97 CHAPTER 4 MODIFIED K-MEANS WITH HYBRID PARTICLE SWARM OPTIMIZATION FOR VECHILE ROUTING PROBLEM The HPSO proposed in the first two chapters consider the deterministic and stochastic VRP with an objective of minimizing the routing cost. The heuristic techniques used in the first two works are route-first and cluster-second heuristic technique. In this chapter, cluster-first and route- second (Beasley 1983) is used as a heuristic method. Clustering is the first phase of this work, the customers are assigned to each vehicle based on its constraint like minimum distance, maximum demand etc. The modified k-means algorithm is proposed in order to take care of clustering with capacity constraint. It uses first-fit decreasing algorithm for packing the customers into clusters based on their demand and distance measure. Then, HPSO algorithm can be applied within the clusters to form the sequences for each vehicle as a second phase. By using this cluster-first and route-second heuristic, the complexity of the problem is also reduced substantially. In this work, modified k-means with HPSO (MK-HPSO) algorithm is proposed. The proposed MK-HPSO is tested with CVRP data set. This work concentrates on further minimizing the total travel cost of vehicles as compared to HPSO. It is also applied to solve MDVRP and tested with MDVRP data sets.
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97
CHAPTER 4
MODIFIED K-MEANS WITH HYBRID PARTICLE
SWARM OPTIMIZATION FOR VECHILE
ROUTING PROBLEM
The HPSO proposed in the first two chapters consider the
deterministic and stochastic VRP with an objective of minimizing the routing
cost. The heuristic techniques used in the first two works are route-first and
cluster-second heuristic technique. In this chapter, cluster-first and route-
second (Beasley 1983) is used as a heuristic method. Clustering is the first
phase of this work, the customers are assigned to each vehicle based on its
constraint like minimum distance, maximum demand etc. The modified
k-means algorithm is proposed in order to take care of clustering with
capacity constraint. It uses first-fit decreasing algorithm for packing the
customers into clusters based on their demand and distance measure. Then,
HPSO algorithm can be applied within the clusters to form the sequences for
each vehicle as a second phase. By using this cluster-first and route-second
heuristic, the complexity of the problem is also reduced substantially. In this
work, modified k-means with HPSO (MK-HPSO) algorithm is proposed. The
proposed MK-HPSO is tested with CVRP data set. This work concentrates on
further minimizing the total travel cost of vehicles as compared to HPSO. It is
also applied to solve MDVRP and tested with MDVRP data sets.
98
4.1 RELATED WORK
Sweep algorithm is a very old method of clustering approach for
CVRP introduced by Gillet and Miller (1974). After which many clustering
techniques such as Fisher and Jaikumar (1981), Petal algorithm etc. are
introduced. A cluster-and-search heuristic to solve the VRP with delivery and
pick-up had been proposed by Ganesh and Narendran (2007). They proposed
a multi-phase constructive heuristic that clusters nodes based on proximity.
Using shrink-wrap algorithm, they orient nodes along a route. Finally,
Generalized Assignment Procedure (GAP) is used to allot the vehicles for
each route. GA is used as an intensive search engine.
Sariklis and Powell (2007) solved the Open VRP (OVRP)
associated with capacity constraints on vehicles with cluster-first
route-second. Their heuristic has two phases. In the first phase, they
constructed clusters of customers taking vehicle capacity into account, then
balanced and improved the clusters by reassigning customers. In the second
phase, open routes are generated by solving a minimum spanning tree
problem. They used penalties to modify the solution and iteratively converted
infeasible solutions to feasible one.
4.2 MODIFIED K-MEANS CLUSTERING WITH HPSO
(MK-HPSO)
In route-first and cluster-second heuristic, the formation of giant
sequence and traversing the whole sequence of dimension n, generally takes
more time as the number of customer increases. To reduce this time
complexity as well as the space complexity, the VRP is now approached with
‘cluster-first and route-second’ method. The customers are partitioned into
number of sub groups; such that the sequences can be easily found within
these subgroups instead of forming the giant route. Modified k-means
99
algorithm is used to partition the large number of customers into clusters.
Here, the PSO is used for finding the sequence within the formed cluster. The
PSO used here is a hybrid PSO as in the proposed HPSO except that there is
no decoding procedure. The output of modified k-means algorithm is
pipelined as input for HPSO. The general cluster-first and route-second
approach used for solving VRP is as shown in Figure 4.1.
