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An Evaluation of ACO and GA TSP in a Supply Chain Network
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 5 (2018) 25350–25357
www.materialstoday.com/proceedings
IConAMMA_2017
An Evaluation of ACO and GA TSP in a Supply Chain Network T Srinivas Raoa* a
Dept of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwavidyapeetham,Bengaluru ,India
Abstract The need for supply chain management solutions in logistics management is the need of the hour. An E commerce based web aggregation model has been developed which aggregates the raw materials in various pockets through a common vehicle which routes the cities and ensures efficient supply at the low cost. In this paper we have developed an GA and ACO algorithm which calculates the shortest routes through which the material can be supplied to the manufacturing units. Our main aim is to compare the two algorithms and evaluate their efficiencies. © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Advances in Materials and Manufacturing Applications [IConAMMA 2017]. Keywords: GA, ACO, Supply chain ,Pockets
1.Introduction In the evolutionary optimization algorithms, the ACO or ant colony optimization, is the technique to find the shortest distance for the vehicle routing problems. In the real scenario the ants find the shortest distance to locate the food destination by laying a chemical substance called pheromone on the trail for finding the food. The pheromone is a communication to other ants to follow them. For weaker destinations the pheromone substance evaporates quickly so that other ants do not follow. When the destination or the source of food is noticed the ants quickly march along the trail and thereby the pheromone concentrations also becomes strong thereby resulting in the shortest route.
* Corresponding author. Tel.: +91-080-251-83700; fax: +91-080-284-40092. E-mail address: [email protected] 2214-7853 © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Advances in Materials and Manufacturing Applications [IConAMMA 2017].
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
25351
By frequent evaporation of the pheromone the local optimization convergence is avoided. The whole process results into a constrained solution space. The GA are computerized search algorithms which mimic genetics and natural selection evolutionary algorithms. Using the technique of GA an array of NP problems can be solved using the technique of GA. GA manipulates strings of information called chromosomes. The fitness function is used to assign a score which is evaluated and assigned to these chromosomes according to the criterion defined by the programmer. The probability of survival is determined by these fitness values during a round of reproduction. A new chromosome is produced from the combination of two or more chromosomes. 2. Related Works A global optimum is found by using the greedy algorithm [6] arriving at each stage thru local optimum choice by following problem metaheuristic. The first algorithm to find an optimum solution of TSP is the nearest neighbour algorithm [12]. We can start the tour from any city and by repeatedly visiting the neighbours we can complete the tour. This continues till the last city has been visited. Though this is not optimum it yields good approximate solutions. In ACO due to numerous local optimum points an optimum solution cannot be reached and also the search space is enormous [13]. Many practical problems can be modelled using Multi travelling sales person (MTSP) [13]. There are no sub tours in TSP and the MTSP which are similar. In the MTSP the m tours get partitioned into n cities and each tour results into a TSP for single salesperson. It is difficult to model MTSP than TSP because it requires assigning salespersons to each cities, also there is a need for optimal ordering of the cities within each salespersons tour. 3. Ant Colony Algorithm Many combinatorial optimization problems can be solved thru ant colony optimization algorithms, starting from traveling sales man problem, job scheduling and many stochastic problems. A near optimal solution can be found thru ACO but still it is far from reaching exact solutions. There is a dynamic change in the graphs in comparison of ACO to genetic algorithm and simulated annealing ; some of the advantages of ACO is that it can adapt to changes in real time and run continuously. In ACO we use a set of ants which form the basis of our study. There are certain rules by which the ant moves from one city to another. Following are the rules laid out for ACO 1] The ants placed in each city must visit only once 2] there is a less probability of choosing a distance city. 3] More the pheromone deposits along the edges more likely the probability to choose the next city. 4] Once the destination to the food source is founds by the ant it starts depositing more pheromone along the edges. This pheromone deposits evaporate after each iteration. The real behaviour of ants depicts the ant colony optimization. The frail and blind ants actually cooperate with one another and this results into a complex behaviour. These ants determine the shortest distance between the source and destination. While on its journey towards source of food it lays a chemical compound called pheromone along the path. The ants are attracted by the deposits of pheromone along the route on which it traverses. In the vehicle routing problem which is the main objective of our study we place artificial ants in each city and in each distribution the algorithm guides by way of probability to choose the next city. The ant colony formula has been adapted from a paper by Dorigo(1997) has been used for our study purpose:
25352
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
The following equation represents the probability to move an ant from route i to route j.
,
(1)
where:
_
,
denotes the probability of ant k that will visit route j from i.
