How to cite: B. P. Silalahi, N. Fathiah, and P. T. Supriyo, “Use of Ant Colony Optimization Algorithm for 

Determining Traveling Salesman Problem Routes", Mantik, vol. 5, no. 2, pp. 100-111, October 2019. 

Use of Ant Colony Optimization Algorithm for Determining 

Traveling Salesman Problem Routes  
 

 

Bib Paruhum Silalahi1, Nurul Fathiah2, Prapto Tri Supriyo3 

Department of Mathematics, IPB University, bibparuhum@gmail.com1 

Department of Mathematics, IPB University, nurulfathiah393@gmail.com2 

Department of Mathematics, IPB University, praptotrisupriyo@gmail.com3 

 
doi: https://doi.org/10.15642/mantik.2019.5.2.100-111 

 

 
Abstrak: Optimisasi Koloni Semut merupakan salah satu metode meta-heuristic yang digunakan 
untuk menyelesaikan masalah optimisasi kombinatorial yang cukup sulit. Algoritma Optimisasi 

Koloni Semut diinspirasi dari perilaku semut dalam dunia nyata untuk membangun jalur terpendek 
antara sumber makanan dan sarangnya. Traveling Salesman Problem adalah suatu permasalahan 

dalam optimisasi. Traveling Salesman Problem merupakan permasalahan untuk mencari jarak 
minimal dari node awal menuju node selanjutnya, dengan setiap node harus didatangi persis satu kali 

dan harus kembali ke node awal.  Traveling Salesman Problem merupakan suatu masalah yang non-
deterministic polynomial-time complete.   Pada penelitian ini didiskusikan penyelesaian Traveling 

Salesman Problem menggunakan algoritma Optimisasi Koloni Semut dan juga menggunakan 
algoritma eksak.  Hasil penelitian menunjukkan bahwa semakin besar ukuran kasus Traveling 

Salesman Problem, waktu eksekusi yang dibutuhkan semakin lama. Kemudian diperoleh hasil  
bahwa waktu eksekusi yang dihasilkan Optimisasi Koloni Semut jauh lebih cepat dibandingkan 

waktu eksekusi metode eksak.  

 

Kata kunci: Algoritma, Optimisasi , Optimisasi Koloni Semut, Traveling Salesman Problem 

 
 

Abstract: Ant Colony Optimization is one of the meta-heuristic methods used to solve combinatorial 

optimization problems that are quite difficult. Ant Colony Optimization algorithm is inspired by ant 
behaviour in the real world to build the shortest path between food sources and their nests. Travelling 

Salesman Problem is a problem in optimization. Travelling Salesman Problem is a problem to find 
the minimum distance from the initial node to the whole node with each node must be visited exactly 

once and must return to the initial node. Travelling Salesman Problem is a non-deterministic 
polynomial-time complete problem. This research discusses the solution of the Traveling Salesman 

Problem using the Ant Colony Optimization algorithm and also using the exact algorithm. The 
results showed that the greater the size of the Traveling Salesman Problem case, the longer the 

execution time required. The results also showed that the execution times of the Ant Colony 
Optimization are much faster than the execution time of the exact method.  

 

Keywords: Algorithm, Optimization, Ant Colony Optimization, Traveling Salesman Problem 

 

 

  

Jurnal Matematika MANTIK 

Volume 5, Nomor 2, October 2019, pp. 100-111 

 

ISSN: 2527-3159 (print) 2527-3167 (online) 

mailto:bibparuhum@gmail.com3
http://u.lipi.go.id/1458103791
http://u.lipi.go.id/1457054096


B. P. Silalahi, N. Fathiah, and P. T. Supriyo 

Use of Ant Colony Optimization Algorithm for Determining Traveling Salesman Problem Routes 

101 

1. Introduction 

Optimization is the study of techniques for finding the maximum or minimum 

value of a problem. Some examples of studies of optimization techniques are 

modifying of steps in the steepest descent method [1] - [3], computational test of 

Newton's method variance algorithm on nonlinear optimization problems without 

constraints [4], comparison of sensitivity analysis on linear optimization using 

optimal partition and optimal basis [5], elaborating a combination method for 

solving nonlinear equations [6], overview of Newton's optimization method [7]. 

Optimization is applied in many aspects of life such as generating star catalogue 

for satellite attitude navigation using density-based clustering [8], but many of 

them are applied to transportation and distribution systems [9] - [11]. 

