Applied mathematics in Engineering, Management and Technology 2 (3) 2014:311-325
www.amiemt-journal.com
Solving the traveling salesman problem in competitive situations
using the game theory
Mohammad Mahdi Mohtadi *
Ph.D. Candidate, Iran university of science and technology
Kazem Nogondarian
Assistant professor, Iran university of science and technology
* Corresponding Author, Tel. +989127588897 , Email:[email protected]
Abstract
This paper deals with examining the traveling salesman problem in competitive situations. The traveling salesman problem is one of the old and classic
problems and its different scenarios and models have been discussed. In this
paper, assuming the presence of a competitor in the environment, we try to
fit the optimal behavior of salesman. For this purpose, after literature review, the game theory is reviewed as a mathematical model for analyzing
the competitive problems. Then different scenarios of traveling salesman
problem have been classified from the perspective of competitive situations
and finally, after determining the investigation scope, one typical problem is
solved using game theory.
Keywords: Traveling salesman, game theory, competitive situations, mathematical modeling, the Nash equilibrium.
1 – Introduction
Traveling salesman problem (TSP) is one of the most basic problems in transportation routing and scheduling
which is discussed due to its importance in combinatorial loptimization as well as in computer science and is
used as a benchmark in most of the optimization methods. In TSP problem, the objective is to find the shortest
route or tour, which crossed a set of cities, and visited each city only once and then returns to the city that it has
started to move. Practical problems are more complex than this structure and have more span and more restrictions.
Diversity of traveling salesman problem has increased over times. Rural traveling salesman, traveling buyer,
Traveling Salesman Problems with Profits, Selective Traveling Salesman Problem, Prize Collection Traveling
Salesman Problem, Traveling Repairman Problem, Covering Tour Problem and On-line Traveling Salesman
Problem are the problems that have been aroused in TSP field.
A focus area of this paper is on the traveling salesman problem in a competitive environment. In this area, eachsales man who meets a node,will allocate all or part of that node request. Competitive environment, as will be
discussed in the next sections have a variety of situations. A special case of this study is a specific type of competition that has been studied in game theory. The main feature of decision making in game situations, is that
before any decide and choice, each player should analyze the response of competitor regarding to his choice and
then make a decision that is best for him. In other words, among various options of decision, regarding to the
response of the opponent and remembering that the competitor make his decision in a same way, he should
choosean alternative, which has most profits for him.
In order to analyze the situation after reviewing the literature in the field of traveling salesman problem, game
theory is introduced as a tool for solving the competitive problems and the algorithm for finding the optimal
behavior in these situations is introduced based on game theory concepts. Then, the traveling salesman problem
is examined from the viewpoint of competition and different scenarios that can be imagined in that are presented. Next, the competitive approach consistent with game theory, is described and modeled and is solved and
validated as an example problem.
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2 –Literature history
The traveling salesman problem was introduced in the eighteenth century and its general form was studied in
1930’s. After that, many articles were written about that. Among thesearticles, on-line TSP, competitive
TSP And MTSP Problems have been reviewed which somehow are connected with the present subject.
In on-line TSP Problems, Theprobleminformationsare obtained over time. Generally,on-time optimization procedure may be suitable for probability problems, which have sequences. Inon-line TSP Problems,Some customers are predetermined. But a group of customers and their request are identified during the service. Ausielloand
et al. in some papers (2001, 2004, 2008) examined this problem in various situations. Blom and et.al (2001),
examined the on-line TSP
Thesalesman of Dust on the line in against fair adversaries and Jaillet& Lu (2011) examined it in terms of its
Services’ flexibility.
Various articles had dealt withmulti-traveling salesman problem (MTSP). This problem is a development
of TSP problem with simultaneous presence of some salesmansin environments. Thesesalesmen Cooperate with
each other and have a common objective function. Several problems have been modeled and solved with this
approach. Angel and et.al (1972) And Orloff (1974) in separate articles, Using the application of
MTSP problem, modeled the problem of schools’ service scheduling. Christofides & Eilon (1969) and Savelsbergh (1995) used this problem for modeling the Pickup and Delivery Problem. Svestka (1973) modeledThe
problem of bank messenger scheduling using this approach.Laporte G, Nobert (1980) presented a cutting planes
algorithm to solve this problem. Ali & Kennington (1986) examined the problem inasymmetric situations. Franca and et.al (1995) examined the MTSP problem by minmax object.Samerkae, Abdolhamid and Takao (1999) presented a competition-based neural network solution for it.
