Complexity
Volume 2018 (2018), Article ID 5754908, 12 pages
https://doi.org/10.1155/2018/5754908

Research Article

A Hybrid Tabu Search Algorithm for a Real-World Open Vehicle Routing Problem Involving Fuel Consumption Constraints

Yunyun Niu,¹ Zehua Yang,¹ Ping Chen,² and Jianhua Xiao³

¹School of Information Engineering, China University of Geosciences, Beijing, Beijing 100083, China
²School of Business, Nankai University, Tianjin 300071, China
³The Research Center of Logistics, Nankai University, Tianjin 300071, China

Correspondence should be addressed to Jianhua Xiao; jhxiao2008@163.com

Received 4 October 2017; Accepted 1 February 2018; Published 28 February 2018

Academic Editor: Danilo Comminiello

Copyright © 2018 Yunyun Niu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Outsourcing logistics operation to third-party logistics has attracted more attention in the past several years. However, very few papers analyzed fuel consumption model in the context of outsourcing logistics. This problem involves more complexity than traditional open vehicle routing problem (OVRP), because the calculation of fuel emissions depends on many factors, such as the speed of vehicles, the road angle, the total load, the engine friction, and the engine displacement. Our paper proposed a green open vehicle routing problem (GOVRP) model with fuel consumption constraints for outsourcing logistics operations. Moreover, a hybrid tabu search algorithm was presented to deal with this problem. Experiments were conducted on instances based on realistic road data of Beijing, China, considering that outsourcing logistics plays an increasingly important role in China’s freight transportation. Open routes were compared with closed routes through statistical analysis of the cost components. Compared with closed routes, open routes reduce the total cost by 18.5% with the fuel emissions cost down by nearly 29.1% and the diver cost down by 13.8%. The effect of different vehicle types was also studied. Over all the 60- and 120-node instances, the mean total cost by using the light-duty vehicles is the lowest.

1. Introduction

Many developing countries are confronted with two problems: how to reduce economic costs and how to develop an environment-friendly society. As a result, researchers as well as companies with integrated logistics expertise tend to adopt new transportation models to counter these problems. Outsourcing logistics operations to the third-party logistics would reduce costs through better resource utilization and operations efficiency in freight transportation. It plays an increasingly important role in freight transportation. In that case, a company can hire vehicles from other companies to deliver its goods. Vehicles do not need return to the company as usual. They are usually described as the open vehicle routing problem (OVRP) [1]. Compared with the vehicle routing problem (VRP), routes in the OVRP model are open; see Figure 1.

Figure 1: Open vehicle routing problem.

The OVRP is NP-hard [2], so heuristics or metaheuristics methods are usually used to deal with it [3], such as the tabu search [4], the neighborhood-based search [5], particle swarm optimization [6], ant colony optimization [7], or evolutionary computing [8]. Hybrid metaheuristic algorithms were also designed to solve the OVRP [9]. Moreover, several variants of the OVRP were studied to model specific practical problems. OVRP with time windows (OVRPTW) was presented by [10] to model the delivery of multiproduct newspapers. The multidepot OVRP (MDOVRP) was first proposed by [11] to model the distribution of fresh meat. The open vehicle routing problem with decoupling points (OVRP-DP) was introduced in [12] to describe open routes performed by more than one carrier. The OVRP with uncertain demands was investigated to deal with nondeterministic customer demands and avoid unsatisfied demands or more extra operation cost [13]. However, among models studied in the past there are rare formulations considering the impact of fuel and carbon emission [14]. This paper is a contribution to this line of research.

Carbon emissions along with freight transportation have hazardous impacts on the environment. How to minimize the fuel consumption becomes a hot topic. In [15], six fuel consumption models were compared with each other and several factors affecting carbon emissions in road transportation were analyzed. In [16], the pollution-routing problem (PRP) was introduced to evaluate the greenhouse gas emissions for the VRP. The total energy consumed on each road can be directly translated into greenhouse gas emissions with the PRP model. Reference [17] considered a heterogeneous vehicle fleet and extended the PRP model. Reference [18] presented a recent review of studies on green freight transportation problems.

This study has been motivated by a real problem in Beijing, China. As outsourcing logistics is playing a more and more important role in China’s freight transportation, many companies hire vehicles from the third-party logistics. Vehicles do not need return to the depot after delivering products to customers. Each route departs from the depot and ends up with one of the customers. These companies are facing the problem of minimizing the total cost, including both fuel consumption cost and driver salaries.

The main contributions of our paper are summarized as follows. (1) Fuel consumption was analyzed in the context of the OVRP; and the green open vehicle routing problem (GOVRP) model was constructed. With the purpose of optimizing the fuel emissions cost in the model, we introduced the comprehensive modal emission model (CMEM) into the OVRP model and extend the OVRP with an objective that accounts for the amount of greenhouse emissions. (2) A hybrid tabu search algorithm involving several neighborhood search strategies was proposed to solve the GOVRP. A modified nearest neighborhood heuristic (mNNH) was proposed to get the initial solution. It considered two factors load and distance when searching for a suitable initial solution. And then four neighborhood search operators were designed to produce the neighborhood of the current solution. (3) Computational experiments were conducted on instances derived from real geographical data of customers in Beijing, China. Both the effect of open routes and the effect of vehicle types were analyzed in respects of reducing total cost.

