Journal of Industrial Engineering
Volume 2017 (2017), Article ID 3019523, 13 pages
https://doi.org/10.1155/2017/3019523

Research Article

Machine Learning-Based Parameter Tuned Genetic Algorithm for Energy Minimizing Vehicle Routing Problem

P. L. N. U. Cooray and Thashika D. Rupasinghe

Department of Industrial Management, University of Kelaniya, Kelaniya, Sri Lanka

Correspondence should be addressed to Thashika D. Rupasinghe; thashika@kln.ac.lk

Received 6 September 2016; Revised 17 December 2016; Accepted 21 December 2016; Published 18 January 2017

Academic Editor: Shu-Chu Liu

Copyright © 2017 P. L. N. U. Cooray and Thashika D. Rupasinghe. 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.

During the last decade, tremendous focus has been given to sustainable logistics practices to overcome environmental concerns of business practices. Since transportation is a prominent area of logistics, a new area of literature known as Green Transportation and Green Vehicle Routing has emerged. Vehicle Routing Problem (VRP) has been a very active area of the literature with contribution from many researchers over the last three decades. With the computational constraints of solving VRP which is NP-hard, metaheuristics have been applied successfully to solve VRPs in the recent past. This is a threefold study. First, it critically reviews the current literature on EMVRP and the use of metaheuristics as a solution approach. Second, the study implements a genetic algorithm (GA) to solve the EMVRP formulation using the benchmark instances listed on the repository of CVRPLib. Finally, the GA developed in Phase 2 was enhanced through machine learning techniques to tune its parameters. The study reveals that, by identifying the underlying characteristics of data, a particular GA can be tuned significantly to outperform any generic GA with competitive computational times. The scrutiny identifies several knowledge gaps where new methodologies can be developed to solve the EMVRPs and develops propositions for future research.

1. Introduction

Background of Research. Vehicle Routing Problem (VRP) can be described as the problem of finding optimal routes for delivery or collection from one to many depots to many customers who are geographically distributed. This problem has been the core for many operations research problems and has many variations. Later, with the focus on sustainable business practices, a novel category of VRP has emerged, known as Green VRP. In this category, the objectives are different from original VRP where it minimizes the travelled distance ultimately, thus reducing the cost. However, GVRP attempts to minimize the impact on environment of routing using different approaches [1]. Energy Minimizing VRP (EMVRP) has been developed to minimize the energy consumption of a fleet while serving all the customers. It has been identified that energy consumption has a direct impact on carbon dioxide emission.

Metaheuristics are used for combinatorial optimization in which an optimal solution is sought over a discrete search space. VRP is one of the most known NP-hard problems; thus, metaheuristics have been widely used to find near-optimal solutions for VRP problems [2].

Machine learning is a subfield of computer science that evolved from the study of pattern recognition and computational learning theory in artificial intelligence. Machine learning explores the construction and study of algorithms that can learn from and make predictions on data. In this research, the intention is to use machine learning to tune the developed metaheuristics to solve the formulated routing problem [3]. The overall objective of this study is to develop a set of genetic algorithms to solve the EMVRP and apply machine learning techniques to tune the developed algorithms to enhance the quality of the solutions. Energy Minimizing VRP has been formulated by Kara et al. [1] with the objective of minimizing energy consumption while serving a distributed set of customers. Though it minimizes the energy consumption, the calculated reduction of carbon dioxide emission has not been calculated; thus, the environmental sustainability of EMVRP has not been stated in the original paper. The study has used CPLEX® for solving moderate sized problems and the computational times were not promising. It is therefore necessary to develop efficient solution procedures for the EMVRP and in this study the authors implement genetic algorithms (GAs) for the genetic EMVRP formulation. The aforementioned GA was enhanced through machine learning techniques to tune its parameters based on the underlying characteristics of the data being used. In the book Tuning Metaheuristics by Birattari [3], it has been mentioned that tuning is crucial to metaheuristic optimization both in academic viewpoint and for practical applications. Nevertheless, relatively less research has been devoted to the issue and shows that the problem of tuning a metaheuristic can be described and solved as a machine learning problem. To the authors’ knowledge, this will be the first instance of machine learning being used as a mechanism to tune the domain of VRP.

2. Literature Review

2.1. Vehicle Routing Problem

Vehicle Routing Problem (VRP) can be described as the problem of finding optimal routes for delivery or collection from one to many depots to many customers who are geographically distributed. VRP is at the core of a huge number of practical applications in the area of transportation. It is one of the most studied classes of problems in operations research (OR) according to Gendreau et al. [4]. The list of variants according to a recent survey by Lin et al. [5] can be classified as Capacitated VRP (CVRP), Time Dependent VRP (TDVRP), Pickup and Delivery Problem (PDP), Multidepot VRP (MDVRP), Stochastic VRP (SVRP), Location Routing Problem (LRP), Periodic VRP (PVRP), Dynamic VRP (DVRP), Inventory Routing Problem (IRP), Fleet Size and Mix Vehicle Routing Problem (FSMVRP), Generalized VRP, Multicompartment VRP (MCVRP), Site Dependent VRP, Split Delivery VRP (SDVRP), Fuzzy VRP, Open VRP (OVRP), VRP with Loading Constraints (VRPLC), and Multiechelon VRP (MEVRP).

