vehicle routing problem application in tour planning ... · simple exchange heuristic was applied....

Vehicle Routing Problem Application in Tour Planning Multimedia System with Metaheuristic Designed Ruey-Maw Chen, Chuin-Mu Wang

Advances on Information Sciences and Service Sciences. Volume 3, Number 2, March 2011

Vehicle Routing Problem Application in Tour Planning Multimedia System with Metaheuristic Designed

1Ruey-Maw Chen, 2Chuin-Mu Wang

*1 National Chin-Yi University of Technology, [email protected] 2 National Chin-Yi University of Technology, [email protected]

doi:10.4156/aiss.vol3. issue2.1

Abstract This work suggests an adaptive tour planning multimedia system designed to meet the tour routing

needs of different tourists. Tour planning is an application of vehicle routing problem. Meanwhile, vehicle routing problem is a ‘traveling salesman problem’ (TSP) type problem. TSP is an example of a combinatorial optimization problem and is known as NP-hard. As studied by many researchers exact, heuristic (approximation) and meta-heuristic algorithms are usually applied for NP problems. A promising meta-heuristic algorithm is proposed to solve the tour routing problems which involves minimization of total traveling time, total travel expenses or the combined total traveling time and expenses. Moreover, a simple exchange local search heuristic is also applied to increase the exploitation (intensification) competence of the scheme. Restated, finding out the optimal planned tour routes using particle swarm optimization (PSO) and increasing performance by exchange heuristic scheme is suggested. Simulation results indicate the PSO with exchange heuristic designed provides a promising strategy, and is efficient for solving tour routing problems. Keywords: Tour Planning, Vehicle Routing Problem, Optimization, Particle Swarm Optimization,

Exchange Heuristic I. Introduction

Due to economic development, the tourism industry grows vigorously day by day. Along with the

demands of the people travelling becoming higher and higher, the governments of various countries also do their utmost to vigorously promote their tourism industries. To make appropriate arrangements for their journey, the collection of travel information (both on quality and quantity) by tourists has become a more and more important issue. Additionally, in tourism activities, the base-type model for the tour has become increasingly evident. For tourism activities in a recreation area, a useful or useless tour route planning can impact heavily on the tour quality. Hence, how the tour route is planed can save time, money, and ensures the tourist’s demands are met is an important issue. Restated, how the visitors centers or hotels located near tourist areas provide information on attractions (scenic spots) around that area is important to the tourist. Also, useful tourist information would help to improve service quality. Similarly, if the visitors center of a national park or popular tourist area can offer favorable tour recommendations or information to the sightseers that would assist in planning for their trip. Currently, most hotels or visitors centers provide certain tour suggestions such as one-day, two-day, multiple-days, or themed tour suggestions. However, that may be inappropriate for some visitors. Hence, to meet the needs of different tourists, an adaptive tour route planning multimedia system is proposed in this work. Restated, it would be very helpful if the trip planning was carried out in accordance with the passenger’s demands; but also helped to improve the quality of service. Therefore, how to supply useful tour-planning information to tourists became important for leisure tours. Restated, good tour planning information is required subject to the traveler’s requirements. Usually, the determined criteria for tour planning may be based on either the traveling time cost or traveling expense cost minimizations which are considered as an overall cost minimization problem with an associated objective function. Moreover, a multi-criteria objective function corresponding to the cost minimization problem is referred to take more objectives such as both traveling time and expense costs into account. Only some studies were conducted in obtaining the optimal tour route in which the tour length is minimized. Nevertheless, the most applied optimization technologies have focused on travelling by air. In Barnhart et al. [1], Graves et al. [2], Brusco et al. [3] and Hoffman and Padberg [4],