_ ^ is the set of routes which are not stayed by ant k in route i. denotes the relative significance of trail of pheromone.
β denotes the relative significance of measured length between the routes . The route chosen as the next route depends upon the existence of the pheromone trail which is given by the probability. The larger weight can be determined by experimenting with the and parameters . The pheromone evaporation is determined by (Dorigo 1991) once the ant exactly visits the route once we can calculate the pheromone deposits made by each ant along the edges. 4.Suggested GA Algorithm The steps in GA algorithm are as follows 1. 2. 3. 4.
Chromosome Setup: The chromosomes can be randomly created if the population is diverse. Good solutions can be produced. Evaluation of Chromosomes: Good chromosomes are evaluated based on the fitness value. Selection of chromosomes: The fittest chromosomes are selected based on how fit they are. Modification of Chromosomes: The modified , recombined genes and pairs of genes are put back into the population.
Chromosome setup: In an arbitrary manner the chromosomes will be initialized. We have setup the chromosome length to be p and each gene is initialized by its index value which is determined by p number of genes. The generated chromosome are as follows
j 1( j ) 2( j ) 3( j ) (p j )
---------- (2)
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
25353
Based on t*he above formulation a X number of chromosomes will be created randomly. The Roulette Wheel Selection and the mechanisms to produce valid crossover operators are discussed herewith.The chromosome encodes each tour. Let us encode, a tour of four cities might be 2,5,1,2,3. In solving a TSP a simple crossover is not possible. Let us see the example in which the crossover occurs at position 3. Parent 1
23456
Parent 2
46325
Child 1
23425
Child 2
46356
In the above problem child 1 visits the routes two times which is a violation of the tsp, and there is no visit done by the Child 2 at all. For producing valid tours a mutation encoding must be used. Partially matched crossover is the simplest to understand. The paper on Buckland (2002, pp. 130-132) suggests the technique as follows: A crossover point is picked.
Parent1 Parent2
2 4
3 6
-----
4 3
5 2
-----
We establish the following mapping. 4 -> 3 5 -> 2 Swapping of the gene takes place in each cycle through each parent gene in the above manner. Iteration No 1 (mapping 4 with 3):
Child1: 24356 Child2: 36425 Iteration No 2 (mapping 5 with 2): Child1: 54326
Child2: 36452
6 5
25354
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
The resulting valid permutations of crossover with no duplicates. Selection of chromosomes: The selection operation lead to x number of fit chromosomes which is based on the minimum fitness value. We mutate the newly obtained chromosomes and will be subjected to the process of, selection and fitness evaluation. The terminating condition is determined by the maximum number of iterations. 5. Implementation of Results Our proposed methodology of GA and ACO is implemented using the platform of MATLAB (MATLAB version 7.10) to determine the optimal supplier routes. A web based agent maintains the, the database containing the detailed data required for the analysis. For our paper we have used the distances between the various cities in the form of latitude and longitude. Total Distance = 351.3765 50 40 30 20 10 0 -10 -20 -30 -20
-10
0
10
20
30
Figure 1Results of TSP using NN Algorithm
40
25355
y
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
Figure 2Results of TSP using ACO Algorithm
Total Distance = 678.1868, Iteration = 9956 80 70 60 50 40 30 20 10 0
0
10
20
30
40
50
60
Figure 3 Results of TSP using GA Algorithm
70
25356
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
Table 1 Comparison Table of GA ,ACO with NN Algorithm No of Cities 30 50 70 96 100
GA 8981 33732 703 326 585
NN 10045 38894 761 351 641
ACO 10404 36303 954.87 521 841
6. Conclusions Better performance is outclassed by GA compared to Nearest neighbor algorithm and Ant colony optimisation as evident from the table. All the raw material aggregation from different pockets is done thru web based model. The main aim of the paper is to determine the shortest route thru which the materials can be dispatched to the manufacturers. Our methodology in adopting the GA has been successfully implemented to solve the optimal vehicle routes. References [1] SevalEne, Nursel, Open loop reverse supply chain network design, Procedia Social and behavioral sciences, Science Direct. 109(2014) 11101115. [2] KananGovindan , Maria Popi, Reverse supply chain coordination by revenue sharing contract: A case for the personal computers industry, European Journal of Operational Research, 233(2014)326-336. [3] TatavarthySrinivasRaoBalkrishnaRao, Naidu,MallikharjunaBabu,A Novel and Efficient Approach for Materials Demand Aggregation using Genetic Algorithm, International Journal of Computer Science and Network Security. 9(2009) 203-212. [4] Vincent, Shih, Multi-start simulated annealing heuristic for the location routing problem with simultaneous pickup and delivery,Applied soft computing. 24(2014)284-290. [5] Ernest D.R, Santibanez , Ali Diabat, Solving a reverse supply chain design problem by improved benders decomposition schemes, An International Journal of Computers & Industrial Engineering. 66 (2013), 889-898. [6] Black, Paul E,greedy algorithm, Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology (NIST). Retrieved 17 August 2012. [7] Applegate, D. ,Bixby, R. M, Chvátal, V. Cook, W. J. (2006), The Traveling Salesman Problem, ISBN 0-691-12993-2.Y. Shyi-Ming Chen, Chih-Yao Chen, Parallelized genetic Ant Colony System for solving the travelling salesman problem, Expert systems with Applications, Elsevier Journal, 38(2011) 3873-3883. [8] Marisa, P. d. B, Simme, D. P. F, RommertD,Reverse Logistics: a review of case studies. Econometric Institute Report,(2002). Boyer, K. K.,Extending the supply chain: Integrating operations and marketing in the online grocery industry. Journal of Operations Management, 23(6) (2005)642. [9] Mollenkopf, D. A,Rabinovich, E, Laseter T. M,Boyer, K. K. , Managing Internet Product Returns: A Focus on Effective Service Operations,Decision Sciences, 38(2) (2007)215.
