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  1. 1. Shenglong YU, Xiaofei YANG, Yuming BO, Zhimin CHEN, Jie ZHANG / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 6, November- December 2012, pp.136-141 A novel particle swarm Optimization algorithm based Fine Adjustment for solution of VRP 1 Shenglong YU , 2 Xiaofei YANG , 2Yuming BO, 1 Zhimin CHEN, 1Jie ZHANG[1] School of Automation, Nanjing University of Science and Technology, Nanjing 2 10094,PR China, [2] Shanghai Radio Equipment Research Institute, Shanghai 200090, ChinaAbstract To solve the problem of easily trapped and electrical engineer Eberhart in 1995. Itinto local optimization and instable is sourced from group behavior theory andcalculation results, a new fine-adjustment enlightened by the fact that bird group and fishmechanism-based particle swarm optimized group will develop toward a correct and properalgorithm applicable to solution seeking of direction as anticipated through the specialVRP model is presented in this paper. This information delivery among individuals. It is aalgorithm introduces the fine-adjustment self-adaption random optimization algorithmmechanism so as to get adapt to the judgment based on population searching. Due to itsbase of function directional derivative value. simplicit y, convenience in actualization andBy adjusting the optimal value and group high calculation speed, it has been widel yvalue, the local searching ability of algorithm applied in route planning, multi-targetin the optimal area is improved. The tracking, data classification, flow planning andexperiment results indicate that the algorithm decision support etc.. However, PSO also haspresented here displays higher convergence some defects. If the initial state forces thespeed, precision and stability than PSO, and population to converge toward the localis a effective solution to VRP. optimal area, then the population may fail t o jump out of the local area and thus be trapped, making application of PSO in VRP solutionKey words: particle swarm, fine adjustment, become difficult.VRP, perturbation, convergence In this paper, a fine adjustment is introduced to PSO, acquiring FT-PSO which will conduce to enhancement of local1. Introduction searching of population in the optimal area at [1] the end of searching. This method, taking the Vehicle routing problem was firstl y optimal value of particles as the center, formsput forward by Dantzing and Ramser in a fine-adjustment area based on the mode [ 2] defined by the algorithm. In the fine1959.VRP can be described as : for a series adjustment area, a fine adjustment particleof loading and unloading places, to plan swarm will form to calculate the fitnessproper vehicle route will facilitate smooth pass function of each fine adjustment particle. Theof vehicle and certain optimal goals can be particle with the largest function value isreached by satisfying specific constraints. selected to replace the optimal value ofGenerally the constraint include: goods updated particles, thus the precision of thedemand, quantity, delivery time, deliver y algorithm is enhanced.vehicles, time constraint etc., while theoptimal goals may consist of shortest distance,lowest cost, the least time and so on. VRP is a 2. Establishment of VRP model [3] The problem of vehicle route is decriedvery important content in logistics as follows: the distribution center is required todistribution researches. A good VRP algorithm deliver goods to l clients (1, 2, , l ) , and theand strategy can bring huge benefit to logisticsdistribution. cargo volume of the i th client Particle swarm optimization algorithm is gi (i  1, 2,  l ) , and the load weight of each [ 4](PSO) is an intelligent optimization vehicle is q , at the same time g i  q . Here wealgorithm put forward by need to find the shortest route which well satisfies the transport requirements.American social psychologist Kennedy is In the mathematical model established 136 | P a g e