Carter and Ragsdale (2006), solved the MSTP problem using genetic algorithm. Qu, Yi and Tang (2007) proposed a columnar competitive model to solve this problem. Talay, Erdogan and Dept (2009) modeled the problem based on Dynamic task selection and robust execution. MengShu & DaiBo (2012) developed a novel method to solve it.
The issue of investigation the competitive traveling salesman (CTSP) in theoretical literature is newly subject
and much less attention has been on it. In this problem, some travelling sales mans are working in environment. The difference of this problem with MTSP is that in MTSP problem, Sales mans are collaborating with
each other and their purpose is to optimizing a common objective function. But in CTSP, Each salesman has a
unique purpose and they want to optimize that which is conflict with other salesmans purpose. Fekete andet.al
(2004) examined this problem for the first time. They examined the problem in the situation of two salesmen,
each seeking to maximize their own interests in competition with an other salesman. In these situations, the
purpose ofgoing through the shortest tour, will be replaced with the purpose to meet the highest nodes before
the competitor meets their and the winner is the person who visit the highest number of nodes. Their paper, examines the competition in on-line situations. In their study, it was assumed that competitors are aware of his
opponent's position in every moment. In addition, the movement of two salesmen is done in turns. With the
mentioned assumptions, the optimal strategy is derived for each of two.
Kendall and Lee (2013) studied the same problem in other situations. They assumed that there are
n Cities and m salesmans andthere is no possibility of collaborating between them andmodeled the problem
based on game theory. Inthat paper, the Hyper-heuristic approach was used. In their study, similar to the general
pattern of Hyper-heuristic models, it was assumed that each salesman has a few solution which choose his path
between them, with respect to the other salesmans. Accordingly, firstly some innovative solutions were developed for each salesmans and then among low-level strategic, searching for the optimum solution was investigated.
As theoretical literature review shows, the Subject of competition in TSP had much less attention and the available articles have examined this subject only from limited. In this paper, this subject is discussed after reviewing game theory as a tool for modeling the problem.
3 - Game Theory
Standard decisions are made in the face of nature, which is considered as a neutral factor. But the decision game
is in the face of an intelligent factor. Competition is a game instance. The competitors considered as an intelli-
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M M Mohtadi and K Nogondarian
gent factor that is seeking for him interests which normally are in conflict with the opposed competitor. In thesesituations, a game will be formed. Each of the game, player and a set of behaviors a player can choose among
them, are called strategy. In real problems, the game can have more than two players. Although the two-players
games have many applications in real problems. Each choice facing players, is called Strategy and the benefits
gained by the players at the end of each game, is called revenue. (Myerson, 1991, Chapter 1)
Most critical question in the game is about the optimal behavior of competitors, or the so-called balance point
of the game. In equilibrium, each player uses the strategy that has the best response to the other players’ strategies. In other words, each of these players, are looking for the solution to this question that in the face of the
competitor decision and understanding the fact that he or him is thinking similarly, he or she should choose
which strategy to bring the most revenue for him or her. The first approach that can be used to solution this
question, is using the principle of rationality. As an initial model, and according to this principle, the strategy
that it’s choosing in the face of every single competitors’ strategy, brings most revenues for the player, is
known as the dominant strategy and its choosing, is preferred. (Brown & Shoham, 2008)
Suppose that in a game, Player 1 has N and player 2 has M different strategies in their face. If player 1, choose
strategy i and player 2 choose strategi j, Uij is the player 1 revenue. If for each M strategies of player 2, we can
find 1 strategy among startegies of player 1 which its revenue is more than strategy k, hence player 1 will never
choose this strategy.
Using this concept, and through eliminating the strategies that are not best responses, we can further restrict the
strategy space of players. But even after eliminating the failed and non-rational strategies, there is no dominant
strategy in many real problems.i.e in Some choosed strategies of the competitor, one strategy andin some other
strategies, another strategy are preferable. In these situations, The Nash equilibrium is noteworthy in order to
achieve the optimal point. This equilibrium is achieved when each player chooses his or her strategy based on a
good faith towards the competitor choice. The Strategies that players choose by this method, from its Nash
equilibrium strategy.