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they focused on the individual's schedule arrangement. Airline network design and frequency of the decision of the aircraft were investigated in Teodorovic et al. [5]. Hence, this study concentrates on solving tour route planning based on the tourist’s demands, which is a TSP-type problem. Meanwhile, TSP is a well known combinatorial problem and confirmed to be an NP-hard problem. Restated, to solve large-scale TSP problems is quite time consuming and impractical. Therefore, many versatile schemes are studied and proposed. Among those schemes, linear programming was a well applied algorithm used to solve the optimal solution, such as Barnhart et al. [1], Hoffman and Padberg [4], and Teodorovic et al. [5]. In other words, heuristic solution methods were applied to solve travel management issues. However, the heuristic method for solving optimization problem was confirmed to be problem specific and un-scalable. Meanwhile, exact algorithms (complete algorithms) fail when the dimension of the problem reaches a certain size. Therefore, meta-heuristic methods; upper level, strategical, scalable, and not problem specific schemes were proposed to solve a wide variety of optimization problems efficiently [6, 7, 8, 9, 10]. Restated, meta-heuristic algorithms have been validated to be an efficient scheme for solving combinatorial optimization problems such as TSP, vehicle routing problems and a variety of scheduling problems. Genetic algorithm (GA) is one of the meta-heuristic methods [8]. Hurley et al. [11] applied GA to investigate the location of tourist attractions to explore the decision-making issues. Hsieh et al. [12] also applied GA to solve adaptive customers’ tour scheduling with traveling time and money costs considered. Recently, ant colony optimization (ACO) and particle swarm optimization (PSO) technologies have become the two most popular meta-heuristic schemes. Hence, they are utilized to solve many TSP based problems. Tiwari1 et al. [13] solved the best process plan in an automated manufacturing environment using ACO. Doerner et al. [14] determined multi-criteria tour planning for mobile healthcare facilities by Pareto-ACO. Meanwhile, Lo et al. [15] also applied ACO with heuristic rules to solve resource-constraint processes scheduling problems. Gambardella, Rizzoli and Oliverio [16] searched for vehicle routing in advanced logistics systems with ACO applied. A discrete particle swarm optimization algorithm for generalized TSP was suggested by Tasgetiren, Suganthan and Pan [17]. Meanwhile, Shi et al. [18] proposed PSO-based algorithms to evaluate the TSP and generalized TSP. Ai and Kachitvichyanukul [19] proposed PSO for a vehicle routing problem with simultaneous pickup and delivery. Additionally, Chen et al. [20] use PSO with heuristic rules to solve RCPSP problems in PSPLIB.

In this study, the tour routing problem of a tour planning multimedia system can be viewed as a special class of the vehicle routing problem (VRP). Restated, in this work, the visitor’s center is viewed as a depot and the scenic spots correspond to the nodes in a vehicle routing problem. A survey of location-routing is presented as in Nagy and Salhi [21], in which related issues (including all variants of VRP), models and methods are given. However, only both exact and heuristic methods are investigated. Hence, the promising meta-heuristic algorithm PSO is given in this work to solve the tour route planning problem. The objectives of this work include minimizing the total travel time, minimizing total travel expenses or minimizing the combined total traveling time and expenses to find the optimal planned tour route. Additionally, to increase the performance of the suggested scheme, a simple exchange heuristic was applied. The exchange heuristic is a local search scheme similar to 2-opt, which enhances the exploitation (intensification) ability in solution space so as to raise performance in discovering an optimal solution. To verify the effectiveness and efficiency, three simulation scenarios were conducted. The simulation tour route planning problem listed in Hsieh et al. [12] was first tested. Then, a tour route planning problem involving more attractions for testing was generated using the random graph generator of Waxman. Finally, a tour planning which includes 17 scenic spots around popular Kenting tourist area in Taiwan was simulated.