Figure 4.1 Proposed Cluster-First and Route-Second Approach
4.2.1 Modified k-means Clustering Algorithm
The general k-means algorithm assigns each point into the cluster
whose center (called as centroid) is nearest to it (MacQueen, 1967). Random
points are selected as centroid initially and the points around it are grouped.
The center is calculated as the average of all the points in the cluster, i.e. the
100
co-ordinates are the arithmetic mean of each dimension separately over all
points in the cluster. This process of computing the centroid and assigning the
points to new centroid proceeds until there is no more change in the formed
clusters or centroids.
The proposed modified k-means algorithm selects the customer
farther from the depot as its initial centroid, rather than selecting randomly.
The number of clusters m to be formed is generally the number of vehicles
available for servicing. This is expressed using Equation (4.1).
= (4.1)
where m is the number of clusters, n is the number of customers, di denotes
the demand / requirement of each customer i and Q is the maximum load or
capacity that a vehicle can service. This will optimize the number of vehicles
required for servicing.
The distances between the customers are calculated using
Euclidean distance formula using Equation (4.2). The customers are assigned
to the centroid similar to first-fit decreasing algorithm (Xia and Tan 2010).
Either distance or demand is not the only criteria for assigning the customer;
it calculates the priority using both the factors using Equation (4.3).
= ( ) + ( ) (4.2)
( ) = (4.3)
where (xi, yi) denotes the location of ith
customer. Based on this priority, the
grouping of customers is done. If customers are assigned based on distance
alone, the number of clusters formed may not be optimal, because the
101
customers with smaller demand may be assigned to the cluster before the
customer with larger demand which may lead to the formation of additional
cluster.
Procedure for modified k-means clustering
The procedure for modified k-means algorithm is explained as
follows.
Input
Customer list C having co-ordinates (xi, yi) and demands di
Output
m clusters and the partitioned customers set C1, C2, … Cm
Procedure
Calculate m using (4.1)
Sort the C into non-increasing order based on their demands
giving the sequence c1>c2>c3…>cn
Calculate the distance of each customer from the depot and
arrange them in non-increasing order based on the distance from
depot, Let it be C1
Select first m customers as the initial centroids from the
arranged list C1
while not converged
Calculate the Euclidean distance measure using (4.2)
between all customer and each of the m centroid
for each customer ci ,,
while ci is not assigned
Identify the nearest centroid for ci
102
Group all unassigned customers ci’s as G having same
centroid as it’s minimum
Calculate the priority value for ci G using (4.3)
If capacity / load constraint is not violated
Assign ci G to Cj based on priority where j is the nearest
centroid
Else
Choose the next nearest centroid as its nearest centroid
end if
end while
end for
Calculate the new centroid from the formed clusters using
Equation (4.4) for each cluster j each assigned with nj number of
customer
= = (4.4)
end while
4.2.2 Hybrid Particle Swarm Optimization for Route Formation
The clusters formed by modified k-means algorithm are given as
input to HPSO where the sequencing is formed for each groups. This HPSO is
to solve TSP, a subset of VRP rather than solving VRP as a whole. The HPSO
represents the solution as permutation encoding. The solution represents a
sequence of customer within each cluster. The solution representation and
fitness evaluation are explained in the following sub sections.
103
4.2.2.1 Solution representation and conversion method
The integer values are used to represent the solutions as proposed in
chapter 2. For example, the customers (5 6 10 15) are clustered as first group
(nj = 4) and it is mapped to (1 2 3 4) to represent the solution. The
permutation of this numbers 1-4 is generated as solution and proposed PSO
finds the optimal sequence of this permutation. The solution represents the
numbers between 1 and nj. The jth
position of ith
particle yij of kth
vehicle is
represented in Equation (4.5)
= 1 1 (4.5)
where represents | | of kth
vehicle. NNH is used as explained in the
section 2.2.2 for generating the initial solutions that simulate fast
convergence. These solutions are converted to continuous values using
Equation (2.3) as particle position value before applying Equations (1.5) and
(1.6) to particles. Elitism of 5% is used in order to preserve the elite particles
in subsequent generations. The ROV conversion and GA operators are same
as in section 2.2.3 and 2.2.4. Further, the route sequence is improved using
hill-climbing with 2-opt local exchange as in section 2.2.6.