The requirement of achieving this equilibrium, is that each player chooses the strategy that has most revenues
for him or her, based onhis or her belief about the competitor choice. To find the mentioned equilibrium
point,the optimal response of player 2 to any strategies of player 1 and also the optimal response of player 1 to
any strategies of player 2 is determined. The Nash equilibrium is the point that two parties have agreed to it implicitly. It means that if the best response of player 2 to the strategy Si of Player 1 is strategy S'j and best response of player 1 to the strategy S'j of Player 2 is strategy Si these paired strategies is pure strategy Nash equilibrium (Porte and et. Al, 2008), (Myerson, 1991, Chapter 2).
In Some games, the pure strategy Nash equilibrium, it means that there is not one unique Nash equilibrium
point or there are more than one pure Nash equilibrium point. In this situation, the mixed strategy Nash equilibrium is used. Mixed strategy represents a mental uncertainty of a player towards the competitor choice. This
uncertainty is represented by probability. Thus a mixed strategy is a mixture of pure strategies of player with a
certain probability distribution. Considering the fact that the game is repeated many times, we can consider
these probabilities as the choosing frequency of each strategy over time by each player and determine the revenue of each player by using that. (Reny &Robson , 2004) (Jeronimo, 2009)
To determine the mixed strategy Nash equilibrium we can use the KKT optimality situation and Lagrange
function., In this situation, after elimination the Non-rational strategies and the formation of the following system of linear equations, we can determine the strategy of each player and its choosing probability.
(1)
(2)
(3)
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(4)
(5)
Where:
Probability of choosing the strategy i By Player 1 (i = 1,2, ..., n)
Probability of choosing the strategy j By Player 2 (j = 1,2, ..., n)
Player 1 Consequence in case of choosing the strategy i By Player 1 and strategy j By Player 2
Player 2 Consequence in case of choosing the strategy i By Player 1 and strategy j By Player 2
In summary it can be said that to determine the equilibrium in a game, at first we must remove the non-rational
strategies sequentially so there is no remain any other non-rational strategy for removing. If 1 strategy remains
for each player, this strategy is introduces as pure strategy Nash equilibrium. Otherwise, by formation of linear
equations system, the mixed probability of choosing each strategy is determined.
If the players’ target functions are specified rather than the revenue matrix, we must identify the whole areas of
justifiable solutions of 1 player, and for each extracted solution, the optimal strategy of opposed player andrevenue amount of both players in both paired strategies should be identified and then, the above measures should
be taken.
4 - Competitive analysis of the traveling salesman problem
Regardless of the competitive aspect of the subject, different structures for the traveling salesman problem Have
been developed that some of them are mentioned in the introduction part. in the model considered in this paper,
the salesman starts from an origin and by meeting a number of nodes, sells a product to them or offers a service
and earns income.
By reviewing the theoretical literature on competition subject and types of situations facing the Competitive
firms and matching them with the requirements of the traveling salesman problem, different scenarios can be
imagined in the face of a traveling salesman.
The first problemwhich is considered in this case,is the monopolistic or competitive being of the space. If there
is at least one competitor in a competitive environment, we should take into account the environment as a competitive environment that is different with monopolistic environment in planning. However, despite with presence a competitor, customers are completely loyal to thesalesmanbrand's reputation the situations will be similar to the monopolistic ones. Given the lack of this loyalty, it is possible that to different reasons, the response
to the competitor behaviour would not be important or the competitor have such thought. if both competitors
want to respond to each other the most critical question which should besolutioned is the possibility or impossibility of achieving cooperation between them. The Cooperation between them can be through the creation of an
organizational unit or nodes division. If two competitors want to plan without the cooperation but by attention
to other possible decisions, a game will be formed. In this game the competitors may collect required informations about the opposed competitor performance which in this case, a dynamic and online game will be formed
or it is possible that at the beginning of the work and simultaneously, each competitor based on to the belief he
or she has about the competitor actions (and by considering this thought about the competitor decision making
method) acts to his or her which a static game will be formed. Both static and dynamic approaches have applications in real world. In this paper, a static approach is considered.
5 - mathematical modeling and structure for traveling salesman problem in competitive situations
5-1 –model assumptions
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In the investigated structure in the present paper, two traveling salesmans with a relatively similar level of ability, operate in a competitive environment. In addition, each of them continually monitor the competitor behavior
and modify their plans accordingly. Because their traveling continuously occur over time so the Nash equilibrium is acceptable as their behavior. Information of both competitors about each other is complete. The structure of the problem of selling a product, consists of some nodes that both distributors return to the station finally. Each node demand is constant and deterministic. Part of the demand for each of these nodes, regardless of
the arrival time of the salesman is provided independently of each other and other part is time-dependent. It
means that each salesman, arrives sooner to a node, will take its demands. The moving origin for both salesmans and the time interval between the origin and nodes and also the distance between the nodes are
clear. Shipping costs are low rather than the income from sale of goods and hence both salesmans put their aims
to maximize the sells and accordingly, each salesman meets all nodes and therefore, we can remove the timeindependent demand from the planning. There is no limitation on the maximum time and capacity of any of the
salesmans.
