This paper addresses a multiple depot, multiple unmanned vehicle routing problem with fuel constraints. The objective of the problem is to find a tour for each vehicle such that all the specified targets are visited at least once by some vehicle, the tours satisfy the fuel constraints, and the total travel cost of the vehicles is a minimum. We consider a scenario where the vehicles are allowed to refuel by visiting any of the depots or fuel stations. This is a difficult optimization problem that involves partitioning the targets among the vehicles and finding a feasible tour for each vehicle. The focus of this paper is on developing fast variable neighborhood descent (VND) and variable neighborhood search (VNS) heuristics for finding good feasible solutions for large instances of the vehicle routing problem. Simulation results are presented to corroborate the performance of the proposed heuristics on a set of 23 large instances obtained from a standard library. These results show that the proposed VND heuristic, on an average, performed better than the proposed VNS heuristic for the tested instances.
1. Introduction
Small unmanned vehicles (UVs) are being used in several environmental monitoring applications [1–6] to collect information such as temperature, moisture, and humidity. These applications require UVs to visit specific target locations and monitor large swathes of land to collect data. Even though there are both economical and operational benefits [7] in using small UVs, they also come with other resource constraints due to their size and limited payload. Typically, small UVs due to their limited fuel capacity may have to revisit the depots (or fuel stations) multiple times for refueling while executing a surveillance mission. Path planning for small vehicles becomes critical in this scenario if the available resources such as fuel must be used as efficiently as possible.
This paper considers a fundamental routing problem that arises in these monitoring applications and is stated as follows: given a set of target locations, fuel stations (or depots) and UVs, find a path for each vehicle such that each target is visited at least once by a vehicle, each vehicle satisfies the fuel constraint as it traverses along its respective path and the travel cost of all the vehicles is a minimum. The travel cost we consider is the total fuel consumed by all the vehicles. To simplify the problem, we assume that the fuel consumed by a vehicle is directly proportional to the distance traveled by the vehicle. The vehicles are expected to refuel at the fuel stations as they run out of fuel. Vehicles are heterogeneous as they are allowed to carry fuel tanks with different capacities. This problem is referred to as the multiple, heterogeneous unmanned vehicle routing problem (MHUVRP).
In the absence of fuel constraints, MHUVRP is a generalization of the traveling salesman problem (TSP) and is NP-hard [8]. The difficulty in solving the TSP is further compounded when multiple vehicles are considered and even more when fuel constraints are imposed on these vehicles. Therefore, the focus of this paper is on developing heuristics that can find good solutions to the MHUVRP as quickly as possible. We accomplish this through the framework of the variable neighborhood search (VNS) and variable neighborhood descent (VND). VNS and VND are metaheuristics [9] used to solve difficult combinatorial and global optimization problems. These are iterative algorithms where in each iteration, the algorithms search through multiple neighborhoods of the current feasible solution to find a feasible solution with lower cost. The use of multiple neighborhoods allows the solution in the VNS and VND heuristics to move away from local optima as quickly as possible.
Routing for UVs has been addressed by several authors in [10–12]. The single vehicle version of the MHUVRP has been addressed by Khuller at al. [13] and Sundar and Rathinam [14]. Authors in [13] present an approximation algorithm for the symmetric version of the problem. An α-approximation algorithm is a polynomial time algorithm that produces a solution whose cost is at most α times the optimal cost for any instance of the problem. Authors in [14] present an approximation algorithm for the asymmetric version of the problem. In addition to an approximation algorithm, 2-opt and 3-opt heuristics were presented in [14] to find good feasible solutions to the single vehicle problem. Computational results [14] showed that the k-opt heuristics in combination with the approximation algorithm can produce near-optimal solutions for single vehicle instances with 25 targets within a couple of seconds of CPU time.
The MHUVRP is also closely related to routing problems with intermediate facilities [15, 16]. Other variants of the MHUVRP have also been studied in the literature. Authors in [17] consider a multiple vehicle TSP incorporating time windows and equity constraints. Approximation algorithms for a heterogeneous multiple vehicle TSP and a Hamiltonian path problem were studied in [18, 19]. Oberlin et al. [20] present a transformation of a heterogeneous multiple vehicle, multiple depot TSP into an asymmetric TSP so that algorithms for the standard TSP can be put to good use.
The contributions of this paper are as follows.
We develop algorithms based on the VNS and VND heuristics proposed by Hansen and Mladenovic [9] for the MHUVRP using multiple neighborhoods.