Travelling Salesman Problem (TSP) is an example of problems in 

optimization. Problems associated with TSP are often encountered in distribution 

systems. Basically, TSP is a problem to find the minimum distance from the 

initial node to the next node with the rule that each node must be visited exactly 

once. Furthermore, after all the nodes visited exactly once, they must return to the 

initial node. One of the characteristics of TSP is non-deterministic polynomial-

time complete (NPC) which implies that there is no optimal solution other than 

having to try all possible solutions [12]. 

The application of TSP in the real world is often used in the case of 

distributing goods and services. Problems that are often encountered relating to 

distribution such as determining travel routes that minimize travel time, distance 

travelled, or operational costs. 

TSP can be solved by exact, heuristic or meta-heuristic methods. One of the 

meta-heuristic methods that can be used to solve the TSP is the Ant Colony 

Optimization algorithm. Some other meta-heuristic methods such as genetic 

algorithm [13], particle swarm optimization [14], firefly algorithm [15][16] and 

interior point method [17], [18]. Ant Colony Optimization (ACO) is adapted from 

the behaviour of the ant colony as a system for finding and collecting food. Ants 

have the ability to find food effectively by finding the shortest path between the 

food source and the nest without using the sense of sight at the beginning of the 

ant, walking random looking for food. Along the way, the ant will leave a trail 

called pheromones. After successfully obtaining food, the ant will return to the 

nest with a trail guide pheromone left behind. Other ants will take food by 

following in the footsteps of pheromones. Pheromones can evaporate. The longer 

the distance of a route, the longer the ants are on the route, so that the pheromones 

on the route will be reduced faster because it evaporates when compared to the 

shorter distance route. This causes the ants to tend to choose the shortest route 

[19]. 

ACO is widely used as a solution to various optimization problems, one of 

which is TSP. This algorithm is compiled using distance data between locations, 

some ants that communicate with their colonies indirectly using the help of 

pheromones. 

 

2. Research Method  

In this research, TSP cases will be solved using the Integer Linear 

Programming (ILP) model and the Ant Colony Optimization (ACO) model. 



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102 

Furthermore, the results of the two models will be compared specifically in the 

aspects of execution time and total travel distance. Both models were executed 

using a laptop with the specifications of ASUS A 455LD i5 8GB RAM. 

 

3. Results and Discussion 

3.1. Travelling Salesman Problem 

TSP is a problem of determining the route when someone will come to all 

locations that have been determined. Starting from the first location and then 

visiting all the locations that have been determined exactly once. Then after that, 

the person/object must return to the first location. What we want to achieve is the 

minimum total travel distance [20]. 

TSP can be formulated as follows, for example: 

𝑛   : number of nodes to be visited,  
𝑐𝑖𝑗 : distance of node 𝑖 to node 𝑗,  

𝑢𝑖  : additional variables when visiting node 𝑖. 
Decision variable: 

𝑥𝑖𝑗 = {
1,  if there is a trip from node 𝑖 to node 𝑗
0,  else

 

The objective function of the TSP is to find a value so that the total distance 

traveled is minimum, that is: 

min 𝑍 = ∑  

𝑛

𝑖=1

∑  

𝑛

j=1

𝑐𝑖𝑗 𝑥𝑖𝑗 . 

 

Constraints:  

1. Nodes must be visited exactly once. 
∑ 𝑥𝑖𝑗

𝑛
𝑖=1 = 1,   ∀𝑗 = 1,2, … , 𝑛 

∑ 𝑥𝑖𝑗
𝑛
𝑗=1 = 1,   ∀𝑖 = 1,2, … , 𝑛 

2. No sub tour formed on the journey in the TSP. This means that there is only 
one tour from TSP. Because returning to the initial location will form a cycle, 

𝑢𝑖 − 𝑢𝑗 + 𝑛𝑥𝑖𝑗 ≤ 𝑛 −  1, 𝑖 ≠ 𝑗, ∀𝑖 = 2, 3, … , 𝑛;  ∀𝑗 = 2, 3, … , 𝑛;  𝑢𝑗 ≥ 0 . 