5-2 - Signs, variables and parameters of the problem
signs, variables and parameters used in the problem are as follows.
Number of demand nodes (central stations are located at node i = 0 ).
:
Demands of Customer i which is provided from distributer that comes sooner.
:
crossing time from node i to j by Salesman 1
crossing time from node i to j by Salesman 2
M:
service time to the customer i by Salesman 1
service time to the customer i by Salesman 2
A very large number.
5-3 - The decision variables in the problem
If the route from node i to j, crosses by Salesman 1, is one and otherwise is zero
If the route from node i to j, crosses by Salesman 2, is one and otherwise is zero
:
:
:
If the salesman 1 arrives sooner to the node i is one and otherwise
is zero.
If the salesman 2 arrives sooner to the node i is one and otherwise
is zero.
arrival time of the salesman 1 to node i
arrival time of the salesman 2 to node i
:
:
5-4 - Mathematical model
Mathematical modeling for the problem using optimization models of operational research is as follows.
In a competitive model, p1 Solves:
(6)
Where p2 Solves:
(7)
s.t.
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(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
Relation 6 is objective function of Salesman 1 and relation 7 is objective function of Salesman 2. In these relations, the objective of each salesman, is modeled as optimization of sales amount of product.
Relations 8 and 9, ensure that the salesman 1 must and only enter and exit through one node into the
node i. Relation 10 means that if the salesman crossed the node i to j, certainly he or she will never retain from
node j to i. these relations are used to prevent the generation of non-logical solutions in computer calculations
and ensures the continuous relation of each vehicle route.
relation 11 calculates the start time of service to node i. based on this relation, time of service to a node is the
sum of time of service during the previous node plus the traveling time from previous node to Considered node.
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Finally, the time of arrival to the previous node should be added to it. relation 12 defines the between the time
origin.
relations 13 to 18 are similar to the relations 8 to 12 for salesman 2. Because this salesman is also has similar
restrictions (although with different parameters) has. Relations 18 to 20 express the competitive situations of the
problem.
These relations ensure that any salesman who is sooner enter a node, will take the node demands. Relations 21
to 24 are also characterize the restrictions related to the variables’ types.
You have to remember that the present model, is not a linear planning model or multi-objective planning model
and you should determine the Nash equilibrium point Using the game theory.
6–Algorithm of solution
The proposed mathematical model is solved with two nested algorithms. The first algorithm is dedicated to
solving the model of game problem that is solved in an iterating manner in steps of the traveling salesman problem which for solving it, genetic and tabu search algorithms have been used separately and coded in MATLAB
software.
6-1 –algorithm of game solution
Generally speaking, if the model is assumed as follows.
(25)
(26)
(27)
(28)
(29)
Then the algorithm for game solution, based on reviewing the available theoretical literature, is as follows.
1 1- This step includes the following sub-steps.
1-1 identify the total area of justifiable solutions for the player 1and call it, set.
(30)
1-2 for every single justifiable solution of player 1, solve the following problem for player 2 and find the strategy corresponds to it and call these strategies set as
.
(31)
(32)
(33)
(34)
1-3 - For all compounds
where
And
,calculate and
ers 1 and 2 .
1-4 Form the revenue matrix using above calculated information.
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values as revenues of Play-
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1- remove the non-rational strategies as below.
2 -1 - for each one of the strategies of player 1, identify the best strategy for player 2 and label it. Continue this
until the last strategy of player 1.
2 -2 - If there is any un-labeled strategy for player 2, remove it and insert i = 0. Otherwise insert i = 1.
2 -3 - for each one of the strategies of player 2, identify the best strategy for player 1 and label it. Continue this
until the last strategy of player 2.
2 -4 - If there is any un-labeled strategy for player 1, remove it and insert j = 0 . Otherwise insert j = 1.
2 -5 - if i * j = 1 Go to Step 3. Otherwise, go to step 3-1.