To understand the effect of choosing different neighborhoods and initial solutions on the quality of the final solutions, we implement the proposed algorithms and test them on large problem instances from the multiple depot vehicle routing library by Cordeau [21]. These results give insights as to which of the neighborhoods provide substantial improvements in the search process and the order in which the neighborhoods must be selected for the VNS/VND algorithms to obtain good solutions for the MHURVP.
2. Problem Statement
Let there be K vehicles denoted by v1,v2,…,vK with fuel capacities denoted by L1,…,LK, respectively. Without loss of generality, we assume that L1≤L2⋯≤LK. Let T denote the set of targets to be visited, and let D denote the set of depots (or fuel stations) that are available. Let V:=T∪D. Each vehicle is initially located at one of the depots. The problem is formulated on a complete undirected graph G=(V,E) where E is the set of edges joining any two vertices in V. Let the amount of fuel (travel cost) required to travel between any two vertices x,y∈V be represented by fxy. The travel costs are assumed to be symmetric and satisfy the triangle inequality; that is, for distinct vertices x,y,z∈V, we have fxy=fyx and fxy+fyz≥fxz. Additionally, for any target t∈T, it is assumed that there is at least one vehicle vi and a depot d∈D such that 2·ftd≤Li. This is a reasonable assumption because target t will be unreachable for any vehicle if this assumption is not true.
A tour for a vehicle is denoted by a sequence of vertices (s,t1,…,tp,s) where ti∈V for i=1,…,p and s∈D is the depot corresponding to the initial location of the vehicle. The cost of traveling a tour is defined as the sum of the cost of traveling all the edges in the tour. A tour satisfies the fuel constraint for its corresponding vehicle if the vehicle does not run out of fuel while traversing its tour. The objective of MHUVRP is to find K tours, one for each vehicle, such that each target is visited at least once by some vehicle, the tours satisfy the respective fuel constraints of the vehicles and the total cost of traveling all the tours is a minimum.
3. Algorithms
Heuristics based on the variable neighborhood search (VNS) are developed in this section for solving the MHUVRP. VNS was proposed by Hansen and Mladenovic in [9, 22]. The main idea of the VNS is to perform a local search systematically using multiple neighborhoods. It explores increasingly distant neighborhoods of the current solution iteratively and jumps from its current solution in the solution space to a new one if and only if an improvement has been made.
The steps of the basic VNS are shown as in Algorithm 1. In Algorithm 1, Nκ (κ=1,…,κmax) denotes a finite set of neighborhood structures. The stopping conditions may include criteria based on the allowable CPU time, maximum number of iterations, or maximum number of iterations between any two improvements. One can observe that the basic VNS heuristic also contains a probabilistic component in the shaking phase. The shaking step is a characteristic feature of the VNS heuristic and it allows the algorithm to get out of a local optimum. The solution obtained from the local search phase is compared to the incumbent solution and is accepted as a new starting point if an improvement can be made; otherwise, it is rejected. Therefore, the VNS procedure is a descent, first-improvement method with randomization. This method can also be used without the randomization phase. Such a method, named as the variable neighborhood descent (VND) heuristic [9], is the same as the VNS heuristic save for the absence of the shaking phase. In the following sections, we describe our algorithms for generating the initial solutions and the improvement procedures in detail.
<bold>Algorithm 1: </bold>Steps of the basic VNS by Hansen and Mladenovic.
Initialization: Select a set of neighborhood structures denoted by Nκ (κ=1,…,κmax) that will be used in
the search; find an initial solution x; choose a stopping condition.
Repeat the following until the stopping condition is met:
(1) Set κ←1.
(2) Repeat the following steps until κ=κmax:
(a) Shaking: Generate a feasible solution x′ at random from the κth neighborhood of x (x′∈Nκ(x));
(b) Local search: Apply a local search method with x′ as initial solution; denote the so obtained
local optimum as x′′;
(c) Move or not: If x′′ is better than incumbent x, set x←x′′ and continue the search with N1 (κ←1);
otherwise set κ←κ+1;
4. Initial Solution: Construction Phase
Construction heuristics are developed by generalizing the approach of the single vehicle algorithm in [13] to multiple vehicles. In particular, we develop two construction heuristics that provide two initial feasible solutions for the MHUVRP. In the following discussion, we summarize the main steps of these construction heuristics and give the details in the appendix.