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
25357
[10] Zee-Sun, Y, Linda, K. G, Developing customer loyalty from e-tail store image attributes,Managing Service Quality, 17(1), 4-22. [11] Shalinising,EjajAslamLodhi, study of variation in tsp using genetic algorithm and its operator comparison, International Journal of Soft Computing and Engineering(IJSCE)ISSN:2231-2307 volume 3, Issue 2 (2013). [12] Applegate, D. L.; Bixby, R. M.; Chvátal, V.; Cook, W. J. (2006), The Traveling Salesman Problem, ISBN 0-691-12993-2 S [13] Chen, Chih-Yao Chen,” Parallelized genetic Ant Colony System for solving the travelling salesman problem”Expert systems with Applications 38(2011) 3873-3883 Elsevier Journal [14] Dorigo,M.&Gambardella, L.M(1997) Ant Colonies for the travelling Salesman Problem. Biosystems,43,73-81
ScienceDirect Materials Today: Proceedings 5 (2018) 25350–25357
www.materialstoday.com/proceedings
IConAMMA_2017
An Evaluation of ACO and GA TSP in a Supply Chain Network T Srinivas Raoa* a
Dept of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwavidyapeetham,Bengaluru ,India
Abstract The need for supply chain management solutions in logistics management is the need of the hour. An E commerce based web aggregation model has been developed which aggregates the raw materials in various pockets through a common vehicle which routes the cities and ensures efficient supply at the low cost. In this paper we have developed an GA and ACO algorithm which calculates the shortest routes through which the material can be supplied to the manufacturing units. Our main aim is to compare the two algorithms and evaluate their efficiencies. © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Advances in Materials and Manufacturing Applications [IConAMMA 2017]. Keywords: GA, ACO, Supply chain ,Pockets
1.Introduction In the evolutionary optimization algorithms, the ACO or ant colony optimization, is the technique to find the shortest distance for the vehicle routing problems. In the real scenario the ants find the shortest distance to locate the food destination by laying a chemical substance called pheromone on the trail for finding the food. The pheromone is a communication to other ants to follow them. For weaker destinations the pheromone substance evaporates quickly so that other ants do not follow. When the destination or the source of food is noticed the ants quickly march along the trail and thereby the pheromone concentrations also becomes strong thereby resulting in the shortest route.
* Corresponding author. Tel.: +91-080-251-83700; fax: +91-080-284-40092. E-mail address: [email protected] 2214-7853 © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Advances in Materials and Manufacturing Applications [IConAMMA 2017].
