Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 29, 1439 - 1454
Multi-Objective Fuzzy Vehicle Routing Problem: A Case Study Radha Gupta1, Bijendra Singh2 and Dhaneshwar Pandey3 1
Sri Bhagawan Mahaveer Jain College of Engineering Bangalore, Karnataka, India 2
School of Studies in Mathematics Vikram University, Ujjain, (M.P.), India 3
Department of Mathematics Dayalbagh University, Agra, (U.P.), India
[email protected],
[email protected] Abstract The Vehicle Routing Problem (VRP) is a well-known problem studied by researchers in Operations Research and it deals with distribution of goods from a depot to a set of customers in a given time period by a fleet of vehicles. The solution of a VRP is a set of minimum cost routes, which satisfy the problem’s constraints, and fulfill customers’ requirements. When the problem is stochastic it becomes all the more difficult to find a solution to an already difficult problem, though the nature inspired algorithms like Genetic Algorithm seems to provide promising results. Fuzzy logic is another concept which deals with uncertainty very effectively. In this study, we have extended the work of Gupta et al [20], on Multi objective Fuzzy Vehicle Routing Problem, which is based on fuzzy logic and genetic algorithm approaches with attributes: maximization of customer’s satisfaction grade, minimization of fleet size, distance minimization and waiting time minimization by incorporating the capacity constraint. The case study of Jain University of Bangalore in Karnataka which provides bus services to pick up and drop students and staff
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from/to home and university is considered to demonstrate the effectiveness of this newly developed model. Mathematics Subject Classification: 90B36, 90C29 Keywords: Fuzzy logic, Genetic Algorithm, Vehicle routing problems, Multi objective VRP.
1. Introduction: In the VRP, the road network is represented by a graph with arcs and vertices. Arcs represent roads and vertices represent road intersections, junctions, customer locations, and the depot. Each arc has an associated cost. Each customer location vertex has an associated number of goods to be delivered. Each vehicle has its own capacity and cost associated with its utilization. There are objectives other than minimizing the transportation cost that may arise in vehicle routing problems such as minimizing the number of vehicles required to serve all customers, balancing the routes, or minimizing the waiting time of a customer. The VRP is an NP-hard problem [15], and so it is difficult to solve. There are many variations on the basic VRP which have their own characteristics. The capacitated VRP (CVRP) is a problem in which all customer demand must be satisfied, all demands are known, and all vehicles have identical capacity and are based at a central depot. The objective of CVRP is to minimize total transportation cost or time [26]. The VRPTW is an extension of CVRP with a time constraint for reaching each customer. The service can be performed only within a specified time interval. A vehicle is permitted to arrive before the opening of the time window, but must wait, at no cost, until service becomes possible. Arrival after the latest time window is not permitted [3]. VRP was first studied by Dantzig & Ramser in 1959 for a Truck Dispatching Problem [7]. Since then there have been many VRP studies reported in literature. Bowerman [2], Kelnhofer [12] and Thangiah [25] applied VRP models to school bus routing. Other applications include inventory and vehicle routing in the beverage, food and dairy industries, distribution and routing in the newspaper industry [11], transit bus services [16], public library systems [9] and grocery delivery [6]. VRP solution methods fall into two main categories: exact methods and heuristics. Branch and Bound and branch-and-cut, are both exact methods which have been proposed for VRP by many researchers [26], [8], [13], [4] and [17]. Heuristics are methods which produce good solutions in practice but do not guarantee optimality. Metaheuristics give better solutions than classical heuristics, but consume more
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computational time. Some of the most popular metaheuristics are simulated annealing (SA), tabu search (TS), genetic algorithms (GA), ant systems (AS), and neural networks (NN). Several researchers use SA, GA and AS to solve VRPTW such as [24], [5, 22]. NN are mostly used in combination with other methods like GA & AS to solve VRP such as [1, 23]. Only Lin [14] has used the concept of fuzzy logic to solve VRPTW. We have used Fuzzy logic with GA to solve a multi objective VRP with capacity and time window constraints.