32 - If only one strategy was remained for each player, introduce this strategy as pure strategy Nash equilibrium
point. Otherwise, go to Step 4.
4 3- With the formation of the corresponding system of linear equations (equations 1 to 5) identify any possible
combination of each strategy.
Two variables i and j, have been defined for the stability of computer algorithms. The presence of these two
variables, provide this guarantee that if in aniteration, none of the two players 1 and 2, have removed strategy,
the remaining strategies, are all rational. Step 1, in turn acts in order to remove the non-rational strategies of
player 2. Because if we would begin with total revenues’ matrix, the justifiable area of this player, should also
be identified. But by considering the fact that any strategy that is not a player response to none of the competitor player strategies, is non-rational. This step, eliminates the non-rational strategies (not necessarily all of
them) of player 2.
6-2 – Solving the problem of linear planning
6-2-1 - Genetic Algorithm
Genetic algorithm considers each solution as a Chromosome and begins its work with Chromosomes as the initial population and at each stage, by changing in present population, creates new generations. This algorithm
performs its work for single-objective problems with encoding, evaluating, combining, mutation and decoding. (Bavi & Salehi, 2008)
Encoding perhaps is the most difficult step of problem solvingby genetic algorithm method and other steps depend on this step. The main methods of encoding include binary coding, permutation coding, value coding and
tree coding.Evaluation function (fitting) is achieved by applying the appropriate transformation on the objective
function. This function evaluates each solution with a numerical value that indicates the level of solution suitability. The most important operator in genetic algorithm, is Crossover operator. The Crossover is a process in
which the older generation chromosomes are combined and mixed together to create a whole new generation of
chromosomes. The pairs that were considered
as a parent in the selection section,interchange their genes in this section and create new members.
Mutation is another operator that creates other possible solutions. In genetic algorithm after a new member is
created in new population, each gene mutates with a constant probability of mutation. In mutation, there is possibility that a gene is removed from the set of genes or a gene is added that had not been in the population.Mutation of a gene means a change in that gene, and various mutation methods are used depending on encoding method.
Decoding is the inverse operation of encoding. At this step, after the algorithm provides the best solution to the
problem, it is necessary to perform the encoding inverse or decoding on solutions to obtain the real version of
the solution.
The parameters of this problem are estimated as in Table 1 by using experimental designs and Taguchi method.
Table 1) parameters of genetic algorithm
Parameter
Parameter value
Npop
20
Maxit
30
Pc
0.4
Pm
0.3
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6.2.2 - tabu search algorithm
Tabu search algorithm at first, moves from an initial response. Then the algorithm chooses the best neighboringsolutionamong the current neighboring solutions. If this solution is not in the tabu list, the algorithm moves
to the neighboring solution. Otherwise, the algorithm, will check a criteria called aspiration criteria. Based on
the aspiration criteria, if the neighboring solution is better than the best solution that has been found so far, the
algorithm will move to it, even if the solution is in the Tabu list. After the algorithm moves to the neighboring
solution, tabu list is updated. i.e. the previous move through that, it achieves to the neighboring solution, is
placed in tabu list to prevent the algorithm to returns to that solution and making a cycle. Actually the tabu list
is a tool intabu search algorithm that by using that, will prevent algorithm from placing in a local optimization. After placing the previous move in tabu list, some moves that previously were placed in tabulist are removed from the list. The time that the moves placed in Tabu list, is determined by a time parameter called tabu
tenure. Move from the current solution to a neighboring solution continues until the termination situation is met.
Different termination situations can be considered for the algorithm. For example, the limitation of the number
of moves to neighboring solution can be a termination situation. (Glover and Laguna, 2002)
Using the Taguchi method, the parameters of this algorithm are estimated as in Table 2.
Table 2) parameters of tabu search algorithm
Parameter
Parameter value
Maxit
100
TL0
0.7
7 - Numerical test
7-1 - small example problem
In order to evaluate the proposed model, at first, one example problem with low number of nodes is explained. Then a series of randomly generated data and the model is solved.
Suppose that there are three demand nodes, which the traveling time between them as well as between each
competitor depot and their demand nodes, are as table below.
As can be seen in the Step 1-2 – of this algorithm, we must solve the mathematical model of one of the playersby assuming the identifying the solution of other player. To solve this problem, both genetic and Tabu search
algorithms are used separately.