 

3.2. Ant Colony Optimization Algorithm  

Ant Colony Optimization (ACO) is one of the meta-heuristic methods used to 

find a combinatorial optimization problem that is quite difficult. The ACO 

algorithm is represented by ant behaviour in the real world to build the shortest 

path between food sources and their nests. Each ant randomly starts its route from 

a node. Each node is visited by ants to form a route, and this is done repeatedly. 
The ant selects the nodes to be visited using the probability function, based on the 

distance of the node, and the number of pheromones found on the side connecting 

the node. Nodes that have smaller distances and have higher levels of pheromones 

will be more likely to be visited by ants [21]. 

Nodes that have been visited by ants will be recorded in memory which we 

call by the name tabulist. This tabulist prevents ants from going to the nodes that 

have been visited. The tabulist will be full when all ants have visited all nodes. 

Then the pheromone update rules are applied. Calculation of the loss of 



B. P. Silalahi, N. Fathiah, and P. T. Supriyo 

Use of Ant Colony Optimization Algorithm for Determining Traveling Salesman Problem Routes 

103 

pheromone levels on each side is done. For shorter routes, the loss of pheromone 

levels will be relatively longer compared to the longer routes. 

The process will be repeated until the tour reaches the maximum value or the 

system is in a stagnant situation, which is a situation where the system is no 

longer looking for other alternative solutions [22]. Figure 1 shows a simple ACO 

flow. 

 

3.3. Steps of Ant Colony Optimization Algorithm   

The flow of the ACO algorithm is shown in Figure 1. The steps taken are as 

follows. 

1. Data The initial data presented is only in the form of 𝑥 and 𝑦 coordinate 
points, so the distance between nodes must be calculated using the Euclidian 

formula: 

𝑑𝑖𝑗 = √(𝑥𝑖 − 𝑥𝑗 )
2 + (𝑦𝑖 − 𝑦𝑗)

2  ,  

        where 

𝑑𝑖𝑗    : the distance between node 𝑖 and node 𝑗 

𝑥𝑖  : 𝑥 coordinate of node 𝑖, 𝑥𝑗    : 𝑥 coordinate of node 𝑗, 

𝑦𝑖  : 𝑦 coordinate of node 𝑖, 𝑦𝑗   : 𝑦 coordinate of node 𝑗. 

2. Determine the next node the ant will pass by calculating the probability using 
the formula:  

𝑃𝑖𝑗
𝑘 (𝑡) =  

[𝜏𝑖𝑗 (𝑡)]
𝛼

∙ [𝜂𝑖𝑗 (𝑡)]
𝛽

∑ [𝜏𝑖𝑗(𝑡)]
𝛼

∙ [𝜂𝑖𝑗 (𝑡)]
𝛽

𝑗 𝜖 𝑁𝑖
𝑘

  , if 𝑗 𝜖 𝑁𝑖
𝑘  

where: 

𝑃𝑖𝑗
𝑘 (𝑡) : the probability of ant 𝑘 run from node 𝑖 to node 𝑗   at time 𝑡, 

𝜏𝑖𝑗(𝑡) : number of ant pheromones from node 𝑖 to node 𝑗 at time 𝑡, 

𝜂𝑖𝑗 (𝑡)  : distance inverse between node 𝑖 and node 𝑗 at time 𝑡, 

𝑁𝑖
𝑘       : the set of points to be visited by ant 𝑘  which is at point 𝑖, 

𝛼:  ant pheromones parameter, 
𝛽: distance control parameter, 
and informing the route, the next node selection criteria use the Roulette 

Wheel Selection algorithm, with the following steps: 

a. determine one random value between 0 and 1, 

b. calculate the cumulative probability value 𝑃𝑖𝑗
𝑘 (𝑡), 

c. compare the random value in step a with each cumulative probability 

value 𝑃𝑖𝑗
𝑘 (𝑡) in step b, 

d. if the cumulative probability value 𝑃𝑖𝑗
𝑘 (𝑡) is greater than the random 

value, then it will be selected as the next node. 

 



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Vol. 5, No. 2, October 2019, pp. 100-111 

104 

 
Figure 1.  The flow of Ant Colony Optimization Algorithm. 

 

3. Repeat step 2 for each ant until all the nodes are selected and form a route. 
4. Calculate the total distance of each node that each ant passes. 
5. Choose the best route from all the routes produced, i.e. the route with the 

minimum total distance obtained in step 4. 