This article is organized as follows. Section 2 presents the studied tour routing problem. In Section 3, the PSO model and exchange heuristic are introduced and applied to solve tour route planning problems. Experiment results are demonstrated and compared with the results from the genetic algorithm in Section 4. Finally, Section 5 gives conclusions and discussions. 2. Tour routing problems

As stated above, the tour routing problem involves finding out the optimal planned tour which has minimum traveling time, minimum traveling expenses, or minimum combined traveling time and

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expenses. Meanwhile, the investigated tour route planning problem is viewed as a vehicle routing problem. Restated, a hotel or recreation service center is regarded as the depot, the scenic spots are considered as the customers (nodes); there are n attractions (scenic spots) near the hotel assumed. Therefore, the tour route planning problem can be configured as finding the best path in a graph. Mathematically, this problem is described as a weighted graph G=(V, E) where the vertices (nodes, spots) are represented by V={v1,v2,...,vn}, and the edges are represented by E={(vi, vj), i≠j}. The v1 represents the hotel, which is the assumed starting point and end point of each tour route. The dij associated with each edge indicates the distance of (vi, vj), which is measured by Euclidean distance between these vertices. For the first situation of minimizing the total traveling time (T), the fitness function can be defined as follows.

rouetetourtheonarejspotandispot

VjiTTTT n

n

i

n

ijj

ij ,,,1122 ,2

(1)

Where Tij indicates the traveling time from spot i to spot j. T12 and Tn1 denote the time from hotel to the first attraction and the time form the final attraction back to hotel respectively. For instance, suppose in a certain recreational area surrounding a hotel operators have five well-known tourist attractions as indicated in Table 1, Hsieh et al. [12]. Table 2 shows the estimated traveling time among attractions surrounding the hotel (symmetric assumed).

Table 1. Attractions near a hotel Spots 1 2 3 4 5 6

Name Hotel Downtown Beach Coral sea

cliffs Ecological park Aquarium

Table 2. estimated traveling time between attractions Attractions 1 2 3 4 5 6

1 0 70 65 30 20 752 70 0 20 25 40 553 65 20 0 20 20 454 30 25 20 0 30 405 20 40 20 30 0 356 75 55 45 40 35 0

The second problem of minimizing the traveling expense (C), and the corresponding fitness function

is defined as listed in Eq. (2)

rouetetourtheonarejspotandispot

VjiCCCC n

n

i

n

ijj

ij ,,,1122 ,1

(2)

Where the Cij is the traveling expense from spot i to spot j. C12 and Cn1 denote the expense from hotel to the first attraction and the expense form the final attraction back to hotel respectively. The estimated traveling expenses among attractions are listed in Table 3.

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Table 3. Estimated traveling expenses among attractions

Attractions 1 2 3 4 5 6 1 0 40 40 20 15 45 2 40 0 15 20 30 35 3 40 15 0 17 17 37 4 20 20 17 0 28 33 5 15 30 17 28 0 35 6 45 35 37 33 35 0

Additionally, the objective function (O) corresponding to combined traveling time and traveling

expenses minimization is designed as follows.

rouetetourtheonarejspotandispotVji

CCCwTTTwO n

n

i

n

ijj

ijn

n

i

n

ijj

ij

,,

),()( 1122 ,2

21122 ,2

1

(3)

Where, w1 and w2 are weighting factors. Restated, the fitness function is the weighted sum of traveling time and traveling expense. Hence, different tour route planning can be determined based on the tourist’s demands through the adjustment of w1 and w2.

3. Particle swarm optimization

In this section, particle swarm optimization and its variant discrete particle swarm optimization algorithms are introduced.

3.1. Particle swarm optimization review

Particle swarm optimization (PSO) was first proposed by Kennedy and Eberhart [10]. It is a multi-agent general meta-heuristic, and can be applied extensively in solving many complex problems. Eberhart and Shi [22] have demonstrated the uniqueness of PSO such as easy implementation and consistency in performance. Recently, PSO has been widely used in the areas where genetic algorithm was used previously since some similarities exist between GA and PSO.

PSO consists of a swarm of particles in a space; the position of a particle is indicated by a vector which presents a solution. PSO is initialized with a population of randomly positioned particles and searches for the best position with best fitness.

In each generation or iteration, every particle moves to a new position and the new position is guided by the velocity (which is a vector), then the corresponding fitness of the particles is calculated. Therefore, velocity plays an important role in searching a solution with better fitness. There are two experience positions used in PSO for updating the velocity; one is the global experience position of all particles, which memorizes the global best solution obtained by all particles; the other is each particle’s individual experience, which memorizes the best position that particle has been moved to. These two experience positions are used in determining the velocity.

