Particle Swarm Optimization with Genetic
Operators for Vehicle Routing Problem
GAZALANAAZ QURESHI
Student, Mtech, 4th sem, Electronics, G.H.R.C.E, Nagpur, Maharashtra, India
[email protected] DR. P. R. BAJAJ
Senior IEEE member, Director, G.H.R.C.E,
Nagpur, Maharashtra, India P. V. PURANIK
Assistant Prof. G.H.R.C.E, Nagpur, Maharashtra, India Abstract :
Vehicle Routing Problem (VRP) is to find shortest route thereby minimizing total cost. VRP is a NP-hard and Combinatorial optimization problem. Such problems increase exponentially with the problem size. Various derivative based optimization techniques are employed for optimization. Derivative based optimization techniques are difficult to evaluate. Therefore parallel search algorithm emerged to solve VRP. In this work, a particle swarm optimization (PSO) algorithm and Genetic algorithm (GA) with crossover and mutation operator are applied to two typical functions to deal with the problem of VRP efficiently using MATLAB software. Before solving VRP, optimization of functions using PSO and GA are checked. In this paper capacitate VRP with time window (CVRPTW) is proposed. The computational result shows generation of input for VRP, optimization of Rastrigin function, Rosenbrock function using PSO and GA.
Keywords: Optimization; Vehicle Routing Problem; PSO; Genetic operators; Rastrigin function; Rosenbrock function.
1. INTRODUCTION
The Vehicle Routing Problem (VRP) is generalized form of the TSP. VRP means determining m vehicle routes, where a route is a tour that begins at the depot, visits a subset of the customers in a given order and returns to the depot.VRP was firstly proposed by Dantzing and Ramser in 1959 [4]. The vehicle routing problem (VRP) is to minimize the total travel distance of a number of vehicles, under various constraints, where every customer must be visited exactly once by a vehicle. There are many variants to vehicle routing problems. One of them is Capacitate VRP with Time Window (CVRPTW). The objective of the problem is to find shortest routes for vehicles to service all the customers, without violating the capacity and travel time constraints of the vehicles and the time windows imposed by the customers. CVRPTW is one of combinatorial & NP-hard optimization problems. Combinatorial means searching for best element in a set of discrete items. NP-hard means Non-deterministic polynomial-time hard, a problem of finding the best solution from all possible solutions where problem increases exponentially with the problem size. CVRPTW can be solved using search algorithm or Meta heuristic.
VRP lies in between The Travelling Salesman Problem (TSP) and The Bin Packing Problem (BPP). Depending upon the restrictions on VRP, the following are the types:
Capacitated VRP – CVRP: vehicles have a limited capacity.
VRP with time windows - VRPTW: Customers have to be supplied within a certain time window.
Multiple Depot VRP - MDVRP: There are many depots to supply the customers.
Split Delivery VRP - SDVRP : Customers may be served by different vehicles.
Stochastic VRP - SVRP: Some values like number of customers, theirs demands, serve time or travel time is random.
Periodic VRP - PVRP: The deliveries of demands may be done in some days.
Particle Swarm Optimization (PSO) is an evolutionary algorithm introduced by Kennedy and Eberhart in 1995 [1]. PSO is an optimization algorithm based on swarm theory. A PSO algorithm is inspired by biological and sociological motivations and can take care of optimality on rough, discontinuous and multimodal surfaces. Also Genetic Algorithm is an evolutionary algorithm introduced by Holland .Genetic Algorithm based on natural evolution. Genetic Algorithm is routinely used to generate useful solutions to optimization and search problems using techniques inspired by natural evolution, such as mutation and crossover.
2. PROBLEM DEFINATION
VRP is to satisfy demands of all customers at the lowest sum of travelling distances by all vehicles according to the problem requirement. Mathematically all the customers have to be grouped into K sections. The kth section is supposed to contain Nk specific customers that are assigned to the kth vehicle. The mathematical for CVRPTW is as follows:
The shortest distance(S) is given as
min ∑
Above equation is the mathematical notation of the optimization objective and overall restraints. Some detailed relations among the variables are as followed [3]:
, 1
For m=1, 2,… … -1
= + t + 0,
∑ = M , 1 =
Where d (m, m+1) originally represents the distance between the and m 1 th customer, yet the task description thinks that this is equal to the time consumed in the journey between the two locations. As a result, we directly adopt d (m, m+1) as the time. is the service time in the mth location for kth vehicle. The total time during the journey of the kth vehicle Tk can be seen as the sum of 3 parts: 1) elapsed time when the kth vehicle arrives at the last customer ; 2) service duration at the th location; 3) time used in the way from the th location to the starting point.