In the first construction heuristic, we first compute a new, modified traveling cost for each vehicle that also includes the fuel constraints of the vehicle. This traveling cost will also include the extra fuel a vehicle may require to visit all the refueling stops when it traverses between any two targets. We compute this new traveling cost based on the approach presented in [13]. For completeness, we have presented the details of this approach in the appendix of this article. Suppose li(x,y) denote the modified cost of traveling from target x to target y for the ith vehicle including the refueling stops. It is easy to verify that these traveling costs monotonically increase with the index of the vehicles since the vehicles are ordered such that the fuel capacity of the vehicles decreases, that is, l1(x,y)≤l2(x,y)⋯≤lK(x,y). Once these traveling costs are computed, the primal-dual heuristic presented in [23] is used to assign the targets to the vehicles. After partitioning the targets to the vehicles, the best single vehicle algorithm available in [14, 24] is used to find a feasible tour for each vehicle. This construction heuristic is referred to as the Approximate Primal Dual (APD) algorithm.
In the second construction heuristic, each target is assigned to the nearest initial depot of the vehicle based on the Euclidean distances between the targets and depots. Once the targets are partitioned among the vehicles, the best single vehicle algorithm available in [14, 24] is used to find a feasible tour for each vehicle. This construction heuristic is referred to as the Voronoi algorithm. Note that the two construction heuristics primarily differ only in the partitioning of the targets.
This section explains the choice of neighborhoods and the main steps in the improvement phase of the VNS algorithm (Algorithm 1) that includes the shaking step, local search, and the move or not step.
5.1. Neighborhood Selection
We use four neighborhoods; three of these are intra-route neighborhoods (2-exchange [25], 3-exchange [25], and depot exchange [24]) and the last one is an inter-route neighborhood (relocate [26]). Any two solutions can be present in an intra-route neighborhood only if the assignment of targets to the vehicles are the same in both the solutions. On the other hand, inter-route neighborhood considers solutions where the assignment of targets to the vehicles are different.
We first formally define all the neighborhoods in the ensuing discussion. A solution S2 is said to be in the k-exchange neighborhood of solution S1 if there is exactly one vehicle vi such that
S2 and S1 may differ only in the tours of vehicle vi, and
the tour for vehicle vi in S2 differs from the tour for vehicle vi in S1 by at most k edges.
A solution S2 is said to be in the depot exchange neighborhood (Figure 1) of solution S1 if there is exactly one vehicle vi such that
S2 and S1 differ only in the tours of vehicle vi, and
the tour for vehicle vi in S2 differs from the tour for vehicle vi in S1 by exactly two edges and one depot in D∖ui where ui denotes the initial depot of vehicle vi.
A solution S2 is said to be in the relocate neighborhood (Figure 2) of solution S1 if there are exactly two distinct vehicles vi and vj satisfying the following conditions:
S2 and S1 differ only in the tours of vehicles vi, vj,
there is exactly one target u such that u is visited by vi in S1 whereas u is visited by vj in S2.
Depot exchange neighborhood; tour for some vehicle vi is shown. The solution S2 is contained in the depot exchange neighborhood of solution S1.
Relocate neighborhood; the solution S2 is in the relocate neighborhood of S1.
5.2. Shaking
The shaking step is the randomization part of the VNS heuristic, the set of neighborhoods that are used in this shaking step form the core of the VNS. The purpose of these neighborhoods is to sufficiently perturb the initial solution (computed in Section 4), while ensuring that this new solution keeps certain aspects of the initial incumbent. In this step, a random feasible solution of the currently selected neighborhood of x is found and denoted as x′. This random selection enables the algorithm to avoid local optima in the local search procedure. For certain neighborhoods, it may be possible that there are no feasible neighbors; that is, the neighborhood of x is empty. In this special case, the shaking step is skipped, and x′ is the same as x. Figure 3(a) shows the solution space where each solution is shown using a black dot, the neighborhood of x is denoted by a circle, and the outcome of the shaking step is denoted as x′.
Shaking step and move or not step of the VNS heuristic.
Shaking step
Move or not step
5.3. Local Search
The output solution (x′) from the shaking procedure is then improved using a local search procedure. The local search examines all the feasible neighbors of x′ in the current neighborhood and finds the solution x′′ which has the least cost. Again, it may be possible that the current neighborhood of x′ is empty. If this is the case, x′′ is set to be the same as x′.
5.4. Move or Not: Acceptance Decision
In this step, the cost of x′′ is compared to the cost of x. The two possible outcomes of interest are when the cost of x′′ is less than the cost of x and when the cost of x′′ is greater than or equal to the cost of x. In the first case, x is set to be x′′ (see Figure 3(b)). When the first outcome is true, the first neighborhood is set as the “current” neighborhood (κ←1 in Algorithm 1), and computations continue with the shaking step. When the second outcome is true, x′′ is forgotten, and the next neighborhood is set as the “current” neighborhood (κ←κ+1 in Algorithm 1). If there is no next neighborhood, that is, the “current” neighborhood is the last neighborhood designated, the algorithm terminates.