Let an N dimension space (the number of dimensions is typically concerned with the definition of the problem) have M particles. For the ith particle (i=1,…,M), its position consists of N components Xi={ Xi1,…, XiN }, where Xij is the jth component of the position. And the velocity of particle i is Vi={Vi1,…,ViN }, particle individual experience is Li={ Li1,…,LiN } denoted by “pbest”. Additionally, G= {G1,…,GN } represents the global best experience shared among all the particles and represented by “gbest”. When updating the jth component of the position and velocity of the ith particle we refer to the equation as shown in Eq. (3).

newnew

new

ijijij

ijjijijijij

VXX

XGrcXLrcVwV )()( 2211 (4a) (4b)

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Where w is an inertia weight used to determine the influence of the previous velocity on the new velocity. The c1 and c2 are learning factors used to derive how the ith particle approaches the position close to the individual experience position or global experience position respectively. Furthermore, the r1 and r2 are random numbers uniformly distributed in interval [0, 1], influencing the tradeoff between the global (swarm’s best experience) and local (particle’s best experience) exploration abilities during search. In PSO the balance between the global and local exploration abilities is mainly controlled by the inertia weights. Suitable selection of the inertia weight w can provide a balance between global and local exploration abilities and thus require less iteration on average to find the optimum. According to Eq. (4b), the particle flies toward a new position. Basically, the operation of generating a new position is similar to the mutation operation of evolutionary computation. Moreover, velocities should not be so high as to cause particles to diverge. Restated, to cap velocities to some maximum is required to prevent overflow, i.e., Vhi∈[Vmin,Vmax]. The procedure of the particle swarm optimization is as follows.

Algorithm: The pseudo-code of PSO with exchange heuristic process

Initialization X and V for each iteration do

for each particle i in the swarm do update new velocity V using Eq. (4a)

subject to Vhi∈ [Vmin, Vmax] update new position new

iX using Eq. (4b)

calculate particle’s fitness f( new

iX )

update pbest (Li) and gbest (G)

exchange heuristic yields new

iX̂

calculate particle’s fitness f( new

iX̂ )

update pbest (Li) and gbest (G) end for

end for generates optimal tour route (π) on the basis of gbest end

3.2. Encoding tour route ordering

In PSO, the position of a particle is indicated by a vector which presents the solution. The first step to applying PSO is the solution representation, i.e., how to map the position vector to the solution is significant for the PSO process. Since the tour route planning problem can be solved by ordering arrangement, the scheme for encoding tour route ordering into the position vector is necessary.

For example, there are 5 attractions (numbered from 1 to 5) surrounding the hotel (denoted number 0); thus the determined tour route could be 2-3-1-5-4. The position vector X has 5 components which are real values for conventional PSO. Restated, PSO is a continuous domain searching algorithm, but in this paper the tour route is discrete. Hence, a mechanism has to be applied to transfer real values into discrete numbers during the process of iteration. In this work, we first simply map real values to integers by ranking the values between 1 and 5. If the result of iteration of position vector X before sorting is

X: 2.4 3.7 1.6 7.2 4.5 After sorting the order is 1.6(1), 2.4(2), 3.7(3), 4.5(4), and 7.2(5). Then the attraction order of the

tour route is as follows

Tour route π: 2 3 1 5 4

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Restated, the recommended tour route ( π ) is 0-2-3-1-5-4-0. Accordingly, the corresponding

objective function (Eq. (1), (2), or (3)) is minimized.

3.3. Exchange Heuristic To refine the performance of the PSO scheme, a simple exchange heuristic was applied. The

exchange heuristic is a local search method similar to the 2-opt or 3-opt local search methods, which promotes the exploitation (contrast to the exploration) ability in a solution neighborhood and raises the probability of obtaining an optimal solution. The property of the well known classical 2-opt exchange heuristic [23], as improvement heuristic, is to modify the current solution by replacing 2 arcs in the tour by 2 new arcs, so as to generate a new improved solution. Exchange heuristics are widely used to improve vehicle routing solutions. In this work, we combined a simple exchange heuristic to improve pbest and gbest based on the best-improvement rule so as to produce a superior tour route. Suppose the set of current solved tour (consisted of n nodes) by particle i is Ti={t1, t2, …, tn}. There are two randomly generated integers ri and rj, 2≦ri, rj≦n. Hence, a new solution is produced by simply interchanging ti and tj when i≠j. For instance, the determined tour route of a particle is as follows.