2. Multi objective Fuzzy Vehicle Routing Problem with Time Windows and capacity constraints (MOFVRP) Gen and Cheng [10] formulated the fuzzy Vehicle routing problem based on the concept of fuzzy due time, where the membership function of fuzzy due time corresponds to the grade of satisfaction of a service time. The objectives considered are maximization of grade satisfaction of customer, minimization of fleet size, total travel distance and total waiting time. Lin [14] formulated a fuzzy vehicle routing and scheduling problem (FVRSP) with five attributes: space utility, service satisfaction, waiting time, delay time, and transportation distance. Gupta et al [18, 19, 20] extended this concept by incorporating the fuzzy service time under time windows and capacity constraints. This paper includes a real time application of the updated mathematical model of fuzzy vehicle routing problems (FVRPTWC) given in Gupta et al [19] which was earlier solved by us [21] using classical techniques. The four objectives considered are: (i) Minimizing the fleet size (ii) Minimizing the total distance traveled, (iii) Maximizing the average grade of customer satisfaction (iv) Minimizing total waiting time over vehicles. The Constraints imposed are: • Service time for each customer lies within a tolerable time for each customer. • Every vehicle is assigned to customers without exceeding its capacity. • Each customer is serviced by one and only one vehicle. • For each customer there are only two customers directly connected with him, one directly reaches him, and another he directly travels to by a vehicle. • Customer is serviced within a given time window.
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The application of this improved model in a pick up/ drop problem where the Bus service is required for College students/faculty to commute between City and University is considered.
3. Optimal Allocation of College Buses in Vehicle Routing Planning of Jain University: A Case Study
Jain University (JU) is a private group of educational institutions in Bangalore, India, that offers degrees in Engineering, Sciences, Arts, commerce, Humanities and Business Administration. The University is widely spread over Bangalore and in outskirts of Bangalore also. The engineering College of Jain University is located at a distance of 40 kms from Bangalore called Jakkasandra. Currently, the JU provides bus services to pick up and deliver staff and students from/to home and university in the morning and evening periods. There are 5 routes currently in service and each route has been established by an intuitive method. A recent survey of JU bus riders revealed widespread dissatisfaction with the service. Riders complained about the limited number of stops and vehicles. JU currently has a fleet of 10 vehicles of which 8 are Volvo buses with a capacity of 60 seats each, 1 is Swaraj Mazda with a capacity of 30 seats, and 1 is tempo traveler with a capacity of 12 seats. Following are the assumptions on which the VRP has been formulated: • Each route will start from JU in the evening and end at a point closer to the last stop. • Each route start from the point where it was parked the previous evening and will end at JU in the morning. • The cost of a route is proportional to the time traveled. • Travel times between each stop are considered to be stochastic in nature due to unpredictable traffic situations or unforeseen circumstances. • Demands at each of the stops are known and it cannot be split. • If the demand at a particular stop is not met then another vehicle may be used to pickup passengers from that stop. • The time taken to serve a passenger is considered to be fuzzy. The constraints in this problem are: • The capacity of the vehicle is strictly enforced; a customer should not stand in the vehicle.
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•
There is a time window for delivering passengers to JU in the morning and picking up in the evening. The overall goal of this research is to develop a plan for the University’s bus service to be able to serve all customers in the most efficient way.
4. A Fuzzy-Genetic Approach for MOFVRP Any real life application of VRP is influenced by many variables that have uncertainties and vagueness. Heuristics are methods which produce good solutions in practice but do not guarantee optimality. Meta heuristics give better solutions and Genetic Algorithm (GA) is one of the most popular Meta heuristic. But application of fuzzy set in VRP is very effective in dealing with the multi objective problem of VRP. Gupta et al [19,20] developed a fuzzy multi objective optimization VRP with time windows and it has been further extended by us by incorporating capacity constraints as follows: k
∑ ai yik ≤ c k , ∀k
i =1
where ck is the capacity of the kth vehicle ai is the starting time of the ith customer
We have used this newly developed model to solve a real life application of pickup/drop service provided by College management to students and faculties. Genetic Algorithm: Inspired by Darwin’s theory of evolution, Genetic Algorithm (GA) is a stochastic search technique based on the mechanism of natural selection and natural genetics. Algorithm begins with a set of solutions (represented by chromosomes) called population. Solutions from one population are taken and used to form a new population. This is motivated by a hope that the new population will be better than the old one. Solutions which are then selected to form a new solutions (offspring) undergo three important operations described below and are selected according to their fitness – the more suitable they are the more chances they have to reproduce.