2.2. Green VRP

Green Logistics has recently received a lot of interest from governments and business organizations. The main reason behind this is that the current logistics practices are not sustainable in the long term and sometimes in the literature there are instances where the improved performance in terms of economic performance has caused a decrease in environmental performance. A classic example can be found in the study done by Yazan et al. [6] where they reengineered the supply chain, but, however, their reengineering outcomes seem to have been unsuccessful with regard to CO₂ emission, since CO₂ emission has been increased after the study, and this shows reengineering failure in environmental sense, although the economic performance has been increased by reengineering the system. With these reasons, the focus is automatically given to sustainable transportation practice which is generally known as Green Transportation. Björklund [7] defines “Green Transportation” as “transportation service that has a lesser or reduced negative impact on health and natural environment when compared with competing transportation services that serve the same purpose.” Under this category, a subcategory has been defined as “Green Vehicle Routing Problems” and they are characterized by the objective of harmonizing the environmental and economic costs by implementing effective routes to meet the environmental concerns and financial indexes [5]. The importance of Green VRP is characterized mainly by CO₂ emission by vehicles since they are one of the prime consumers of petroleum products. The sector contributes to 15% of overall greenhouse gas (GHG) emissions and 23% of overall CO₂ emissions, which is the highest found GHG [8]. The different variants of Green VRP have been introduced to the literature which have various objectives to support sustainable VRP. Those can be listed as Pollution Routing Problem (PRP), Green Vehicle Routing Problem (GVRP), and VRP in Reverse Logistics (VRPRL) [5].

2.3. Energy Minimizing Vehicle Routing Problem

Under the GVRP aimed problems, one problem introduced in 2007 by Kara et al. [1] is Energy Minimizing VRP (EMVRP). EMVRP has been identified in the survey by Lin et al. [5] under the category of GVRP. The objective of EMVRP is to reduce the energy consumption and energy has been justified to be the product of distance travelled by the load of the vehicle. Concretely, EMVRP aims at minimizing the sum of the product of load and distance for each arc. However, in the original paper, the authors have not justified the CO₂ emission reduction of EMVRP. Furthermore, the problem has been solved using CPLEX 8.0 for small sized instances and it is mentioned that the CPU times over moderate sized problems have not been optimized. Thus, this raises the need for developing efficient heuristics for solving moderate and large instances.

EMVRP is mathematically formulated first by Kara at el. [1] as in the following:

 Objective function is

 Constraints are as follows:

The cost of traversing an arc () is the product of the distance between the nodes and and weight on this arc. Constraints (2) and (3) ensure that vehicles are used. Constraints (4) and (5) are the degree constraints for each node. Constraint (6) is the classical conservation of flow equation balancing inflow and outflow of each node, which also prohibits any illegal subtours. Constraint (7) initializes the flow on the first arc of each route; the cost structure of the problem necessitates such initialization. Capacity restrictions are considered and force to zero when the arc () is not on any route. Then, the constraint produces lower bounds for the flow on any arc.

2.4. Metaheuristics for Solving VRP

Metaheuristics are categories of heuristics used for solving optimization problems, mostly NP-hard problems, which cannot be optimally solved within feasible time. Lenstra and Kan [9] have shown that all the Vehicle Routing Problems are NP-hard and cannot be solved in polynomial time. In the literature, there are many instances where metaheuristics have been applied to solve VRP. Among them, the most applied path-based and population-based metaheuristics can be selected using the categorized bibliography done by Gendreau et al. [4].

2.4.1. Genetic Algorithm (GA)

Genetic algorithms (GAs) are adaptive methods which may be used to solve a wide variety of optimization problems. They are based on the genetic processes of biological organisms. Over many generations, natural populations evolve according to the principles of natural selection and “survival of the fittest.” By mimicking this process, genetic algorithms are able to “evolve” solutions to real world problems, if they have been suitably encoded. The basic principles of GAs were first laid down rigorously by Holland [10]. GA is more applicable for solving an optimization problem where the optimum result is derived from a large random dataset like the scenario in the research problem.

2.5. Machine Learning for Metaheuristics

Machine learning is an area of computer science that evolved from the study of pattern recognition and computational learning theory in artificial intelligence. It explores the construction and study of algorithms that can learn from and make predictions on data. When metaheuristics are developed, the algorithms can be tuned for better performance (Birattari [3]). Each metaheuristic has already set parameters that have to be initialized before the execution of the algorithm. The metaheuristics adaptation requires the adjustment of these parameters according to the problem at hand. This is known as parameter tuning. An appropriate initial parameter setting has a noteworthy impact on the solving progress, such as the exploitation or exploration rate of the search space, and therefore the quality of the solution [11]. Tuning metaheuristics using machine learning techniques is a novel approach and there is not much literature on the area.

3. Methodology

3.1. Research Approach

In the first phase of the study, Green VRP methodologies will be systematically reviewed to identify important characteristics of problem solving methodologies [12]. In the second phase of the study, GA-based metaheuristics are developed to solve the routes which minimize the energy consumption of a vehicle fleet while serving a set of customers. In the third phase, the applicability of machine learning techniques for parameter tuning of metaheuristics will be tested (Figure 1).

Figure 1: Flow diagram of the research study.

Phase 1: Systematic Review of the Literature. During this phase, a thorough review of the literature on the Green VRP problem formulations and solving methodologies was carried out. The knowledge gathered from the systematic review is intended to be used for identifying and justifying the metaheuristics to be used in solving the EMVRP.

Phase 2: Development of the GA. With the knowledge gathered from Phase 1, a genetic algorithm will be developed to solve the EMVRP. Design of experiment (DOE) will be carried out to identify the parameters and the levels for each parameter to reduce the bias. The data for the algorithms will be obtained from CVRPLib [13], an online benchmark repository of VRP instances. CVRPLib contains data that is prominently used in VRP literature with number of cities or customers that request goods to be delivered to them and the location and the size of the demand.

Phase 3: Development of the Parameter Tuned GA Using Machine Learning Techniques. The GA’s parameters will be tuned using clustering techniques based on the inherent characteristics of the data being used. The tuned GA will be compared with results from untuned GA from Phase 2. Recommendations will be drawn for using machine learning as a mechanism for parameter tuning within the context of VRP.