Table 3) example data in a problem with three demand nodes (demand and nodes’ distance)
node
Demand
Depot 1
Depot 2
1
2
3
1
20
7
9
0
12
6
2
25
17
21
12
0
12
3
30
7
11
6
12
0
Each player has 3!i.e. 6 strategies in his face that are shown and namedin table below. In addition, the arrival
time of each competitor in each node is calculated based on the values listed in the above table and these values
are shown in table below.
Table 4) the strategy in the face of the players andarrival time(node/ player/strategy)
Player 1
Player 2
Route
Name
1
2
3
1
2
3
0-1-2-3
A
7
19
31
9
21
33
0-1-3-2
B
7
25
13
9
27
15
0-2-1-3
C
29
17
35
33
21
39
0-2-3-1
D
35
17
29
39
21
33
0-3-1-2
E
13
25
7
17
29
11
0-3-2-1
F
31
19
7
35
23
11
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In calculating the times listed in above table, the time of depot node is considered zero. For example, if player 1
chooses the route 0-1-2-3, he or she arrives at node 1 at time 7, arrives at node 2 at time 19 and arrives at node 3
at time 7. While player 2 by choosing the same strategy arrives at node 1 at time 9, arrives at node 2 at time 21
and arrives at node 3 at time 33. It means that he or she arrives at each node after the competitor and subsequently will lose the sell to player 1. Revenue matrix is shown in the following table. These are the revenue of
player 1 and since the game is with constant sum, the revenue of player 2 is derived from the difference between these values from 75 (total nodes’ demand).
Player 1
Table 5)players revenue matrix
Player 2
A
B
C
D
E
F
A
75
45
75
75
45
45
B
50
75
50
50
45
20
C
25
25
75
45
25
45
D
55
25
55
75
25
25
E
30
55
50
50
75
50
F
55
55
75
75
55
75
According to the explained algorithm, first we must remove non-rational strategies respectively. By reviewing
the strategies of player 2,we observe that he or she does not choose the strategies C and D in response to any
strategies of player 1,hence these strategies are considered as non-rational and are eliminated. Similarly,by examining the strategies of player 1 to the 4 remain strategies of player 2, the strategies C and D are considered as
non-rational and are eliminated. In the next iteration, there is no non-rational strategy for both players and hence
the revenue matrix is formed as below figure.
Player 1
Table 6) players’ revenue matrix after removing non-rational strategies
Player 2
A
B
E
F
A
75
45
45
45
B
50
75
45
20
E
30
55
75
50
F
55
55
55
75
This game has no pure strategy Nash equilibrium and therefore we must obtain the mixed strategy Nash equilibrium, using equations 1 to 5. These equations for this problem are as follows:
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Which by Solving the above system of linear equations, the probability values of choosing each strategy for
each player, are obtained as below table.
Table 7) mixed strategy Nash equilibrium and Players’ revenue
Probability
Strategy
Player 1
Player 2
A
0.27
0.33
B
0.13
0.27
E
0.19
0.40
F
0.41
0
Total revenues
48.36
26.63
By repeating the game, the revenue of each player goes to the amounts stated in this table. The reason of higher
revenue of player 1, is better position of him or her depot than demand nodes compared to the player 2.
7-2 - Solving the average problems
In order to validate the model, in this section, the problem is solvedwith a range of modeling data. Parameter
values are generated using the following random functions. Furthermore, the amounts of service time to the
both salesmans’ customers have been considered similarly.
Table 8) parameters of the problem
U [8 0000 25000]
U [0.1 0.3]
For suitable estimating the traveling time between the nodes, the location of demand points in the city, was estimated using a random function. To this end, separately a depot for each salesman is identified and demand
points are jointly determined.
Table 8) Coordinates of Depots’ locations and the demand points
U[0 20]
U[0 20]
U[0 20]
U[0 20]
U[0 20]
U[0 20]
The traveling speed of the vehicle is considered 20 and by assuming that the distances are geometrical, the time
parameters are estimated as follows.
The purpose of the above calculation is presence of a logical relationship between Depots’ distances and their
distribution in the region.
In the first stage, a number the smaller size problems, were solved with method of branch and bound and the
results were compared with the approximate solution to ensure the proper functioning of approximate algorithms.