Variables:
K: the number of vehicles;
Q: capacity of each vehicle (all vehicles have the same capacity); : Index of each vehicle;
: Length travelled by the vehicle;
Nk: the number of customers assigned to the vehicle; M: the number of all customers;
: X coordination of the customer corresponding to the kth vehicle; : Y coordination of the customer corresponding to the kth vehicle; : Demand amount of the customer corresponding to the kth vehicle; : Arriving time of the customer corresponding to the vehicle; T1mk: ready time of the customer corresponding to the kth vehicle; T2mk: due time of the customer corresponding to the vehicle;
: The time when the vehicle returns to the original starting location; T20: the total consuming time limit;
(i, j): Distance between and jth location for the vehicle. 3. PARTICLE SWARM OPTIMIZATION
In PSO there are particles exploring the search space for best solution. Each particle has its position and velocity. Each particle has fitness values evaluated by the fitness function to be optimized. For updating particles there are two best values. Personal best (pBest) is the first one. pBest is the best position a particle has achieved in history. Global best (GBEST) is the second one. GBEST is the best position tracked by the particle swarm optimizer in the population. The particle updates its velocity and positions after finding the two best values, with following equation (a) and (b).
V [ ] =w* V [ ] + c1 * Rand () * (pBest [ ] – Present [ ]) + c2 * Rand () * (gBest [ ] - Present[ ])(a) Present [ ] = Present [ ] + V [ ] (b)
where,
V [ ] is the particle velocity,
Present [ ] is the current particle (solution), pBest [ ] and gBest [ ] are defined as stated before, Rand () is a random number between (0, 1)
c1, c2 are learning factors. Usually c1 = c2 = 2, w = inertia factor PSO algorithm [6]:
For each particle Initialize particle END
Do
For each particle Calculate fitness value
If the fitness value is better than the best fitness value (pBest) in history set current value as the new pBest
End
Choose the particle with the best fitness value of all the particles as the gBest For each particle
Calculate particle velocity according equation (a) Update particle position according equation (b) End
While maximum iterations or minimum error criteria is not attained. 4. GENETIC ALGORITHM AND OPERATORS
Genetic Algorithm is directed search algorithms involves biological evolution. In Genetic Algorithm concept of natural selection and Genetic inheritance is used. In GA, the first generation consists of initial population of randomly generated candidate solutions. Fitness function is applied to the initial population and any subsequent offspring. The crossover operator recombines the parental genes. The mutation operator alters the genotype of an individual in a random way.
GA [7]: {
Initialize population; Evaluate population;
While Termination Criteria Not Satisfied {
Evaluate population; }
}
5. COMPUTATIONAL RESULTS
The work is executed by first testing two benchmark functions using PSO algorithm: Here x is a D dimensional vector representing the solution to be evaluated.
Also D= length(x)
The two functions used are 1) Rastrigin function:
fit = 10*D + sum(x.^2 - 10*cos (2.*pi.*x)); For i=1:1: D-1,
2) Rosenbrock function:
f=0; f = f + 100*((x (i+1) - x(i)^2)^2) + (1 - x(i))^2;
Rosenbrock (Dejong2) are uni-modal function, while Rastrigin are multimodal function which has a great of local minimum.
Observation for PSO: 1) For Rastrigin function:
Avg. fitness = 5.21e-005(2.73e-005) Avg. time = 1.62e-001(3.00e-002)
The Avg. fitness is the optimized value of Rastrigin function 2) For Rosenbrock function:
Avg. fitness = 6.48e-005(2.53e-005) Avg. time = 1.40e-001(5.22e-002)
The Avg. fitness is the optimized value of Rosenbrock function Observation for Genetic algorithm using GA Toolbox:
For Rosenbrock function (Dejong2):
For solving VRP random locations of cities need to be generated or benchmarks can be used. Following observation shows the generation of random locations of cities:
0 10 20 30 40 50 60 70 80 90 100
0 20 40 60 80 100 120
Generation
F
it
n
e
s
s va
lu
e
Best: 0.035567 Mean: 0.056656
6. CONCLUSION
A PSO with genetic operators and mathematical programming model for the capacitate vehicle routing problem are suggested in this study. Optimization using PSO and Genetic algorithm are verified by using test functions. In computational study input for VRP is generated. This work can be extended to solve CVRPTW.
References
[1] J.P. Castro, D. Landa-Silva, and J.A. Moreno, “Exploring Feasible and Infeasible Regions in the Vehicle Routing Problem with Time Windows Using a Multi- Objective Particle Swarm Optimization Approach”, Nature Inspired Cooperative Strategies for Optimization (NICSO 2008) Studies in Computational Intelligence Volume 236, 2009, pp 103-114.
[2] Lin Lu, Qi Luo, Jun-yong Liu, Chuan Long, “An Improved Particle Swarm Optimization Algorithm”, Granular computing, 2008.IEEE international conference.
[3] Matlab Central.
[4] S. Geetha, G. Poonthalir and P. T. Vanathi, “A Hybrid Particle Swarm Optimization with Genetic Operators for Vehicle Routing Problem”, Journal of Advances in Information Technology, vol. 1, no. 4, November 2010.
[5] VRP Web http://neo.lcc.uma.es/radi-aeb/WebVRP/. [6] www.particleswarm.net/.