2. Mathematical Model of the GOVRP

The CMEM was developed by [19, 20] to calculate fuel emissions. In this section, we introduced the CMEM to the OVRP and formulated the mathematical model of the GOVRP. The fuel consumption (in liters) of vehicle type is given by [21]

where , , , and are constraints. is the total vehicle weight, is the distance, and is the vehicle speed. Three terms of are referred to as the engine module, the weight module, and the speed module, respectively. Notations and their default values are listed in the following two tables [17, 22]. Common parameters of vehicles are listed in Table 1. Specific parameters for different vehicle types are listed in Table 2.

Table 1: Common parameters of vehicles.

Notation
Description
Typical values

Fuel-to-air mass ratio
1

Gravitational constant (m/s²)
9.81

Air density (kg/m³)
1.2041

Coefficient of rolling resistance
0.01

Efficiency parameter for diesel engines
0.45

Fuel and CO₂ emissions cost (RMB/liter)
12.0165

Driver wage (RMB/s)
0.0111

Heating value of a typical diesel fuel (kj/g)
44

Conversion factor (g/s to L/s)
737

Vehicle drive train efficiency
0.45

Lower speed limit (m/s)
0

Upper speed limit (m/s)
27.8 (or 100 km/h)

Acceleration (m/s²)
0

Table 2: Specific parameters for different vehicle types.

Notation
Description
Light duty
Medium duty
Heavy duty

Curb weight (kg)
3500
5500
14,000

Maximum payload (kg)
4000
12,500
26,000

Engine friction factor (kj/rev/liter)
0.25
0.20
0.15

Engine speed (rev/s)
38.34
36.67
30.0

Engine displacement (liter)
4.5
6.9
10.5

Coefficient of aerodynamics drag
0.6
0.7
0.9

Frontal surface area (m²)
7.0
8.0
10.0

Let be the set of customers and the depot, and let be the set of arcs. Node 0 denotes the depot, so the customer set is . represents the capacity of vehicles. Variables means the demand of customer . Variables is the distance from node to node . Variables are the total amount of flow on arc . The mathematical model of GOVRP is defined as follows:

Formula (2)–(5) describes the objective of the GOVRP, where terms (2)–(4) calculate the total fuel consumption cost and term (5) measures the total driver wages. More specifically, is the fuel emissions cost per liter, and terms (2)–(4) calculate the fuel consumption costs induced by the engine module, the weight module, and the speed module, respectively. denotes the total time spent on a route where is the last served customer.

The binary variable is equal to 1 if there is a vehicle traveling on arc ; otherwise, is equal to 0. Constraints (6)–(9) ensure that each customer is served by one vehicle and it is served only once. Vehicles do not return to the deport. If is the last served customer, constraint (8) can be written as . Otherwise, it can be written as . Constraints (10) and (11) define flows. The binary variable is equal to 1 when a vehicle travels on arc at speed .

3. Hybrid Tabu Search Algorithm for the GOVRP

In this study, a hybrid tabu search algorithm including several neighborhood search strategies was designed to solve the GOVRP. The flowchart of this algorithm is shown in Figure 2. The detailed steps are listed as follows.

Figure 2: The flowchart of our algorithm.

Step 1. The initial setup includes initializing an empty tabu list and an empty candidate list, creating an initial solution, setting that initial solution as the best solution to date. A modified nearest neighborhood heuristic (mNNH) is used to gain the initial solution. We will elaborate on it specifically in Section 3.1.

Step 2. If the stopping condition is satisfied, the search process stops, and the best solution is returned. Otherwise, it turns to Step 3.

Step 3. The solution space is explored each step from a current solution to the accepted solutions in the neighborhood by using four efficient neighborhood search algorithms. For each neighbor of the current solution , the optimal speed on each arc of their routes is set by using the speed optimization algorithm according to [23].

Step 4. The neighboring solutions are checked for tabu elements. We search for the best solution and update the candidate list.

Step 5. If the best local candidate is better than the current best solution, update the current best solution. The local optimal solution is added to the tabu list. Then, turn to Step 2.

In this paper, the stopping condition is the maximum number of iterations . When our algorithm is iterated times, the algorithm is terminated.

3.1. The Initial Solution

We use a modified nearest neighborhood heuristic (mNNH) to obtain an initial solution. The mNNH considers two factors load and distance when searching for a suitable initial solution. In a delivery system, a vehicle can reduce load after servicing customer . We define

where is defined as the unrouted customer set.

The mNNH builds routes one by one according to the following steps. The first route begins with an unrouted customer , which can be calculated by the following equation:

Then, we calculate between the current node and the other unrouted nodes. We choose the node with the greatest current value as the next node if the current route does not violate the vehicle capacity constraint. We update the current node and search for the next node in the same way. When no customer can be assigned to the route, a new route is started. When all the customers are already routed, the process stops. The algorithm is described as Algorithm 1.

Input: ,

Output:

(1) Set , ,

(2) while    do

(3) Select a car

(4) Set

(5) while   and   do

(6) Set , ,

(7) ,

(8) Set

(9) return  

Algorithm 1: A modified nearest neighborhood heuristic.

3.2. The Neighborhood Search

The neighborhood of the current solution is first obtained by four neighborhood search operators. They cannot violate the capacity constraint during this process as before. Then, the optimal speed on each route is set according to [23].

(1) Random Operator (RO). It randomly selects one node from the solution and then randomly finds a possible position for it.