Tour routeπ: 2 3 1 5 4

The corresponding fitness function is )(f . Assume that the two randomly generated numbers are 2

and 4, and then a new tour route after exchange would be

Tour route π’: 2 5 1 3 4

The fitness function for the new tour route is )( f . The pbest and gbest are then updated

accordingly, when )( f < )( f .

4. Simulation cases and results

The parameters setting used in the simulation are Vmax=4, Vmin=-4, c1=1, c2=1, and w=0.7.

Meanwhile, there are 20 particles used in the simulation. The simulation tour route planning problem listed in Hsieh et al. [12] was first tested. Those cases are displayed in Tables 1, 2, and 3. Various situations of tourist demand were experimented; first: traveling time minimization, second: traveling expense minimization, third: weighted traveling time and traveling expense minimization. Meanwhile, to compare the efficiency of the PSO algorithm and the exchange local search, 50 runs were simulated with 500 iterations for each run. The average deviation was calculated based on the simulation results of the 50 tries. The average deviation is defined as follows.

%100

optimum

optimumaveragedeviationAverage (5)

In Eq. (5), the average indicates the average fitness of 50 tries; the optimum is the minimum fitness

obtained in simulation for the simulated case. The experimented outcomes are shown in Table 4. According to the experiment results, the optimal results yield by GA [12] can also be obtained using PSO based schemes.

Table 4. Simulation results of different schemes GA [12] PSO PSO w/ exchange Min T 175 175 175 Min C 135 135 135 Min O 317 317 317

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Meanwhile, situation 4 in [12] was tested to verify the adaptation of the algorithm to the tourist’s

demands. In which, the tourist designates the no. 6 spot to be the first scenic spot to visit. The simulation results of applying proposed schemes in this work, say PSO and PSO with exchange heuristic is displayed in Table 5.

Table 5. Simulation results of applying PSO with and without exchange heuristic Demand Minimum

fitness Average Fitness

Average deviation

w/oexchange

w/exchange

w/oexchange

w/ exchange

Min T 200 200.8 200.2 0.4% 0.1% Min C 145 145.42 145 0.29% 0% Min O 345 346.92 345.48 0.557% 0.139%

To further verify the efficiency of applying exchange heuristic, a tour route planning problem

involving more attractions for testing was generated using the random graph generator of Waxman [24]. The generator randomly distributes 10 nodes over a rectangular area with size 20×20 as listed in Table 6. The cost of each edge is set to the rounded distance between its corresponding nodes. The node 1 is assumed to be the start and stop point. Experiments were conducted on three cases; the Case 1 tour route is to visit all attractions. The simulation result of the shortest tour route is 1-10-4-7-6-8-2-9-3-5-1 (1-5-3-9-2-8-6-7-4-10-1) and the associated shortest distance is 43. Case 2 only chose 4 attractions, in this case spots 2, 3, 6, and 9 to visit. The obtained optimal rout with the shortest distance (33) is 1-3-9-2-6-1 (1-6-2-9-3-1). Meanwhile, the first visiting attraction was designated to be the node 2. Then the other 3 attractions, 3, 6, and 9, are selected to be visited as Case 3. The simulation results of Case 3 have the shortest tour route 1-2-6-9-3-1 (1-2-3-9-6-1) with distance 40. The computed average deviations of the results with and without exchange heuristic applied are listed in Table 7. The results of applying exchange heuristic are superior to that obtained without heuristic applied as demonstrated.