6. Computational Analysis
The VNS/VND heuristic proposed in Section 5 was implemented using Visual C++ and the elements of the Standard Template Library (STL). Simulations were performed on a 2.4 GHz AMD Phenom machine. The problem instances on which the simulations were performed derives from the Cordeau’s benchmark instances [21] for the multiple depot vehicle routing problem. Additional information was added to these benchmark instances to satisfy the requirements of the MHUVRP as follows.
The number of customers in the benchmark instances was used as the number of targets for the MHUVRP.
The depot locations in the benchmark instances were used as the initial depot locations of all the vehicles.
The number of vehicles in the MHUVRP was set to be equal to the number of depots in the benchmark instances.
The fuel capacity of the kth vehicle in an instance was computed using the formula Lk:=S-15(k-1) where S is the size of the surveillance area. S was set to be equal to maxi,j∈V|xi-xj| where xi denotes the x coordinate of vertex i (rounded to the nearest 100 units).
The above modifications resulted in 23 distinct instances (listed in Table 1) on which the computational study was performed. The number of targets and vehicles in these instances varied from 50 to 360 and 2 to 9, respectively. We use a maximum allowable time of 1000 secs as the stopping condition for the heuristics.
Comparison of the initial feasible solutions.
Instance
Targets
Vehicles
APD Cost (Cost_{apd})
Voronoi Cost (Costv)
Costv/Costapd (%)
p01
50
4
2483.04
2474.2
99.64
p03
75
5
2170.27
3666.57
168.95
p04
100
2
2569.91
4354.53
169.44
p05
100
2
2588.21
2645.91
102.23
p06
100
3
5089.49
4158.92
81.72
p07
100
4
5759.59
5798.38
100.67
p08
249
2
40044.1
59003.2
147.35
p09
249
3
37783.8
51802.7
137.10
p10
249
4
34990.6
50382.6
143.99
p11
249
5
34610.9
71234.3
205.81
p12
80
2
5917.04
12618
213.25
p15
160
4
23619.6
9628.57
40.77
p21
360
9
77354.6
19131.9
24.73
pr01
48
4
6964.58
9940.97
142.74
pr02
96
4
7625.7
7854.82
103.00
pr03
144
4
21303.9
29336.8
137.71
pr04
192
4
14659.2
19382
132.22
pr05
240
4
9780.66
24196.3
247.39
pr06
288
4
24778.5
9144.85
36.91
pr07
72
6
4026.09
10006.3
248.54
pr08
144
6
9340.81
14959.8
160.16
pr09
216
6
12207.2
16921.7
138.62
pr10
288
6
9119.63
30491.9
334.35
6.1. Quality of the Initial Feasible Solutions
Table 1 shows the cost of the initial feasible solutions produced by the two construction heuristics. On an average, these results show that the initial feasible solution produced by the APD algorithm was superior to the initial feasible solution produced by the Voronoi partitioning algorithm for a majority of instances.
6.2. Analysis on the Best Sequence of Neighborhoods
In this section, we first study the effect of the choice of the neighborhoods used in the VNS and VND algorithms. The % improvement in the quality of solutions using an algorithm is defined as 100×((CostI-Costalg)/CostI) where Cost_{alg} denotes the travel cost of the final solution produced by the algorithm and CostI denotes the travel cost of the initial solution produced by a construction heuristic. As there are two construction heuristics used, we evaluate the effect of the sequence of neighborhoods used on both the initial solutions obtained using the construction heuristics. Table 2 shows the average improvement in the quality of the solutions using a sequence of 4 neighborhoods. We only report the sequences of the neighborhoods that produced the best improvements. In general, the order of the neighborhoods did not significantly affect the improvement in the quality of the solutions.
Influence of 4 neighborhoods sequences on improving quality.