3.2. Data Collection

The data required for the execution of algorithms will be acquired from CVRPLib: a Vehicle Routing Problem library available online at http://vrp.atd-lab.inf.puc-rio.br/index.php/en/. This is a prominently used library used in many research studies found in the literature. These data files include collection of cities to be served with and coordinates, required demand for each city, and the capacity of the vehicle.

3.3. Parameter Tuning Using Machine Learning

Tuning is crucial for metaheuristic optimization from both academic and practitioners’ standpoint. Nevertheless, relatively less research has been devoted to the issue. Using the machine learning perspective, it is possible to give a formal definition to tuning of an algorithm.

4. Implementation

4.1. Problem Definition

In this section, the problem formulation of EMVRP has been described. The idea of EMVRP is stated in the study by Kara et al. [1]. Some important assumptions of this problem formulation are as follows:

(1)A vehicle can only carry a load which is less than its capacity.(2)When loading a vehicle, the cumulative demand required by the cities that the particular vehicle will serve needs to be considered. This is important because if not, a vehicle might bring back the left carriage in its return trip back to the depot.(3)There is only one depot.(4)The demands and capacities are measured using the same measurement.(5)The weight of the vehicle is not considered since it will be the same over the fleet.

Figure 2 illustrates the above statements in a diagram.

Figure 2: Overview of a trivial problem instance.

A, B, C, and D are cities that need to be served with demands of 100, 100, 50, and 100, respectively. The maximum capacity of the vehicle is 200. In the first trip (or first vehicle) which will serve A and B, the vehicle will be loaded with 200 considering demands of A and B and after serving it will return back to the depot. And in the second trip (or second vehicle), it will be loaded with only 150 considering that it has to serve C and B where cumulative demand is 150 and after serving it will return back to the depot.

4.2. The Proposed GA

The implementation of the GA uses a fully object oriented design as there are three models: city, tour, and population. The implementation of GA is shown in Algorithm 1.

Input

Read list of cities to be served with demands from CVRP Lib file

Read vehicle capacity from text file

Population size ()

Number of generations ()

//Initialization

Create a tour with a random order of cities

Do this for times to create a population

Get the best tour from the population

Save as the elite

initial energy = energy consumption of the fittest tour in first population

//Genetic algorithm

//Run for    times

Loop1

//Run for times

Loop2

                                //Tournament selection

                                Select a random set of tours from the population

                                Get the fittest and return

                                //Crossover

                                Parent1 = tournament selection ()

                                Parent2 = tournament selection ()

                                child = Crossover (Parent1, Parent2)

                                //Mutation

                                Swap random two cities in the child

                 endLoop2

                //New population is created

                Get the fittest

                Replace previous elite if fittest is better than elite

endLoop1

Get the elite

Final energy = energy consumption of elite

Reduction of energy = (initial energy − final energy)/initial energy 100%

Print “Reduction of energy”

Algorithm 1: Overview of basic genetic algorithm developed for the EMVRP.

4.3. Parameter Tuning of Metaheuristics Using Machine Learning

GAs need to be parameter tuned for better performance [3]. The main parameters considered in this study are mutation rate, size of the population, and number of generations. Larger sizes of the population and number of generations are deemed to yield better results as this will increase the scope of the search paradigm. However, the optimum setting for the rate of mutation cannot be identified without a scientific approach. In this study, -means clustering algorithm from the machine learning literature has been used as a mechanism to tune the developed metaheuristics. -means is one of the widely used unsupervised learning algorithms which solves the well-known clustering problem. For this purpose, a customized -means clustering algorithm has been developed to read the data files from the CVRPLib and cluster them using the total accumulated demand of the cities and total number of cities to be served. Then, different mutation rates have been applied for different clusters of the data files and the energy minimization percentages are calculated. The hypothesis is as follows: different mutation rates work better on different clusters of CVRPLib. Detailed description of this experiment is elaborated in the next section along with the DOE.

5. Results

As mentioned in a previous section, the data for the experiment is taken from CVRPLib. In particular, the dataset used is from Uchoa et al. [13]. This repository contains 100 data files with the number of cities and various demands required by each city. All the experiments are carried out using a single computer with the following features and configurations.

Computational Specification of the Computer

 Processor: Intel® Core™ i5-4200 CPU @ 2.50 GHz Installed memory (RAM): 4.00 GB System type: 64-bit operating system, x64-based processor Pen and touch: No pen or touch input available for this display

To analyze the performance of GA, the experiment has been designed for the selected 100 data files (Table 1).

Table 1: Design of experiment (DOE) for the genetic algorithm.

Selected parameter
Values

Mutation rates
0.0010, 0.0025, 0.0050, 0.0075, 0.0100

Size of the population
50

Size of the generations
100 and 1000

Replications
5 per one mutation rate, population size, and number of generations

Note that these mutation rates, size of the population, and number of generations are the most widely used values found in the literature. The outcome of GA is energy reduction percentage compared to initial energy consumption of the tour. This is calculated as

5.1. Analysis of Results of the Developed GA for EMVRP

As depicted in Table 2 using any mutation rate with number of generations = 100, the energy reduction percentage is greater than 20%. The best energy reduction is achieved when the mutation rate is at 0.0025. The average computational time taken for all the 100 files is shown in Table 2.

Table 2: Results of the genetic algorithm.

Selected parameter
Value for the parameter of the genetic algorithm

Size of the population
50

Size of the generations
100
1000

Mutation rates
0.001
0.0025
0.005
0.0075
0.01
0.001
0.0025
0.0050
0.0075
0.01

Average energy reduction percentage (%)
26.64
26.70
24.54
22.54
21.42

50.04
42.95
37.60
34.04

Average computational time (100 files)
1044.6 seconds

energy reduction percentage.

As depicted in Table 2 using any mutation rate with number of generations of 1000, the energy reduction percentage is greater than 30%. The best energy reduction is achieved when the mutation rate is at 0.001 which is very promising being more than 50%. The computational time for all the 100 files is less than seven minutes. Therefore, it is around 4 seconds per file which is very much promising. To identify whether the differences between the averages with the changes in the mutation rates were significant, statistical analyses were carried out using the Freedman test.

