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Table 4: HGTS for SDVRP with Stop nodes 7. Conclusions and further research In this work, it has been introduced a new variant of the vehicle routing problem where ¢

Table 4 HGTS for SDVRP with Stop nodes 7. Conclusions and further research In this work, it has been introduced a new variant of the vehicle routing problem where ¢

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Real life situations need to adapt the CVRP model to their own characteristics and, thus, penalties associated with partial services if the customers are not fully served, etc.
Real life situations need to adapt the CVRP model to their own characteristics and, thus, penalties associated with partial services if the customers are not fully served, etc.
Figure 2: Split Delivery Vehicle Routing Problem A first study on this saving was carried out in [19] using a heuristic approach. Fox A first study on this saving was carried out in [19] using a heuristic approach. For problem sets of 75, 115 and 150 demand points and a fleet of vehicles with capacity fixed at Q = 160 units, the average number of vehicles saved was 14.07 and the average percentage distance gained for this set of problems was 11.24% for demands between 0.7Q and 0.9Q. They also showed empirically that if the demands are less than 0.1Q, there is no (or very little) advantage in solving the problem as a SDVRP versus a CVRP. These savings are theoretically studied in [6]: the cost of an optimal CVRP solution is compared with the cost of an optimal SDVRP solution by computing an upper bound on the ratio between the solution of both problems. It is shown that savings can be at most 50% and that this bound is tight (i.e., there exist instances where the value of the optimal solution of the CVRP is exactly twice the optimal value of the SDVRP). solution cost is 15.
Figure 2: Split Delivery Vehicle Routing Problem A first study on this saving was carried out in [19] using a heuristic approach. Fox A first study on this saving was carried out in [19] using a heuristic approach. For problem sets of 75, 115 and 150 demand points and a fleet of vehicles with capacity fixed at Q = 160 units, the average number of vehicles saved was 14.07 and the average percentage distance gained for this set of problems was 11.24% for demands between 0.7Q and 0.9Q. They also showed empirically that if the demands are less than 0.1Q, there is no (or very little) advantage in solving the problem as a SDVRP versus a CVRP. These savings are theoretically studied in [6]: the cost of an optimal CVRP solution is compared with the cost of an optimal SDVRP solution by computing an upper bound on the ratio between the solution of both problems. It is shown that savings can be at most 50% and that this bound is tight (i.e., there exist instances where the value of the optimal solution of the CVRP is exactly twice the optimal value of the SDVRP). solution cost is 15.
Now, consider that the use of the stop node is allowed. This means that a vehicle might o the stop node if the route leg length is close to L. The optimal solution is also performed wo routes, but this time with a cost of 12. In the first tour, vehicle 1 visits nodes 1 and 2, t top node s and finally node 3. Note that, at node 2 the route leg length is 4 (co1 + c12 =: f it decides to go to node 3 and then from node 3 to the depot, then the route leg leng vill exceed L. If the vehicle goes to the stop node, the route leg length is 5 (< L). Then, esumes the next route leg from the stop node and, because the remaining capacity on t ehicle is 1, the last visited customer on this route is node 3 before returning to the dep vith a leg length equal to 4 (cs3-++¢309). This route has two legs, both with length less than ‘ehicle 2 visits the fourth customer with a route length of 2 (Figure 3d). A cost of 1 m Now, consider that the use of the stop node is allowed. This means that a vehicle might g We assume a cost of one unit when the stop node is used. First, consider the case where Vehicle 2 visits the fourth customer with a route length of 2 (Figure 3d). A cost of 1 must
Now, consider that the use of the stop node is allowed. This means that a vehicle might o the stop node if the route leg length is close to L. The optimal solution is also performed wo routes, but this time with a cost of 12. In the first tour, vehicle 1 visits nodes 1 and 2, t top node s and finally node 3. Note that, at node 2 the route leg length is 4 (co1 + c12 =: f it decides to go to node 3 and then from node 3 to the depot, then the route leg leng vill exceed L. If the vehicle goes to the stop node, the route leg length is 5 (< L). Then, esumes the next route leg from the stop node and, because the remaining capacity on t ehicle is 1, the last visited customer on this route is node 3 before returning to the dep vith a leg length equal to 4 (cs3-++¢309). This route has two legs, both with length less than ‘ehicle 2 visits the fourth customer with a route length of 2 (Figure 3d). A cost of 1 m Now, consider that the use of the stop node is allowed. This means that a vehicle might g We assume a cost of one unit when the stop node is used. First, consider the case where Vehicle 2 visits the fourth customer with a route length of 2 (Figure 3d). A cost of 1 must
Constraints (7) say that a customer must be visited by at least one vehicle while cor traints (8) impose that exactly K vehicles leave the depot (if the model verifies the triangulz 1equality these constraints are not needed). Constraints (9) indicate that if vehicle k visit ode p, then it must leave it and (10) are the classical subtour elimination constraints. Not hat the subset D must have at least one stop node because the existence of subtours forme y only customer nodes are avoided by constraints (14)-(17). Constraints (11) guarant« hat the quantity delivered by each vehicle does not exceed the demand of the node. Co1 traints (12) limit to Q the maximum load of each vehicle while constraints (13) ensure thé he entire demand of each node is satisfied. Finally, constraints (14) to (17) and (20) are th oute leg length constraints.