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
25351
By frequent evaporation of the pheromone the local optimization convergence is avoided. The whole process results into a constrained solution space. The GA are computerized search algorithms which mimic genetics and natural selection evolutionary algorithms. Using the technique of GA an array of NP problems can be solved using the technique of GA. GA manipulates strings of information called chromosomes. The fitness function is used to assign a score which is evaluated and assigned to these chromosomes according to the criterion defined by the programmer. The probability of survival is determined by these fitness values during a round of reproduction. A new chromosome is produced from the combination of two or more chromosomes. 2. Related Works A global optimum is found by using the greedy algorithm [6] arriving at each stage thru local optimum choice by following problem metaheuristic. The first algorithm to find an optimum solution of TSP is the nearest neighbour algorithm [12]. We can start the tour from any city and by repeatedly visiting the neighbours we can complete the tour. This continues till the last city has been visited. Though this is not optimum it yields good approximate solutions. In ACO due to numerous local optimum points an optimum solution cannot be reached and also the search space is enormous [13]. Many practical problems can be modelled using Multi travelling sales person (MTSP) [13]. There are no sub tours in TSP and the MTSP which are similar. In the MTSP the m tours get partitioned into n cities and each tour results into a TSP for single salesperson. It is difficult to model MTSP than TSP because it requires assigning salespersons to each cities, also there is a need for optimal ordering of the cities within each salespersons tour. 3. Ant Colony Algorithm Many combinatorial optimization problems can be solved thru ant colony optimization algorithms, starting from traveling sales man problem, job scheduling and many stochastic problems. A near optimal solution can be found thru ACO but still it is far from reaching exact solutions. There is a dynamic change in the graphs in comparison of ACO to genetic algorithm and simulated annealing ; some of the advantages of ACO is that it can adapt to changes in real time and run continuously. In ACO we use a set of ants which form the basis of our study. There are certain rules by which the ant moves from one city to another. Following are the rules laid out for ACO 1] The ants placed in each city must visit only once 2] there is a less probability of choosing a distance city. 3] More the pheromone deposits along the edges more likely the probability to choose the next city. 4] Once the destination to the food source is founds by the ant it starts depositing more pheromone along the edges. This pheromone deposits evaporate after each iteration. The real behaviour of ants depicts the ant colony optimization. The frail and blind ants actually cooperate with one another and this results into a complex behaviour. These ants determine the shortest distance between the source and destination. While on its journey towards source of food it lays a chemical compound called pheromone along the path. The ants are attracted by the deposits of pheromone along the route on which it traverses. In the vehicle routing problem which is the main objective of our study we place artificial ants in each city and in each distribution the algorithm guides by way of probability to choose the next city. The ant colony formula has been adapted from a paper by Dorigo(1997) has been used for our study purpose:
25352
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
The following equation represents the probability to move an ant from route i to route j.
,
(1)
where:
_
,
denotes the probability of ant k that will visit route j from i.
_ ^ is the set of routes which are not stayed by ant k in route i. denotes the relative significance of trail of pheromone.
β denotes the relative significance of measured length between the routes . The route chosen as the next route depends upon the existence of the pheromone trail which is given by the probability. The larger weight can be determined by experimenting with the and parameters . The pheromone evaporation is determined by (Dorigo 1991) once the ant exactly visits the route once we can calculate the pheromone deposits made by each ant along the edges. 4.Suggested GA Algorithm The steps in GA algorithm are as follows 1. 2. 3. 4.
Chromosome Setup: The chromosomes can be randomly created if the population is diverse. Good solutions can be produced. Evaluation of Chromosomes: Good chromosomes are evaluated based on the fitness value. Selection of chromosomes: The fittest chromosomes are selected based on how fit they are. Modification of Chromosomes: The modified , recombined genes and pairs of genes are put back into the population.
Chromosome setup: In an arbitrary manner the chromosomes will be initialized. We have setup the chromosome length to be p and each gene is initialized by its index value which is determined by p number of genes. The generated chromosome are as follows
j 1( j ) 2( j ) 3( j ) (p j )
---------- (2)
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
25353
Based on t*he above formulation a X number of chromosomes will be created randomly. The Roulette Wheel Selection and the mechanisms to produce valid crossover operators are discussed herewith.The chromosome encodes each tour. Let us encode, a tour of four cities might be 2,5,1,2,3. In solving a TSP a simple crossover is not possible. Let us see the example in which the crossover occurs at position 3. Parent 1
23456
Parent 2
46325
Child 1
23425
Child 2
46356
In the above problem child 1 visits the routes two times which is a violation of the tsp, and there is no visit done by the Child 2 at all. For producing valid tours a mutation encoding must be used. Partially matched crossover is the simplest to understand. The paper on Buckland (2002, pp. 130-132) suggests the technique as follows: A crossover point is picked.