  2. 2. Shenglong YU, Xiaofei YANG, Yuming BO, Zhimin CHEN, Jie ZHANG / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 6, November- December 2012, pp.136-141 here, cij denotes the transportation cost from point i to point j , including the distance, time yik  0or1, i  0,1, , l;k  1, 2, , m  (9) and expenses etc.. The distribution center is number as 0, clients numbered i (i  1, 2, , l ) , 3. Particle swarm algorithm [5] vehicle numbered k , at most m vehicles can Particle swarm algorithm essentially be used. After finishing the transportation, the is enlightened by the modeling and simulation vehicle will return back to the distribution center. research results of many bird populations, The variables are defining as follows: wherein the modeling and simulation algorithm 1vehiclekdriveto jfromi mainly utilizes the mode of biologist Hepper . It xijk   (1) is an optimal algorithm based on group 0otherwise intelligent theory that will generate group 1Cargo transportation task of itobefinishedbyk intelligent to guide the optimal searchingyik   (2) through cooperation and competition of particles, 0otherwise [6] The vehicle route problem is dealing and has been widely applied now. PSO with the integer planning with constraints, that is, algorithm can be expressed as: initializing a the sum of the clients task on each vehicle route particle swarm with quantity of m, wherein the can not exceed the capacity of this vehicle. number of iteration n , and the position of the The total minimal transportation cost after i th particle X i  ( xi1 , xi 2 ,  xin ) , the finishing the service is: speed Vi  (vi1 , vi 2 , vin ) . During each iteration, l l m particles are keeping upgrade its speed and m i n   c j xi k Z i j position through individual i 0 j  0 k 1 extremum P  ( P1 , P 2 , P ) i i i in and global m l R  max( gi yik  q, 0) (3) extremum G  ( g1 , g 2 ,  g n ) , accordingly k 1 i 1 [7 ] reaching optimization solution seeking . The where R takes a very large positive number as upgrade equation is the punishment coefficient. Supposing all the vehicles used satisfying the load requirements, then we have the following vk 1  c0vk  c1 ( pbestk  xk )  c2 ( gbestk  xk ) (10) expression: xk 1  xk  vk 1 (11) l where v k denotes the particle speed, x k is the g y i 1 i ik  q,k  1, 2, , m (4) position of the current particles, pbestk is the Giving that only one vehicle is assigned to serve position where particles find the optimal each customer only once, that is, solution, gbestk is the position of the optimal solution of the population, c 0 , c1 and m 1i  1, 2, , l  yik  mi  0 (5) c 2 denote the population recognition coefficient, k 1  c 0 usually is a random number in interval (0,1), In the model the vehicle reaching and leaving each client are the same, which is expressed as c1 , c 2 is a random number [8] in (0,2). follows: l 4 Particle swarm algorithm improvement x i 0 ijk  ykj , j  0,1, 2, l ;k (6) 4.1 Mechanism of fine adjustment According to the flow of PSO, after calculation of all parties, pbest and a gbest l x will be acquired, wherein the quantity and value ijk  yki ,i  0,1, 2, l ;k (7) j 0 of pbest depends on the total number of Given the variable value constraint as 0 or 1, and population. In fine adjustment, gbest is the expression is: adjusted as the object of fine adjustment with the goal of searching the optimal solution in the xijk  0or1, i, j  0,1, , l;k  1, 2, , m  (8) neighboring area of gbest . 137 | P a g e
  3. 3. Shenglong YU, Xiaofei YANG, Yuming BO, Zhimin CHEN, Jie ZHANG / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 6, November- December 2012, pp.136-141 Whereas the excellent performance of divides the area and range by the defined methodPSO in fine adjustment principle, we should and forms a fine-adjustment area based on themake full use of the existing advantages in the mode defined by the algorithm. In the fineadjustment. Fine adjustment is not necessarily adjustment area, a fine adjustment particleimplemented in each generation unless meeting swarm will form to calculate the fitness functionspecific