Scheme
Construction heuristic
N1
N2
N3
N4
Average improvement (%)
Average computation time (sec)
VND
APD
2-opt
Relocate
3-opt
Depot
33.92
78.04881
VND
APD
2-opt
Relocate
Depot
3-opt
33.79
77.33881
VND
APD
2-opt
3-opt
Relocate
Depot
33.76
197.7543
VNS
APD
Relocate
2-opt
3-opt
Depot
27.52
70.8876
VNS
APD
Relocate
2-opt
Depot
3-opt
27.39
75.12725
VNS
APD
Relocate
Depot
2-opt
3-opt
26.99
75.44715
VND
Voronoi
Depot
Relocate
3-opt
2-opt
38.08
310.7845
VND
Voronoi
Relocate
3-opt
Depot
2-opt
37.94
304.7469
VND
Voronoi
Depot
Relocate
2-opt
3-opt
36.85
203.9593
VNS
Voronoi
Relocate
Depot
2-opt
3-opt
42.22
221.0236
VNS
Voronoi
Relocate
Depot
3-opt
2-opt
38.23
741.5726
VNS
Voronoi
Depot
Relocate
3-opt
2-opt
37.52
799.8579
Table 3 shows the average improvement in the quality of the solutions using a sequence of 3 neighborhoods. These results show that the VND heuristic provides more improvement in the quality of the solutions, on an average, compared to the VNS heuristic. We also note that majority of the best sequences has 3-opt neighborhood as the last search neighborhood. In general, we found that the addition of the 3-opt neighborhood produced high quality solutions for the problem at the expense of computation time.
Influence of 3 neighborhoods sequences on improving quality.
Scheme
Construction heuristic
N1
N2
N3
Average improvement (%)
Average computation time (sec)
VND
APD
2-opt
Relocate
3-opt
33.99
483.1837
VND
APD
2-opt
3-opt
Relocate
33.86
595.6045
VND
APD
Relocate
3-opt
2-opt
33.49
89.5769
VNS
APD
Relocate
2-opt
3-opt
27.52
71.3513
VNS
APD
Relocate
3-opt
2-opt
26.77
178.1168
VNS
APD
2-opt
Relocate
3-opt
25.87
79.254
VND
Voronoi
Relocate
Depot
3-opt
42.59
323.1872
VND
Voronoi
Relocate
2-opt
3-opt
36.24
225.4593
VND
Voronoi
Depot
Relocate
3-opt
35.97
891.1418
VNS
Voronoi
Relocate
2-opt
3-opt
37.37
241.9253
VNS
Voronoi
Relocate
3-opt
Depot
37.18
298.7079
VNS
Voronoi
Depot
Relocate
3-opt
36.93
795.2278
Table 4 shows the average improvement in the quality of the solutions using a sequence of 2 neighborhoods. This table provides a clear picture of the overall improvement capability of the different neighborhoods. These results show that the relocate and 3-opt neighborhoods appear most frequently followed by the 2-opt, indicating that the neighborhood combinations containing 3-opt or relocate are the most effective, even though they take an order or two of magnitude longer to run than the other neighborhood combinations. They also provide the best improvement in the quality of solutions on an average. Finally, Table 5 shows the effect of choosing a single neighborhood in the improvement of the quality of the solutions. Results in Table 5 shows a clear trade-off in the quality of the solutions produced by an algorithm versus the computation time the algorithm requires. In terms of solution quality, algorithms using the 3-opt neighborhood performed the best while using the maximum amount of computation time. On the other hand, the algorithms using only the relocate neighborhood ran quickly while producing an improvement of up to 17.56%.
Influence of 2 neighborhoods sequences on improving quality.
Scheme
Construction heuristic
N1
N2
Average improvement (%)
Average computation time (sec)
VND
APD
Relocate
2-opt
32.82
417.9984
VND
APD
2-opt
Relocate
32.15
422.4113
VND
APD
Relocate
3-opt
31.99
495.1695
VNS
APD
Relocate
3-opt
22.76
170.846
VNS
APD
3-opt
Relocate
21.24
173.4163
VNS
APD
3-opt
2-opt
20.38
121.7078
VND
Voronoi
Relocate
3-opt
39.42
753.4336
VND
Voronoi
3-opt
Relocate
33.73
1109.158
VND
Voronoi
Relocate
2-opt
30.32
11.05305
VNS
Voronoi
Relocate
3-opt
33.93
503.4305
VNS
Voronoi
3-opt
Relocate
29.22
479.8828
VNS
Voronoi
Relocate
2-opt
27.92
7.714526
Influence of one neighborhood on improving quality.
