This research considers the time-dependent vehicle routing problem (TDVRP). The time-dependent VRP does not assume constant speeds of the vehicles. The speeds of the vehicles vary during the various times of the day, based on the traffic conditions. During the periods of peak traffic hours, the vehicles travel at low speeds and during non-peak hours, the vehicles travel at higher speeds. A survey by TCI and IIM-C (2014) found that stoppage delay as percentage of journey time varied between five percent and 25 percent, and was very much dependent on the characteristics of routes. Costs of delay were also estimated and found not to affect margins by significant amounts. This study aims to overcome such problems arising out of traffic congestions that lead to unnecessary delays and hence, loss in customers and thereby valuable revenues to a company. This study suggests alternative routes to minimize travel times and travel distance, assuming a congestion in traffic situation. In this study, an efficient GA-based algorithm has been developed for the TDVRP, to minimize the total distance travelled, minimize the total number of vehicles utilized and also suggest alternative routes for congestion avoidance. This study will help to overcome and minimize the negative effects due to heavy traffic congestions and delays in customer service. The proposed algorithm has been shown to be superior to another existing algorithm in terms of the total distance travelled and also the number of vehicles utilized. Also the performance of the proposed algorithm is as good as the mathematical model for small size problems.
Vehicle travel times in the cities and urban areas vary due to various reasons and factors, like congestions in traffic, accidents, road repairs, movement of important personalities who need high security, and weather conditions. If these travel time variations are ignored during the process of developing route plans for pick-up and/or delivery vehicles, the development of route plans may be inefficient in terms of the vehicles travelling into congested urban traffic conditions. Due to these travel time variations, in some cases the vehicles waste valuable time in traffic jams and customers have to wait unreasonably long without having any reliable information about the actual arrival times of vehicles. In these circumstances, it becomes difficult to satisfy the time windows during which the demand nodes must be visited.
In addition, insertion of new demands for pick-up that arise after completion of route planning, in the planned vehicle routes in real time, may result in significant savings. Considering time-dependent travel times as well as information regarding demands that arise in real time in solving vehicle routing problems can reduce the costs of ignoring the changing environment.
The basic VRP relies on the assumption that travel times remain constant throughout the whole planning horizon. In reality, however, travel times may vary during the day. The variation in travel times may result from predictable events such as congestion during rush hours or from unpredictable events such as accidents (Ichoua et al. [
According to Ichoua et al. [
The vehicle routing problem with time windows (VRPTW) models many real- world optimization problems in logistics, supply chain management, where we need to have distribution and collection of materials in geographically distributed facilities, from suppliers and to customers. But in reality, the classical VRPTW model is often not adequate to model the real-world situation, because of the assumption of constant travel times between customer sites or locations. Time-varying factors, such as traffic conditions whether they have a significant impact on the actual travel time also need to be considered. According to Eglese et al. [
Hence, the total time taken for travel between two locations is dependent on the specific departure time. Hence to take these external influences into consideration, the VRPTW is extended and is studied as time-dependent vehicle routing problem with time windows (TDVRP). The driving time that changes with the time of the day, is suitably represented by a time-dependent function.
Time-dependent travel times (or travel speeds) are either simulated or derived from traffic monitoring systems. When analyzing the data from traffic monitoring systems, one can observe that most travel times follow a certain pattern during the day. This helps to predict future traffic situations. Especially travel time variations during rush hour congestion show a high predictability. This is proven by Eglese et al. [
The literature relating to the TDVRP is rather scarce when compared to other VRP variants. However, Ichoua et al. [
The first problem mentioned is the time-dependent shortest path problem which was introduced in the late 1950s (Ford and Fulkerson [
Most VRPs assume that the travel times between depots and customers are deterministic and constant (Kok et al., [
According to Ichoua et al. [
The first category refers to models based on simple travel time functions. In this case, multiplier factors are used to integrate time-dependency.
Other TDVRP models assume discrete travel times (Malandraki and Daskin [
Ahn and Shin [
all travel times, service times and waiting times. The authors developed a model based on a mixed integer linear programming (MILP) formulation. The heuristics are tested on 32 randomly generated problems which consist of 10 to 25 customers.