4.2.2.2 Fitness function
The particles represent the sequence (Rj) within each cluster, Cj.
Then, the Rj is evaluated to find the total cost (Dj) from the service point
(depot in case of VRP) using Equation (4.6), and the overall fitness function
is given in Equation (4.7).
=1
(4.6)
(4.7)
where a1 and an are the first and last customer in each cluster,
represents the location where vehicles are stationed and costab represent the
distance or time to move from a to b.
104
Procedure for HPSO to form a routing sequence of customers
The algorithm for HPSO is explained as follows.
Input
m clusters and the partitioned customers set C1, C2, … Cm
Customer list C with n number of co-ordinates (xi, yi) and
demands di
Output
m sequence of customers
Procedure
Initialize the parameters of HPSO
For each cluster
Initialize I solutions (Yi) as a population, using NNH.
While not the termination condition met
Calculate the fitness value for each individual
using (4.6)
Arrange the particles in the ascending order based
on fitness value
Perform HC with 2-opt local exchange as in
section 2.2.6.
Convert Yi into Xi, the particles position value
using Equation (2.3)
Set or update the pbest, Pi=Xi, if fitness of Xi <
fitness of Pi
Set or update the gbest, Pg=Pi, if fitness of Pi <
fitness of Pg
Repeat the following steps for 95% of particles
Decrease the inertia using equation (2.9)
105
Update the velocity and the position of
each ith
particle using (1.5) and (1.6)
Apply GA operators as in section 2.2.4
Convert the particle position value Xi into
solution Yi using smallest position value as
in section 2.2.3
End
End while
End for
Calculate the overall fitness using Equation (4.7)
The overall flow of proposed MK-HPSO is shown in Figure 4.2.
Figure 4.2 Proposed MK-HPSO for VRP
106
4.3 CAPACITATED VEHICLE ROUTING PROBLEM
CVRP is considered for testing the proposed MK-HPSO of
clustering and then routing. In this, CVRP is formulated based on cluster-first
and route-second methodology.
4.3.1 Problem Definition
The CVRP is defined as follows.
Objectives
Minimize the distance travelled by each vehicle
Constraints
Load of each vehicle should not exceed the given vehicle
capacity
Each customer is serviced exactly once
Each vehicle route starts and ends at depot
The problem is given with a set of
Customers: c1, c2, c3 … cn
Demands : d1, d2, d3 … dn
Vehicles : v1, v2, v3 … vm
Capacity : Q
where ci C are the set of customers distributed in the Euclidean plane (xi,
yi) whose distances are symmetric, the demand (di) and capacity (Q) of
vehicle are positive integers. The Costij is the travelling cost/distance between
customer i to customer j.
107
The n customers are grouped to form m clusters, based on their
demand and location (x, y). The set C is partitioned into m number of subsets
Ci. Then
, for i = 1, . . m, (4.8)
= , for i, j = 1, . . .m and i j (4.9)
C = C (4.10)
The Equation (4.9) further indicates that no customer is serviced by
more than one vehicle. A customer ci is included to a subset only if the
summation of customer demands in that subset is less than or equal to the
capacity of the vehicle as in Equation (4.11).
, = 1 (4.11)
The number of customers in each cluster is denoted by n1, n2, ...,
nm, such that
(4.12)
The routing is formed within each cluster Cj with Dj as its route
cost. It is calculated using Equation (4.13).
= (4.13)
where a1 and an are the first and last customer in each group, represents
the location where vehicles are stationed and represent the
time/distance from a to b. Then, the overall objective of CVRP is
(4.14)
108
4.4 ANALYSIS OF PROPOSED MK-HPSO
The cluster-first and route-second is analyzed both theoretically and
empirically. The following subsections explain the analysis made in detail.
4.4.1 Theoretical Analysis
The NP-hard VRPs are decomposed into multiple TSP as detailed
in the above sub sections. Let n be the number of customers and m be the
number of vehicles available to service the customers. Then, the space
complexity is analyzed based on the size of the particles used by MK-PSO,
that takes the size of nj (<n) which is more efficient than the size of HPSO in
chapter 1. The modified k-means algorithm requires storing n customers and
m centroids of clusters. The space required is shown in Table 4.1.
Table 4.1 Space Complexity
Algorithm Space required
HPSO for VRP O(n)
MK-HPSO
Modified k-means
HPSO for TSP
O(n+m)
O(nj)
The size of particle in the proposed work is nj, which automatically
reduces the space needed for storing and to perform other operations. The
time complexity of the proposed MK-HPSO is depicted in Table 4.2.