Scheme
Construction heuristic
N1
Average improvement (%)
Average computation time (sec)
VND
APD
3-opt
22.96
1462.62
VND
APD
2-opt
21.96
420.1842
VND
APD
Relocate
14.08
1.017391
VNS
APD
3-opt
17.32
109.5311
VNS
APD
2-opt
10.36
0.75705
VNS
APD
Relocate
8.13
0.30115
VND
Voronoi
3-opt
21.29
835.2096
VND
Voronoi
Relocate
17.56
1.6659
VND
Voronoi
2-opt
15.56
4.2504
VNS
Voronoi
3-opt
19.92
459.5533
VNS
Voronoi
2-opt
13.19
3.118789
VNS
Voronoi
Relocate
8.87
0.399421
6.3. Effect of the Construction Heuristics on the Final Cost
Table 6 shows the effect of the two construction heuristics on the quality of the final solutions obtained by the VNS and VND algorithms. In this table, Costvf and Costapdf denote the cost of the final solution obtained by the algorithms using the Voronoi and the APD heuristic, respectively. The table shows the average values of ((Costvf-Costapdf)/Costapdf)×100 for some of the best combination of neighborhoods. For a majority of instances, these results show that the choice of the construction heuristic, on an average, does not significantly affect the quality of the final solutions produced by the algorithms. Figures 4 and 5 show the initial and the best solution obtained using the proposed algorithms for the instance pr05.
Comparison between the final solutions produced by the algorithms.
Scheme
N1
N2
N3
N4
(Costapdf-Costvf)/Costapdf (%)
VNS
Relocate
2-opt
Depot
3-opt
9.90
VND
Depot
2-opt
Relocate
3-opt
4.42
VND
Depot
2-opt
Relocate
3-opt
4.42
VNS
Relocate
3-opt
Depot
−18.64
VND
3-opt
Relocate
Depot
1.80
VNS
Relocate
2-opt
3-opt
8.73
VND
Relocate
2-opt
3-opt
5.85
VNS
Relocate
2-opt
−1.52
VND
Relocate
2-opt
6.47
VNS
3-opt
2-opt
−35.45
VND
Relocate
3-opt
2.10
VNS
Relocate
−51.36
VND
2-opt
−3.04
VND
Relocate
−34.98
Initial solutions for the instance pr05 using the APD and Voronoi partitioning heuristics.
The best solution obtained for the instance pr05 using the proposed heuristics.
6.4. The Best Algorithm with the Neighborhood Combinations
In this subsection, we specify the best combination of the neighborhoods obtained for the instances and the corresponding cost of the final solutions obtained by the algorithms. For a single neighborhood, the VND heuristic with 3-opt (Table 7) performed the best with the initial solution provided by the APD heuristic. This heuristic improved the quality of the solutions by 22.96% on an average. For two neighborhoods, the VND heuristic with relocate as the first neighborhood and 3-opt as the second neighborhood (Table 8) performed the best with the initial solution provided by the Voronoi heuristic. This heuristic improved the quality of the solutions by 39.42% on an average. We found that the relocate neighborhood in combination with either 2-opt or 3-opt were effective in improving the quality of the solutions substantially.
Results using the VND heuristic with 3-opt.
Instance
Initial cost (Costi)
Final cost (Costf)
(Costi-Costf)/Costi (%)
Computation time (sec)
p01
2483.04
1942.48
21.77
7.368
p03
2170.27
1172.44
45.98
4.822
p04
2569.91
1301.3
49.36
15.816
p05
2588.21
1468.28
43.27
30.267
p06
5089.49
2904.23
42.94
180.685
p07
5759.59
4456.7
22.62
195.286
p08
40044.1
40044.1
0.00
192.995
p09
37783.8
37783.8
0.00
142.009
p10
34990.6
34990.6
0.00
374.885
p11
34610.9
34610.9
0.00
30.816
p12
5917.04
2064.31
65.11
25.047
p15
23619.6
3105.85
86.85
5842.052
p21
77354.6
76936.9
0.54
9000
pr01
6964.58
6885.05
1.14
11.20039
pr02
7625.7
7625.7
0.00
9.709453
pr03
21303.9
21303.9
0.00
29.87566
pr04
14659.2
13465.5
8.14
943.9597
pr05
9780.66
8137.73
16.80
286.9558
pr06
24778.5
14108.8
43.06
32975.92
pr07
4026.09
2151.59
46.56
2.17765
pr08
9340.81
7385.68
20.93
178.9951
pr09
12207.2
9294.9
23.86
480.1001
pr10
9119.63
9119.63
0.00
50.35585
Results using the VND heuristic with relocate and 3-opt.