Hill and Benton [
Ichoua et al. [
Fleischmann et al. [
Haghani and Jung [
Woensel et al. [
Maden et al. [
Balseiro et al. [
Ehmke et al. [
In this section, the reviewed papers are summarised in the
In this section, the mathematical model which is developed in this research to solve the TDVRP and the genetic algorithm for the TDVRP are discussed in detail. Usinga flow-arc formulation [Desrochers et al. [
Article | Variant of TDVRP | Objective (Min.) | Solution method |
---|---|---|---|
Kuo et al. (2009) | TDVRP | Total route duration | Tabu search |
Kok et al. (2012) | TDVRP with hard TW | Total route duration | Dijkstra algorithm and restricted dynamic programming heuristic |
Balseiro et al. (2011) | TDVRP with hard TW | First number of routes, second total route duration | Ant colony algorithm with insertion heuristics |
Figliozzi (2012) | TDVRP with hard and soft TW | First number of routes, second total route duration | Iterative route construction and improvement heuristic |
Hagani & Jung (2005) | Dynamic vehicle routing problem with time-dependent travel times | Total distance travelled | Genetic Algorithm |
0 which is a function of the departure time from customer i. The set of available vehicles is denoted by K.
The cost per unit of route duration is denoted as Ct; the cost per unit distance travelled is denoted Cd. It is assumed that the problem is feasible, i.e. it is always feasible to serve any individual customer starting from the depot. The primary objective function for the TDVRP is the minimization of the number of routes or the number of vehicles utilized. The optimal number of routes is unknown. A secondary objective is the minimization of total time or distance travelled. There are two decision variables in this formulation; x i j k is a binary decision variable that indicates whether vehicle k travels between customers i and j. The real number decision variable y i k indicates service start time for customer i served by vehicle k.
The mathematical model of TDVRPTW is given as below.
min imize ∑ k ϵ K ∑ j ϵ C x 0 j k (1)
min imize C d ∑ k ϵ K ∑ ( i , j ) ϵ A d i j k x i j k + C t ∑ k ϵ K ∑ j ϵ C ( y n + 1 k − y 0 k ) x 0 j k , (2)
Subject to:
∑ i ϵ C q i ∑ j ϵ V x i j k ≤ q max , ∀ k ϵ K (3)
∑ i ϵ C ∑ j ϵ V x i j k = 1 , ∀ i ϵ C (4)
∑ i ϵ V x i l k − ∑ i ϵ V x i j k = 0 , ∀ i ϵ C , ∀ k ϵ K (5)
x i 0 k = 0 , x n + 1 , i k = 0 , ∀ i ϵ V , ∀ k ϵ K (6)
∑ j ϵ V x 0 j k = 1 , ∀ k ϵ K (7)
∑ j ϵ V x j , n + 1 k = 1 , ∀ k ϵ K (8)
e i ∑ j ϵ V x i j k ≤ y i k , ∀ i ϵ C , ∀ k ϵ K (9)
l i ∑ j ϵ V x i j k ≥ y i k , ∀ i ϵ C , ∀ k ϵ K (10)
x i j k ( y i k + g i + t i , j y i k + g i ) ≤ y j k , i ϵ C , ∀ ( i , j ) ϵ A , ∀ k ϵ K (11)
x i j k ϵ { 0 , 1 } , ∀ ( i , j ) ϵ A , ∀ k ϵ K (12)
The objectives of the TDVRPTW are defined by (1) and (2) respectively. The constraints are defined as follows: the total demand in a particular route should not exceed the vehicle capacity (3); all the customers must be served and each customer must be served exactly once (4); a vehicle that arrives at a customer should also depart from that customer (5); routes must start at the depot and end at the depot (6); each vehicle leaves from the depot and returns to the depot exactly once, (7) and (8) respectively; service times must satisfy time window start (9) and ending (10) times specified; service start time must allow for travel time between customers (11). Decision variables type is indicated in (12).
