Proceedings of the 2016 International Conference on ...ieomsociety.org/ieom_2016/pdfs/643.pdf ·...

Proceedings of the 2016 International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia, March 8-10, 2016

Let a CPSO+ algorithm explore your high-quality shortest paths: an effective chaos-enhanced PSO-based strategy

Ali Asghar Heidari*, Rahim Ali Abaspour School of Surveying and Geospatial Engineering, College of Engineering

University of Tehran Tehran, Iran

as_heidari, [email protected]

Abstract—In this research, the shortest path problem (SPP) is investigated to evaluate a new competent heuristic strategy. The SPP is a key interested issue in operational research. Industrial experts frequently should tackle this problem in diverse procedures such as transportation, routing and communications. Due to the inherent complexity of SPP, metaheuristics can find satisfactory trajectories in a rational time. For this purpose, an attuned chaotic particle swarm optimizer, which is entitled CPSO+, is proposed here to realize the SPP in a more operational way. In addition, the influence of chaos is evaluated and discussed specifically to investigate the quality of obtained paths. Hence, some chaos-based signals such as Zaslavskii, Tinkerbell, Lozi and Burgers are investigated based on qualitative comparisons. Then, the best chaotic pattern is identified and picked out to provision the disordered particle motions in PSO approach. These patterns are utilized to enhance the exploration also exploitation styles of the normal PSO. The CPSO+ technique is evaluated against PSO and several methods from literature. The Obtained computational outcomes exemplify that the effectiveness of the suggested methodology is desirable based on quality measurements. The offered PSO approach with Zaslavskii sequence can effectively obtain the optimal routes compared to other implemented strategies.

Keywords— Shortest Path Problem; Particle Swarm Optimisation; Chaos; Meta-heuristics; Quality Assessment; Chaotic Signal

I. INTRODUCTION

The shortest path problem (SPP) is defined by an identified subjective directed graph, which can be investigated by using its edges and vertices [1]. The crucial goal of SPP is to discover any possible shortest route in that network [2, 32]. The SPP is a crucial task in operational research. Industrial specialists often should handle SPP in transportation planning or traffic routing [3]. Up to now, several search strategies have been developed to tackle the SPP such as Dijkstra [4] or Bellman–Ford methods [5]. In 2014, Trevizan et al. developed depth-based SPPs [6]. Pulido et al. realized multi-objective version of SPP by considering lexicographic favorites [7]. Takaoka proposed a noteworthy technique in information distribution in all pairs SPP [8]. There is also a challenge to offer more efficient methods to realize SPP [3]. Due to the inherent complexity of SPP; metaheuristics can also discover the superior trajectories during a rational time [9]. Research also confirms the profitable utilization of PSO method [10] and neural networks (NN) [2]. For instance, the genetic algorithm (GA) has been implemented before to investigate the SPP [9]. Their results also exposed that GA can outperform NN methods in solving SPP. In 2008, PSO was applied to tackle SPP and attained solutions exposed that the PSO based path are superior to routes of GA strategy [10]. In 2010, an effective ant colony optimizer (ACO) has been utilized to evaluate SPP [11].

The effect of random motions on search procedures can be delivered by using chaotic maps [12, 33]. Up to 2015, Different works substantiated that chaos relations can enhance the performance of heuristics approaches [13]. Xu et al. [14] have offered a better-quality artificial bee colony (ABC) with disorder of chaos to handle aerial route planning. The results confirmed that the adjusted ABC outperforms in tackling that task. Gandomi et al. [15] affirmed that the bat algorithm (BA) combined via sinusoidal wave instead of its pulse ratio is the finest chaotic version of BA. In 2013, Jothiprakash applied chaotic waves to effectually initialize the GA and differential evolution (DE) for explaining water resource systems [16]. In 2014, Gandomi et al. [13] improved krill herd (KH) technique with chaos. In 2015, Fister et al. [17] studied several firefly strategies (FA) coupled with chaotic patterns. Up to now, these series have been combined with certain procedures such as GA [18], harmony search (HS) [19] also PSO technique [20]. In 2011, Chuang et al. developed an improved chaotic PSO for clustering tasks [21]. In 2013, Gandomi et al. proposed a chaotic accelerated PSO (CAPSO) [20]. The results affirmed that the CAPSO version is able to outperform immature PSO.