Constraints (7) say that a customer must be visited by at least one vehicle while cor traints (8) impose that exactly K vehicles leave the depot (if the model verifies the triangulz 1equality these constraints are not needed). Constraints (9) indicate that if vehicle k visit ode p, then it must leave it and (10) are the classical subtour elimination constraints. Not hat the subset D must have at least one stop node because the existence of subtours forme y only customer nodes are avoided by constraints (14)-(17). Constraints (11) guarant« hat the quantity delivered by each vehicle does not exceed the demand of the node. Co1 traints (12) limit to Q the maximum load of each vehicle while constraints (13) ensure thé he entire demand of each node is satisfied. Finally, constraints (14) to (17) and (20) are th oute leg length constraints.
with the greater saving cost. We use a different criterion because a route with a lower saving then delete ifrom all route in B and introduce it in r* at position (g + 1)"’. Demands
with the greater saving cost. We use a different criterion because a route with a lower saving then delete ifrom all route in B and introduce it in r* at position (g + 1)"’. Demands
Table 1: Solution of the SDVRP with different algorithms Problem set P11 is found in [11], [12] and [2] as P6. We do not report the solutio1 of P12 (or P7) because it is not reported in [8]. Previous results show that MA seem: to be the most efficient algorithm for solving the SDVRP because they improve almost al the solutions of TABUSPLIT, especially for demands lower than Q/2. SS reaches bette improving percentages on the instances with original demands P5, P11 and P2(0.01,0.1) P3(0.01,0.1), P5(0.3,0,7) and P11(0.3,0.7). TSVBA also is a very good algorithm. Not that MA and TSVBA do not use the minimum number of vehicles but SS and HGTS dc it. Because of that we present two comparisons: column 11 presents the algorithm that get: the best improving percentage among all algorithms (based on columns 4, 6, 8 and 10) anc column 12 presents the best improving percentage using the minimum number of vehicle:
Table 1: Solution of the SDVRP with different algorithms Problem set P11 is found in [11], [12] and [2] as P6. We do not report the solutio1 of P12 (or P7) because it is not reported in [8]. Previous results show that MA seem: to be the most efficient algorithm for solving the SDVRP because they improve almost al the solutions of TABUSPLIT, especially for demands lower than Q/2. SS reaches bette improving percentages on the instances with original demands P5, P11 and P2(0.01,0.1) P3(0.01,0.1), P5(0.3,0,7) and P11(0.3,0.7). TSVBA also is a very good algorithm. Not that MA and TSVBA do not use the minimum number of vehicles but SS and HGTS dc it. Because of that we present two comparisons: column 11 presents the algorithm that get: the best improving percentage among all algorithms (based on columns 4, 6, 8 and 10) anc column 12 presents the best improving percentage using the minimum number of vehicle:
Table 2: Improving percentage respect to TABUSPLIT the number of stop nodes in the routes that use at least one stop node. Columns 6 and 7 present the mean objective value over 5 runs and the minimum value of those, respectively. The objective value includes the cost of using the stop nodes, set as 8000 for every stop node. Finally, column 8 shows the increase of the cost for the new constraints with respect to the best solutions of the HGTS for the SDVRP in Table 1. Note that P3, P4 and P5 with demands with parameters (0.10,0.30) do not use the stop nodes. All the tested instances use more than one stop node, and also, some of them present routes that use more than one stop node. The total cost increases between 0.69% and 9.74% respect with the best solution of the SDVRP.
Table 2: Improving percentage respect to TABUSPLIT the number of stop nodes in the routes that use at least one stop node. Columns 6 and 7 present the mean objective value over 5 runs and the minimum value of those, respectively. The objective value includes the cost of using the stop nodes, set as 8000 for every stop node. Finally, column 8 shows the increase of the cost for the new constraints with respect to the best solutions of the HGTS for the SDVRP in Table 1. Note that P3, P4 and P5 with demands with parameters (0.10,0.30) do not use the stop nodes. All the tested instances use more than one stop node, and also, some of them present routes that use more than one stop node. The total cost increases between 0.69% and 9.74% respect with the best solution of the SDVRP.
Figure 12: Solution of P1(0,0) with Stop Nodes From the results, it can be observed that the HGTS outperforms many tested instance with respect to very efficient algorithms with the advantage that our heuristic considers th minimum number of vehicles needed to serve all the demands (as SS does). Thus HGTS gets better improving percentages with respect to TABUSPLIT in 16 instances among al the compared algorithms and 27 instances with respect to SS. When the stop nodes ar needed, the algorithm obtains a solution by inserting these nodes to avoid the violation o the maximum route leg length L. We cannot compare our results with other algorithm because this variant of vehicle routing problem is new in the literature. From the results, it can be observed that the HGTS outperforms many tested instances
Figure 12: Solution of P1(0,0) with Stop Nodes From the results, it can be observed that the HGTS outperforms many tested instance with respect to very efficient algorithms with the advantage that our heuristic considers th minimum number of vehicles needed to serve all the demands (as SS does). Thus HGTS gets better improving percentages with respect to TABUSPLIT in 16 instances among al the compared algorithms and 27 instances with respect to SS. When the stop nodes ar needed, the algorithm obtains a solution by inserting these nodes to avoid the violation o the maximum route leg length L. We cannot compare our results with other algorithm because this variant of vehicle routing problem is new in the literature. From the results, it can be observed that the HGTS outperforms many tested instances
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EngineeringComputer ScienceTabu SearchVehicle routingVehicle Routing ProblemCapacitated Vehicle Routing Problem
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