During real-time traffic situations, the assumption of constant speed will not hold good. Hence we need to find out alternate routes, to minimize the distance travelled and the number of vehicles utilized as well as to minimize the total time taken to cover the entire tour. The problem lies in finding an efficient GA-based meta-heuristic which is more efficient than existing algorithms. A genetic algorithm to solve the TDVRP is developed. The problem that has been considered is a pick-up or delivery vehicle routing problem to various customers or from the various supplier sites with time windows in which we consider multiple vehicles with different capacities, with real-time variations in travel times between the different nodes. To load and start the time dependent VRP variant in HeuristicLab, we first need a valid file to load the time-dependent VRP data from. We use the data from the Solomon (CVRPTW) instances and add the travel time matrices. The dimension of the travel time matrix must match the number of cities. For the travel times, random values between 0.5 and 1.0 were generated using random function in Excel. The performance of the genetic algorithm for TDVRP is evaluated by comparing its results with one more existing algorithm for the TDVRPTW developed by Hashimoto et al. [
A modified version of the Genetic Algorithm formulation developed for the VRPTW solution developed by Nandakumar and Panneerselvam [
The representation of the chromosome as a set of genes representing the nodes in a route is shown in
In the
A crossover operator is a major process of producing offspring from the current population. There are many methods for crossover operation according to different problems. In this paper, “Random Sequence Insertion-Based Crossover” method is used, which is explained in the next section.
The Vehicle Routing Problem with Time Windows is solved using the Genetic Algorithm (GA) with multi-chromosome representation. It is used for finding a (near) optimal solution to a variation of the TDVRPTW by setting up a GA to search for the shortest route (distance), taking into account additional constraints, and minimizing the number of vehicles used.
The steps of the proposed genetic algorithm (SNRPGA2) for the time dependent vehicle routing problem with time windows are presented below. A schematic view of the crossover operation used in this algorithm is shown in
The steps of the Random Sequence Insertion-based Crossover (RSIX) method are presented below.
Step 1: Two chromosomes from the chromosome pool are randomly chosen as parents.
Step 2: Generate two crossover points, which will lead to three chromosome segments in each chromosome as shown in
Step 3: Next, swapping operation of the middle crossover genes segment takes place, as in
Step 4: Next, taking into account the VRPTW constraints, with each customer (node) allowed to be visited only once, assuming that ∀ ( i , j ) , d0i < d0j + dji triangle property and accommodating time window constraint with possible waiting time, use the data from the Solomon (CVRPTW) and add the travel time matrices. For the travel times, random values between 0.5 and 1.0 have been generated. Then retain the gene segment that underwent crossover, and then remove the gene (node) with same value, in their parent chromosome, such as in
Step 5: This results in obtaining two new offspring having gene segments that underwent crossover and they are added into the next generation as shown in
・ The new offspring are tested for fitness values.
・ The smaller the “fitness-value”, the stronger road chromosome is obtained.
Mutation to ChromosomesAfter the crossover is carried out on each chromosome, mutation operation is carried out to improve the chromosome. During the mutation operation, two randomly chosen genes are selected and the mutation operator changes their value into other possible values. Mutation helps to prevent the genetic algorithm from converging to local optima. Other genetic algorithms parameters can also influence the GA efficiency. Crossover probability which is specified in the GA, determines the rate at which the crossover occurs. Mutation probability that is specified is used to find how often the mutation is applied on the offspring. If no mutation happens, the respective chromosome is replaced in the population without any change. The population size defines the number of individuals in the population. The results of application of the mutation operation on the offspring shown in the
According to Sivasankaran and Sahabudeen [
This algorithm determines the minimized total distance travelled and the number of vehicles utilized to serve all the nodes during the tour as per each chromosome/offspring. The items that are considered while evaluating the fitness
function value are as listed below.
1) Each vehicle starts at the depot or warehouse, travels to a set of customer nodes and ends at the depot.
2) Except for the depot, each customer node is visited exactly once by the vehicle.
3) This algorithm uses a special, multiple-chromosome (parents) genetic representation to code solutions into individual offspring, which offer better routes and solutions.
4) Special genetic operators like selection, crossover and mutation are used.
5) The number of vehicles used is minimized using this algorithm.
6) Additional constraints have to be satisfied.
Minimum number of vehicles to cover all the nodes.
Minimizing the maximum distance travelled by each vehicle.
