Y. Tan, Y. Shi, and K.C. Tan (Eds.): ICSI 2010, Part I, LNCS 6145, pp. 523–532, 2010. © Springer-Verlag Berlin Heidelberg 2010

Solving Vehicle Assignment Problem Using Evolutionary Computation

Marina Yusoff1, Junaidah Ariffin2, and Azlinah Mohamed1

1 Faculty of Computer and Mathematical Sciences Universiti Teknologi MARA 40450 Shah Alam, Selangor

Malaysia {marinay,azlinah}@tmsk.uitm.edu.my

2 Flood-Marine Excellence Centre Faculty of Civil Engineering Universiti Teknologi MARA 40450 Shah Alam, Selangor

Malaysia junaidahariffin@yahoo.com

Abstract. This paper examines the use of evolutionary computation (EC) to find optimal solution in vehicle assignment problem (VAP). The VAP refers to the allocation of the expected number of people in a potentially flooded area to various types of available vehicles in evacuation process. A novel discrete particle swarm optimization (DPSO) algorithm and genetic algorithm (GA) are presented to solve this problem. Both of these algorithms employed a discrete solution representation and incorporated a min-max approach for a random initialization of discrete particle position. A min-max approach is based on minimum capacity and maximum capacity of vehicles. We analyzed the per-formance of the algorithms using evacuation datasets. The quality of solutions were measured based on the objective function which is to find a maximum number of assigned people to vehicles in the potentially flooded areas and cen-tral processing unit (CPU) processing time of the algorithms. Overall, DPSO provides an optimal solutions and successfully achieved the objective function whereas GA gives sub optimal solution for the VAP.

Keywords: Discrete Particle Swarm Optimization, Evacuation Process, Evolutionary Computation, Genetic Algorithm, Vehicle Assignment Problem.

1 Introduction

Manual process of vehicle assignment adopted by the National Security Council (NSC) of Malaysia had reported uneven distribution of transport, non-timely assis-tance, and poor coordination at operational level [1]. The above is evident during one of the worst flash floods in Kota Tinggi district in Johor State of Malaysia where more than ten thousand flood victims need to be evacuated to the relief centers. A solution to the above problem is vital to assist the NSC and other related agencies in managing and monitoring the evacuation process. This paper addresses the solution

524 M. Yusoff, J. Ariffin, and A. Mohamed

for the assignment of vehicle to the expected number of people at the potentially flooded areas using datasets from selected flash flood events.

Evolutionary computation (EC) algorithms for solving a combinatorial problem have been reported in various types of assignment problem [2][3]. Most of them com-pare discrete particle swarm optimization (DPSO) and genetic algorithm (GA) to obtain solution. In most cases DPSO provides better solution in comparison to genetic algorithm (GA). Limited research was found related to vehicle assignment problem (VAP) in emergency evacuation processes [4]. In previous investigations several approaches; mathematical method, exact algorithms, and heuristic algorithms [5] have been used to solve only evacuation process but fail to consider assignment of vehicles in solving evacuation processes. Nevertheless, DPSO has given near optimal solution to VAP using min-max approach based on minimum capacity of vehicles and average capacity of vehicles using small data sets [6]. This paper would however introduces the use of min-max approach with emphasis on the minimum capacity of vehicles and maximum capacity of vehicles.

It was discovered that GA is very much competitive to that of particle swarm opti-mization (PSO) [3]. Two evolutionary computation (EC) algorithms namely DPSO and GA are addressed to seek for VAP solution. Solutions obtained through DPSO and GA with respect to their fitness value and CPU processing time will be discussed. Fitness value refers to the maximum number of people that were assigned to vehicles divided by the number of expected people at each potentially flooded area. Details of the calculation of the fitness value are explained in Section 3.

The organizational structure of this paper is as follows. Section 2 presents PSO and the novel DPSO is discussed in Section 3. Discussion on the computational results are made in Section 4. In Section 5 we discuss the analysis of results of the algorithms. Section 6 presents conclusion and recommended future work.

2 Particle Swarm Optimization

PSO is a population-based stochastic approach grouped under swarm intelligence [7] to solve continuous and discrete problems. It was introduced by Kennedy and Eber-hart [8]. PSO indicates the velocity and position of particles in multi-dimensional space. PSO is able to explore regions of the search space and exploit the search to refine a feasible solution. Many researchers have demonstrated the advantages of using PSO to solve several types of problems [2][9][10][11].