(2) High-Cost-Node Improvement Operator (HCNIO). As illustrated in Figure 3, the operator tries to reassign the high cost node , because the total distance between its preceding customer and its following customer is the longest. Customer can be calculated as follows:

where is the preceding customer and is the following customer.

Figure 3: High-cost-node improvement operator.

We compute the best possible position for node based on

where is the cost for inserting node between customer and customer and is the set of currently routed customers.

(3) Short-Route Improvement Operator (SRIO). The operator tries to combine or delete short routes by reassigning nodes belonging to those routes. A threshold is set to evaluate the length of a route. Only if the length of a route is shorter than the threshold , the nodes in that route can be reassigned to other positions. The process can be illustrated in Figure 4 (). First, arcs in the shorter routes are deleted and then isolated nodes are assigned to other longer routes.

Figure 4: Short-route improvement operator.

The best possible position for each unrouted node is also calculated by (19). By using (20), the most suitable customer is selected to be inserted in the route repeatedly. Unrouted nodes will be inserted to the current routes one by one.

(4) Random-Route-Improvement Operator (RRIO). Each route in the current solution has a chance of being destroyed. In that case, customers in that route should be reassigned. The best possible position for each node is calculated by (19). Then best customer to be inserted in the route is selected by using (20) repeatedly. All other unrouted customers will be inserted into the sequence one by one. As shown in Figure 5, two routes included in the dotted oval are destroyed; nodes in those routes are reassigned in possible positions.

Figure 5: Random-route-improvement operator.

For each neighbor of the current solution , the optimal speed on each route is set. According to [23], the optimal speed is

4. Computational Analysis

Experiments are run with data derived from real geographical distances of customers in Beijing. Three smaller classes with 10, 20, and 30 customers and four larger classes with 60, 80, 100, and 120 customers are generated. Each class includes 10 instances. All the customers involved in this study are shown in Figure 6. The algorithmic parameters values are given in Table 3.

Table 3: Algorithmic parameters values.

Notation
Description
Typical values

The maximum iteration number
500

The length of Tabu list
5

The number of RO in each step
40

The number of HCNIO in each step
5

The threshold in SRIO in each step
3

Each route has a chance to be improved
50%

Figure 6: Customer nodes.

4.1. Effect of Objectives Min TD, Min FEC and Min DC

Three different objectives, Min TD (total distance), Min FEC (fuel emissions cost), and Min DC (driver cost), are used to minimize the total distance, fuel emissions cost, and driver cost, respectively. We obtained solutions with different cost components on each performance measure and analyzed the effect of different objectives. Experiments were conducted on 10-, 20-, and 30-node instances by using light-duty vehicles. The mean result of each instance collected over ten runs is reported; see Tables 4, 5, and 6. The columns display total distance (TD), execution time (ET), fuel emissions cost (FEC), driver cost (DC), total cost (TC), and CO₂ emissions (CE). Table 7 presents the average results of each class.

Table 4: Cost components with Min TC.

Instances
TD (m)
ET (s)
FEC (RMB)
DC (RMB)
TC (RMB)
CE (kg)

BJ10_01
160000
1.52
641.04
1427.23
2068.27
123.77

BJ10_02
215400
1.21
860
1669.73
2529.73
166.04

BJ10_03
148200
1.43
598.82
1515.92
2114.74
115.61

BJ10_04
125700
1.32
512.95
1604.03
2116.98
99.03

BJ10_05
131600
1.39
527.49
1261.2
1788.69
101.84

BJ10_06
195700
1.28
800.49
1731.86
2532.35
154.55

BJ10_07
161200
1.51
629.72
1462.42
2092.15
121.58

BJ10_08
160500
1.46
643.06
1598.9
2241.96
124.15

BJ10_09
174300
1.29
708.01
1687.92
2395.93
136.69

BJ10_10
196900
1.32
794.67
1724.37
2519.03
153.42

BJ20_01
207050
1.78
847.71
2872.04
3719.75
163.67

BJ20_02
327870
1.89
1313.52
3411.58
4725.1
253.6

BJ20_03
214080
1.84
869.09
2954.44
3823.52
167.79

BJ20_04
255540
2.01
1012
2740.72
3752.73
195.39

BJ20_05
276920
1.88
1118.67
3355.55
4474.22
215.98

BJ20_06
289340
2.18
1144.59
3143.29
4287.89
220.98

BJ20_07
303290
1.74
1257.06
3656.81
4913.87
242.7

BJ20_08
272120
1.8
1118.93
2995.08
4114.01
216.03

BJ20_09
310000
1.93
1250.4
3023.52
4273.93
241.41

BJ20_10
273670
2.25
1086.06
2945.31
4031.37
209.68

BJ30_01
383030
2.52
1594.7
4489.58
6084.28
307.89

BJ30_02
413850
2.62
1653.91
4497.14
6151.06
319.32

BJ30_03
425160
2.27
1763.89
4932.42
6696.31
340.55

BJ30_04
422810
2.55
1735.05
4702.76
6437.8
334.98

BJ30_05
348850
2.82
1405.68
4627.92
6033.6
271.39

BJ30_06
459680
2.28
1907.54
4772.35
6679.89
368.28

BJ30_07
460900
2.51
1900.86
4826.39
6727.26
367

BJ30_08
423510
2.38
1742.33
4625.07
6367.4
336.39

BJ30_09
447060
2.35
1869.31
4768.26
6637.57
360.9

BJ30_10
281950
2.59
1141.17
4007.94
5149.11
220.32

Table 5: Cost components with Min FEC.