Parent1 Parent2
2 4
3 6
-----
4 3
5 2
-----
We establish the following mapping. 4 -> 3 5 -> 2 Swapping of the gene takes place in each cycle through each parent gene in the above manner. Iteration No 1 (mapping 4 with 3):
Child1: 24356 Child2: 36425 Iteration No 2 (mapping 5 with 2): Child1: 54326
Child2: 36452
6 5
25354
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
The resulting valid permutations of crossover with no duplicates. Selection of chromosomes: The selection operation lead to x number of fit chromosomes which is based on the minimum fitness value. We mutate the newly obtained chromosomes and will be subjected to the process of, selection and fitness evaluation. The terminating condition is determined by the maximum number of iterations. 5. Implementation of Results Our proposed methodology of GA and ACO is implemented using the platform of MATLAB (MATLAB version 7.10) to determine the optimal supplier routes. A web based agent maintains the, the database containing the detailed data required for the analysis. For our paper we have used the distances between the various cities in the form of latitude and longitude. Total Distance = 351.3765 50 40 30 20 10 0 -10 -20 -30 -20
-10
0
10
20
30
Figure 1Results of TSP using NN Algorithm
40
25355
y
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
Figure 2Results of TSP using ACO Algorithm
Total Distance = 678.1868, Iteration = 9956 80 70 60 50 40 30 20 10 0
0
10
20
30
40
50
60
Figure 3 Results of TSP using GA Algorithm
70
25356
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
Table 1 Comparison Table of GA ,ACO with NN Algorithm No of Cities 30 50 70 96 100
GA 8981 33732 703 326 585
NN 10045 38894 761 351 641
ACO 10404 36303 954.87 521 841
6. Conclusions Better performance is outclassed by GA compared to Nearest neighbor algorithm and Ant colony optimisation as evident from the table. All the raw material aggregation from different pockets is done thru web based model. The main aim of the paper is to determine the shortest route thru which the materials can be dispatched to the manufacturers. Our methodology in adopting the GA has been successfully implemented to solve the optimal vehicle routes. References [1] SevalEne, Nursel, Open loop reverse supply chain network design, Procedia Social and behavioral sciences, Science Direct. 109(2014) 11101115. [2] KananGovindan , Maria Popi, Reverse supply chain coordination by revenue sharing contract: A case for the personal computers industry, European Journal of Operational Research, 233(2014)326-336. [3] TatavarthySrinivasRaoBalkrishnaRao, Naidu,MallikharjunaBabu,A Novel and Efficient Approach for Materials Demand Aggregation using Genetic Algorithm, International Journal of Computer Science and Network Security. 9(2009) 203-212. [4] Vincent, Shih, Multi-start simulated annealing heuristic for the location routing problem with simultaneous pickup and delivery,Applied soft computing. 24(2014)284-290. [5] Ernest D.R, Santibanez , Ali Diabat, Solving a reverse supply chain design problem by improved benders decomposition schemes, An International Journal of Computers & Industrial Engineering. 66 (2013), 889-898. [6] Black, Paul E,greedy algorithm, Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology (NIST). Retrieved 17 August 2012. [7] Applegate, D. ,Bixby, R. M, Chvátal, V. Cook, W. J. (2006), The Traveling Salesman Problem, ISBN 0-691-12993-2.Y. Shyi-Ming Chen, Chih-Yao Chen, Parallelized genetic Ant Colony System for solving the travelling salesman problem, Expert systems with Applications, Elsevier Journal, 38(2011) 3873-3883. [8] Marisa, P. d. B, Simme, D. P. F, RommertD,Reverse Logistics: a review of case studies. Econometric Institute Report,(2002). Boyer, K. K.,Extending the supply chain: Integrating operations and marketing in the online grocery industry. Journal of Operations Management, 23(6) (2005)642. [9] Mollenkopf, D. A,Rabinovich, E, Laseter T. M,Boyer, K. K. , Managing Internet Product Returns: A Focus on Effective Service Operations,Decision Sciences, 38(2) (2007)215.
T Srinivas Rao / Materials Today: Proceedings 5 (2018) 25350–25357
25357
[10] Zee-Sun, Y, Linda, K. G, Developing customer loyalty from e-tail store image attributes,Managing Service Quality, 17(1), 4-22. [11] Shalinising,EjajAslamLodhi, study of variation in tsp using genetic algorithm and its operator comparison, International Journal of Soft Computing and Engineering(IJSCE)ISSN:2231-2307 volume 3, Issue 2 (2013). [12] Applegate, D. L.; Bixby, R. M.; Chvátal, V.; Cook, W. J. (2006), The Traveling Salesman Problem, ISBN 0-691-12993-2 S [13] Chen, Chih-Yao Chen,” Parallelized genetic Ant Colony System for solving the travelling salesman problem”Expert systems with Applications 38(2011) 3873-3883 Elsevier Journal [14] Dorigo,M.&Gambardella, L.M(1997) Ant Colonies for the travelling Salesman Problem. Biosystems,43,73-81