Considering "No Free Lunch" (NFL) theorem [22], there is no metaheuristic best appropriate for tackling many Optimisation tasks including SPP. Hence, NFL encourages investigators for improving current strategies or inventing new metaheuristics [23]. This also inspires our efforts to advance a well-established PSO by chaos. In this work, a new PSO variant (CPSO+) is proposed to realize the SPP. The effect of chaos is assessed to improve the quality of paths. Hence, some chaos equations are examined to recognize the best disordered PSO approach. These measures are employed to increase the exploration/exploitation of a modified PSO. The CPSO+ method is evaluated for tackling SPP in detail. It is expected that modified CPSO+ algorithm with some maps attains preferable paths in view of quality. The evaluated results may affirm the superiority and efficiency of the suggested optimizer in solving SPP.

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II. PSO ALGORITHM

PSO algorithm is nominated by Eberhart and Kennedy in 1995 [20]. PSO technique can be regarded as an adaptable, well-founded metaheuristic, which is originated from a principled social-psychological conceptualization; a crowd of fowls (particles) attuned by exploiting prior more fruitful areas [21]. In each replication, each explorer travels toward the preceding local best and overall finest locations. The location of the ith particle at the tth repetition is employed to assess the quality and superiority of that search element. The location and velocity vectors of each particle are updated in all iterations. This process continues until the algorithm reaches to the global optimum. The main equations of PSO are as:

1 1 2 2( 1) . ( ) ( ( )) ( ( ))i i i i g iv t v t c p x t c p x tω ϕ ϕ+ = + − + − (1) ( 1) ( ) ( )i i ix t x t v t+ = + (2)

where xi denotes the position of ith member, vi shows the particle i’s velocity value, t is iteration number, pi denotes the earlier best point, pg denotes the global best location, w shows the inertia weightiness and c1, c2, φ1 and φ2 represent social constants and random values inside [0, 1], respectively. A revised PSO method (CFM) proposed by Clerc in 1999 [24], which employs a constriction factor (CF) as mentioned below:

1 1 2 2( ) ( ) ( ( )) ( ( ))i i i i g iv t v t c p x t c p x tχ ϕ ϕ⎡ ⎤= + − + −⎣ ⎦ (3)

( 1) ( ) ( )i i ix t x t v t+ = + (4) 2 0.5

1 22 / 2 ( 4 ) , where , 4,c cχ ϕ ϕ ϕ ϕ ϕ= − − − = + > (5)

where χ shows the constriction factor. Clerc et al. substantiated that via adjusting φ, the convergence behaviors of PSO may be regulated [24]. The CFM is usually is utilized to confine the velocity inside the restrictions. In this paper, the CFM is employed to realize SPP.

A. Chaotic PSO strategyThe description of chaos may be explained as a topological mixing bounded aperiodic procedure recognized in nature or

dynamic systems [25]. The landscape of chaotic performances is noticeably unsystematic and consistent [33]. The noteworthy consequence of randomization on PSO paradigm can be delivered by employing chaotic arrangements during Optimisation steps. Based on ergodicity, chaotic maps can competently avert PSO from untimely convergence to local bests by enhancing the propagation of the particles. In this study, chaos has been realized to enrich the excellence of PSO-based routes. First, the utilized chaos equations that launch disordered waves are explained mathematically and then the PSO is reorganized. The utilized chaotic maps are briefly reviewed below.

(a) Tinkerbell mapThe Tinkerbell sequence [26] can be given as

2 21n n n n nX X Y aX bY+ = − + + (6)

1 2n n n n nY X Y cX dY+ = + + (7)

The operational parameters for this map are a=0.9, b=-0.6, c=2, d=0.5. In addition, the primary position are located at X0=0, Y0=0 [26].

(b) Zaslavskii mapThe Zaslavskii signal [27] is defined by

1 1( ) mod(1)n n nX X v aY+ += + + (8)

1 cos(2 ) rn n nY X e Yπ −

− = + (9)Here, [ ]1 1.0512, 1.0512nY + ∈ − .