Instances
TD (m)
ET (s)
FEC (RMB)
DC (RMB)
TC (RMB)
CE (kg)

BJ10_01
160000
1.5
641.04
1427.23
2068.27
123.77

BJ10_02
215400
1.21
860
1669.73
2529.73
166.04

BJ10_03
148200
1.56
598.82
1515.92
2114.74
115.61

BJ10_04
126500
1.37
511.94
1607.36
2119.3
98.84

BJ10_05
132750
1.47
519.04
1265.98
1785.02
100.21

BJ10_06
195500
1.31
800.42
1731.02
2531.45
154.54

BJ10_07
161200
1.57
629.72
1462.42
2092.15
121.58

BJ10_08
161900
1.5
640.76
1604.72
2245.47
123.71

BJ10_09
174300
1.3
708.01
1687.92
2395.93
136.69

BJ10_10
196900
1.33
794.67
1724.37
2519.03
153.42

BJ20_01
207400
1.76
849.35
2873.49
3722.84
163.98

BJ20_02
328330
1.95
1309.37
3413.49
4722.86
252.8

BJ20_03
214900
1.97
872.32
2957.85
3830.17
168.42

BJ20_04
255610
2.18
1011.35
2741.01
3752.36
195.26

BJ20_05
277600
1.94
1120.84
3358.38
4479.21
216.4

BJ20_06
289930
2.25
1139.38
3145.75
4285.12
219.98

BJ20_07
304560
1.79
1262.84
3662.1
4924.93
243.81

BJ20_08
273430
1.88
1118.78
3000.53
4119.3
216

BJ20_09
309870
2.04
1247.69
3022.98
4270.67
240.89

BJ20_10
273750
2.25
1082.62
2945.65
4028.27
209.02

BJ30_01
382650
2.42
1589.04
4488
6077.04
306.79

BJ30_02
413260
2.74
1652.82
4494.69
6147.51
319.11

BJ30_03
421230
2.27
1747.69
4916.08
6663.77
337.42

BJ30_04
422830
2.43
1736.24
4702.84
6439.08
335.21

BJ30_05
352760
2.75
1417.82
4644.18
6062
273.74

BJ30_06
458930
2.19
1903.61
4769.23
6672.84
367.53

BJ30_07
458810
2.49
1884.59
4817.7
6702.3
363.86

BJ30_08
426290
2.32
1748.32
4636.63
6384.94
337.54

BJ30_09
452450
2.3
1890.76
4790.68
6681.44
365.05

BJ30_10
281430
2.6
1135.95
4005.78
5141.73
219.32

Table 6: Cost component with Min DC.

Instances
TD (m)
ET (s)
FEC (RMB)
DC (RMB)
TC (RMB)
CE (kg)

BJ10_01
159400
1.44
644.62
1424.74
2069.36
124.46

BJ10_02
215400
1.21
860
1669.73
2529.73
166.04

BJ10_03
148200
1.47
598.82
1515.92
2114.74
115.61

BJ10_04
125200
1.39
518.15
1601.95
2120.11
100.04

BJ10_05
131600
1.39
530.81
1261.2
1792
102.48

BJ10_06
195620
1.3
800.46
1731.52
2531.99
154.54

BJ10_07
161200
1.45
629.72
1462.42
2092.15
121.58

BJ10_08
160400
1.48
646.39
1598.48
2244.87
124.8

BJ10_09
174300
1.29
709.63
1687.92
2397.55
137.01

BJ10_10
196900
1.32
794.67
1724.37
2519.03
153.42

BJ20_01
206800
1.74
846.78
2871
3717.78
163.49

BJ20_02
329150
1.82
1327.82
3416.9
4744.72
256.36

BJ20_03
215720
1.81
879.27
2961.26
3840.53
169.76

BJ20_04
256490
1.99
1018.96
2744.67
3763.63
196.73

BJ20_05
280140
1.85
1134.25
3368.94
4503.19
218.99

BJ20_06
292080
2.1
1158.64
3154.69
4313.33
223.7

BJ20_07
303580
1.7
1259.44
3658.02
4917.46
243.16

BJ20_08
272670
1.78
1123.32
2997.37
4120.69
216.88

BJ20_09
310710
1.89
1254.84
3026.47
4281.31
242.27

BJ20_10
271600
2.18
1076.31
2936.7
4013.01
207.8

BJ30_01
381430
2.35
1589.86
4482.93
6072.79
306.95

BJ30_02
420450
2.47
1689.85
4524.59
6214.44
326.26

BJ30_03
427000
2.15
1773.68
4940.07
6713.75
342.44

BJ30_04
422740
2.35
1748.59
4702.47
6451.05
337.6

BJ30_05
352590
2.57
1423.67
4643.47
6067.14
274.87

BJ30_06
458020
2.17
1904.47
4765.45
6669.91
367.69

BJ30_07
459540
2.33
1897.68
4820.74
6718.42
366.38

BJ30_08
430970
2.21
1784.59
4656.09
6440.68
344.55

BJ30_09
445190
2.18
1864.22
4760.49
6624.71
359.92

BJ30_10
282940
2.44
1146.82
4012.06
5158.88
221.42

Table 7: Cost components with different objectives.