Improvements for PSO have been developed to measure the performance of PSO for various types of problems [12][13] and across standard benchmark datasets [10][12][14]. For example, the canonical PSO applies inertia weight in updating veloc-ity to simulate the social behavior of birds. After two years of the PSO development, a discrete problem had concentrated on the initial work involving discrete binary PSO introduced by [15]. They proposed a new way of updating the position of particles to accommodate a discrete binary problem. This approach was then improved in several studies based on a benchmark dataset [16][17] and a real-world situation [2][18] of the discrete problem. When compared to other optimization methods, the performance of PSO is competitive to the genetic algorithm [2]. This demonstrates that PSO, with its

Solving Vehicle Assignment Problem Using Evolutionary Computation 525

global search capability and local exploitation, is a promising approach to finding the optimum solution and, therefore, it should be further investigated.

3 A Novel Discrete Particle Swarm Optimization

Discrete particle position from [6] is adopted to represent the number of vehicles for each potentially flooded area. The initialization of these values is randomly generated based on the expected number of vehicles allocated for the flood victims from the potential flooded areas. We introduced min-max approach for the discrete particle position generation as demonstrated in Equation 1, Equation 2, and Equation 3. The minimum discrete position values were initialized taken into account the minimum capacity whereas the maximum discrete position value refers to the maximum capac-ity of vehicles as shown in Equation 1 and Equation 2, respectively. Equation 3 shows the calculation of the initial discrete particle position.

Pmin = T / Cmin (1)

Pmax = T / Cmax (2)

where, Pmin is the minimum position value, Pmax is the maximum position value, T is the total number of people in the potentially flooded area, Cmin is the minimum capac-ity of vehicle, and Cmax is the maximum capacity of vehicle.

D = Pmin + r (Pmax - Pmin) (3)

where D is the initial discrete position value for DPSO (genes in GA), and parameter r is the random value in the range of [0,1]. DPSO algorithm as illustrated in Figure 1, starts with the initialization of the number of particles. Particles are represented in the form of matrices, Xij, where i is the number of dimension of particle or the size of swarm and j is the number of the potentially flooded area. Step 3 declares parameters; W is the inertia weight and C1 and C2 are the acceleration constant parameters. Step 4 initializes velocity, V, using minimum velocity (Vmin) and maximum velocity (Vmax) of each particle using Equation 4. Step 5 involves the initialization of discrete position value using Equation 3 as mentioned in Section 2.3. The calculation of Pbest is shown in step 8 and it is referring to Equation 5.

V= Vmin + r (Vmax - Vmin ) (4)

Pbest = ta / tl (5)

where, ta is the number of people assigned to the respective vehicles, meanwhile tl is the total number of people in all potentially flooded area. Pbest is the personal best of the ith particle, and Gbest is the fitness value from the best position derived from all particles in the swarm. Line number 11 and 12 are the calculation of new velocity value and new discrete particle position using Equation 6 and Equation 7, respectively [19].

Vij(new) = W x Vij(old) + C1 x r (Pbest - Xij(old)) + C2 x r x (Gbest -Xij(old)) (6)

526 M. Yusoff, J. Ariffin, and A. Mohamed

Xij(new) =Xij(old) + Vij(new) (7)

The next three lines, tries to find discrete particle positions that will lead to the best value of Gbest. Lines17 to 20 are steps for the decision of the assignment of people to vehicles depending on the selected discrete particle position. Line 20, involves assign and re-assigned people until all the discrete particle position are allocated to vehicles. Finally, Pbest (new) and Gbest (new) are determined. The number of iteration of these algo-rithm starts from line 9 until line 23 and stop upon satisfying the stopping condition.

1.Begin2. Initialize number of particles3. Declare W, C1 and C24. Initialize Vmin and Vmax and calculate Velocity, V using a

random initialization5. Initialize the available vehicles with a random capacity6. Initialize D using min-max approach7. Assign vehicle according to the initialize discrete

particle position, D8. Calculate Pbest and Gbest value for each particle9. Do10. For each particle11. Calculate V(new)12. Calculate D(new)13. If Gbest (new) is more than or equal Gbest (old)14. employ discrete particle position, D15. else16. employ discrete particle position, D (new)17. If (D is equals or less than the available vehicles)18. Assign people to vehicles according to the number of

the employed discrete position, D19. else20. Assign and re-assigned people until all the employed

discrete particle position are allocated to vehicles

21. Calculate Pbest (new)22. Calculate Gbest (new)23. While (stopping condition is reached)24.End

Fig. 1. A Novel DPSO Algorithm

4 Computational Results

This section presents the computational results to examine the quality of solution of the DPSO algorithm and GA in VAP. The quality of solution on two aspects were evaluated; fitness value and CPU processing time. The following sub sections discuss the experiment setup and results of DPSO and GA according to the aspect of the quality of solutions.