Hypotheses

 Null Hypothesis . Means of minimized energy percentage for different mutation rates are the same Alternative Hypothesis . Means of minimized energy percentage for different mutation rates are not the same.

The value = 0.000 <0.05 and we reject the null hypothesis and post hoc analysis has been carried out on the detailed dataset (Table 3) to determine which mutation rate has the highest mean percentage of reduction in energy.

Table 3: Detailed results of the GA.

CVRPLib file name
Rate of mutation

0.0010
0.0025
0.0050
0.0075
0.0100

X-n1001-k43.vrp
12.80103
11.88827
10.85936
9.694029
9.556848

X-n101-k25.vrp
32.02678
35.75923
37.56841
40.00939
35.55532

X-n106-k14.vrp
30.27717
33.81137
34.26686
35.76881
33.72598

X-n110-k13.vrp
33.95242
40.20993
42.49106
39.00524
36.37393

X-n115-k10.vrp
50.81193
52.37917
55.09787
54.92929
53.33978

X-n120-k6.vrp
42.11564
45.09899
43.3998
44.35186
42.59954

X-n125-k30.vrp
26.52183
29.63255
31.01679
27.39243
27.4007

X-n129-k18.vrp
31.66396
33.715
34.8625
32.79096
32.22348

X-n134-k13.vrp
37.75922
42.5249
42.52067
42.57679
40.28671

X-n139-k10.vrp
37.36237
38.82001
38.74307
36.62869
35.8474

X-n143-k7.vrp
37.54207
43.38776
41.28395
37.84237
37.53593

X-n148-k46.vrp
31.93441
33.53385
34.04095
32.45357
30.09326

X-n153-k22.vrp
49.45064
49.81333
54.32997
52.78865
45.06589

X-n157-k13.vrp
42.86481
46.98111
45.72171
42.56459
42.47363

X-n162-k11.vrp
34.27922
38.17874
38.77056
37.83853
35.89787

X-n167-k10.vrp
31.62204
35.85662
37.1133
33.45995
32.22809

X-n172-k51.vrp
28.99557
34.80488
36.56064
29.37898
27.77682

X-n176-k26.vrp
39.44419
41.23805
40.69566
39.15282
36.88177

X-n181-k23.vrp
30.22022
33.84771
32.57614
30.78921
27.88417

X-n186-k15.vrp
34.75806
35.94411
33.87898
29.67616
30.61732

X-n190-k8.vrp
34.53248
36.59408
36.81079
36.82402
32.81114

X-n195-k51.vrp
31.78228
32.85653
29.30497
26.42445
22.69677

X-n200-k36.vrp
27.0885
28.60889
25.96263
24.49728
23.69031

X-n204-k19.vrp
31.70243
33.92431
30.44077
31.17991
27.19798

X-n209-k16.vrp
31.60467
32.05206
29.85411
27.84361
24.32678

X-n214-k11.vrp
32.55193
35.04465
33.91079
29.86931
64.77444

X-n219-k73.vrp
20.10571
22.6770
20.8841
19.1462
17.90529

X-n223-k34.vrp
30.40860
30.14028
29.89211
25.98539
20.03798

X-n228-k23.vrp
45.04471
45.64562
45.38162
40.09223
35.30641

X-n233-k16.vrp
31.90803
33.2655
33.22402
25.19938
25.45072

X-n237-k14.vrp
32.55161
32.68141
31.20284
30.6823
28.61488

X-n242-k48.vrp
23.91807
26.61617
20.90652
19.31902
14.68955

X-n247-k47.vrp
35.5378
38.53135
36.33797
30.11682
29.20752

X-n251-k28.vrp
30.01463
28.47438
24.65875
24.20706
22.11476

X-n256-k16.vrp
33.25466
34.19156
34.25381
28.92623
25.92442

X-n261-k13.vrp
29.70967
30.71204
28.49667
24.99286
23.82931

X-n266-k58.vrp
23.88774
26.97132
22.6137
17.39035
16.23727

X-n270-k35.vrp
27.88478
28.69578
24.80018
22.34739
20.61391

X-n275-k28.vrp
31.94181
32.22524
31.55025
29.16687
26.01753

X-n280-k17.vrp
36.40002
40.28805
37.492
31.2107
32.18607

X-n284-k15.vrp
29.24362
29.22297
25.84051
24.31867
22.31105

X-n289-k60.vrp
26.03244
24.4524
21.37549
17.71002
16.30786

X-n294-k50.vrp
27.29334
26.12614
22.5941
18.20006
16.88417

X-n298-k31.vrp
27.38218
25.41562
22.39082
18.86125
19.93721

X-n303-k21.vrp
27.91977
31.20337
25.23259
24.35885
20.68154

X-n308-k13.vrp
33.39066
36.73234
35.50009
32.08833
31.20307

X-n313-k71.vrp
24.28404
23.18796
19.81391
18.23606
16.86009

X-n317-k53.vrp
32.08766
31.86606
28.42572
25.00814
25.18776

X-n322-k28.vrp
25.12437
27.05396
23.15435
21.11874
17.37640

X-n327-k20.vrp
26.26653
26.12335
22.98445
21.70285
20.14563

X-n331-k15.vrp
29.44784
28.36366
27.57544
25.02837
22.02214

X-n336-k84.vrp
22.6318
22.90019
18.02802
16.15295
14.52159

X-n344-k43.vrp
24.32391
25.13196
22.02908
21.00029
17.65428

X-n351-k40.vrp
27.68809
26.01819
23.83853
22.73273
18.48367

X-n359-k29.vrp
21.84092
21.23801
17.35729
15.69808
13.77700

X-n367-k17.vrp
38.19213
33.30166
31.86441
27.12176
24.79421

X-n376-k94.vrp
20.09361
20.08725
17.43900
14.55602
15.48883

X-n384-k52.vrp
23.58085
23.29087
18.4361
17.41201
14.50117

X-n393-k38.vrp
25.30698
22.72104
19.75677
17.32702
16.88916

X-n401-k29.vrp
23.39231
21.57439
18.24951
18.13344
16.87411

X-n411-k19.vrp
39.58560
37.58242
31.96097
28.89972
25.96533

X-n420-k130.vrp
22.02357
21.37115
17.82149
15.26686
13.76679

X-n429-k61.vrp