109
Table 4.2 Time Complexity
Procedures in proposed
MK-HPSO
Time
complexity
Modified k-means O(knm)
Particle Conversion O(nm)
Genetic Operation O(nm)
HC with 2-opt local exchange O(mn2)
The time complexity is calculated for a single iteration. The
modified k-means spend time in computing the distance is O(m). The
reassignment and its overall complexity is O(knm). Each particle in PSO is
converted to positional value and vice versa in O(n). This particle conversion
is performed for all m vehicles as O(nm). The genetic operation crossover and
mutation performed for each particle is again O(n) for one vehicle, but the
occurrence of this operation depends on the probability value. The 2-opt
operator in HC takes O(mn2) computational time for m vehicles.
4.4.2 Empirical Analysis
The proposed MK-HPSO is tested with CVRP problem instances of
Christofides et al and Augerat et al The characteristics of Christofides et al
problem instances are shown in Table 2.11.
The MK-HPSO is implemented in MatLab 7.0.1 and parameters for
PSO are set as in Table 2.7. The number of clusters or vehicles required for
routing is calculated using Equation (4.1) and it is shown in Table 4.3. This
shows that the modified k-means algorithm packs the customers with the
optimal number of vehicles. Table 4.3 shows the time taken for clustering and
also the average load of a vehicle. The average load of a vehicle shows that
the vehicles are utilized to its maximum capacity.
110
Table 4.3 Results Obtained After Clustering
Problem
Instance
No. of
Vehicles
Clustering
Time (sec)
Average
vehicle load
1 5 1 155.40
2 10 2 136.40
3 8 2 182.25
4 12 4 186.25
5 16 5 185.50
11 7 2 196.42
12 10 2 181.00
After clustering, the clusters formed are given as input to the
HPSO. In which, the routing is formed for each clusters. Figure 4.3 shows the
comparison of MK-HPSO cost with BKS for Chrisofieds et al problem
instances.
Figure 4.3 Cost Comparison of MK-HPSO with BKS for Christofieds
et al Problem Instances
111
Figure 4.3 shows that the cost obtained are almost nearer to the best
known solution. The obtained route cost along with mean over 10 runs and
also its deviation from BKS is depicted in Table 4.4.
Table 4.4 Comparison of MK-HPSO Route Cost with BKS Cost
Problem
InstanceBKS
MK-HPSO
costRPD Mean
1 524.61 524.61 0.00 528.13
2 835.26 836.82 0.18 860.26
3 826.14 827.25 0.13 838.89
4 1028.42 1030.55 0.20 1064.03
5 1291.29 1301.42 0.78 1330.64
11 1042.11 1042.11 0.00 1045.99
12 819.56 819.56 0.00 821.29
The first phase of clustering enhances the solution quality of PSO.
Because of which, the problem with clustered customers provides optimal
solution. The deviation from BKS is 0 for the problem instance 1, when the
number of customer is 50. When customers are clustered in nature, the
cluster-first and route-second approach works well and the problem instances
11 and 12 are clustered in nature, so the deviation is 0 in the proposed work.
For other problem instances, the RPD is on an average 0.32. The obtained
MK-HPSO cost is compared with other existing PSO method. Table 4.5
shows the comparison with Ai et al (2009) and Marinakis et al (2010)
112
Table 4.5 MK-HPSO Cost Compared with Other Existing PSO Method
Problem
Instance
BKS
Cost
Ai (SR-2) MK-HPSO
CostTime
in secCost
Time
in sec
1 524.61 524.61 24 524.61 7
2 835.26 844.42 57 836.82 13
3 826.14 829.40 101 827.25 13
4 1028.42 1048.89 223 1030.55 20
5 1291.29 1323.89 413 1301.42 25
11 1042.11 1052.34 93 1042.11 15
12 819.56 819.56 88 819.56 15
The proposed PSO shows better performance than Ai and
Kachivichyanukul (2009) both in terms of route cost and computational time.
The test is further made with Augerat et al data sets. The
characteristics of Problem set are shown in Table 4.6.
Table 4.6 Characteristics of Augerat et al Problem Instances
Problem Instance
Number
Problem
instance
No.
customers
Vehicle
capacity
No.