Solving Vehicle Assignment Problem Using Evolutionary Computation 527

4.1 Experimental Setup

Table 1 shows the list of parameter for this experiment. The inertia weight of 0.9 was selected based on the recommendation of [20]. Statistical analysis were used to con-firm the result inclusive of descriptive measurement; average, standard deviation, minimum and maximum. A paired t-tests were used to analyze the relationship be-tween DPSO algorithm and GA based on 95 percents of confidence level (α=0.05). The dataset was gathered from the major flooded areas in Kota Tinggi district in Johor state that were badly affected in the December 2006 and January, 2007. Table 2 de-picts the number of flooded areas and its total number of flood victims. Various types of vehicles of different standard capacities were used in the evacuation operation. Sixty seven of the available vehicles with a total of 650 in capacity were identified to be used before and during flash flood in this district.

Table 1. List of parameters

Parameter Value W 0.9 C1 2.5 C2 1.5 Available vehicles 67 Initial Vmin -4 Initial Vmax 4 Sequence of vehicle capacity Random Stopping condition while (Gbest < 0.6)

Table 2. List of flooded areas and its flood victims’ number

Flash flood event Number of flood victims Number of flooded areas

December , 2006 13112 26

January, 2007 5352 35

4.2 Performance of DPSO and GA According to Its Fitness Values

Figure 2 shows the fitness values tabulated from a different number of populations tested for DPSO and GA using dataset of December, 2006. It involves 26 flooded areas with a total of 13,112 flood victims. Generally, the scatter graph in Figure 2(a) and 2(b) present a similar pattern in fitness value for both GA and DPSO. It is appar-ent that the GA has resulted in the lowest value of fitness compared to DPSO. As can be seen in figures 2(a), 2(b), and 2(c) the fitness values are in the range of 0.82 to 0.94. This confirms that GA gives sub optimal solution for the VAP. Contrary to expectations, the fitness values generated using DPSO, in figure 2(a), 2(b), and 2(c) indicate that the performance of DPSO is better compared to GA for the three popula-tions. This result is significant with a significant value equals to 0.00 on paired t-test performance. A strong evidence of good solution to DPSO was found in both figures, 2(b) and 2(c). These figures demonstrate that the fitness values remained steady between 0.96 and 1.00. In summary, DPSO had given optimal solution with all

528 M. Yusoff, J. Ariffin, and A. Mohamed

(a)

(b)

(c)

Fig. 3. Performance of DPSO and GA based on the generated fitness value (a) 10 populations, (b) 20 populations, (c)30 populations

people successfully assigned to the vehicles at the potential flooded. An indication of the above is illustrated at the fitness value or Gbest equals to one as can be seen in figure 2(b) and 2(c). Based on the above results, it is recommended for DPSO to use 30 populations for optimal solution. Other results are shown in Table 3 to confirm this argument.

Table 3 demonstrates that the fitness value for DPSO gave similar results for the both datasets on all experiments. DPSO gives higher average of fitness value com-pared to GA The maximum value of GA obtained is 0.92 for December, 2006 and 0.91 for January, 2007. The good performance of DPSO to GA is supported by the

Solving Vehicle Assignment Problem Using Evolutionary Computation 529

results of a paired t-test with a significant value equal to 0.00 (α < 0.05). The overall performance of DPSO concludes that the objective function is achieved. The result from [6] had reported less than 0.95 for the average of fitness value. Thus, the use of min-max approach using minimum capacity and maximum capacity of vehicles in the initialization of discrete particle position gives better results with the assignment of larger number of people.