22.76805
19.75252
15.07036
12.97871
14.51224

X-n439-k37.vrp
23.90484
25.4478
22.03636
19.64785
19.02882

X-n449-k29.vrp
22.58562
18.94513
16.51374
14.83471
13.86315

X-n459-k26.vrp
25.01886
25.35202
21.53826
19.56277
19.13728

X-n469-k138.vrp
15.98851
14.69173
13.00686
10.68891
10.80067

X-n480-k70.vrp
20.17462
18.28826
13.21342
13.50108
12.19322

X-n491-k59.vrp
22.35715
17.40456
16.12807
13.92029
12.32098

X-n502-k39.vrp
29.43752
28.93300
25.58213
23.71366
22.40697

X-n513-k21.vrp
20.66056
22.19829
17.66711
17.38231
15.59318

X-n524-k137.vrp
29.25552
25.00689
21.94620
18.42015
16.28576

X-n536-k96.vrp
22.00046
21.26785
15.21787
14.84602
13.92225

X-n548-k50.vrp
20.21664
19.10249
16.73358
17.28269
15.87240

X-n561-k42.vrp
19.32241
17.75355
14.06087
12.41879
12.15678

X-n573-k30.vrp
24.01248
23.39799
19.72642
18.85806
16.65078

X-n586-k159.vrp
17.84853
14.28692
13.70556
10.91328
10.95873

X-n599-k92.vrp
17.81388
15.09005
12.12366
11.29734
10.65991

X-n613-k62.vrp
19.58326
15.61245
13.32296
11.98591
11.20262

X-n627-k43.vrp
21.24216
18.5817
15.72306
13.56177
15.04474

X-n641-k35.vrp
17.08436
16.39485
13.36082
13.64283
11.30660

X-n655-k131.vrp
15.48288
14.88927
12.54741
12.25737
11.85345

X-n670-k126.vrp
28.71788
26.85190
20.31016
19.55446
17.42389

X-n685-k75.vrp
18.45748
15.06038
12.07450
10.94300
11.51439

X-n701-k44.vrp
16.57985
12.76328
11.40417
9.379679
8.427837

X-n716-k35.vrp
20.86056
19.27732
16.90755
14.85372
15.44707

X-n733-k159.vrp
17.96192
15.17854
12.20108
10.38958
8.888177

X-n749-k98.vrp
15.74652
14.80900
10.89733
11.00262
9.564129

X-n766-k71.vrp
27.54911
24.8529
19.96212
16.19431
13.99631

X-n783-k48.vrp
16.50255
11.88705
10.22504
12.07379
10.41174

X-n801-k40.vrp
17.87667
16.77141
14.99016
13.30332
13.53483

X-n819-k171.vrp
14.42093
13.74995
11.63282
11.58539
11.08715

X-n837-k142.vrp
13.10938
10.77977
9.634830
7.912295
9.019517

X-n856-k95.vrp
16.6359
15.19706
13.46041
13.24301
12.79089

X-n876-k59.vrp
16.99766
12.49846
13.28471
10.75942
11.86574

X-n895-k37.vrp
15.00296
14.3762
13.11793
11.61722
10.35098

X-n916-k207.vrp
11.61902
10.47219
8.532556
7.726192
7.366569

X-n936-k151.vrp
25.03095
24.13244
17.19300
14.26762
14.49740

X-n957-k87.vrp
14.77510
14.11095
12.44374
12.33696
11.75502

X-n979-k58.vrp
14.69893
12.25920
11.47749
9.918681
9.964106

According to the box and whisker plot, mutation rate of 0.0025 is showing the highest mean value; hence, it can be concluded with 95% confidence interval that 0.0025 will generate better energy reductions for EMVRP when using population size of 50 and generations size of 100 (Figure 3).

Figure 3: Box and whisker plots of untuned GA.

5.2. Analysis of Performance of Parameter Tuned GA

As mentioned in the previous section, before applying the developed genetic algorithm, data on the CVRPLib has been clustered using -means clustering algorithm into three clusters (Cluster 0, Cluster 1, and Cluster 2). The idea of clustering is to check the impact of characteristics of data on parameter tuning of the GA. The parameter considered for tuning is the rate of mutation. As for the experimental study, the 100 files used in the earlier experiment have been used. The clusters are created using number of cities to be served and total accumulated demand of a particular tour. To analyze the performance of GA on different clusters of data, the design of experiment has been utilized (Table 4).

Table 4: Design of experiment of tuned GA with CVRPLib data clusters.

Selected parameter
Values

Number of data files
100

Mutation rates
0.0010, 0.0025, 0.0050, 0.0075, 0.0100

Replications
5 per one mutation rate

Size of the population
50

Size of the generations
100

The rationale behind the experiment in Table 4 is to assess whether there is a significant difference among the reductions of energy percentages when the mutation rate is changed.

5.3. Results of CVRPLib Data Cluster 0

 Null Hypothesis . Means of minimized energy percentage for different mutation rates are the same. Alternative Hypothesis . Means of minimized energy percentage for different mutation rates are not the same.

The value = 0.000 <0.05 and we reject the null hypothesis and post hoc analysis has been carried out on the detailed dataset (Table 5) to determine which mutation rate yields the highest percentage in reduction of energy.