Vehicles
1 A-n32-k5 31 100 5
2 A-n33-k5 32 100 5
3 A-n37-k5 36 100 5
4 A-n45-k6 45 100 6
5 A-n60-k9 59 100 9
6 B-n31-k5 30 100 5
7 B-n41-k6 40 100 6
8 B-n68-k9 67 100 9
9 E-n22-k4 21 6000 4
10 E-n30-k3 29 4500 3
11 E-n51-k5 50 160 5
12 P-n22-k2 21 160 2
13 P-n60-k15 59 80 15
14 P-n101-k4 100 350 4
113
The clusters formed by modified k-means algorithm are shown in
Table 4.7. The clustering done for each vehicle utilizes the capacity of vehicle
efficiently.
Table 4.7 Results After Clustering
Problem InstanceClustering time
(seconds)
Average vehicle
load
A-n32-k5 1 82
A-n33-k5 1 89.2
A-n37-k5 1 81.4
A-n45-k6 2 98.83
A-n60-k9 2 92.11
B-n31-k5 1 82.4
B-n41-k6 1 94.5
B-n68-k9 2 93
E-n22-k4 1 5625
E-n30-k3 1 4250
E-n51-k5 2 155.4
P-n22-k2 1 154
P-n60-k15 6 75.6
P-n101-k4 2 364.5
The route cost obtained after routing by PSO is shown in Figure 4.4
along with BKS.
114
Figure 4.4 Cost comparison of MK-HPSO with BKS for Augerat et al
problem instances
Table 4.8 shows the BKS and RPD obtained from BKS and also
average cost over 10 runs.
Table 4.8 Comparison of MK-HPSO Route Costs with BKS
Problem Instance BKS cost MK-HPSO Cost RPD Mean
A-n32-k5 784 784 0.00 790
A-n33-k5 661 661 0.00 672
A-n37-k5 669 669 0.00 695
A-n45-k6 944 944 0.00 960
A-n60-k9 1354 1355 0.07 1390
B-n31-k5 672 672 0.00 680
B-n41-k6 829 829 0.00 854
B-n68-k9 1272 1275 0.20 1309
E-n22-k4 375 375 0.00 378
E-n30-k3 534 534 0.00 547
E-n51-k5 521 521 0.00 549
P-n22-k2 216 216 0.00 220
P-n60-k15 968 970 0.20 998
P-n101-k4 681 683 0.29 701
115
For almost all data sets, the optimal cost is obtained and for data set
60 and above there is a small deviation from the BKS. The time taken on an
average is 8.14 seconds. The MK-HPSO cost and computational time is
compared with other existing PSO methods. It is shown in Table 4.9.
Table 4.9 Comparison of MK-HPSO with other Existing PSO Methods
Problem
instance
BKS
cost
Chen Ai (SR-2) MK-HPSO
CostTime
in secCost
Time
in secCost
Time
in sec
A-n32-k5 784 - - - - 784 6
A-n33-k5 661 661 32 661 32 661 6
A-n37-k5 669 - - - - 669 7
A-n45-k6 944 - - - - 944 8
A-n60-k9 1354 1354 309 1355 40 1355 12
B-n31-k5 672 - - - - 672 6
B-n41-k6 829 - - - - 829 8
B-n68-k9 1272 1272 344 1274 50 1275 12
E-n22-k4 375 - - - - 375 4
E-n30-k3 534 534 28 534 16 534 4
E-n51-k5 521 528 301 521 22 521 8
P-n22-k2 216 - - - - 216 3
P-n60-k15 968 - - - - 970 16
P-n101-k4 681 694 978 683 86 683 14
The comparison of CPU time taken by MK-HPSO with other PSO
methods is shown in Figure 4.5.
116
Figure 4.5 CPU time of MK-HPSO compared with other PSO methods
for Augerat et al Problem Instances
Figure 4.5 clearly shows that the MK-HPSO works better than
other method in terms of CPU time. The MK-HPSO shows better
performance than HPSO proposed in chapter 2. The efficiency of MK-HPSO
is compared with HPSO by calculating the RPD of HPSO with respect to
MK-HPSO. It is calculated using Equation (2.16) and projected in Table 4.10
and Table 4.11 for Christofides et al. and Augerat et al problem instances
respectively.