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(1) Selection Operator: (i) Preference is given to better individuals, allowing them to pass on their genes to the next generation. (ii) The goodness of each individual depends on the fitness, which is determined by an objective function. (2) Crossover Operator: (i) Process of combining the genes is performed by using order crossover as follows: Parent 1
1
2
3
4
5
6
7
8
9
Child 1
2
9
8
4
5
6
7
3
1
Child 2
1
3
7
4
5
6
8
9
2
Parent 2
2
9
5
8
4
7
3
1
6
(3) Mutation Operator: (i) It prevents falling of all solutions in the population into a local optimal solution. Mutation randomly changes the offspring resulted from crossover. 2 9 8 4 5 6 7 3 1 Offspring 1 Offspring after mutation
1 9
8
4
5
6
7
3 2
Fuzzy VRP: It is based on concept of fuzzy due time where membership function of fuzzy time corresponds to the grade of satisfaction of a service time. The objective is to maximize the average grade of satisfaction over customers. The conventional approach considers the tolerance of customers & represents it as a time window. The grade of satisfaction (or membership function) of service µi(t) can be
defined for any service time (t>0) as µi(t) = where (ei, li) represents the earliest and the latest start time of customer i and t is the time at which the customer is served.
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µ
Zero Satisfaction
Zero Satisfaction
Full satisfaction
ei
Time Window
The Fuzzy approach represents customer’s preference as a triangular fuzzy number (TFN) and defines the membership function as:
=
where (ei, ui, li) represents the earliest, fuzzy due time and the latest start time of customer i and t is the time at which the customer is served. Full satisfaction
µ
Zero Satisfaction ei
ti
Fuzzy
li
due
time
t
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The data required for this study is furnished below: Table 1: Transit distance (km) between Engineering College and various stops
Destination College C1 C2 C3 C4 C5 C6 C7 C8 C9
College
C1
C2
C3
C4
C5
C6
C7
C8
C9
0
1.7
11.7
8.4
12.5
9.2
10
12.5
10
10.8
1.7 11.7 8.4 12.5 9.2 10.0 12.5 10.0 10.8
0 10 6.7 10.8 7.5 8.4 10.8 8.4 9.2
10 0 0 0 0 0 0 0 0
6.7 0 0 4.2 0.8 1.7 4.2 1.7 2.5
10.8 0 4.2 0 0 0 0 0 0
7.5 0 0.8 0 0 0 0 0.8 1.7
8.4 0 1.7 0 0 0 2.5 0 0
10.8 0 4.2 0 0 2.5 0 0 0
8.4 0 1.7 0 0.8 0 0 0 0
9.2 0 2.5 0 1.7 0 0 0 0
Table 2: Due service time schedules Destinations C1 C2 C3 (Customers) e1 u1 l1 e2 u2 l2 e3 u3 l3 Service time 4:00 5:30 6:30 5:00 5:30 6:00 5:15 5:45 6:15 Required time Destinations C4 C5 C6 (Customers) e4 u4 l4 e5 u5 l5 e6 u6 l6 Service time 5:30 6:00 6:30 5:40 6:10 6:40 5:50 6:20 6:50 Required time Destinations C7 C8 C8 (Customers) e7 u7 l7 e8 u8 l8 e9 u9 l9 Service time 6:00 6:30 7:00 6:05 6:30 7:00 6:15 6:45 7:15 Required time
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5. Experimental Results The basic setting of weights for the evaluation function and the parameters (pop_size, max_gen) for genetic algorithms is given in Table 5.1. Based on the above settings, the population size was varied from 100 to 180. The average results over 10 runs for each case are reported in Table 5.2. We get a better fitness corresponding to population size of 120, in which we achieve the best value for all the attributes with little compromise on slightly more waiting time. The results show that when the population size is 100, the total fitness is best but the distance traveled is more and the number of vehicles increases by one. Based on the above settings, the weights of objective functions (1) were varied considering population size to be 140, as shown in Table 5.3 to investigate how they impact on the final decision. From the results we can see that emphasis on the objective of distance minimization will lead to a much better solution (with total fitness of 0.8017). The chromosome and the best route with arrival time corresponding to each setting have been displayed in Table 5.4. The second column represents the chromosomes as a sequence of customers generated randomly. The third column gives the number of vehicle used to serve the customers. The last column shows the path which each vehicle will follow starting from depot, satisfying the time windows of the customers and coming back to depot.