Instance
Initial Cost (Costi)
Final Cost (Costf)
(Costi-Costf)/Costi (%)
Computation time (sec)
p01
2474.2
1096.8
55.67
1.012
p03
3666.57
1253.25
65.82
16.18
p04
4354.53
1762.64
59.52
206.734
p05
2645.91
1465.84
44.60
38.875
p06
4158.92
1829.36
56.01
22.65
p07
5798.38
2224.81
61.63
48.702
p08
59003.2
59003.2
0.00
518.078
p09
51802.7
51802.7
0.00
557.311
p10
50382.6
50382.6
0.00
479.043
p11
71234.3
67496.8
5.25
9570.828
p12
12618
1573.43
87.53
8.731
p15
9628.57
2398.17
75.09
23.964
p21
19131.9
5543.1
71.03
58.40544
pr01
9940.97
9940.97
0.00
2.577926
pr02
7854.82
2425.53
69.12
16.12673
pr03
29336.8
22436
23.52
4844.833
pr04
19382
18776.1
3.13
508.4298
pr05
24196.3
17398.5
28.09
4531.525
pr06
9144.85
5088.55
44.36
252.208
pr07
10006.3
5294.35
47.09
2.493464
pr08
14959.8
8907.27
40.46
142.704
pr09
16921.7
5272.41
68.84
100.4253
pr10
30491.9
30491.9
0.00
46.63218
For three neighborhoods, the VND heuristic with relocate, depot exchange, and 3-opt neighborhoods (in the given sequence) performed the best with the initial solution provided by the Voronoi heuristic. In particular, they improved the quality of the solutions by 42.59% as shown in Table 9. For four neighborhoods, the VNS heuristic with relocate, depot exchange, 2-opt, and 3-opt neighborhoods (in the given sequence) performed the best with the initial solution provided by the Voronoi heuristic. This heuristic improved the quality of the solutions by 42.22% as shown in Table 10. In general, using four neighborhoods as compared to three neighborhoods only increased the computation time while not providing any substantial improvements in the solution quality.
Results using the VND heuristic with relocate, depot exchange and 3-opt.
Instance
Initial Cost (Costi)
Final Cost (Costf)
(Costi-Costf)/Costi
(%)
Computation time (sec)
p01
2474.2
1096.8
55.67
0.962
p03
3666.57
1253.25
65.82
16.196
p04
4354.53
1634.81
62.46
186.685
p05
2645.91
1465.84
44.60
38.83
p06
4158.92
1829.36
56.01
22.633
p07
5798.38
2224.81
61.63
48.9
p08
59003.2
59003.2
0.00
515.802
p09
51802.7
51802.7
0.00
554.835
p10
50382.6
50382.6
0.00
476.858
p11
19327.5
4599.05
76.20
212.3943
p12
12618
1573.43
87.53
8.829
p15
9628.57
2398.17
75.09
24.106
p21
19131.9
5391.13
71.82
89.22671
pr01
9940.97
9940.97
0.00
2.634846
pr02
7854.82
2435.17
69.00
14.46503
pr03
29336.8
22475.2
23.39
4365.047
pr04
19382
18776.1
3.13
541.5995
pr05
24196.3
17403.7
28.07
4190.062
pr06
9144.85
5165.15
43.52
235.299
pr07
10006.3
5510.2
44.93
2.12073
pr08
14959.8
8689.09
41.92
179.7975
pr09
16921.7
5289.71
68.74
94.15305
pr10
30491.9
30491.9
0.00
46.17499
Results using the VNS heuristic with relocate, depot exchange, 2-opt and 3-opt.
Instance
Initial Cost (Costi)
Final Cost (Costf)
(Costi-Costf)/Costi (%)
Computation time (sec)
p01
2474.2
694.271
71.94
0.232
p03
3666.57
1143.62
68.81
4.915
p04
4354.53
1551.03
64.38
58.451
p05
2645.91
1476.75
44.19
19.571
p06
4158.92
1474.53
64.55
12.117
p07
5798.38
1464.33
74.75
26.84
p08
59003.2
59003.2
0.00
519.381
p09
51802.7
51802.7
0.00
559.957
p10
50382.6
50346.4
0.07
470.61
p11
71234.3
65498.1
8.05
1910.843
p12
12618
2385.06
81.10
53.939
p15
9628.57
2409.39
74.98
11.693
p21
19131.9
5492.56
71.29
78.08324
pr01
9940.97
9940.97
0.00
2.691766
pr02
7854.82
2107.41
73.17
14.26489
pr03
29336.8
22494.6
23.32
974.1806
pr04
19382
18603.2
4.02
213.7126
pr05
15198.4
9801.09
35.51
484.938
pr06
9144.85
5240.83
42.69
98.146
pr07
10006.3
4827.39
51.76
1.632319
pr08
14959.8
8357.23
44.14
64.22047
pr09
16921.7
4676.42
72.36
168.083
pr10
30491.9
30473.4
0.06
47.34827
7. Conclusion
The effects of different neighborhoods and partitioning heuristics on benchmark instances for the MHUVRP were examined. The neighborhoods included the 2-opt, 3-opt, depot exchange, and the relocate neighborhoods. Simulation results showed that, on an average, the construction heuristics did not affect the quality of the final solutions obtained by the algorithms despite the fact that the primal-dual heuristic produced better initial solutions compared to the Voronoi heuristic. Overall, the VND heuristic produced better solutions compared to the VNS heuristic. This result shows that including the shaking step in the VNS may not always yield better results. In general, heuristics with the 3-opt neighborhood provided the best solutions for majority of the instances. However, 3-opt neighborhoods can be replaced with 2-opt neighborhoods whenever possible, to obtain solutions of similar quality while reducing the computation time.