Table 5: Average percentage of reduction of energy for Data Cluster 0.

CVRPLib file name
Rate of mutation

0.0001
0.0025
0.005
0.0075
0.01

(X-n101-k25,5147.0,101.0)
30.2795
36.2461
37.7101
37.9236
35.2168

(X-n106-k14,13011.0,106.0)
32.0238
32.833
36.2804
35.2223
30.4027

(X-n110-k13,13827.0,110.0)
33.0007
37.1898
41.2574
38.6756
40.5764

(X-n115-k10,15362.0,115.0)
49.5449
50.5952
55.6424
54.8956
51.0182

(X-n120-k6,15481.0,120.0)
41.4965
43.7106
45.6016
45.7964
41.0161

(X-n125-k30,21017.0,125.0)
24.5716
28.4982
30.0036
28.8937
26.8316

(X-n129-k18,21682.0,129.0)
31.0658
33.7182
34.406
33.0458
32.7406

(X-n134-k13,29902.0,134.0)
40.4161
44.4603
43.73
43.9817
41.7856

(X-n139-k10,30941.0,139.0)
37.7481
38.4019
41.7096
38.2775
35.7362

(X-n143-k7,38416.0,143.0)
36.1435
42.5054
39.74
39.2316
36.9879

(X-n148-k46,39233.0,148.0)
28.6224
35.0103
33.176
32.0835
26.2372

(X-n153-k22,42301.0,153.0)
47.0119
49.6091
50.3011
50.222
46.9372

(X-n157-k13,42457.0,157.0)
44.5515
46.5625
46.7771
43.414
42.4649

(X-n162-k11,54651.0,162.0)
33.8149
38.9795
42.114
36.896
34.6667

(X-n167-k10,55887.0,167.0)
34.3191
39.1482
34.7953
33.867
30.1161

(X-n172-k51,63979.0,172.0)
30.9868
34.2055
31.901
29.9517
25.5072

(X-n176-k26,67611.0,176.0)
37.8435
43.655
40.7705
40.2538
38.4579

(X-n181-k23,67791.0,181.0)
32.5043
32.0691
34.2579
31.2722
29.6108

(X-n186-k15,81644.0,186.0)
32.5559
34.7171
34.9303
32.4279
28.6864

(X-n190-k8,82687.0,190.0)
35.493
35.8647
38.8699
35.8953
32.1755

(X-n195-k51,91895.0,195.0)
30.3231
32.635
32.6071
27.5018
25.8458

According to the box and whisker plot, mutation rate of 0.05 is showing the highest mean in the percentage reduction in energy. Hence, it can be concluded with 95% confidence interval that 0.05 generates better energy reductions for Cluster 0 (Figure 4).

Figure 4: Box and whisker plot for CVRPLib Data Cluster 0.

5.4. Results of CVRPLib Data Cluster 1

 Null Hypothesis . Means of minimized energy percentage for different mutation rates are the same. Alternative Hypothesis . Means of minimized energy percentage for different mutation rates are not the same.

The value = 0.000 <0.05 and we reject the null hypothesis and post hoc analysis has been carried out on the detailed dataset (Table 6) to determine which mutation rate has the highest mean.

Table 6: Average percentage of reduction of energy for Data Cluster 1.

CVRPLib file name
Rate of mutation

0.0001
0.0025
0.005
0.0075
0.01

(X-n200-k36,106158.0,200.0)
25.8922
28.244
25.4307
24.8176
21.5926

(X-n204-k19,121293.0,204.0)
31.7862
33.3268
31.0833
27.8423
28.1407

(X-n209-k16,122840.0,209.0)
29.6146
33.0459
27.9114
27.624
25.0044

(X-n214-k11,133196.0,214.0)
32.4836
32.4988
34.1668
28.6341
27.2371

(X-n219-k73,133414.0,219.0)
20.4221
21.3139
20.8364
18.7089
37.9443

(X-n223-k34,134648.0,223.0)
30.2742
33.6129
30.2504
23.4108
24.9498

(X-n228-k23,138126.0,228.0)
42.3674
48.8541
42.1286
38.2606
35.9466

(X-n233-k16,148219.0,233.0)
31.7538
35.2122
30.4283
26.9163
27.2217

(X-n237-k14,148455.0,237.0)
34.5302
33.6068
70.7867
32.2174
29.0436

(X-n242-k48,149779.0,242.0)
25.3818
26.5639
22.9681
18.7777
17.4704

(X-n247-k47,155979.0,247.0)
36.489
39.7815
35.6563
30.8481
28.9909

(X-n251-k28,157846.0,251.0)
27.3622
31.0231
26.4525
22.6824
20.6356

(X-n256-k16,177360.0,256.0)
32.9859
33.1934
33.5661
31.6681
25.8602

(X-n261-k13,190741.0,261.0)
31.9163
29.6783
28.0676
24.9275
21.9564

(X-n266-k58,192756.0,266.0)
24.7841
23.4658
21.208
17.4536
17.1201

(X-n270-k35,213175.0,270.0)
27.9878
28.8698
26.5334
22.7852
21.0692

(X-n275-k28,213449.0,275.0)
32.2499
31.4036
29.4528
29.5828
26.7106

(X-n280-k17,216682.0,280.0)
34.7643
38.5678
36.3531
34.8625
25.8963

(X-n284-k15,218209.0,284.0)
30.5936
27.6397
25.0705
25.1467
22.4253

(X-n289-k60,234183.0,289.0)
23.5243
24.7778
21.3323
16.7366
14.785

According to the box and whisker plot, mutation rate of 0.0025 is showing the highest median value; hence, it can be concluded with 95% confidence interval that 0.0025 will generate better energy reductions for Cluster 1 (Figure 5).

Figure 5: Box and whisker plot, Cluster 1.