6. Update the pheromone with the following formula: 
𝜏𝑖𝑗 (𝑡) = (1 − 𝜌)𝜏𝑖𝑗(𝑡) + Δ𝜏𝑖𝑗(𝑡) 

with  Δ𝜏𝑖𝑗(𝑡) = {
1

𝐿𝑘
,  if ant 𝑘 run from node 𝑖 to node 𝑗

0 ,  else
 

where: 

 𝜏𝑖𝑗 (𝑡)   :  number of ant pheromones from node 𝑖 to node 𝑗 at time 𝑡, 

 Δ𝜏𝑖𝑗 (𝑡) :  changes in pheromone values in the node (𝑖, 𝑗)  made by ant 𝑘, 

 𝐿𝑘           : travel length of ant 𝑘, 
 𝜌           : global pheromone evaporation parameter with  0 <  𝜌 ≤ 1, 
pheromone updates are carried out with the aim of updating the amounts of 

pheromones left by ants as a result of evaporation. The number of 

pheromones that have been updated will be used as information for ants in 

the next iteration. 

7. This algorithm is terminated if the stopping criteria have been reached, i.e. 
when the number of iterations has been reached. 

8. Return to step 2 if the stopping criteria have not been reached.  
 

  



B. P. Silalahi, N. Fathiah, and P. T. Supriyo 

Use of Ant Colony Optimization Algorithm for Determining Traveling Salesman Problem Routes 

105 

3.4. Solving by Using Integer Linear Programming (ILP) 

Accomplishment by using ILP is carried out with the help of LINGO 11.0 

software. The data used are 𝑥 and 𝑦 coordinate which indicate the location of the 
nodes in Cartesian coordinates. The data are hypothetic, obtained from a random 

function. The 𝑥 and 𝑦 coordinate data are presented in Table 1. From the data in 
Table 1, the distance between nodes is searched using the euclidian formula to 

produce a distance matrix. 

Case I 

The results of execution using ILP for the case of 30 nodes required execution 

time of 1 minute 34 seconds with the number of iterations of 436.649 and 

obtained a minimum total travel distance 388.371 km. 

Case II 

The results of execution using ILP for the case of 35 nodes required 11 

minutes 04 seconds of execution time with the number of iterations 3.351.017 and 

obtained a minimum total travel distance 427.584 km. 

Case III 

The results of execution using ILP for the case of 38 nodes required execution 

time of 6 hours 38 minutes 14 seconds with the number of iterations 142.214.825 

and the result obtained a minimum total travel distance 430.251 km. 

 

3.5. Solving by Using Ant Colony Optimization Algorithm   

In this work, the helping of a mathematics software is used to find the 

solution of the TSP using the ACO algorithm. Each case uses different parameter 

values α, β, and m (number of ants). The value of α is at the interval   0<α≤1,  

β>0, and m>0 [23]. The values of α, β, dan m, which can produce the smallest 

total approach distance will be presented. 

Case I 

In case I, the experiments are carried out using parameters with values as 

presented in Table 2. After conducting several experiments, the smallest total 

approach distance is 392.8014 km with execution time 11.836 s in 500 iterations. 

The route obtained is as shown in Figure 2. 

Case II 

In case II, the experiments are carried out using parameters with values, as 

shown in Table 3. After conducting several experiments, the smallest total 

approach distance was 463.4509 km with an execution time 21.340 s in 700 

iterations. Figure 3 shows the route obtained. 

Case III 

In case III, experiments will be conducted by using parameters with the 

values presented in Table 4. After conducting several experiments, the smallest 

total approach distance is 464.7083 km with an execution time of 38.00 s in 800 

iterations. The route obtained is presented in Figure 4. 