(c) Lozi mapThis sequence [28] is described as:

1 1n n nX a X bY+ = − + (10)



1n nY X+ = (11)

The parameters are a=1.7 and b=0.5. The initial point: X0 = -0.1, Y0 = 0.1.

(d) Burgers map This signal can be formulated as [29]:

21n n nX aX Y+ = − (12)

1n n n nY bY X Y+ = + (13)

The parameters are a=0.75, b=1.75. The initial condition: X0=-0.1, Y0=0.1.

Here, the hybrid chaotic PSO (CPSO+) approaches are presented. In CPSO+, the effect of randomization is attained by employing chaotic sequences in PSO. The crossbred operators attempt to enrich the exploration/exploitation of PSO by using ergodicity of chaos. These mechanisms moreover assist PSO to move from local best more competently. In CPSO, velocity vector is updated as

1 1 2 2 3( 1) ( ) ( ) ( ( )) ( ) ( ( ))i i i i g iv t CS v t C t CS p x t C t CS p x t+ = × + × − + × − (14)

1 1, 1, 1, max( ) ( ( ) ( / ))i f iC t C C C t t= + − × (15)

2 2, 2, 2, max( ) ( ( ) ( / ))i f iC t C C C t t= + − × (16)

where, based on chaotic relations, CS1, CS2 and CS3 returns some chaotic values inside [0, 1], [0.5, 2.5], [2.5, 0.5], respectively, Cf and Ci indicate the limitations of utilized coefficients. After some initial tests, the proper values have been selected as C1,i = 2.5, C2,i=0.5, and C1,f =0.5, C2, f =2.5. Hence, there will be a better tradeoff amongst exploration and exploitation. The CPSO+ with Tinkerbell, Zaslavskii, Lozi and Burgers signals is symbolized via CPSO+1, CPSO+2, CPSO+3 and CPSO+4, respectively.

III. CPSO+ ALGORITHM FOR SOLVING SPP In this paper, the designed CPSO is implemented to tackle SPP. One the key task in the SPP is the approach that one should

encode the graph for applying an appropriate form of network based route into a particle. In this work, the priority based encoding is used for all compared optimizers, which has been discussed in [9]. A cost function can be utilized in order to assess and judge the quality of the obtained results. In this problem, the core objective is to exploit a path with minimum cost. The cost equation is obtained via

( ) 11

1Cost , t = ( ), ( 1)iN i i

i tkjf PP j k PP j

−−

== = +∑ (17)

where PPi symbolizes the set of consecutive node IDs for ith search agent, Ni = PPi denotes the overall number of nodes that are located on a trajectory and Costtk represents the cost of the division that links node t to node k [10]. The path optimality (PO) ratio represents the percentage of time required for a searching algorithm to exploit a solution which is the overall best in the shortest path scenario. The path failure (PF) ratio can be defined as the opposite of PO ratio definition. Hence, it can asymptotically represent the probability (PR) that the obtained path by an optimizer is not optimal. In addition, CFM algorithm is utilized as a conventional PSO to tackle SPP. In the next section, the evaluation results are discussed.

IV. SIMULATION RESULTS AND DISCUSSIONS In the implementation, first, 20 nodes are distributed to create the preliminary topology of road network. Then, the number

of nodes is increased to 70. In addition, when it is necessary, the population size can be equal to node numbers. The figured routes are compared to the Dijkistra’s paths as a reference approach [9]. The numbers of 20 to 70 nodes are randomly generated to evaluate the performance of the CPSO+ variants and PSO for dissimilar random topologies. The tests are compared with regard to the resulting quality, time and convergence speed. In these implementations, first, the best chaotic wave is selected and then the finest PSO alternative will be compared to other well-established techniques. The chaotic PSO with that chosen map should be capable to perform with lower PF ratios in different network configurations. Hence, the best chaotic signal can be determined based on the PF ratio. In Fig. 1, PSO with 4 different signals is compared to its basic version with regard to the PF criterion.