7) Time windows are defined for each customer node (e.g. unloading/loading times) and time-dependent travel times are introduced, by using a travel time matrix.
8) The single route chromosome is then assigned to multiple sub chromosomes of smaller routes consisting of a set of customers from the original chromosome. Each route is assigned a vehicle to visit each customer node on the route to meet the customer’s demand in the route.
The variables used are the number of vehicles, the number of customers to be serviced by the vehicles, vehicle capacities, maximum distance travelled by each vehicle which is to be minimized. The objective is to minimize the number of vehicles used and the total distance travelled by the vehicles, in servicing the customers. The time of the day during which the vehicles are sent to the supplier sites for the pick-ups is also considered in this research.
The steps of the proposed genetic algorithm (SNRPGA2) for the time dependent vehicle routing problem with time windows are presented below.
Step 1: Input the following:
Number of customer nodes (n)
Number of vehicles (k)
Capacity of the vehicles (a)
Set Generation Count (GC) = 1
Maximum number of generations to be carried out (MNG) = 1000
Step 2: Generate a random initial population (L) of 100 (N) chromosomes (suitable solutions routes for the problem).
Step 3: Evaluate the fitness function f(x) of each chromosome in the population L.
Step 4: Selection.
Sort the population L by the objective function (fitness function) value in the ascending order, since the objective of the study is minimization of the total distance travelled. Copy a top 30% of the population to form a subpopulation S rounded to the whole number. Smaller fitness value is preferred here.
Step 5: Randomly select any two unselected parent chromosomes from the subpopulation S. Let them be c1 and c2 using tournament selection.
Step 5.1: Perform two-point random Cross-Over using the random sequence insertion-based crossover (RSIX) for the TDVRPTW described in the earlier section among the chromosomes c1 and c2 to obtain their offspring d1 and d2 by assuming a crossover probability of 0.7.
Step 5.2: Perform mutation on each of the offspring using a mutation probability of 0.3.
Step 5.3: Evaluate the fitness function with respect to the total distance travelled and number of vehicles utilized value for each of the offspring d1 and d2.
Step 5.4: Replace the parent chromosomes c1 and c2 in the population with the offspring d1 and d2, respectively, if the fitness function of the offspring is less than that of the parent chromosomes.
Step 6: Increment the generation count (GC) by 1
i.e., GC = GC + 1
Step 7: If GC ≤ MNG, then go to step 4, else go to step 8.
Step 8: The topmost chromosome in the last population serves as the solution for implementation.
Print the tour along with the total distance travelled and number of vehicles used.
Step 9: Stop.
In this section a detailed presentation of the experimental results is given, for the time-dependent vehicle routing problem (TDVRP).
In this section, a comparison is made among the proposed algorithm SNRPGA2 and an existing algorithm TDVRPTW algorithm in terms of number of vehicles utilized using a complete factorial experiment. The existing algorithm was an iterated local search algorithm proposed by Hashimoto et al. [
The model of ANOVA is given as below:
Y i j k = μ + A i + B j + A B i j + e i j k
Problem Class (Factor A) | Algorithm (Factor B) | ||
---|---|---|---|
Replication of Class | TDVRP | SNRPGA2 | |
1. Random 1 | 1 | 9 | 9 |
2 | 14 | 9 | |
3 | 13 | 10 | |
4 | 10 | 9 | |
2. Clustered 1 | 1 | 10 | 8 |
2 | 10 | 8 | |
3 | 10 | 8 | |
4 | 10 | 8 | |
3. Random Clustered 1 | 1 | 10 | 10 |
2 | 11 | 11 | |
3 | 11 | 11 | |
4 | 14 | 12 | |
4. Random 2 | 1 | 4 | 3 |
2 | 4 | 3 | |
3 | 3 | 3 | |
4 | 3 | 3 | |
5. Clustered 2 | 1 | 3 | 3 |
2 | 3 | 3 | |
3 | 3 | 3 | |
4 | 3 | 3 | |
6. Random Clustered 2 | 1 | 4 | 3 |
2 | 4 | 3 | |
3 | 3 | 4 | |
4 | 3 | 3 |
where,
Yijk is the number of vehicles utilized w.r.t the kth replication under the ith treatment of factor A (Problem Size) and the jth treatment of factor B (Algorithm).