Table 4.10 Comparison of MK-HPSO with HPSO for Christofides et al
problem instances
Problem
Instance
BKS
Cost
HPSO MK-HPSO
RPDCost Time in Sec Cost
Time in
Sec
1 524.61 524.61 9 524.61 7 0.00
2 835.26 850.53 22 836.82 13 1.61
3 826.14 828.12 52 827.25 13 0.10
4 1028.42 1050.56 150 1030.55 20 1.90
5 1291.29 1328.75 324 1301.42 25 2.05
11 1042.11 1050.19 84 1042.11 15 0.76
12 819.56 819.56 48 819.56 15 0.00
117
The HPSO has an average of 0.91% deviation when compared to
MK-HPSO. The solution representation and the decoding procedure are
simple in the proposed work that substantially reduces the computational time
when compared to HPSO. On an average the time taken by proposed work is
15.43 seconds.
Table 4.11 Comparison of MK-HPSO costs with proposed HPSO for
Augerat et al problem instances
Problem
Instance
BKS
cost
HPSO MK-HPSO RP
DCost Time in Sec Cost Time in Sec
A-n32-k5 784 784 8 784 6 0.00
A-n33-k5 661 661 8 661 6 0.00
A-n37-k5 669 669 9 669 7 0.00
A-n45-k6 944 944 11 944 8 0.00
A-n60-k9 1354 1368 18 1355 12 0.95
B-n31-k5 672 672 7 672 6 0.00
B-n41-k6 829 829 10 829 8 0.00
B-n68-k9 1272 1281 21 1275 12 0.46
E-n22-k4 375 375 5 375 4 0.00
E-n30-k3 534 534 8 534 4 0.00
E-n51-k5 521 522 13 521 8 0.19
P-n22-k2 216 216 5 216 3 0.00
P-n60-k15 968 970 18 970 16 0.00
P-n101-k4 681 685 34 683 14 0.29
For Augerat et al problem instances, the HPSO has 0.13% of
deviation on an average when compared to MK-HPSO. The time taken is also
less when compared to HPSO.
118
The MK-HPSO is also tested for MDVRP data set. The
characteristics of problem instances are shown in Table 2.12 and initial
clusters formed are shown in Table 2.13. These clusters formed by general k-
means algorithm are given as input for MK-HPSO to form the routes for each
vehicle of a depot and total cost obtained along with BKS and is shown in
Table 4.12 along with RPD from BKS. It also compares the results obtained
by GA methods.
Table 4.12 Comparison of MK-HPSO Cost with BKS and GA Methods
After Routing for MDVRP
Problem
InstanceBKS
GenClust
(Thangiah, &
Salhi, 2001)
GA (Ombuki
& Hanshar,
2004)
MK-HPSO
Cost RPD
1 576.86 591.73 622.18 579.37 0.43
2 473.53 473.55 480.04 474.98 0.30
3 641.18 694.49 706.88 680.23 5.74
4 1001.49 1062.38 1024.78 1015.54 1.38
5 750.26 754.84 785.15 760.12 1.29
6 876.5 976.02 908.88 894.64 2.02
7 885.69 976.48 918.05 915.29 3.23
8 4437.58 4812.52 4690.18 4598.03 3.48
9 3900.13 4284.62 4240.08 4322.34 9.76
10 3663.00 4291.45 3984.78 3668.26 0.14
11 3554.08 4092.68 3880.65 3601.48 1.31
12 1318.95 1421.94 1318.95 1319.20 0.01
13 1318.95 1318.95 1318.95 1319.60 0.04
14 1360.12 1360.12 1365.69 1360.12 0.00
15 2505.29 3059.15 2579.25 2595.34 3.46
16 2572.23 2719.98 2587.87 2671.91 3.73
17 2708.99 2894.69 2731.37 2764.66 2.01
18 3702.75 5462.90 3903.85 3815.27 2.94
19 3827.06 3956.61 3900.61 3856.70 0.76
20 4058.00 4344.81 4097.06 4109.45 1.25
21 5474.74 6872.11 5926.49 5566.83 1.65
22 5702.06 5985.32 5913.59 5917.34 3.63
23 6095.36 6299.04 6145.58 6146.28 0.82
119
The MK-HPSO has 2.14% of deviation from BKS. For the problem
instance 14, the optimal cost is obtained by MK-HPSO. For other problems,
near optimal solution is obtained.
Table 4.13 Comparison of MK-HPSO with HPSO for MDVRP