Fig. 1. Comparison of different CPSO methods with PSO with respect to the PF ratio

It can be realized from Fig .1 that the PF ratio of the CPSO+2 is lower than other methods in different topologies. Hence, the overall optimality of the paths of the CPSO+2 is higher than those of the other investigated versions. For instance, in the case of 70 nodes, the PF ratio of PSO is 0.11 (89% PO), while the PF value for CPSO+1 to CPSO+4 is about 0.53, 0.091, 0.14 and 0.48, respectively, which indicates that the CPSO+2 can still outperform other variants in the networks with more nodes. Based on the results, it also can be observed Zaslavskii map is the most appropriate chaotic signal that can be employed to increase the optimality of PSO based paths. For example, the results of chaotic PSO with Zaslavskii wave are better than PSO with PR< 0.12 in networks with 50 nodes. It also can be realized from Fig. 1 that Tinkerbell chaotic pattern cannot enhance the quality of computed routs. Also Lozi sequence is the subsequent best choice for increasing the PO values of routs. It is also noticeable that some of the chaotic maps such as Tinkerbell and Burgers cannot enrich the excellence of the unique PSO. Hence, just some selected waves can be utilized to improve the PSO method.

All the found results can be seen in Table I. Based on the records of Table I, the average PF of "CPSO+2" is about 0.035 in comparison with the PSO which is 0.044. Hence, the proposed CPSO+2 is superior to PSO with PR < 0.09. Meanwhile, standard deviation (SD) of PF ratio (probability) for the suggested CPSO+2 is lower than others. Hence, the CPSO+2 can still retain robustness by changing nodes with respect to quality of paths or PO. This advantage affirms that the employed map can effectively increase the exploration and exploitation competences of PSO in tackling SPP.

TABLE I. COMPARISON OF THE QUALITY OF OBTAINED PATHS

Results Algorithms Name PSO CPSO+1 CPSO+2 CPSO+3 CPSO+4

Used Map / Tinkerbell Zaslavskii Lozi Burgers Average RF 0.044 0.267 0.035 0.056 0.225 SD Value 0.035 0.1749 0.0296 0.0388 0.1696

In Fig .2, the average CPU time of PSO based variants are compared in detail. For a rational assessment, the population size was fixed to 25.

Fig. 2. Comparison of different CPSO methods with standard PSO with respect to the CPU time

From Fig .2, it is clear that the required computational time for CPSO+2 is lower than PSO and others in all topologies. With more nodes, the CPSO+2 can vividly demonstrate a better search compared to PSO. In these experiments, the worst total CPU time for "CPSO+2" is 3.7 (s) but the best total time of studied PSO is not better than 5.36 (s). As it can be seen in Fig .2,



CPU times of CPSO+1, CPSO+3 and "CPSO+3" will be increased with more than 50 nodes, dramatically. In opposition, the "CPSO+2" can outperform others with less excessive CPU burden with regard to its time curve. It can be known from these results that the revised PSO with Zaslavskii map is the finest version amongst other alternatives with regard to its time and quality of solutions.

After determining the CPSO+2, its solutions are compared to the PSO, GA with indirect route encoding [9], L-GPS as a new hybrid of GA and PSO offered in 2015 [30], GA-PSO technique, as another robust hybrid method [31]. For L-GPS and GA-PSO the priority-based road coding [10] is utilized. In addition, the prerequisite parameters of GA, PSO, L-GPS and GA-PSO are cautiously selected according to [9, 10] in tackling SPP. In Fig .3, the attained paths are compared based on PF ratio.

Fig. 3. Comparison of CPSO+2 with other algorithms with respect to PF ratio

The final results also substantiate the exploration/exploitation capacity of CPSO+2 against other optimizers. Based on Fig . 3, the GA-PSO outperforms L-GPS and GA in all examined topologies. The exciting results also show that PSO is able to overtake its hybridized variants with GA in tackling SPP. The average and SD of the PF results are exposed in Table II. From Table II, it can be recognized that the average PO ratio for CPSO+2 (97%) is less than GA (75%), L-GPS (78%) and GA-PSO strategies (85%). Based on results, the SD values vividly substantiate the robustness of the proposed CPSO+02 in competition with those optimizers.

TABLE II. COMPARISON OF THE QUALITY OF COMPUTED PATHS

Results Algorithms Name PSO GA CPSO+2 L-GPS GA-PSO

Average RF 0.04464 0.25909 0.03564 0.22182 0.15636 SD Values 0.0351 0.17541 0.02968 0.16774 0.15233

To assess the overall running time, each approach is investigated by a fixed population size of 25. Fig .4 shows the average time of evaluated GA, PSO, L-GPS and GA-PSO methods. It can be detected that CPSO+2 outperforms other techniques as well. It also can be seen that L-GPS performs better than GA and GA-PSO for more complex networks.