Table 5.1: Basic Setting of Function parameters and GA parameters
ρ1 0.5
ρ2 0.3
ρ3 0.1
ρ4 0.1
Pop_size
max_gen
100
500
WT 4.695 5.450 4.232 6.065 2.638
DT 47.6 43.4 44.2 43.4 47.6
Table 5.2: Results with varied pop size Pop_size 100 120 140 160 180
Chromosome [8 1 4 9 6 3 5 2 7] [5 2 8 4 6 7 1 3 9] [4 5 9 1 2 6 3 8 7] [9 5 8 2 4 7 3 6 1] [4 6 3 9 7 5 2 1 8]
SG 0.2153 0.1580 0.1943 0.3385 0.3427
FS 7 6 6 6 7
TF 0.8229 0.7820 0.7726 0.7564 0.7869
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Table 5.3: Results with varied weights SettIngs 1 2 3 4 5
ρ1
ρ2
ρ3
ρ4
SG
WT
DT
FS
TF
0.85 0.05 0.05 0.05 0.50
0.05 0.85 0.05 0.05 0.30
0.05 0.05 0.85 0.05 0.10
0.05 0.05 0.05 0.85 0.10
0.1744 0.2788 0.3052 0.3594 0.1943
4.5489 6.6338 3.9874 5.3913 4.232
43.4 48.4 44.2 44.2 44.2
6 6 6 6 6
0.8017 0.7221 0.6632 0.4282 0.7726
•
SG – Satisfaction grade
FS - Fleet size (# of Vehicles)
•
WT - Waiting time (in minutes)
TF - Total fitness
•
DT – Distance traveled (in kms)
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Table 5.4: Results with chromosomes and corresponding routes Settings Chromosome Pop_size [customers] 1
[8 1 4 9 6 3 5 2 7]
100
2
[5 2 8 4 6 7 1 3 9]
120
3
[4 5 9 1 2 6 3 8 7]
140
4
[9 5 8 2 4 7 3 6 1]
160
5 180
[4 6 3 9 7 5 2 1 8]
Fleet Vehicle Routes and Time at which the size (k) customer is serviced (0 indicates depot) 7 k=1 {0 5 8 0} & [4:00 6:12 5:12 ] k=2 {0 1 0} & [4:00 5:43 ] k=3 {0 4 7 0} & [4:00 5:03 6:35 ] k=4 {0 9 0} & [4:00 4:54 ] k=5 {0 6 0} & [4:00 5:10 ] k=6 {0 3 0} & [4:00 4:50 ] k=7 {0 2 0} & [4:00 6:24 ] 6 k=1 {0 5 8 0} & [4:00 5:09 5:59 ] k=2 {0 2 4 0} & [4:00 5:15 5:52 ] k=3 {0 6 0} & [4:00 4:55 ] k=4 {0 3 7 0} & [4:00 6:01 4:08] k=5 {0 1 0} & [4:00 5:36] k=6 {0 9 0} & [4:00 5:38 ] 6 k=1 {0 4 7 0} & [4:00 4:59 6:38 ] k=2 {0 5 8 0} & [4:00 5:29 6:09 ] k=3 {0 9 0} & [4:00 5:24 ] k=4 {0 1 0} & [4:00 6:00 ] k=5 {0 2 6 0} & [4:00 5:13 6:02 ] k=6 {0 3 0} & [4:00 5:10 ] 6 k=1 {0 9 0} & [4:00 4:27 ] k=2 {0 5 8 0} & [4:00 4:44 6:16 ] k=3 {0 2 4 0} & [4:00 5:18 6:37 ] k=4 {0 3 7 0} & [4:00 6:10 4:09 ] k=5 {0 6 0} & [4:00 4:18 ] k=6 {0 1 0} & [4:00 5:31 ] 7 k=1 {0 4 7 0} & [4:00 5:25 6:33 ] k=2 {0 6 0} & [4:00 5:08 ] k=3 {0 3 0} & [4:00 5:33 ] k=4 {0 9 0} & [4:00 5:06 ] k=5 {0 5 8 0} & [4:00 5:34 6:37 ] k=6 {0 2 0} & [4:00 5:53 ] k=7 {0 1 0} & [4:00 6:01 ]
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Graphical Representation: The first network diagram below shows that 9 customers are served by 9 different vehicles whereas the second network shows that the use of MOFVRP can reduce the number of vehicles to 6 with a little compromise on distance traveled and the customer’s satisfaction grade hence saving the cost of three extra vehicles. The graphical representation shows the variation in different objectives corresponding to the total fitness if the population size is taken to be 80.
e(6) d(4) c(8)
d(4)
12.5
f(7)
10
c(8)
12.5 b(2)
a(5)
10
g(1) a(5)