 
Table 1. The data in each case 

Case I Case II Case III 

x y x y x y 

81.91 37.47 81.91 37.47 81.91 37.47 

83.16 49.92 83.16 49.92 83.16 49.92 



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106 

Case I Case II Case III 

x y x y x y 

84.76 40.37 84.76 40.37 84.76 40.37 

75.27 63.61 75.27 63.61 75.27 63.61 

38.10 28.80 38.10 28.80 38.10 28.80 

23.28 95.00 23.28 95.00 23.28 95.00 

77.49 15.56 77.49 15.56 77.49 15.56 

39.22 22.32 39.22 22.32 39.22 22.32 

62.88 32.60 62.88 32.60 62.88 32.60 

36.63 65.29 36.63 65.29 36.63 65.29 

17.02 41.47 17.02 41.47 17.02 41.47 

89.50 30.82 89.50 30.82 89.50 30.82 

17.02 52.23 17.02 52.23 17.02 52.23 

13.25 100.12 13.25 100.12 13.25 100.12 

73.20 52.75 73.20 52.75 73.20 52.75 

34.08 89.16 34.08 89.16 34.08 89.16 

55.29 90.36 55.29 90.36 55.29 90.36 

41.42 25.66 41.42 25.66 41.42 25.66 

53.31 90.58 53.31 90.58 53.31 90.58 

42.66 31.08 42.66 31.08 42.66 31.08 

54.49 56.85 54.49 56.85 54.49 56.85 

99.85 60.28 99.85 60.28 99.85 60.28 

63.39 100.56 63.39 100.56 63.39 100.56 

21.86 77.04 21.86 77.04 21.86 77.04 

35.09 36.91 35.09 36.91 35.09 36.91 

56.86 61.57 56.86 61.57 56.86 61.57 

83.89 52.24 83.89 52.24 83.89 52.24 

53.04 56.22 53.04 56.22 53.04 56.22 

34.06 37.48 34.06 37.48 34.06 37.48 

66.57 25.84 66.57 25.84 66.57 25.84 

  34.56 20.48 34.56 20.48 

  63.63 63.48 63.63 63.48 

  56.74 22.22 56.74 22.22 

  15.90 62.84 15.90 62.84 

  88.86 79.06 88.86 79.06 

    73.48 82.00 

    85.58 55.79 

    90.92 54.49 

 

Table 2. The results of case I. 

The Number 

of Iterations 
𝛼 𝛽 𝑚 

Total Distance  

(in km) 

300 1 5 50 425.1656 

350 1 5 50 417.1519 

400 1 5 50 407.1231 

450 1 5 50 399.8285 

500 0.9 5 50 420.6379 

500 1 5 50 392.8014 

 

 



B. P. Silalahi, N. Fathiah, and P. T. Supriyo 

Use of Ant Colony Optimization Algorithm for Determining Traveling Salesman Problem Routes 

107 

 

Figure 2.  Route approach in case I. 

 

Table 3. The results of case II. 

The Number 

of Iterations 
𝛼 𝛽 𝑚 

Total Distance  

(in km) 

300 0.7 1 50 535.0748 

300 1 5 50 497.4685 

400 1 5 50 481.1290 

500 1 5 50 472.6510 

600 1 5 50 470.1542 

700 1 5 50 463.4509 

 

 

Figure 3.  Route approach in case II. 



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108 

Table 4. The results of case III. 

The Number 

of Iterations 
𝛼 𝛽 𝑚 

Total Distance 

 (in km) 

400 0.9 5 50 543.4537 

400 0.9 5 70 494.852 

500 0.9 5 70 487.6729 

600 1 5 70 488.1252 

700 0.9 5 70 481.1320 

800 0.9 5 70 464.7083 

 

 

Figure 4.  Route approach in case III. 

 

3.6. Results comparison 

ILP and ACO execution times are shown in Table 5. From the table, it can be 

seen that ACO execution time is much faster than ILP execution time. For Case I, 

ACO execution time is eight times faster than ILP. In Case II, ACO execution 

time is 31 times faster than ILP execution time. ACO execution time for Case III 

is 629 times faster than ILP. 

Table 6 shows the total distance produced by each method. ILP method 

produces optimal values whereas ACO produces a value approach. In Case I, the 

percentage of distance difference is 1%. While in Case II and Case III have a 

percentage difference of 8%. 

 
Table 5. Comparison of execution time. 

Case 

Execution Time  

(hours: minutes: seconds) 

ILP ACO 

I 0:01:34 0:0:11.836 

II 0:11:04 0:0:21.340 

III 6:38:14 0:0:38.000 

 

  



B. P. Silalahi, N. Fathiah, and P. T. Supriyo 

Use of Ant Colony Optimization Algorithm for Determining Traveling Salesman Problem Routes 

109 

Table 6.  Comparison of the total distance. 

Case 

Total Distance  

(in km) 

ILP ACO 

I 388.371 392.8014 

II 427.584 463.4509 

III 430.251 464.7083 

 

4. Conclusion 

Based on the results of this study, it can be seen that the larger the cases that 

must be solved, the longer the execution time required. It can also be seen that the 

execution time produced by ACO is much faster than the execution time of the 

exact method.  From the three cases, the percentage of distance difference of the 

first case is 1 %, while the two other cases have a percentage difference of 8%. 

 

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