µ is the overall mean of the response variable.
Ai is the effect of the ith treatment of factor A (Problem Size) on the response variable.
Bj is the effect of the jth treatment of factor B (Algorithm) on the response variable.
ABij is the interaction effect of the ith Problem Size and jth Algorithm on the response variable.
eijk is the random error associated with the kth replication under the ith Problem Size and the jth Algorithm.
In this model, Factor A (Problem Size/Problem Class) is a random factor and the Factor B (algorithm) is a fixed factor. Since the factor A is a random factor, the interaction factor ABij is also a random factor. The replications are always random and the number of replications under each experimental combination is 4. The derivation of the expected mean square (EMS) is given in Panneerselvam [
The alternative hypotheses of the model are as given below.
H1: There are significant differences between the different pairs of treatments of Factor A (Problem Size) in terms of the number of vehicles utilized
H1: There are significant differences between the different pairs of treatments of Factor B (Algorithm) in terms of the number of vehicles utilized.
H1: There are significant differences between the different pairs of interaction between Factor A and Factor B in terms of number of vehicles utilized.
The results of ANOVA of the data given in the
Source of Variation | Sum of Squares | Degrees of freedom | Mean sum of squares | Calculated F ratio | F ratio (α = 0.05) | Significant F |
---|---|---|---|---|---|---|
Algorithm (B) | 3.521 | 1 | 3.521 | 8.311 | 4.12 | Significant |
Problem Size (A) | 551.604 | 5 | 110.321 | 260.430 | 2.47 | Significant |
Problem Size × Algorithm (A × B) | 5.604 | 5 | 1.121 | 2.646 | 2.47 | Significant |
Error | 15.250 | 36 | 0.424 | |||
Total | 575.979 | 47 |
From the ANOVA results shown in
The standard error used in this test is computed as shown below using the mean sum of squares of the interaction terms (Problem Size × Algorithm) and the number of replications under each of the algorithms (24).
S E = ( MSS A B ÷ n ) 0.5 = ( 1.121 ÷ 24 ) 0.5 = 0.216
The least significant ranges (LSR) are calculated from the significant ranges of Duncan’s multiple range tests table for α = 0.05 and 36 degrees of freedom as shown in
The proposed GA-based meta-heuristic for the time-dependent vehicle routing
No. of treatments-1 (j) | Significant Range | Standard Error | LSR = Significant Range × Standard Error |
---|---|---|---|
2 | 2.872 | 0.216 | 0.6204 |
problem with time windows (SNRPGA2) is compared with one other existing meta-heuristics, viz. the algorithm developed for the TDVRPTW by Demir [
Problem Class (Factor A) | Algorithm (Factor B) | ||
---|---|---|---|
Class | Replication of Class | ALNS (Demir) | TDVRPTW (SNRPGA2) |
Random 1 | 1 R1 | 971 | 831 |
2 R1 | 932 | 675 | |
3 R1 | 948 | 717 | |
4 R1 | 1048 | 664 | |
Clustered 1 | 1 C1 | 822 | 593 |
2 C1 | 826 | 567 | |
3 C1 | 827 | 585 | |
4 C1 | 827 | 580 | |
Random Clustered 1 | 1 RC1 | 1207 | 867 |
2 RC1 | 1114 | 771 | |
3 RC1 | 1258 | 847 | |
4 RC1 | 1457 | 811 | |
Random 2 | 1 R2 | 740 | 525 |
2 R2 | 701 | 688 | |
3 R2 | 731 | 590 | |
4 R2 | 794 | 506 | |
Clustered 2 | 1 C2 | 585 | 422 |
2 C2 | 585 | 430 | |
3 C2 | 586 | 432 | |
4 C2 | 586 | 430 | |
Random Clustered 2 | 1 RC2 | 777 | 762 |
2 RC2 | 783 | 573 | |
3 RC2 | 923 | 732 | |
4 RC2 | 962 | 606 |
Source of Variation | Sum of Squares | Degrees of freedom | Mean sum of squares | Calculated F ratio | F ratio (α = 0.05) | Significant F |
---|---|---|---|---|---|---|
Algorithm (B) | 697,352.302 | 1 | 697,352.302 | 146.138 | 4.12 | Significant |
Problem Size (A) | 1,313,751.984 | 5 | 262,750.397 | 55.062 | 2.47 | Significant |
Problem Size × Algorithm (A × B) | 105,970.302 | 5 | 21,194.060 | 4.441 | 2.47 | Significant |
Error | 171,787.094 | 36 | 4771.864 | |||
Total | 2,288,861.682 | 47 |
The model of ANOVA is given as below:
Y i j k = μ + A i + B j + A B i j + e i j k
where,
Yijk is the total distance travelled w.r.t the kth replication under the ith treatment of factor A (Problem Size) and the jth treatment of factor B (Algorithm).