requirements. Directional derivative is of each fine adjustment particle. The particleintended to be used here as the determinant with the optimal adjustment is selected tobenchmark. Determination methods are compare with Yg and replace the optimal valueintroduced as follows: of updated particles and becomes a new gbestGiven Y the function of X , if its fitness excels Yg . X  [ x1 , x2 , , xn ] , the directional derivative ofY at g direction is defined as: 5. Algorithm improvement and VRP Y solving Dg Y  (12) || X || 5.1 Model improvement In a specific generation j , the target Particle swarm algorithm is a solution [9]function fitness Yg corresponding to the overall algorithm for continuous space, whereas VRP involves integer planning, thus the model shouldoptimal value of the population shows the be improved to construct particles in theposition of gbest in the design space of j algorithm so as to solve the constraint issue.generation that can be expressed as: Letting X v of particle denotes the number ofgbest j  x1j ex1  x2j ex2 (13) vehicle serving each task point, X r theSupposing the population going through p delivery order of this node. Particle fitnessiterations, gbest arrives at the new position, function is the minimal cost value satisfying thethen the displacement vector of these two constraints.positions are: The key of this paper lies in finding a proper expression with which the particlesg  ( x1j  p  x1j )u  ( x2j  p  x2 )v (14) correspond to the model. For this, the particle The difference of the fitness of these twoobjective functions is swarm optimized VRP issue is constructed as a space with (l  m  1) dimensionsY  Ygj  Ygj  p (15) corresponding to l tasks. Numbering of clients Using Equation A, the directional derivativeof function Yg in j  p th generation is: is represented by natural number, i denotes the i th clients and 0 denotes the delivery center. For Y the VRP with l clients and m vehicles,Dg Y  (16) || g || m 1 of 0 are inserted into the client series After calculation, if Dg Y acquired is which will be then divided into m sections,larger, it means the population is performing each of which denotes the route of searching and in a variable situation. Orotherwise the distribution of population particles Each particle corresponds to aare gathering in the area where the current l  m 1 -dimensional vector."population locates for regional searching, whilethe movement of population is also slow 5.2 Algorithm procedurerelatively. In this case, the algorithm determines The VRP solution by using thewhether to initiate fine adjustment mechanism algorithm presented here consists of theby using Dg Y . following procedures: Step 1: given the proper proportion of random particle number q , execute the determined4.2 Operation of fine adjustment The core concept of fine adjustment is to values p of fine adjustment operation and ftake gbest as the center. More specifically, it Dcr of directional derivative as well as the 138 | P a g e
  4. 4. Shenglong YU, Xiaofei YANG, Yuming BO, Zhimin CHEN, Jie ZHANG / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 6, November- December 2012, pp.136-141particle number n of population.  8 (number 1, 2, ,), and the cargo volume of each task g (unit: ton), loading (or unloading)Step 2: When c  p  1 , if the directional time Ti (unit: hour) and the time rangederivative Dg Y of f g is less than or equal to [ ETi ,LTi ] of each task are given in table 1. fDcr , then implement step (3) and (4) for fine These tasks are to be finished by three vehiclesadjustment, or otherwise if Dg Y is more than with capacity of 8 ton respectively from parking 0. The Distance between the parking and each fDcr , implement step (5). task point and the intervals between these tasks points are given in Table 2. Supposing theStep 3: Perform fine adjustment on gbesti , use driving time of vehicle is negativelyequation (17) to generate fine-adjustment proportional to distance, and the average drivingparticles that are randomly generated