Instances
Objective
TD (m)
ET (s)
FEC (RMB)
DC (RMB)
TC (RMB)
CE (kg)

10-node
Min FEC
167265
1.41
670.44
1569.67
2240.11
129.44

Min DC
166822
1.37
673.33
1567.83
2241.15
130

Min TC
166950
1.37
671.63
1568.36
2239.98
129.67

20-node
Min FEC
273538
2
1101.45
3112.12
4213.57
212.66

Min DC
273894
1.89
1107.96
3113.6
4221.57
213.91

Min TC
272988
1.93
1101.8
3109.83
4211.64
212.72

30-node
Min FEC
407064
2.45
1670.68
4626.58
6297.27
322.56

Min DC
408087
2.32
1682.34
4630.84
6313.18
324.81

Min TC
406680
2.49
1671.44
4624.98
6296.43
322.7

On average of 10-node class, the fuel emissions cost represents about 30% of the total cost. For 20- and 30-node classes, the fuel emissions cost accounts for approximately 26% of the total cost. The carbon dioxide emissions increase with the number of customers. Only considering the driver cost in the objective leads to the most carbon dioxide emissions and the poorest total cost performance. Only considering the fuel emissions gets solutions with the lowest carbon dioxide emissions, but a higher total cost. Min TC as an objective yields the lowest total cost and the shortest total distance.

4.2. Effect of Open Routes

In this subsection, we compared the total cost of open routes with that of closed routes. Experiments were also conducted on smaller instance classes by using light-duty vehicles. The mean results of closed routes collected over ten runs are listed in Table 8. The average results of each class are reported in Table 9 to illustrate the effect of open routes. As shown in Table 9, open routes reduce the total cost by 18.5% with the fuel emissions cost down by nearly 29.1% and the diver cost down by 13.8% over all the instances.

Table 8: Cost components with closed routes.

Instances
TD (m)
ET (s)
FEC (RMB)
DC (RMB)
TC (RMB)
CE (kg)

BJ10_01
243300
1.35
973.42
1773.67
2747.08
187.94

BJ10_02
287900
1.23
1143.69
1971.25
3114.95
220.81

BJ10_03
233000
1.28
929.49
1868.6
2798.09
179.46

BJ10_04
193600
1.33
774.88
1886.42
2661.31
149.61

BJ10_05
190300
1.33
752.29
1505.32
2257.61
145.24

BJ10_06
306900
1.26
1216.62
2194.33
3410.94
234.89

BJ10_07
225500
1.19
914.3
1729.84
2644.14
176.52

BJ10_08
222400
1.33
883.91
1856.33
2740.24
170.65

BJ10_09
250200
1.29
1001.88
2003.58
3005.46
193.43

BJ10_10
285200
1.3
1133.7
2091.6
3225.3
218.88

BJ20_01
296880
1.75
1191.94
3245.63
4437.57
230.13

BJ20_02
432490
1.74
1733.98
3846.68
5580.66
334.78

BJ20_03
302470
1.93
1227.85
3322.04
4549.9
237.06

BJ20_04
350360
1.8
1408.36
3135.07
4543.43
271.91

BJ20_05
397350
1.75
1601.07
3856.4
5457.48
309.12

BJ20_06
404900
1.76
1627.02
3623.9
5250.92
314.13

BJ20_07
453910
1.79
1833.88
4283.23
6117.1
354.06

BJ20_08
377970
1.71
1526.82
3435.3
4962.12
294.78

BJ20_09
458430
1.79
1842.5
3640.83
5483.32
355.73

BJ20_10
380810
1.75
1533.55
3390.9
4924.45
296.08

BJ30_01
544130
2.53
2213.58
5159.58
7373.16
427.37

BJ30_02
602750
2.32
2434.58
5282.76
7717.34
470.04

BJ30_03
603420
2.25
2446.55
5673.79
8120.33
472.35

BJ30_04
574210
2.51
2328.31
5332.41
7660.72
449.52

BJ30_05
495510
2.46
2006.16
5237.86
7244.02
387.33

BJ30_06
636960
2.23
2584.8
5509.64
8094.44
499.04

BJ30_07
666890
2.43
2711.67
5683.09
8394.76
523.54

BJ30_08
588990
2.25
2385.82
5313.28
7699.1
460.63

BJ30_09
640480
2.22
2600.55
5572.68
8173.23
502.08

BJ30_10
394830
2.35
1599.07
4477.4
6076.47
308.73

Table 9: Cost components of different types of routes.

Instances
Type of routes
TD (m)
ET (s)
FEC (RMB)
DC (RMB)
TC (RMB)
CE (kg)

10-node
Open
166950
1.37
671.63
1568.36
2239.98
129.67

Closed
243830
1.29
972.42
1888.09
2860.51
187.74

20-node
Open
272988
1.93
1101.8
3109.83
4211.64
212.72

Closed
385557
1.78
1552.7
3578
5130.7
299.78

30-node
Open
406680
2.49
1671.44
4624.98
6296.43
322.7

Closed
574817
2.36
2331.11
5324.25
7655.36
450.06

4.3. Results for Larger Scale Instances

Experiments were also conducted on the four largest sets with the objective Min TC. The results collected over ten runs are reported in Table 10 by using light-duty vehicles. Notations BS, MS, WS, SD, and ET in columns represent the best solution, mean solution, the worst solution, standard deviation, and execution time, respectively. As shown in Table 10, the total cost increases with the number of customers.

Table 10: Computational results on larger scale instances.