5.5. Results of CVRPLib Data Cluster 2

 Null Hypothesis . Means of minimized energy percentage for different mutation rates are the same. Alternative Hypothesis . Means of minimized energy percentage for different mutation rates are not the same.

The value = 0.000 <0.05 and we reject the null hypothesis and post hoc analysis has been carried out on the detailed dataset (Table 7) to determine which mutation rate has the highest mean.

Table 7: Average percentage of reduction of energy for Data Cluster 2.

CVRPLib file name
Rate of mutation

0.0001
0.0025
0.005
0.0075
0.01

X-n1001-k43.vrp
12.827504
11.715805
9.88541
10.448598
8.5876013

X-n294-k50.vrp
25.533544
25.854934
19.708729
18.366514
16.064914

X-n298-k31.vrp
25.725572
25.218533
22.642264
18.163332
16.76971

X-n303-k21.vrp
27.279439
32.206786
26.958772
22.055703
22.331427

X-n308-k13.vrp
37.079786
37.304107
35.981044
30.050856
30.860439

X-n313-k71.vrp
24.308694
23.646315
19.792177
16.502028
15.505046

X-n317-k53.vrp
30.92691
31.23127
28.24586
25.473993
25.535702

X-n322-k28.vrp
25.964802
25.395741
22.888762
21.486427
19.010605

X-n327-k20.vrp
25.915225
28.059972
23.76868
21.030628
20.912205

X-n331-k15.vrp
27.524263
27.977593
25.065238
24.569392
22.226545

X-n336-k84.vrp
24.156591
20.834126
17.204415
13.497206
14.088004

X-n344-k43.vrp
26.146141
23.651449
20.511471
15.331793
15.746694

X-n351-k40.vrp
26.616442
27.407423
23.158878
18.557153
18.888495

X-n359-k29.vrp
24.174821
23.677236
16.568104
16.769655
16.189016

X-n367-k17.vrp
37.507069
36.93659
33.44967
28.866205
28.357283

X-n376-k94.vrp
19.829914
18.384511
16.634847
15.279178
14.862174

X-n384-k52.vrp
22.773069
21.093078
18.998565
15.645967
14.923587

X-n393-k38.vrp
24.073648
23.581917
18.655457
18.284506
13.701024

X-n401-k29.vrp
22.379471
22.160249
18.760097
16.242995
15.404726

X-n411-k19.vrp
40.168322
37.421562
31.271868
27.884874
30.411745

X-n420-k130.vrp
21.897389
21.619467
16.981261
17.189068
13.689977

X-n429-k61.vrp
20.845623
17.722107
18.414669
15.042775
13.250864

X-n439-k37.vrp
25.2498
24.724619
23.048158
20.482543
19.241802

X-n449-k29.vrp
22.699744
18.136894
16.015191
14.389112
14.01591

X-n459-k26.vrp
25.5812
25.288133
21.133864
19.248823
18.753649

X-n469-k138.vrp
16.436619
14.524521
11.857235
11.05651
10.132928

X-n480-k70.vrp
18.390041
17.002514
14.246622
14.659912
12.984294

X-n491-k59.vrp
23.596872
19.402038
15.007576
13.256561
14.380208

X-n502-k39.vrp
28.250292
28.665693
25.821766
23.507696
22.389436

X-n513-k21.vrp
22.116418
19.963564
18.940326
16.178493
14.949564

X-n524-k137.vrp
28.219495
26.851244
20.376538
17.192628
16.736019

X-n536-k96.vrp
23.001011
20.568751
16.76173
16.488653
14.923288

X-n548-k50.vrp
20.281627
19.318569
16.651976
14.723431
14.636057

X-n561-k42.vrp
19.084868
17.423921
14.60639
15.02745
12.550507

X-n573-k30.vrp
24.638549
21.655251
19.526443
17.813234
16.501781

X-n586-k159.vrp
17.464901
15.448383
12.99529
13.177791
12.257662

X-n599-k92.vrp
17.846692
15.155531
13.045469
12.320552
10.514622

X-n613-k62.vrp
17.128463
16.110833
13.362154
12.063181
11.614861

X-n627-k43.vrp
21.567104
17.753015
14.741696
14.799587
13.825805

X-n641-k35.vrp
17.920677
17.321866
14.078229
12.711317
12.318776

X-n655-k131.vrp
15.962562
13.796249
12.337187
11.534519
10.876943

X-n670-k126.vrp
27.925595
25.990998
19.859656
16.643033
14.84004

X-n685-k75.vrp
19.123097
15.352256
13.352629
11.422474
12.132536

X-n701-k44.vrp
14.817606
13.860182
10.16125
11.001818
9.2017285

X-n716-k35.vrp
22.93946
18.60758
16.68929
14.603138
16.022459

X-n733-k159.vrp
17.752462
14.108984
10.329737
12.149165
9.6394828

X-n749-k98.vrp
16.501435
12.987899
11.393273
10.471188
9.3638983

X-n766-k71.vrp
25.594234
25.868375
21.526236
18.644464
14.898724

X-n783-k48.vrp
16.315407
12.343775
11.520721
9.696421
10.501402

X-n801-k40.vrp
16.916469
15.886679
13.943067
13.577301
12.386437

X-n819-k171.vrp
14.896745
14.002887
12.156429
11.572758
10.604485

X-n837-k142.vrp
12.668758
10.664722
8.3282448
9.0021583
7.7984541

X-n856-k95.vrp
15.931693
16.107941
14.412979
12.662221
12.718295

X-n876-k59.vrp
17.512326
14.775116
10.77695
11.211426
11.920959

X-n895-k37.vrp
14.636802
12.045023
11.659649
12.339187
11.006958

X-n916-k207.vrp
11.556926
10.198883
8.571053
8.233212
7.1698279

X-n936-k151.vrp
27.588469
24.401292
17.427975
14.002693
14.131898

X-n957-k87.vrp
15.921267
14.435585
13.276304
12.420921
10.195919

X-n979-k58.vrp
15.763138
12.940315
10.150284
10.668937
9.7730485

According to the box and whisker plot, mutation rate of 0.001 is showing the highest mean value; hence, it can be concluded with 95% confidence interval that 0.001 will generate better energy reductions for Cluster 1 (Figure 6).