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node
Table 9) comparison of the exact and approximate solutions for some small problems
Theexact algorithm
Genetic Algorithm
Tabu search algorithm
Player 1
Player 2
Player 1
Player 2
Error*
Player 1
Player 2
Error*
3
24853
25260
24942
25171
0.35
24999
25114
0.58
3
19994
18812
20166
18640
0.92
20114
18692
0.64
4
44632
17720
44536
17816
0.54
44566
17786
0.37
4
38559
20143
38750
19952
0.95
38723
19979
0.81
5
22260
32742
22513
32489
0.78
22037
32965
0.69
5
18084
61954
17356
62682
1.21
17489
62549
0.99
5
40700
33971
40981
33690
0.83
41049
33622
1.03
6
65889
53117
66753
52253
1.63
66500
52506
1.53
6
60261
19837
60503
19595
1.22
59965
20133
1.49
6
47687
78363
49054
76996
1.77
48584
77466
1.16
7
65385
58208
64136
59457
2.16
64540
59053
1.46
7
57003
62658
58048
61613
1.68
57739
61922
1.18
7
78880
43668
79720
42828.26
1.93
79547
43002
1.53
* percentage
ingenerated responses by each approximation algorithm, the average relative distance of 1 and 2 players respect
to the response of player 1 is recorded as the response error. As can be seen in this table the error of approximate method respect to the exact solution, is acceptable.
As you can see in this plot, the error of tabu search algorithm is generally less than the genetic algorithm.
At the next stage, some larger problems are solved using the mentioned Meta-heuristic algorithms and their results with the calculation times in minutes are shown in the following table.
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Applied mathematics in Engineering, Management and Technology 2 (3) 2014
M M Mohtadi and K Nogondarian
Table 10) comparison of the approximate solution of some average problem
Genetic Algorithm
Tabu search algorithm
Nodes
Deviation
Player 1
Player 2
Time
Player 1
Player 2
Time
8
17710
29513
8
17788
29435
6
0.35
8
18777
28446
8
18535
28688
6
1.07
8
17583
29640
8
17841
29382
6
1.17
8
17662
29561
8
17670
29553
6
0.03
9
17447
29776
15
17720
29503
11
1.24
9
18866
28357
15
18670
28553
11
0.87
9
18819
28404
15
18665
28558
11
0.68
9
18934
28289
15
18642
28581
11
1.29
10
19032
28191
31
18702
28521
19
1.45
10
17284
29939
32
17733
29490
19
2.05
10
19097
28126
32
18727
28496
19
1.63
11
17211
30012
68
17645
29578
41
1.98
11
17198
30025
68
17732
29491
40
2.44
11
17236
29987
67
17666
29557
41
1.96
12
19392
27831
102
18785
28438
73
2.66
12
17077
30146
99
17658
29565
71
2.67
12
17071
30152
101
17604
29619
72
2.44
13
16781
30442
167
17457
29766
109
3.12
13
19513
27710
165
18777
28446
108
3.21
13
16858
30365
165
17469
29754
108
2.82
13
19571
27652
167
18789
28434
109
3.41
14
16861
30362
379
17332
29891
254
2.18
14
16853
30370
379
17456
29767
254
2.78
14
16867
30356
377
17394
29829
253
2.43
14
16820
30403
377
17448
29775
253
2.90
15
64928
42741
691
62426
45243
473
4.85
15
56857
42476
688
59060
40273
471
4.53
15
46298
45165
689
44268
47195
472
4.44
15
43869
32308
691
45692
30485
473
4.90
Deviation of the responses generated by two algorithms, similar to the previous method based on the changes in
search Tabu search algorithm with respect to the genetic algorithm, are calculated and are shown in the table
that are within the acceptable range.
These deviations are shown in the plot below, that according to the previous results, the results of the Tabu
search algorithm
look more logical.
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Applied mathematics in Engineering, Management and Technology 2 (3) 2014
M M Mohtadi and K Nogondarian
Furthermore, it was not possible to solve the problems with more than 15 nodes and in a time-frame, which
means that the algorithm is unable to solve them.
7 – Summary and conclusions
In this paper, the traveling salesman problem in competitive situations was examined. In this investigation
firstly, the research history was studied and then, the game theory was introduced as a mathematical model for
examining the competitive problems. In following, the various types of structure for traveling salesman were
investigated in various scenarios. These scenarios have been validated using the Delphi technique. In next section, the mathematical modeling of the scenario consistent with game theory is presented. Finally, this model
was solved using the exact and approximate algorithms and their results are compared with each other. The approximate algorithms, which are used, are genetic and Tabu search algorithm sand the calculation results, show
that the computational error is within a reasonable range. Tabu search algorithm, generated solutions that are
more suitable and less time was needed.