and evenly speed of each vehicle is 50kh/h, than the drivingdistributed in the fine-adjustment area centered time from i to j is tij  d ij / 50 . Taking theby gbesti . distance between each point as the expensextd  gbestd  l  [rand ()  0.5] (17) cij  dij (i, j  0,1, ,8) , and the unitStep 4: For the fine-adjustment instance punishment of exceeding time windowgenerated, its fitness function values Yt are is c1  c2  50 , wherein n  80 ; w  1 ;calculated and arranged in order to select theoptimal find-adjustment instance and the c1  1.5 ;c2  1.5 ,given the maximal iterationcorresponding optimal function fitness. If there number 200, each of the three algorithm runsis an optimal function fitness in the original 1000 times and the operation results are given inpopulation, then it can be replaced by the Table 3.optimal fin-adjustment particles and fitnessfunction values. Table 1 Cargo volume, loading and unloadingStep 5: Calculate the fitness function value Yi in time and time window at each distribution pointaccordance with the known target function; Task 1 2 3 4 5 6 7 8make a comparison between the fitness Yi of numbeach particle at the current position and the eroptimal solution Pbesti , if P  Pbesti , then i Freig 2 1, 4. 3 1. 4 2. 3 ht 5 5 5 5new fitness function value can replace the volumprevious optimal solution, and the new particles ereplace the previous ones, e.g., P  Pbesti , i ( gi )xi  xbesti , or otherwise implement step (7). Ti 1 2 1 3 2 2. 3 0.8Step 6: Compare the optimal fitness Pbesti of 5 ETi , 1, 4, 1, 4, 3, 2, 5, 1.5,each particle with the optimal fitness gbesti of 4 6 2 7 5. 5 8 4 LTi 5all particles, if Pbesti  gbesti , then replace theoptimal fitness of all particles with the one of Table 2 Distance between the parking andeach particles, at the same time reserve the each task point and the intervals betweencurrent state of particles, e.g., these task points gbesti  Pbesti , xbesti  xbesti . 0 1 2 3 4 5 6 7 8Step 7: use equation (10) and (11) to update the 0 0 40 60 75 90 20 10 16 80speed and position of particles. 0 0 0Step 8: determine the end condition, and exit if 1 40 0 65 40 10 50 75 11 10the condition is satisfied, or otherwise transfer to 0 0 0Step 2 for operation until the iterations are 2 60 65 0 75 10 10 75 75 75finished or the preset precision is satisfied. 0 0 3 75 40 75 0 10 50 90 90 156. Simulation experiment 0 0 4 90 10 10 10 0 10 75 75 10 Experiment 1: Experimental analysis of 0 0 0 0 0the example in literature[10] is performed. In 5 20 50 10 50 10 0 70 90 75this example, there are 8 transportation tasks 139 | P a g e
  5. 5. Shenglong YU, Xiaofei YANG, Yuming BO, Zhimin CHEN, Jie ZHANG / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 6, November- December 2012, pp.136-141 0 0 0 Figure 1: Optimal solution curve of two6 10 75 75 90 75 70 0 70 10 algorithms 0 07 16 11 75 90 75 90 70 0 10 Experimental results show that the 0 0 0 successful rate of the algorithm presented here is8 80 10 75 15 10 75 10 10 0 higher than that of other algorithms. While 0 0 0 0 0 enhancing the precisions, it does not show any significant increase in average searching time. Taking time and precision into consideration, theTable 3 Comparison of different results algorithm here is of more practical value.gained by different algorithmAlgorithm Search Average success rate search time Experiment 2: inspect the situation /(%) /s with multiple dimensions. Model and parameterBasic PSO 72.3 0.34 are referred to literature [8]. Clients are 2Gen etic 54.7 0.28 distributed in the interval [0,100] and dividedalgorithm into high, middle and low ends based on theirPredatory 93.8 0.55 needs, e.g., U (1, 9)  low ,searchalgorithm U (5,15)  middle ,FA-PSO 96.1 0.39 U (10, 20)  high . Given the anticipation of n E[ Di ] the cargo on the vehicle as f   , 1250 gQ i 1 PSO where g denotes the quantities of vehicles. 