Table 3. Performance of DPSO and GA according to its fitness value

DPSO GA Dataset

Min Max Avg Std. dev

Min Max Avg Std. dev

December, 2006

0.97 1.00 0.99 0.01 0.83 0.92 0.86 0.03

January, 2007

0.97 1.00 0.99 0.01 0.82 0.91 0.87 0.02

4.3 Performance of DPSO and GA According to Its CPU Processing Time

This section presents the second aspect of quality solution which is CPU processing time. Both algorithms were tested to see its performance using the same data set as used in the previous section. There is an increase of CPU processing time for both GA and DPSO as shown in Table 4. This indicates that the number of people influences the CPU processing time. It can be seen from the table that the average score of CPU processing time for GA is fewer than DPSO for December, 2006. The minimum proc-essing time for DPSO is about 18 seconds while the minimum processing time for GA is about 12 seconds. However, DPSO gives better CPU processing time for January, 2007 in about 12 seconds.

Table 4. Performance of DPSO and GA according to its CPU processing time

DPSO with CPU processing time (milliseconds)

GA with CPU processing time (milliseconds)

Data set

Min Max Avg Std. dev

Min Max Avg Std. dev

December, 2006

18794 65328 55188.40 10632.76 12090 72195 40522.66 16651.44

January, 2007

8949 77248 12827.26 9792.59 10424 35370 27341.96 5533.78

5 Analysis of Results

This section discusses the analysis of the results demonstrated in the previous section. We divide the discussion into two perspectives; DPSO perspective and GA perspective.

530 M. Yusoff, J. Ariffin, and A. Mohamed

5.1 DPSO Perspective

The introduction of min-max approach establishes the initialization of minimum dis-crete particle position and maximum discrete position vehicles for each potentially flooded area yields a limitation in search space for each particle. This approach starts with the random initialization of discrete particle position as shown in equation 3. Prior to this initialization, the minimum discrete particle position and the maximum discrete position are calculated (see equation 1 and equation 2) for each potentially flooded area. Thus, the assignment of the expected number of people at each potential flooded area is within the range of the required number of vehicles. To obtain optimal solution (Gbest equals to 1), the generated discrete particle position must be able to accommodate the number of people with different capacity of vehicles. Re-assignment of the same vehicles is carried out with the aim to accommodate the dis-crete particle position. As illustrated in the above results, optimal solution can be obtained using 30 populations, but improvement is required for CPU processing time in conjunction with velocity clamping procedure.

The use of inertia weight equals to 0.9 proposed by [20], confirms to be suitable for the DPSO in solving the VAP for optimal solution. Inertia weight and random value in the range of [0,1] are used in the calculation of new velocity for each particle lead to a small range of the new generation of velocity. In addition, the initialization of random velocity also has an implication to the calculation of a new velocity value. A small range of velocity between 0.4 and -0.4, inertia weight, and random value (r) contributes to a small difference for the new generation of discrete particle position when compared to the discrete particle position generated in [6].

5.2 GA Perspective

Analyses have proven that GA produced sub optimal solution for vehicle assignment problem (VAP). The above is however contrary to the report by [3] that GA is able to give competitive results in comparison to DPSO in benchmark problems. The evi-dence is as demonstrated in Table 3 where the average of fitness values are 0.87 and 0.86 for December, 2006 and January, 2007, respectively.

Although GA applied the same solution representation as DPSO, GA theoretically has difficulty to obtain good solution. The crossover process has shown that the dis-crete particle position is not updated to a small range, and depends on the crossover point with changing range of discrete values. This can be illustrated in the following example, in an incident where gene is supposedly to be 6, where 6 vehicles are ex-pected to accommodate 40 people at the potential flooded area. During a crossover, this value may change to 2. With 2 vehicles, the new gene value will not be able to accommodate this people. However, this problem can be improvised with a hybrid of GA and DPSO to improve the results.

6 Conclusion and Future Works

This paper introduces a novel DSPO in solving the VAP. With the adoption of a min-max approach the DPSO provides an optimal solution for the VAP. Assignment of people at the potential flooded areas can be made possible using this approach.

Solving Vehicle Assignment Problem Using Evolutionary Computation 531

The findings show that the novel DSPO gives better solution compared to DPSO with average capacity [6] and GA. In general the proposed DPSO has shown significant performance while GA yields sub optimal solution. Nevertheless, CPU processing time needs to be improved to obtain a much faster results. This could be made possi-ble with the introduction of velocity clamping. Different parameter values should be considered in future research, and to be experimented on a large evacuation datasets.

Acknowledgements. We gratefully acknowledge the help of Prof. Engelbreght from the University of Pretoria. This study has been made possible by the support of the Ministry of Science and Technology Malaysia through the Science fund and Univer-sity Technology MARA.

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