1.7
f(7)
25
g(1)
3.4
h(3)
16.8
b(2)
20
11.7
23.4
8.4
h(3)
9.2
e(6)
10.8
i(9)
21.6 i(9)
Chromosome S=(a,b,c,d,e,f,g,h,i)
Optimal routing
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5. Conclusions This algorithm provides many compromised strategies to College management which helps them taking a decision in any given situation. Looking at that day’s need and situation they can very well plan the priorities of the goals. Reducing the fleet size really helps the management in saving Institution’s transportation cost. At present College is running 10 buses to meet the requirement whereas our study shows that the number can be easily brought down to 6 or 7 buses thereby saving the cost, fuel charges, maintenance expenses and driver’s salary of at least 3 buses. The solutions given above are the best solutions under given circumstances from management’s point of view. In case any visiting faculty or VIP demands an early service or doesn’t want to wait at all, an extra vehicle can be arranged for that person which will cost more to the management but customer’s satisfaction grade will be high that will help the management in retaining the good faculties and future business.
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References [1] Baran, B. & Schaerer, M. (2003). A Multiobjective Ant Colony System for VRP eith Time Windows. Proceedings of the 21st IASTED International Conference on Applied Informatics, IASTED, Innsbruck, Austria, February 2003, 97-102. [2] Bowerman, B., Hall, B. & Calamai, P. (1995). A Multiobjective Optimization Approach to Urban School Bus Routing: Formulation and Solution Method. Transportation research – Part A, 29 A, 107-123. [3] Braysy, O. & Gendreau, M.(2005). Vehicle Routing Problems with Time Windows, part I: Route Construction and Local Search Algorithms. Transportation Science, 39(1), 104-118. [4] Brad, J., Kontoravdis, G. & Yu, G. (2002). A Branch-and-Cut Procedure for the Vehicle Routing Problem with Time Windows. Transportation Science, 36(2), 250269. [5] Bullnheimer, B., Hartl, R. & Strauss, C. (1997). Applying the Ant System to the Vehicle Routing Problem. Paper presented at 2nd International Conference on Metaheuristics, Sophia-Antipolis, France. [6] Burchett, D. & Campion, E. (2002). Mix Fleet Vehicle Routing Problem – An application of Tabu Search in the Grocery Delivery Industry. Research Paper, University of Canterbury, New Zealand. [7] Dantzig, G.B. & Ramser, J.H. (1959). The Truck Dispatching Problem. Management Science, 6, 80-91. [8] Fisher, M. (1994). Optimal Solution of Vehicle Routing Problems Using Minimum K-Trees. Operations Research, 42, 626-642. [9] Foulds, L., Wallace, S., Wilson, J. & Sagvolden, L. (2001). Bookmobile Routing and scheduling in Buskerud County, Norway. Proceedings of the 36th Annual Conference of the Operational Research Society of New Zealand, Christchurch, New Zealand , December 2001, 67-77.