Instance
BS (RMB)
MS (RMB)
WS (RMB)
SD
ET (s)

BJ60_01
11061.79
11254.42
11417.58
92.8
5.37

BJ60_02
10309.83
10393.21
10492.87
62.43
4.88

BJ60_03
11574.38
11790.03
11933.82
94.03
5.06

BJ60_04
12094.39
12212.64
12462.25
95.54
5.14

BJ60_05
11236.35
11320.09
11431.91
52.74
5.44

BJ60_06
11883.87
12040.48
12249.58
94.39
4.89

BJ60_07
12600.43
12745.03
12867.3
92.99
4.74

BJ60_08
11939.78
12060.86
12173.83
59.37
4.73

BJ60_09
11583.85
11727.88
11922.82
95.35
4.7

BJ60_10
12938.11
13121.2
13364.03
114.64
4.74

BJ80_01
14787.03
15036.55
15227.3
127.09
7.56

BJ80_02
14018.34
14220.78
14411.5
115.1
7.18

BJ80_03
14714.36
14964.93
15139.74
125.11
7.72

BJ80_04
13872.84
14065.74
14550.33
199.54
7.6

BJ80_05
14613.9
14788.56
15078.73
148.41
7.52

BJ80_06
16857.87
17148.49
17572.08
217.41
6.83

BJ80_07
15371.73
15632.16
15925.82
170.46
7.1

BJ80_08
14273.97
14484.07
14815.66
183.64
7.39

BJ80_09
14526.1
14743.32
14969.09
122.54
7.19

BJ80_10
14736.48
14942.92
15284.86
137.4
7.09

BJ100_01
19330.25
19617.94
20069.92
203.78
10.22

BJ100_02
20194.9
20701.34
21013.24
215.65
10.33

BJ100_03
20081.75
20286.97
20556.28
138.13
10.38

BJ100_04
18483.02
18689.28
18893.14
114.28
9.87

BJ100_05
18776.99
18983.5
19256.85
165.22
10.01

BJ100_06
17478.28
17764.38
18119.14
187.27
10.5

BJ100_07
18451.61
18775.37
18938.18
167.43
10.49

BJ100_08
18527.85
18762.86
18941.62
142.11
10.52

BJ100_09
17458.07
18048.33
18385.66
259.49
10.92

BJ100_10
20847
21099.27
21541.13
201.04
10

BJ120_01
21699.63
21954.27
22209.26
135.47
15.38

BJ120_02
22803.9
23135.32
23736.67
273.1
16.09

BJ120_03
22074.76
22569.41
22921.39
270.03
17.46

BJ120_04
22380.9
22560.26
22764.57
138.1
16.91

BJ120_05
22602.33
23308.71
23666.93
289.5
14.39

BJ120_06
22656.5
22975.25
23166.4
145.56
14.44

BJ120_07
21156.14
21711.99
22170.6
301.69
14.37

BJ120_08
20837.63
21069.17
21418.09
211.61
15.66

BJ120_09
21865.67
22195.93
22498.88
170.26
14.43

BJ120_10
22705.98
23054.33
23556.8
217.54
14.4

4.4. Results with Different Vehicle Types

Experiments were conducted on the 60- and 120-node instances by using different types of vehicles. In this subsection, the analysis focuses on which type of vehicle is the most suitable to minimize the total cost. The specific vehicle parameters are listed in Table 2. Results with different vehicle types are listed in Tables 11 and 12. Over all the instances in the two classes, the mean total cost by using the light-duty vehicles is the lowest. For 120-node instances, the mean total cost by using medium-duty vehicles increases by 4.8% and the result by using heavy-duty vehicles rises 23.6%. For 60-node instances, the mean total cost by using medium-duty vehicles increases by 7.7% and the value by using heavy-duty vehicles rises 28%.

Table 11: Computational results on the 60-node instances by using different vehicles.

Type of vehicles
Instance
BS (RMB)
MS (RMB)
WS (RMB)
SD
ET (s)

Only light duty
BJ60_01
11061.79
11254.42
11417.58
92.8
5.37

BJ60_02
10309.83
10393.21
10492.87
62.43
4.88

BJ60_03
11574.38
11790.03
11933.82
94.03
5.06

BJ60_04
12094.39
12212.64
12462.25
95.54
5.14

BJ60_05
11236.35
11320.09
11431.91
52.74
5.44

BJ60_06
11883.87
12040.48
12249.58
94.39
4.89

BJ60_07
12600.43
12745.03
12867.3
92.99
4.74

BJ60_08
11939.78
12060.86
12173.83
59.37
4.73

BJ60_09
11583.85
11727.88
11922.82
95.35
4.7

BJ60_10
12938.11
13121.2
13364.03
114.64
4.74

Only medium duty
BJ60_01
12449.13
12616.53
12823.93
109.15
4.51

BJ60_02
11328.74
11557.08
11780.96
125.14
4.19

BJ60_03
12959.56
13222.5
13390.12
121.83
4.42

BJ60_04
13411.61
13630.23
13870.71
147.61
4.4

BJ60_05
12515.46
12678.95
12843.89
87.75
4.63

BJ60_06
13037.75
13302.09
13413.95
100.83
4.29

BJ60_07
12360.13
12425.3
12579.82
63.71
4.22

BJ60_08
12942.29
13064.34
13360.38
121.13
4.49

BJ60_09
12112.23
12348.68
12563.22
122.72
4.43

BJ60_10
12832.96
12974.52
13105.98
88.75
4.03

Only heavy duty
BJ60_01
14787.24
14995.59
15334.52
176.58
9.38

BJ60_02
13418.08
13700.42
13911.3
143.14
7.91

BJ60_03
15469.9
15746.78
16052.25
183.26
7.54

BJ60_04
16164.2
16411.77
16729.41
172.11
7.78

BJ60_05
14772.72
14937.7
15269.37
154.91
7.98

BJ60_06
15893.22
16093.82
16272.75
133.43
7.41

BJ60_07
14441.09
14606.23
14757.43
103.53
7.46

BJ60_08
15088.73
15473.2
15638.19
152.24
7.61

BJ60_09
14521.68
14811.72
14972.54
119.53
7.22

BJ60_10
14961.08
15164.55
15321.96
109.22
6.88

Table 12: Computational results on the 120-node instances by using different vehicles.