This paper mainly focused on developing heuristics for the MHUVRP. However, there are currently no algorithms in the literature that can find optimal solutions to the problem relatively quickly for the large instances considered in this paper. Therefore, future work for the MHUVRP can focus on finding good lower bounds and optimal solutions for these instances.
Appendix
The purpose of this computation is to find a path for every vehicle vk to travel from any target x∈T to any other target y∈T such that the path can be a part of the feasible tour for vk, satisfies all the refueling constraints for vk, and is of minimum travel cost. Given a vehicle vk, we note that the maximum amount of fuel available for the vehicle when it reaches target x in any tour is Lk-mindfdx. Also, in any feasible tour, there must be at least mindfyd units of fuel left when the vehicle reaches target y so that the vehicle can continue to visit other vertices along its tour. Define Fx:=mindfdx (also Fx=mindfxd since the travel costs are symmetric) for any x∈T. The first step of the algorithm finds a feasible path of least cost (also referred as the shortest path) such that the vehicle starts at target x with at most L-Fx units of fuel and ends at target y with at least Fy units of fuel. If there is enough fuel available for the vehicle vk to travel from x to y (or, if Lk-Fx-Fy≥fxy), the vehicle can directly reach y from x while respecting the fuel constraints. In this case, we say that the vehicle vk can directly travel from x to y and the shortest path (also referred to as the direct path) is denoted by PATHk(x,y):=(x,y). The cost of traveling this shortest path is just fxy. If the vehicle vk cannot directly travel from x to y (if Lk-Fx-Fy<fxy), the vehicle must visit some of the depots on the way before reaching target y. In this case, we find a shortest path using an auxiliary graph, (V′,E′), defined on all the depots and the targets x,y, that is, V′=D∪{x,y} (illustrated in Figure 6).
The first step of the algorithm: the solid edges represent the shortest path PATHk(x,y) for the vehicle vk, to travel from target x to target y, and the cost of traveling this path is denoted by lk(x,y).
An edge is present in this graph only if traveling the edge can satisfy the fuel constraint. For example, as the vehicle vk has at most Lk-Fx units of fuel to start with, the vehicle can reach a depot d from x only if fxd≤Lk-Fx. Therefore, E′ contains an edge (x,d) if the constraint fxd≤Lk-Fx is satisfied. Similarly, the vehicle can travel from a depot d to target y only if there are at least Fy units of fuel remaining after the vehicle reaches y. Therefore, E′ contains an edge (d,y) if the constraint fdy≤Lk-Fy is satisfied. In summary, the following are the edges present in E′:(A.1)E′:={{(x,d):∀d∈D,fxd≤Lk-Fx}∪{(d1,d2):∀d1,d2∈D,fd1d2≤Lk}∪{(d,y):∀d∈D,fdy≤Lk-Fy}.
Any path starting at x and ending at y in this auxiliary graph will require the vehicle vk to carry at most Lk-Fx units of fuel at target x, satisfy all the fuel constraints, and reach target y with at least Fy units of fuel left. Also, we let the cost of traveling any edge (i,j)∈E′ to be equal to fij (as defined in Section 2). Now, we use Dijkstra’s algorithm [27] to find a shortest path to travel from x to y. This shortest path (also referred to as the indirect path using intermediate depots) can be represented as PATHk(x,y):=(x,d1,d2,…,y). Let the length of this path be represented by lk(x,y). This new cost function is computed for every vehicle between every pair of targets and is denoted by lk where k=1,…,n.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This material is based upon work supported by the National Science Foundation under Grant no. 1015066 and the Air Force Office of Scientific Research under Grant no. FA9550-10-1-0392.
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