Figure 6: Box and whisker plot, Cluster 2.

From the results obtained for these three clusters, it has been significantly shown that mutation rates will work best for particular clusters shown in Table 8.

Table 8: Best mutation rates for data clusters.

Data cluster identified
Best mutation rate

Cluster 0
0.0500

Cluster 1
0.0025

Cluster 2
0.0010

These results depict the importance of identifying and using different parameter settings when designing GAs to solve EMVRP instances which have different characteristics. As it has been noted in first set of results obtained just using the GA, the best mutation rate has been 0.001 for any problem instance. However, when CVRPLib data was clustered into three distinctive clusters and extensive parameter tuning was carried out, the best mutation rates were different. Thus, it can be concluded that machine learning techniques such as clustering have improved the solution quality immensely compared with a generic untuned GA.

6. Conclusions

This study adds new knowledge to the VRP literature in several ways. Initially, the systematic review which was carried out on the problem solving methods of Green VRP can be seen as the first study to focus on that segment. Then, developing two classes of GAs to solve the EMVRP using the CVRPLib data repository can be seen as another. The EMVRP is the problem which looks at finding routes of vehicles which use the least amount of energy when serving a set of cities or customers. In EMVRP, the energy is generalized to be equal to a function of load and distance. As the first instance of using metaheuristics for EMVRP, untuned and tuned genetic algorithms have been developed and detailed analyses have been carried out through rigorous data analyses. The developed algorithms have been proven to generate competitive solutions within reasonable computational times as depicted in the analyses. Further, this study looked at the GA parameter tuning as a prominent design aspect in coming up with flexible and adaptive GAs which are highly dependent upon the inherent characteristics of problem instances. The authors have utilized different mutation rates to improve the performance of the GAs using a very novel approach of applying -means clustering, the machine learning technique. It has been proved that identifying different clusters within data has a significant impact on improving the performance of GA.

The following highlights the contributions made through this study:

(1)A systematic review of the literature on the Green VRP solving methods(2)First instance of solving EMVRP using genetic algorithms(3)First instance of the applicability of machine learning techniques for parameter tuning of metaheuristics

This study can be further worked on by relaxing the assumptions considered in EMVRP formulation with the use of heterogeneous vehicles, considering the actual weight of the vehicle and continuous loads. Furthermore, by developing a path-based metaheuristic such as Tabu search to solve EMVRP and comparing its performance against the population-based GAs, this study can be easily extended by developing a more realistic cost function to reduce the energy. Finally, researching the applicability of other machine learning-based mechanisms to augment metaheuristics would be another interesting and important research area for future research.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

References

I. Kara, B. Y. Kara, and M. Kadri Yetis, “Energy minimizing vehicle routing problem,” in Combinatorial Optimization and Applications, vol. 4616 of Lecture Notes in Computer Science, pp. 62–71, Springer, Berlin, Germany, 2007.

M. Gendreau, “Metaheuristics in vehicle routing,” in Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES '12), 2012.

M. Birattari, Tuning Metaheuristics: A Machine Learning Perspective, vol. 197, Springer, Berlin, Germany, 2009.

M. Gendreau, J. Y. Potvin, O. Bräumlaysy, G. Hasle, and A. Løkketangen, “Metaheuristics for the vehicle routing problem and its extensions: a categorized bibliography,” in The Vehicle Routing Problem: Latest Advances and New Challenges, pp. 143–169, Springer US, 2008.

C. Lin, K. L. Choy, G. T. S. Ho, S. H. Chung, and H. Y. Lam, “Survey of green vehicle routing problem: past and future trends,” Expert Systems with Applications, vol. 41, no. 4, pp. 1118–1138, 2014.

D. M. Yazan, A. M. Petruzzelli, and V. Albino, “Analyzing the environmental impact of transportation in reengineered supply chains: a case study of a leather upholstery company,” Transportation Research Part D: Transport and Environment, vol. 16, no. 4, pp. 335–340, 2011.

M. Björklund, “Influence from the business environment on environmental purchasing—drivers and hinders of purchasing green transportation services,” Journal of Purchasing and Supply Management, vol. 17, no. 1, pp. 11–22, 2011.

I. T. F. Leipzig, Reducing Transport Greenhouse Gas Emissions: Trends & Data, Background for the 2010 International Transport Forum, Berlin, Germany, 2010.

J. K. Lenstra and A. H. G. R. Kan, “Complexity of vehicle routing and scheduling problems,” Networks, vol. 11, no. 2, pp. 221–227, 1981.

J. H. Holland, Adaptation in Natural and Artificial Systems: An Introductoryanalysis with Applications to Biology, Control, and Artificial Intelligence, University of Michigan Press, Ann Arbor, Mich, USA, 1975.

F. Dobslaw, “A parameter tuning framework for metaheuristics based on design of experiments and artificial neural networks,” in Proceedings of the International Conference on Computer Mathematics and Natural Computing (WASET '10), 2010.

P. L. N. U. Cooray and T. D. Rupasinghe, “An analysis of methodologies for solving green vehicle routing problem: a systematic review of literature,” in Proceedings of the Conference on Research for Transport and Logistics Industry, Colombo, Sri Lanka, 2016.

E. Uchoa, D. Pecin, A. Pessoa, M. Poggi, A. Subramanian, and T. Vidal, “New benchmark instances for the capacitated vehicle routing problem,” Research Report Engenharia de Produção, Universidade Federal Fluminense, 2014.