1200 FA-PSO 8  10  15 1150 E ( Di )   11 . Letting 3 1100 f  max{0, f  1} denotes the anticipation of route failures. Q  11n / (1  f ) , where n cost 1050 represents the number of client nodes to be 1000 served. In the experiment, f  0.7 and f  0.9 . Taking 950 n  20,30, 40,50, 60 respectively, the 900 improvement rate is compared with greedy algorithm, The simulation result is shown in the 850 following table: 0 20 40 60 80 100 120 140 160 180 200 iteration numberTable 2: Comparison of VRP solution performance by different algorithmparameters Improvement rate /% Time/s (n, Q, f ) PSO ACO FA-PSO CPSO PSO ACO FA-PSO CPSO(20,129,0.7) 4.21 4.46 4.81 2.04 1.83 1.82(20,116,0.9) 4.47 4.72 5.20 1.63 1.51 1.54(30,194,0.7) 4.13 4.37 4.76 6.52 6.04 6.08(30,174,0.9) 4.61 4.83 5.28 5.96 5.42 5.41(40,259,0.7) 5.07 5.38 5.85 16.27 15.11 14.88(40,232,0.9) 4.68 4.98 5.36 12.76 11.80 11.52(50,324,0.7) 5.50 5.80 6.43 68.25 62.39 62.33(50,289,0.9) 5.18 5.52 6.01 62.20 55.94 56.05(60,388,0.7) 4.76 5.02 5.29 114.64 103.87 104.92(60,347,0.9) 4.68 4.94 5.14 105.53 95.27 95.98Average 4.72 5.03 5.40 Seeing from the simulation result, to 140 | P a g e
  6. 6. Shenglong YU, Xiaofei YANG, Yuming BO, Zhimin CHEN, Jie ZHANG / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 Vol. 2, Issue 6, November- December 2012, pp.136-141solve VRP by FA-PSO achieves better effect 507-519.than ACO and PSO with an average [4] Ying Li, Bendu Bai, Yanning Zhang.improvement rate of 5.40% by contrast with the Improved particle swarm optimizationprimitive solution (greedy algorithm solution). algorithm for fuzzy multi-class SVM.Besides, regarding the operation time, the Journal of Systems Engineering andimproved algorithm requires the least operation Electronics.2010,21(3): 509-513.time, showing that the convergence of the [5] Z. Fang, G. F. Tong, X. H. Xu.algorithm here has been improved in solvingVRP. Particle swarm optimized particle filter. Control and Decision, 2007.7. Conclusion 273-277. Through analysis of the shortcomingsof PSO, a fine adjustment-based algorithm [6] ZHANG W , LIU Y T. AdaptiveFA-PSO, which is suitable for VRP solving, is particle swarm optimization forproposed here. This algorithm enhances theeffectiveness of initial samples and at the same reactive power and voltage control intime solve PSOs problem of being easily power systems. Lecture Note intrapped into local optimization, therebyimproving the global searching ability. The Computer Science , 2005 , 449~452.experimental result of VRP model indicates that [7] Bergh F van den, Engelbrecht A P. Athe algorithm presented here features betterspeed and precision than the previous algorithm Study of Particle Swarm Optimizationand thus is of good application value in solving Particle Trajectories. InformationVRP. Sciences. 2006, 176(8): 937-971.9. Acknowledgement [8] Halil Karahan. DeterminingThis paper was supported by the National rainfall-intensity-duration-frequencyNature Science Foundation of China (No. relationship using Particle Swarm61104196), the National defense pre-research Optimization. KSCE Journal of Civilfund of China (No. 40405020201), the Engineering.2012,16(4):667-675.Specialized Research Fund for the Doctoral [9] Xudong Guo,Cheng Wang, RongguoProgram of Higher Education of China (No. Yan. An electromagnetic localization200802881017), the Special Plan of method for medical micro-devicesIndependent research of Nanjing University of based on adaptive particle swarmScience and Technology of China (No. optimization with neighborhood2010ZYTS051). search. Measurement.2011,44(5):852-858.References [10] Jun LI, Yaohuang GUO. Optimization [1] Claudia Archetti;Dominique Feillet;Michel Gendreau,etl. of logistics delivery vehicle Complexity of the VRP and SDVRP. scheduling theory and methods.2001 Transportation Research Part C: Emerging Technologies. 2011,19(5): 741-750. [2] Eksioglu B, Vural A V, Reisman A. The Vehicle Routing Problem:A Taxonomic Review. Computers & Industrial Engineering, 2009, 57(4): 1472-1483. [3] Zagrafos K G, Androutsopulos K N. A Heuristic Algorithm for Solving Hazardous Materials Distribution Problems. European Journal of Operational Research, 2004, 152(2): 141 | P a g e