µ is the overall mean of the response variable total distance travelled
Ai is the effect of the ith treatment of factor A (Problem Size) on the response variable.
Bj is the effect of the jth treatment of factor B (Algorithm) on the response variable.
ABij is the interaction effect of the ith Problem Size and jth Algorithm on the response variable.
eijk is the random error associated with the kth replication under the ith Problem Size and the jth Algorithm.
In this model, the factor A is a random factor and the factor B is a fixed factor. Since the factor A is a random factor, the interaction factor is also a random factor. The replications are always random and the number of replications under each experimental combination is k. The derivation of the expected mean square (EMS) is given in Panneerselvam [
The alternative hypothesis of the model is stated as below:
H1: There are significant differences between the different pairs of treatments of Factor A (Problem Size) in terms of the total distance travelled.
H1: There are significant differences between the different pairs of treatments of Factor B (Algorithm) in terms of the total distance travelled.
H1: There are significant differences between the different pairs of interaction between Factor A and Factor B in terms of total distance travelled.
From the ANOVA results shown in
The standard error used in this test is computed as shown below using the mean sum of squares of the interaction terms (Problem Size × Algorithm) and the number of replications under each of the algorithms (24). The treatment means for the Factor B (Algorithm) in terms of the total distance travelled are arranged in the descending order from left to right. The standard error for the performance measure is calculated using the formula and found to be 22.92 One can notice the fact that the mean sum of squares of the interaction term AB is used in estimating the standard error (SE), because the F ratio for the factor “Algorithm” is obtained by dividing its mean sum of squares by the mean sum of squares of the interaction term ABij (Panneerselvam [
The least significant ranges (LSR) are calculated from the significant ranges of Duncan’s multiple range tests table for α = 0.05 and 5 degrees of freedom as shown in
S E = ( MSS A B ÷ n ) 0.5 = ( 21194.060 ÷ 24 ) 0.5 = 29.72
From the Duncan’s Multiple Range Test performed above as shown in
No. of treatments-1 (j) | Significant Range | Standard Error | LSR = Significant Range × Standard Error |
---|---|---|---|
2 | 2.872 | 29.72 | 85.356 |
to the existing algorithm based on ALNS used in this study for comparison, in terms of total distance travelled.
In this section a comparison is done between the proposed algorithm SNRPGA2 for the TDVRP and the mathematical model for the TDVRP for the two measures, distance travelled and the number of vehicles used, which was solved using LINGO 15.0 optimization software.
A full factorial design is conducted for the comparison of the proposed GA- based algorithm, SNRPGA2 for the TDVRPTW with the mathematical model for the total distance travelled, whose data is given in
The model of ANOVA is given as below:
Y i j k = μ + A i + B j + A B i j + e i j k
where,
Yijk is the total distance travelled w.r.t the kth replication under the ith treatment of factor A (Problem Size) and the jth treatment of factor B (Algorithm).
µ is the overall mean of the response variable total distance travelled.
Ai is the effect of the ith treatment of factor A (Problem Size) on the response variable.
Bj is the effect of the jth treatment of factor B (Algorithm) on the response variable.