References
Ali, A.I., Kennington, J.L. (1986). The asymmetric m-traveling salesmen problem: A duality based branch-and-bound algorithm. Discrete Appl Math, 13, 259–276
Ausiello, G., Bonifaci, V., & Laura, L. (2008). The on-line prize-collecting traveling salesman problem. Information
Processing Letters, 107(6), 199-204.
Ausiello, G., Demange, M., Laura, L., &Paschos, V. (2004). Algorithms for the on-line quota traveling salesman problem.
Information Processing Letters, 92(2), 89-94.
Ausiello, G., Feuerstein, E., Leonardi, S., Stougie, L., &Talamo, M. (2001). Algorithms for the on-line travelling salesman.
Algorithmica, 29(4), 560-581.
Bavi O., Salehi M. (2008). Genetic Algorithms & Optimization of Composite Structure, Abed Publication, Tehran.
Blom, M., Krumke, S. O., de Paepe, W. E., &Stougie, L. (2001). The online TSP against fair adversaries. INFORMS Journal on Computing, 13(2), 138-148.
Carter, E., Ragsdale, C. T. (2006). A new approach to solving the multiple traveling salesperson problem using genetic
algorithms. Eur J Oper Res, 175, 246–257.
Christofides, N., Eilon, S. (1969). An algorithm for the vehicle dispatching problem. Res Quart, 20,309–18.
Fekete, S.P., Fleischer, R., Fraenkel, A., & Schmitt, M. (2004). Traveling salesmen in the presence of competition. Theoretical Computer Science, 313, 377– 392.
Franca, P. M., Gendreau, A., Laporte, G., & Muller, F. M. (1995). The m-traveling salesman problem with minmax object.
Trans Sci, 29(3), 267-275.
Jeronimo, G., Perrucci, D., &Sabia, J. (2009). A parametric representation of totally mixed Nash equilibria. Computers and
Mathematics with Applications, 58, 1126-1141.
324
Applied mathematics in Engineering, Management and Technology 2 (3) 2014
M M Mohtadi and K Nogondarian
Glover, F. & Laguna, M. (2002), Tabu Search, In Handbook of Applied Optimization, Pardalos P.M. &Resende, M.G.C.
(Eds.), Oxford University Press, 194-208.
Jaillet, P., & Lu, X. (2011). Online traveling salesman problems with service flexibility. Networks, 58(2), 137-146.
Kendall, G., & Li, J. (2013). Competitive travelling salesmen problem: A hyper-heuristic approach. Journal of the Operational Research Society, 64(2), 208-216.
Laporte, G., Nobert, Y. (1980). A cutting planes algorithm for the m-salesmen problem. J Oper Res Soc, 31, 1017–1023.
Layton Brown, K., Shoham, Y. (2008). Essentials of Game Theory: A Concise Multidisciplinary Introduction, Morgan and
Claypool Publishers.
MengShu, H., DaiBo, L. (2012). A novel method for solving the multiple traveling salesmen problem with multiple depots.
Chinese science Bulletin, 57(15), 1886-1892.
Myerson, R.B. (1991). Game Theory: Analysis of Conflict. Cambridge: Harvard University Press.
Porter, R., Nudelman, E., &YoavShoham, Y. (2008). Simple search methods for finding a Nash equilibrium. Games and
Economic Behavior. 63(2), 642-662.
Qu, H., Yi, Z., Tang, H. J. (2007). A columnar competitive model for solving multi-traveling salesman problem. Chaos
SolitonFract, 31, 1009–1029
Reny, Philip J. & Robson, Arthur J., 2004. "Reinterpreting mixed strategy equilibria: a unification of the classical and
Bayesian views," Games and Economic Behavior, Elsevier, vol. 48(2), pages 355-384, August.
Samerkae, S., Abdolhamid, M., Takao, N. (1999). Competition-based neural network for the multiple traveling salesmen
problem with minmax objective. ComputOperat Res, 26,395–407.
Savelsbergh, M. W. P., (1995). The general pickup and delivery problem. Trans Sci, 29(1), 17–29.
Svestka, J. A., Huckfeldt, V. E. (1973). Computational experience with an m-salesman traveling salesman algorithm. Manage Sci, 19, 790–799.
Talay, S. S., Erdogan, D. R., &Dept, N. (2009). Multiple traveling robot problem: A solution based on dynamic task selection and robust execution. IEEE/ASME Trans Mechatronics, 14, 198–206.
325