This paper proposes a heuristic algorithm to solve the vehicle routing problem (VRP). The algorithm uses a 2-phase approach: 1) customers are clustered based on their location and demand, and 2) routes are determined to service the customers in each cluster. The algorithm was tested on problems with up to 10,000 customers and showed good computation time but suboptimal solutions. Increasing the vehicle capacity improved the algorithm's performance. The paper concludes that adding an improvement method like 2-opt could further enhance the solution quality.

1. 1
A feasible solutionalgorithmfor a
primitivevehicle routing problem
Cem Recai Çırak
31 December 2015
Abstract
This paper is written to give brief information about
a feasible solution algorithm offer for a primitively
defined vehicle routing problem. The paper broaches
to subject with a short review about on a vehicle
routing problem solution algorithms. After that;
definition of the problem at issue, choice of the type
of offered algorithm, brief explanation of the
algorithm, simulation results on pseudo-randomly
generated artificial problems and comparative
interpretation about the performance of the
algorithm are follows explicated.
1. Introduction
The vehicle routing problem (VRP) is a
combinatorial optimization and integer
programming problem seeking to service a
number of customers with a fleet of
vehicles. Proposed by Dantzig and Ramser
in 1959, VRP is an important problem in the
fields of transportation, distribution, and
logistics. [1] The Vehicle Routing Problem
(VRP) is a general name given to an entire
type of problems in which a set of routes for
a fleet of vehicles based at single or multiple
depots must be determined for a number of
physically dispersed customers. The
objective of the VRP is to deliver a set of
customers with known demands on
minimum-cost vehicle routes starting and
finishing at a depot.
A representation of a characteristic input
for a VRP problem and one of its possible
solution outputs can be seen below in
Figure 1.
Figure 1: An instance of a VRP and its solution
Approximately all of the most commonly
used techniques to solve Vehicle Routing
Problems are heuristics and metaheuristics
since there is no exact algorithm which
guarantees to find the optimal solution
within admissible computing time while the
number of customers are getting larger.
This is due to the issue that VRP is an NP-
Hard class problem and its exact algorithm
time complexity T(n)=O(cn). Classification of
solution techniques and some of the most
commonly used solution algorithms are
listed.
Exact Approaches: Exact approaches
compute every possible solution until the
best (optimal) solution is found. This
technique uses divide and conquer strategy
by applying a branch and bound or branch
and cut algorithms on a K-tree. Customers
are set as leaves of K-tree. Then problem is
divided into small sub-problems and solved.

2. 2
Heuristics: Heuristic methods perform a
relatively limited exploration of the search
space and get a not too bad but much faster
solution by applying some experience-
based guesses. Heuristics are divided into
two parts as constructive methods and 2-
phase algorithms.
In constructive methods, a feasible solution
is provided by checking whether it satisfies
the cost requirement and do not improve
upon. Some reputed constructive
algorithms are below:
 Clark and Wright savings
 Matching-based savings
 Multi-route improvement heuristics
 Kinderwater and Savelsbergh
 Thomson and Psaraftis
 Van Breedam
In 2-phase algorithms, customers are set as
nodes and then divided into clusters and
the routes are determined in which there
are paths connecting the clusters
themselves as a feedback loop. 2-phase
algorithms are divided into two types:
 Cluster first, route second
 Taillard
 The Petal algorithms
 The Sweep algorithms
 Fisher and Jaikumar
 Route first, cluster second
In many versions of 2-phase algorithms, the
main improvements are included in the
applied clustering technique.
Meta-Heuristics: Meta-heuristics performs
a complete search in the most appropriate
regions of the solution area. Meta-
heuristics gets better quality solutions in
comparison with heuristics. Some reputed
meta-heuristics algorithms are below:
 Ant algorithms
 Constraint programming
 Genetic algorithms
 Tabu search
2. Problem Definition
A very simplified Single Depot Multiple
Route Vehicle Routing Problem is defined as
follows:
 There is a single depot at the origin.
 There are n customers with pseudo-
randomly generated coordinates and
demands with via computer
programming. Their coordinates are
defined with a radius between 1 and
100 and an angle between 0° and 359°
with respect to the depot at the origin
(0,0). Their demands are defined
between 1 and 100.
 All customers connected with a
complete weighted graph and only one
edge connects any two of customers.
Weights of edges are defined as directly
equal to the geometric distance
between two customers which are
consisting the edge.
 Unlimited number of trucks are
available with identical capacities.
However, a truck can be used if and only
if it is necessarily needed. In other
words, if total demand exceeds the
maximum truck capacity, then usage of
another truck is allowed.
 Each customer on the network must be
visited exactly one times. Therefore,
their demands must be satisfied at
once.
 Total cost is assumed equal to the sum
of the weights of used edges.
Figure 2: Pseudo generated VRP scheme