Problem Size (Number of Nodes) | Replication | Problem instance | Method | |
---|---|---|---|---|
SNRPGA2 | Mathematical Model | |||
10 | 1 | C102 | 35.754 | 35.754 |
2 | C103 | 35.754 | 35.754 | |
15 | 1 | C102 | 87.815 | 87.815 |
2 | C103 | 84.608 | 84.6079 | |
20 | 1 | C102 | 113.684 | 113.67 |
2 | C103 | 103.55 | 103.50 | |
25 | 1 | C102 | 132.181 | 132.179 |
2 | C103 | 124.45 | 124.44 | |
30 | 1 | C102 | 142.9 | 142.89 |
2 | C103 | 135.171 | 135.161 | |
35 | 1 | C102 | 207.848 | 207.82 |
2 | C103 | 188.99 | 188.41 |
ABij is the interaction effect of the ith Problem Size and jth Algorithm on the response variable.
eijk is the random error associated with the kth replication under the ith Problem Size and the jth Algorithm.
In this model, the factor A is a random factor and the factor B is a fixed factor. Since the factor A is a random factor, the interaction factor is also a random factor. The replications are always random and the number of replications.
The alternative hypothesis of the model is stated as below:
H1: There are significant differences between the different pairs of treatments of Factor A (Problem Size) in terms of the total distance travelled.
H1: There are significant differences between the different pairs of treatments of Factor B (Algorithm) in terms of the total distance travelled.
H1: There are significant differences between the different pairs of interaction between Factor A and Factor B in terms of total distance travelled.
The ANOVA results are shown in the
From the ANOVA results in
Once again a full factorial design is conducted for the comparison of the proposed GA-based algorithm, SNRPGA2 for the TDVRPTW with the mathematical model for the number of vehicles used, whose data is given in
From the
Source of Variation | Sum of Squares | Degrees of freedom | Mean sum of squares | Calculated F ratio | F ratio (α = 0.05) | Sig. |
---|---|---|---|---|---|---|
Problem Size (A) | 59,325.510 | 5 | 11,865.102 | 237.658 | 3.11 | Significant |
Algorithm (B) | 0.021 | 1 | 0.021 | 0.000 | 4.75 | Insignificant |
Algorithm * Size (AB) | 0.073 | 5 | 0.015 | 0.000 | 3.11 | Insignificant |
Error | 599.102 | 12 | 49.925 | |||
Total | 59,924.71 | 23 |
Problem Size (Number of Nodes) | Replication | Problem instance | Method | |
---|---|---|---|---|
SNRPGA2 | Mathematical Model | |||
10 | 1 | C102 | 1 | 1 |
2 | C103 | 1 | 1 | |
15 | 1 | C102 | 2 | 2 |
2 | C103 | 2 | 2 | |
20 | 1 | C102 | 2 | 2 |
2 | C103 | 2 | 2 | |
25 | 1 | C102 | 3 | 3 |
2 | C103 | 3 | 3 | |
30 | 1 | C102 | 3 | 3 |
2 | C103 | 3 | 3 | |
35 | 1 | C102 | 4 | 4 |
2 | C103 | 4 | 4 |
In this research, a GA based meta-heuristic is developed using Random Sequence-based Insertion Crossover (RSIX) method (SNRPGA) for solving the time dependent vehicle routing problem (TDVRP) with time windows. Next, through a full factorial experiment with two factors, viz. “Problem Size” and “Algorithm”, it is proved that there are significant differences among the algorithms in terms of number of vehicles utilized (Kumar and Panneerselvam [
The proposed genetic algorithm is compared with the mathematical model and it is found that there is no significant difference in the results obtained by using the proposed GA and the mathematical model, in terms of each of the performance measures, viz. the number of vehicles used and the total distance travelled.
This study can be useful for planning the supplier site pickups by e-commerce companies, taking into consideration of traffic conditions during different periods of the day with time window requirements of the suppliers. Future researchers can implement the TDVRP using other meta-heuristics and compare the efficiencies of the various meta-heuristics. The Solomon’s benchmark instances are got from SINTEF [
Kumar, S.N. and Panneerselvam, R. (2017) Development of an Efficient Genetic Algorithm for the Time Dependent Vehicle Routing Problem with Time Windows. American Journal of Operations Research, 7, 1-25. http://dx.doi.org/10.4236/ajor.2017.71001