Table 6. Random generated by Waxman (1988) x-coord. y-coord. 1 2 3 4 5 6 7 8 9 10

8.8443 6.7351 1 0 7 4 8 3 11 8 11 5 2 6.6774 13.124 2 7 0 5 14 4 12 12 11 3 8 5.3301 7.897 3 4 5 0 11 3 14 12 13 3 5 15.1 2.0135 4 8 14 11 0 10 10 4 11 13 6 8.3259 9.2383 5 3 4 3 10 0 11 9 10 3 4 19.009 11.127 6 11 12 14 10 11 0 6 2 14 10 16.732 6.0465 7 8 12 12 4 9 6 0 7 12 6 17.918 12.615 8 11 11 13 11 10 2 7 0 13 10 5.1056 10.486 9 5 3 3 13 3 14 12 13 0 7 10.422 6.1778 10 2 8 5 6 4 10 6 10 7 0

Table 7. Simulation results of applying PSO with and without exchange heuristic Shortest

distance Average distance

Average deviation

w/o exchange

w/ exchange

w/o exchange

w/ exchange

Case 1 43 47.04 43.6 9.39% 1.4% Case 2 33 35.18 34.40 6.61% 4.24% Case 3 40 40.74 40.26 1.85% 0.65%

Furthermore, the third scenario which includes 17 scenic spots around popular Kenting tourist area

in Taiwan was simulated. The supposed simulation situation was based on a customer’s one-day tour requirements. The tour planning requirements are as follows; starting and ending at Youth activity center, the first spot to visit is the Kenting forest recreation area where customers can absorb phytoncid and exercise in the morning; the last attraction is Guanshan where they can see the beautiful evening

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glow before returning to the lodge. Meanwhile, there are four scenic attractions planned to visit; they are Jialeshuei, Chuhuo, Eluanbi, and Maobitou spots. The simulation results are displayed in Fig. 1. Figure 1(a) displays all possible routes; Figs. 1(b) and 1(c) demonstrate the obtained other non-shortest route orders. Figure 1(d) shows the shortest route order. The suggested best tour order is Youth activity center Kenting forest recreation area (150 min) Jialeshuei (50 min) Eluanbi (40 min) Maobitou (40 min)Chuhuo (40 min) Guanshan Youth activity center.

(a) All possible tour route orders

(b) Tour route example, not shortest

(c) Tour route example, not shortest

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(d) The shortest (best) tour route order

Figure 1. Simulation results of one-day tours 5. Conclusions and discussions

An adaptive tour planning multimedia system is vital to tourism. In this study, a promising meta-heuristic algorithm is proposed to provide a suggested tour on a multimedia system to tourists at a visitor’s center or hotel. The recommended adaptive tour is generated based on the tourist’s demands, which involve minimization of total traveling time, total travel expenses or the combined total traveling time and expenses. Moreover, a simple exchange local search heuristic is also applied to increase the performance of the scheme.

The proposed particle swarm optimization with exchange heuristic integrated system outperforms pure particle swarm optimization as listed in Tables 5 and 7. The demonstrated simulation results in Table 5 illustrate the average deviations are from 0.4% down to 0.1%, from 0.29% down to 0%, and from 0.557% down to 0.139% for Min T, Min C, and Min O cases respectively. Meanwhile, the average deviations are from 9.39% down to 1.4%, from 6.61% down to 4.24%, and from 1.85% down to 0.65% for cases 1, 2, and 3 respectively as shown in Table 7. The simulation results indicate that much more improved results are obtained when applying exchange heuristic in PSO. Restated, applying exchange heuristic to a local search improves the quality of the tour planning solution. The supplied multimedia system sample is as indicated in Fig. 2. This multimedia system gives either the default suggested tours or the planned tour route according to the visitor’s input.

However, the simulation scenarios in this work focus on only one-day tour route planning. To be more practical, two-day tours or three-day tours will be further studied. Meanwhile, an adequate and attractive multimedia system would be designed in the near future. Moreover, the requirements input from tourists will become more and more complex; hence other heuristics can be further studied and applied. Additionally, some other schemes applied on solving VRP such as in [25, 26, 27] can also be adopted to increase the efficiency.

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Figure 2. Adaptive tour planning system

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