equation represents that the demand of all nodes have to be met. 11., 2,3,...nij iijx bzin=− ≤=∑ The equations provide the balance of variables (starting value s for a vehicle S) in the case node serves as an in-ter-depot. 110, 2,3,...nnij jiiiijxyjn= =≠− +≥=∑∑ The equations represent the self-service of the node, in the case the node is on the route of a vehicle L. 11nijjy=≥∑, 1, 2, ...in= The equations represent that each node must be served. 11nnij jijjyy= ==∑∑, 1, 2, ...in=,ij≠ The equations rep resent that the number of incoming and the number of outcoming routes in and out of the node is identical. ( )1. .(1)0njii ijsjsz bzyx=+−− =∑,1, 2, ...in=,2, 3, ...jn=,ij≠ The equations provide the accumulation of the load of the vehicle S. iSzg≤, 1, 2, ...in= The equations represent that the capacity of vehicle S must not be exceeded 1nii iijjw qbx== −⋅∑, 2, 3, ...in= The equations represent the balance of quantity of goods of vehicl e S. 11.nnjiij jiiijyyx w= =≠=∑∑ , 2, 3, ...jn= The equations provide that the amount of goods for ve-hicles S cannot exceed the quantity of goods that is transported into the node by a vehicle L. .ji jiiyy yz=, , 1, 2, ...ij n=, ij≠ The equations provide the balance of cumulative number of unloaded goods transported on the edge (i, j). { }0, 1ijy∈, , 1, 2, ...ij n= { }0, 1ijx∈, , 1, 2, ...ij n= 0ijyy ≥, , 1, 2, ...ij n= 0iu≥, 1, 2, ...in= 0iw≥, 2, 3, ...in= 0iz≥, 1, 2, ...in= 0iq≥, 1, 2, ...in= 5. Empirical Results The principle of heterogeneous fleet vehicle routing problem with selection of inter-depots is illustrated on 8013 Copyright © 20 SciRes. Č. Zuzana ET AL. TI three de monstrative instances. Firstly, cost per vehicle L and vehicle S were considered in proportion ten to six, L (10), S (6 ). Later, in order to demonstrate the model functionality, it was considered to set significantly higher costs per vehicle L, (value 100) and to set significa nt l y lower costs per vehicle S (value 1). This should lead to preference of vehicles S (if it is possible to fulfill the de-mands due to the restriction limit of the vehicle). In the third case, high cost fo r vehicle S (100) and low cost for vehicle L (1) were preferred, which should essentially resul t s in the preference of vehicle L, i.e. the model works as classical CVRP. Model input data, used in all three demonstrative in-stances: n – number of nodes = 10 b – vector of demands = (0, 5, 2, 2, 20, 10, 6, 3, 9, 10) gL – capacity of the vehicle L = 50 gS – capacity of the vehicle S = 15 D – matrix of minimal distances between all the pairs of nodes (Table 1) Table 1 For each demonstrative instance the input data were modified in different cost adjustments: 1) Proportional costs adjustment: cL – cost per unit of the vehicle L = 10 cS – cost per unit of the vehic le S = 6 2) The higher cost of operating a large capacity vehicle compared to a small capacity vehicles: cL – cost per unit of the vehicle L = 100 cS – cost per unit of the vehic le S = 1 3) The higher cost of operating a small capacity vehicles compared to a large capacity vehicle: cL – cost per unit of the vehicle L = 1 cS – cost per unit of the veh ic le S = 1 00 Proposed problems were solved using software GAMS. Solutions of the model for various demonstrative in-stances: 1. Insta nc e with proportional cost values (cL = 10, cS = 6). Computed value f = 9103.8. Solution is depicted on Figure 1, dashed lines represent the transportation by vehicle L, and full line represents the route of ve-hicles S. Figure 1 2. Insta nc e with significantly higher cost of operating vehicle L (cL = 100, cS = 1). Computed value f = 17757.5. Solution is depicted on Figure 2, dashed lines represent the route of vehicle L and full lines represent the route of vehicles S. The value of objec-tive function f has significantly increased. It is caused by the fact that delivery cannot be made only by vehicle type S because of the demand of node 5 exceeds the capacity of the vehicle S. At the same time, assumption that the vehicle S can finish its rout e in other as in starting i nt er -depot is met, how-ever the number of incoming and outgoing vehicles of type S must be identical (node 1 and 5). Figure 2 3. Case with significantly higher cost of operating ve-hicles S (cL = 1, cS = 100). Computed value f = 1152.2. Solution is depicted on Figure 3. Dashed lines represent the transportation by L type vehicle, D1234567891010 11148.5 216.782.5 207.840.8 182.5 105.1 118.82111 077.5 301.633.5 125.9 151.88078.9 121.9348.5 77.50265.267 169.289.3137.5 132.2 145.94216.7 301.6 265.20273.1424.5 182.1 374.5 259.1272.8582.533.567 273.10151.4123.3105 79.1 93.46207.8 125.9 169.2 424.5 151.40248.691.5 184.4 228.1740.8 151.889.3 182.1 123.3 248.60223.3 145.9159.68182.580137.5 374.510591.5 223.30117.9 161.69105.178.9 132.2 259.179.1 184.4 145.9 117.9046.210 118.8 121.9 145.9 272.893.4 228.1 159.6161.646.201 6 4 8 5 9 3 10 7 2 1 3 4 6 5 9 8 2 7 10 8113 Copyright © 20 SciRes. Č. Zuzana ET AL. TI 1 3 4 6 5 8 7 2 10 9 the vehicles S are not in use (as in classical CVRP). Figure 3 6. Conclusion The paper is focused on the heterogeneous fleet vehicle routing problem with selection of inter -depots. It in-volves not only optimization of distribution on two stag-es, but also the choice between two types of vehicles based on cost optimization. The model deals with one central warehouse and n – 1 delivery points (each of them can serve as an inter-depot). In the central ware-house, large capacit y vehicles L are available. Small ca-pacity vehicles S are available at the endpoints and also at the central warehouse. The solution was presented on three demonstrative in-stance s. The first case is close to real ratio of vehicle costs (proportional cost per vehicle L (10) and vehicle S (6)). Furthermore, also specific cases with extreme cost ratios have been considered, which should verify a "rea-sonable" behavior of the model. Obtained results confirm the feasibility of usage of the model which allows utili-zation of the existing storage capacities at delivery points (inter-depots), and could lead to a more efficient distri-butio n. 7. 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