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[10] Gen, M. and R. Cheng, “Genetic Algorithms and Engineering Design”, John Wiley and Sons, 1996. [11] Golden, B., Assad, A. & Wasil, E. (2002). Routing Vehicles in the Real World: Applications in the Solid Waste, Beverage, Food, Dairy, and Newspaper Industries. In The Vehicle Routing Problem, Toth, P. and Vigo, D. (eds), SIAM Monographs on Discrete Mathematics and Applications, Philadelphia, 245-286. [12] Kelnhofer-Feeley, C., Hu, Q., Chien, S. Spasovic, L. & Wang, Y. (2001). A Methodology for Evaluating of School Bus Routing: A case Study of Riverdale, new Jersey, Transportation Research Board 80th Annual Meeting, Washington DC, January 2001. [13] Lee, E., & Mitchell, J. (1998). Branch and Bound Methods for Integer Programming. The Encyclopedia of Optimization, Kluwer Academic Publishers. [14] Lin, J., “A GA-based Multi-objective Decision Making for optimal Vehicle Transportation”, Journal of information science and engineering, 24: 237-260, 2008. [15] Machado, P., Tavares, J., Pereira, F. && Costa, E. (2002). Vehicle Routing Problem: Doing it the Evolutionary way. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’2002), 690. [16] Nurcahyo, G., Alias, R. Shamsuddin, S. & Sap, M. (2002). Vehicle Routing Problem for Public Transport: A Case Study. Proceedings of the 2002 International Technical Conference On Circuits/Systems, Computers and Communications, Phuket, Thailand, 1180-1183. [17] Ortega, F., & Wolsey, L. (2003). A Branch-and-Cut Algorithm for the singleCommodity, Uncapacitated, Fixed Charge Network Flow Problem. Networks, 41(3), 143-158. [18] R. Gupta, B. Singh and D. Pandey, Genetic Algorithm Based Vehicle Routing Problems: The State of the Art, Vikram Mathematical Journal, Ujjain, India, 26 (2006), 89-110. [19] R. Gupta, B. Singh and D. Pandey, A Genetic Approach for Fuzzy Vehicle Routing Problems with Time Windows, communicated to Indian Academy of Mathematics, Indore, India, (2009).
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[20] R. Gupta, B. Singh and D. Pandey (2009). Fuzzy Vehicle Routing Problem with Uncertainty in Service Time, International Journal Of Contemporary Mathematical Sciences, Bulgaria, 5(11), 2009, 497-507. [21] R. Gupta, B. Singh and D. Pandey, Time Dependent Vehicle Routing Problem with stochastic Demand and Undefined Service Time, Varahmihir Journal of Mathematical Sciences, Sandipani Academy, Ujjain, India, 8(2), 2008, 51-60. [22] Reimann, M., Doerner, K. & Hartl, R. (2003). Analyzing a unified Ant System for the VRP and some of its Variants. S. Cagnoni et al (eds): EvoWorkshops Springer-Verlag Berlin Heidalberg 2003, LNCS 2611, 300-310. [23] Restori, M. (2004). An Application of VRP Algorithms with Original Modifications. Presented paper at the IIE International Conference in Houston, TX in May, 2004. [24] Stewart, P. & Tipping, J. (2002). Scheduling and Routing Grass Mowers Around Christchurch. 37th Annual ORSNZ Conference, University of Auckland, NZ, November (2002). [25] Thangiah, S., Wilson, B. Pitluga, A. & Mennell, W. (2004). School Bus Routing in Rural School Districts. Artificial Intelligence and Robotics lab. Computer Science Department, Slppery Rock University, Slippery Rock, PA. [26] Toth, P. & Vigo, D. (2002). The Vehicle Routing Problem. SIAM, Philadelphia. Received: November, 2009