Type of vehicles
Instance
BS (RMB)
MS (RMB)
WS (RMB)
SD
ET (s)

Only light duty
BJ120_01
21699.63
21954.27
22209.26
135.47
15.38

BJ120_02
22803.9
23135.32
23736.67
273.1
16.09

BJ120_03
22074.76
22569.41
22921.39
270.03
17.46

BJ120_04
22380.9
22560.26
22764.57
138.1
16.91

BJ120_05
22602.33
23308.71
23666.93
289.5
14.39

BJ120_06
22656.5
22975.25
23166.4
145.56
14.44

BJ120_07
21156.14
21711.99
22170.6
301.69
14.37

BJ120_08
20837.63
21069.17
21418.09
211.61
15.66

BJ120_09
21865.67
22195.93
22498.88
170.26
14.43

BJ120_10
22705.98
23054.33
23556.8
217.54
14.4

Only medium duty
BJ120_01
23290.23
23541.68
23691.38
124.27
12.99

BJ120_02
23133.82
23473.76
23710.98
178.16
12.55

BJ120_03
23320.12
23573.14
23908.95
186.7
12.9

BJ120_04
23117.27
23545.71
23757.64
194.07
13.01

BJ120_05
23687.6
24099.96
24385.48
207.61
13.08

BJ120_06
23273.19
23621.69
23870.13
162.68
12.49

BJ120_07
23136.08
23257.6
23424.7
91.21
13.13

BJ120_08
22752.22
23085.88
23320.75
200.2
13.35

BJ120_09
23171.61
23279.47
23450.24
85.55
12.65

BJ120_10
23476.59
23845.67
24258.76
222.02
12.58

Only heavy duty
BJ120_01
27565.54
27974.16
28341.07
204.51
20.49

BJ120_02
27337.33
27628.98
28138.69
243.32
18.79

BJ120_03
26855.88
27379.48
27991.72
327.29
20.89

BJ120_04
27580.86
27834.25
28040.53
168.47
21.69

BJ120_05
28189.69
28500.43
28755.03
186.67
18.96

BJ120_06
27679.42
27933.79
28172.32
143.93
19.15

BJ120_07
26969.4
27668.76
28014.39
286.72
21.03

BJ120_08
26617.8
27268.75
27742.87
315.44
22.64

BJ120_09
27000.07
27382.75
27631.08
187.61
20.21

BJ120_10
27663.24
27887.53
28107.04
126.69
20.54

5. Conclusions

In this study, the GOVRP was introduced as a formulation considering the impact of fuel emissions for the third-party logistics. The problem was to construct open routes for vehicles to visit all customers with the vehicle capacity constraints. The objective was to minimize the total cost composing of the fuel emissions cost and the driver cost.

A hybrid tabu search algorithm was designed to deal with the GOVRP instances. Experiments were conducted on 60 instances derived from real geographical data of customers in Beijing. We analyzed the cost components with different objectives and compared open routes with closed routes in respects of reducing the total cost. A homogenous fleet was considered in this paper. Computational results showed that vehicle type influenced the total cost and it should be changed according to the instance’s size. In the future, a heterogeneous fleet of vehicles can be used to minimize the total cost. Other types of the OVRP, such as OVRPTW and MDOVRP, can be solved by our algorithm with some changes.

Nowadays, several novel computing techniques are used to deal with complex problems. Some of them are bioinspired models, such as membrane-inspired evolutionary algorithms [24–26] and probe machine [27]. Their nondeterministic distributed parallel frameworks have been proved to improve the performance of optimization algorithms [28]. Most of them can be used to solve real-life problems [29]. We hope that more competitive results for our GOVRP instances can be obtained by using those algorithms in the future.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 61502012, 61772290, and 61373066); the Beijing Natural Science Foundation (Grant no. 4164096); the Humanity and Social Science Youth Foundation of Ministry of Education of China (Grant no. 13YJC630010); the Science and Technology Development Strategy Research Program of Tianjin (Grant no. 16ZLZXZF00030); and the Collaborative Innovation Center for China Economy.

References

L. Schrage, “Formulation and structure of more complex/realistic routing and scheduling problems,” Networks, vol. 11, no. 2, pp. 229–232, 1981.

S. Raff, “Routing and scheduling of vehicles and crews: the state of the art,” Computers and Operations Research, vol. 10, no. 2, pp. 63–211, 1983.

J. Brandao, “A tabu search algorithm for the open vehicle routing problem,” European Journal of Operational Research, vol. 157, no. 3, pp. 552–564, 2004.

U. Derigs and K. Reuter, “A simple and efficient tabu search heuristic for solving the open vehicle routing problem,” Journal of the Operational Research Society, vol. 60, no. 12, pp. 1658–1669, 2009.