Fig. 4. Comparison of CPSO+2 with other strategies with respect to the CPU time

Fig .5 illustrates a road network with 20 nodes and some weighted edges, which can be found in [9].



Fig. 5. A test network with 20 nodes (The node IDs are exposed in each

node, while other values show the cost of the edges) [9]

With population size of 20, the routes are attained for implemented optimizers. The costs of obtained shortest paths by CPSO+, PSO, GA, L-GPS and GA-PSO are equal to 142, 187, 142, 187 and 234. The cost of the optimal path in this network is 142, which found by Dijkstra’s technique. Figs .6, Fig .7 and Fig .8 demonstrate the obtained routs for employed techniques.

Fig. 6. Obtained path of PSO and L-GPS algorithm

Fig. 7. Obtained path of GA-PSO algorithm

Fig. 8. Obtained path of CPSO+2, GA and Dijkistra’s technique



Fig. 6 indicates that the costs of the best routes of the PSO and L-GPS are equal to 187 units. However, these optimizers often revealed suboptimal results. The achieved paths by GA-PSO strategy is exposed in Fig .7. Based on Fig .7, the cost of GA-PSO approach is not better than 234 units, which is not satisfactory in comparison with routes gotten by Dijkstra’s procedure (142 units). The calculated results of Dijkstra’s process, GA algorithm and CPSO+ technique are also exposed in Fig. 8. Based on Fig .8, it can be recognized that the answers attained by CPSO+ are in accordance with the acquired paths of Dijkstra’s algorithm as the best results. In addition, Fig. 8 reveals that the utilized GA can compute the best roads, which is in accordance with results declared by Ahn et al. [9]. Hence, the proposed CPSO+ is competent to discover the best route as GA algorithm (142 units), while the PSO, L-GPS and GA-PSO algorithms were often in competition to achieve suboptimal edges. These results affirm that the PSO with Zaslavskii wave outperforms as a competence SPP solver.

V. CONCLUSIONS

This research investigated an enriched chaos based PSO (CPSO+) technique to tackle the shortest path problems more competently. Decreasing adaptive operators have been designed for improving the exploration/exploitation of PSO via chaos. Then, the offered approach has been evaluated against PSO, GA, GA-PSO and L-GPS strategies. The better-quality performance of CPSO+ has been experienced via Zaslavskii map. The results of PF ratio are very promising with regard to obtained paths. The time comparison also confirms that the CPSO+ is an effective optimizer in realizing SPP. Hence, the Obtained outcomes exemplify that the effectiveness of the suggested CPSO+ is satisfactory according to the quality measurements with regard to the other algorithms. In future, CPSO+ approach can be evaluated in tackling different kind of routing tasks. There are also other chaotic systems that can be employed in PSO algorithm.

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BIOGRAPHY Ali Asghar Heidari is currently an Exceptional Talented Ph. D. Student in Geospatial Information System (GIS) Engineering at School of Surveying and Geospatial Engineering, College of Engineering, University of Tehran, Tehran, Iran (Born in 1989). He earned B.S. and M.S. Degrees at University of Tehran as an Exceptional Talented Student in Geospatial and GIS Engineering, honorably. He RecentlyReceived a Brilliant Scholar Admission at Ph. D. Degree at University of Tehran. Up to 2015, He has Published Several Journal andConference Papers. His Leading Study Interests include Chaos, Optimisation, Metaheuristics, Spatial Analysis, Spatial Modeling, PathPlanning and Evolutionary computation.

Rahim Ali Abaspour is an Honored Fulltime Assistant Professor at School of Surveying and Geospatial Engineering, College of Engineering, University of Tehran, Tehran, Iran. He Successfully Fulfilled His B.S., M.S. and Ph. D. in Geospatial and GIS Engineering at University of Tehran (UT). Up to 2015, He has Published Many Journal and Conference Papers. His Main Research Interests are Meta-heuristics, GIS, Spatial Analysis, Spatial Modeling, LBS, VGI, Planning and Evolutionary computation.


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