3. 3
n: Number of customers
m: Number of trucks used
i = 1, 2, …, n;
j = 1, 2, …, n;
k = 1, 2, …, m
C0: Depot
C1, C2, …, Ci, …, Cn: Customers
T1, T2, …, Tk, …, Tm: Trucks
M: Maximum truck capacity
Di: Demand of Ci,
where 0 < Di < 100, ∀i
Eij: Edge between Ci and Cj,
where i ≠ j, ∀i,j
Rk: Route of Tk,
which is a combination of Eijs
Wij: Weight of Eij,
where 0 < Wij < 200, ∀i,j
Wk: Total weight of Rk, the sum
of Wijs that belongs to Eijs
which are consisting Rk
C(Wk): Total cost function,
𝐶(𝑊𝑘) = ∑ 𝑊𝑘
𝑚
𝑘=1
Minimize total cost function:
min ∑ 𝑊𝑘
𝑚
𝑘=1
3. Solution Algorithm
A 2-phase heuristic algorithm is used for
solution. In first phase customers are
assumed as nodes and they are assigned
into a cluster. In second phase all clusters
are routed inside and minimized total
weights of each route are computed. Then
total cost function is calculated as sum of
total weights of all routes.
First phase algorithm works as:
1. Sort all customer according to their
angle with ascending order.
2. Then add sorted customers in order
into a cluster and subtract the
added customer’s demand from the
maximum truck capacity.
3. If remaining truck capacity is less
than the added customer’s demand,
remove back that last added
customer from the cluster.
4. Then create a new cluster and add
the removed customer into new
cluster and continue to add sorted
customers in order.
5. Check step 3 after each addition
operation and repeat steps 3 and 4
until all customers assigned into a
cluster.
Figure 3: Phase 1, clustered graph scheme

4. 4
Second phase algorithm works as:
1. Start with first cluster. Add depot to
the cluster route as starting point.
Add all customers into an adjacent
list. Then sort all customers in the
cluster with respect to their
distances from the depot in
ascending order.
2. Select the closest customer and
assign it as current customer. Add
current customer to the cluster
route. Add its distance from the
depot to total weight of route. Then
mark current customer as visited.
3. Search in adjacent list except the
ones marked as visited. Find the
closest adjacent to current
customer and assign it as next
customer. Add next customer to the
cluster route. Add distance between
next customer and current
customer to total weight of route.
Then assign next customer as
current customer. Then repeat this
step until all adjacent list marked as
visited.
4. After all adjacent list is visited. Add
depot to the cluster route as
finishing point. Add depot distance
from the current customer to total
weight of route.
5. Apply first four steps to all clusters.
Then sum all total weight of routes
and find total cost function value.
Figure 4: Phase 2, routed graph scheme
4. Simulation Results
Average results are obtained over 1000
simulations for each different case, for
maximum truck capacity 500.
Table 1: Results for Maximum Truck Capacity 500
#*
Multiple Route Single Route
Cost Time #** Cost Time
10 532 1,6 6 496 1,9
20 1083 2,2 7 686 2,7
50 2438 4 8 1087 4,1
100 4954 7,8 8 1520 8,3
200 9996 13,1 9 2130 20
500 23811 30,2 9 3243 73,9
1000 40832 62,1 9 4523 235
10000 162200 654 9 12885 17515
Average results are obtained over 1000
simulations for each different case, for
maximum truck capacity 5000.
Table 2: Results for Maximum Truck Capacity 5000
#*
Multiple Route Single Route
Cost Time #** Cost Time
10 509 1,9 10 496 1,7
20 682 2,4 20 686 2,7
50 1070 4,4 50 1087 4,4
100 1692 8,3 65 1520 8,4
200 3168 16 83 2130 19,6
500 7907 39,1 85 3243 76,4
1000 15425 77,9 93 4523 225
10000 68479 817 97 12885 16330
* Number of customers
** Number of customers per route
Since the time complexity of overall
algorithm 𝑇(𝑛, 𝑚) = 𝑂(
𝑛2
𝑚
). This algorithm
performs approximately under linear time
complexity

5. 5
5. Conclusion
Graph 1: Cost minimization comparison
Graph 2: Computation time comparison
As seen in Graph 1, while number of
customers is getting larger, multiple route
algorithm’s cost minimization performance
gets worse in comparison with single route
algorithm for maximum truck capacity 500.
However, if truck capacity increased (i.e. to
5000), multiple route algorithm’s
performance gets significantly much better.
As seen in Graph 2, while number of
customers is getting larger, multiple route
algorithm’s cost minimization performance
gets so much better in comparison with
single route algorithm for maximum truck
capacity 500. Also when truck capacity
increased, there is be no significant
increment in computation time of multiple
route algorithm.
Although, the offered algorithm has very
successful time complexity, its cost
minimization performance is not good
enough. Therefore, addition of 2-opt
algorithm to the end of this algorithm as
third step, may lead a significant
improvement in its cost minimization
performance also.
6. References
[1] Vehicle Routing Problem. (available on 31
December 2015). Retrieved from
http://neo.lcc.uma.es/vrp/