E. E. Zachariadis and C. T. Kiranoudis, “An open vehicle routing problem metaheuristic for examining wide solution neighborhoods,” Computers and Operations Research, vol. 37, no. 4, pp. 712–723, 2010.

S. A. Mirhassani and N. Abolghasemi, “A particle swarm optimization algorithm for open vehicle routing problem,” Expert Systems with Applications, vol. 38, no. 9, pp. 11547–11551, 2011.

X.-Y. Li, P. Tian, and S. C. H. Leung, “An ant colony optimization metaheuristic hybridized with tabu search for open vehicle routing problems,” Journal of the Operational Research Society, vol. 60, no. 7, pp. 1012–1025, 2009.

S. Yu, C. Ding, and K. Zhu, “A hybrid GA-TS algorithm for open vehicle routing optimization of coal mines material,” Expert Systems with Applications, vol. 38, no. 8, pp. 10568–10573, 2011.

P. P. Repoussis, C. D. Tarantilis, O. Braysy, and G. Ioannou, “A hybrid evolution strategy for the open vehicle routing problem,” Computers and Operations Research, vol. 37, no. 3, pp. 443–455, 2010.

R. Russell, W.-C. Chiang, and D. Zepeda, “Integrating multi-product production and distribution in newspaper logistics,” Computers & Operations Research, vol. 35, no. 5, pp. 1576–1588, 2008.

C. D. Tarantilis and C. T. Kiranoudis, “Distribution of fresh meat,” Journal of Food Engineering, vol. 51, no. 1, pp. 85–91, 2002.

R. Atefi, M. Salari, L. C. Coelho, and J. Renaud, “The open vehicle routing problem with decoupling points,” European Journal of Operational Research, vol. 265, no. 1, pp. 316–327, 2018.

C. Erbao, L. Mingyong, and Y. Hongming, “Open vehicle routing problem with demand uncertainty and its robust strategies,” Expert Systems with Applications, vol. 41, no. 7, pp. 3569–3575, 2014.

K. Fleszar, I. H. Osman, and K. S. Hindi, “A variable neighbourhood search algorithm for the open vehicle routing problem,” European Journal of Operational Research, vol. 195, no. 3, pp. 803–809, 2009.

E. Demir, T. Bektaş, and G. Laporte, “A comparative analysis of several vehicle emission models for road freight transportation,” Transportation Research Part D: Transport and Environment, vol. 16, no. 5, pp. 347–357, 2011.

T. Bektaş and G. Laporte, “The pollution-routing problem,” Transportation Research Part B: Methodological, vol. 45, no. 8, pp. 1232–1250, 2011.

Ç. Koç, T. Bektaş, O. Jabali, and G. Laporte, “The fleet size and mix pollution-routing problem,” Transportation Research Part B: Methodological, vol. 70, pp. 239–254, 2014.

E. Demir, T. Bektaş, and G. Laporte, “A review of recent research on green road freight transportation,” European Journal of Operational Research, vol. 237, no. 3, pp. 775–793, 2014.

M. Barth, T. Younglove, and G. Scora, “Development of a Heavy-duty Diesel Modal Emissions and Fuel Consumption Model,” Tech. Rep. UCB-ITSPRR-2005-1, California PATH Program, Institute of transportation Studies, University of California at Berkeley, California, Calif, USA, 2005.

M. Scora and G. Barth, “Comprehensive Modal Emission Model (CMEM), Version 3.01, User Guide,” Tech. Rep., 2006, http://www.cert.ucr.edu/cmem/docs/CMEM_User_Guide_v3.01d.pdf.

M. Barth and K. Boriboonsomsin, “Real-world carbon dioxide impacts of traffic congestion,” Transportation Research Record, no. 2058, pp. 163–171, 2008.

Man, Trucks in Distribution Transport, http://www.mantruckandbus.co.uk/en/trucks/start_trucks.html.

E. Demir, T. Bektas, and G. Laporte, “An adaptive large neighborhood search heuristic for the Pollution-Routing PROblem,” European Journal of Operational Research, vol. 223, no. 2, pp. 346–359, 2012.

G. Zhang, H. Rong, F. Neri, and M. J. Pérez-Jiménez, “An optimization spiking neural P system for approximately solving combinatorial optimization problems,” International Journal of Neural Systems, vol. 24, no. 5, pp. 1–16, 2014.

G. Zhang, H. Rong, J. Cheng, and Y. Qin, “A population-membrane-system-inspired evolutionary algorithm for distribution network reconfiguration,” Chinese Journal of Electronics, vol. 23, no. 3, pp. 437–441, 2014.

X. Zhang, J. Li, and L. Zhang, “A multi-objective membrane algorithm guided by the skin membrane,” Natural Computing, vol. 15, no. 4, pp. 597–610, 2016.

J. Xu, “Probe machine,” IEEE Transactions on Neural Networks and Learning Systems, vol. 27, no. 7, pp. 1405–1416, 2016.

G. Zhang, J. Cheng, M. Gheorghe, and Q. Meng, “A hybrid approach based on differential evolution and tissue membrane systems for solving constrained manufacturing parameter optimization problems,” Applied Soft Computing, vol. 13, no. 3, pp. 1528–1542, 2013.

G. Zhang, M. J. Prez-Jimnez, and M. Gheorghe, Real-Life Applications with Membrane Computing, Springer, Berlin, Germany, 2017.