A Competitive Many-period Postman Problem With Varying Parameters

Transcription

1 Applied Mathematical Sciences, vol. 8, 2014, no. 146, HIKARI Ltd, A Competitive Many-period Postman Problem With Varying Parameters Xeniya Grigorieva St.Petersburg State University Faculty of Applied Mathematics and Control Processes University pr. 35, St.Petersburg, , Russia Oleg Malafeev St. Petersburg State University Faculty of Applied Mathematics and Control Processes University pr. 35, St.Petersburg, , Russia Copyright c 2014 Xeniya Grigorieva and Oleg Malafeev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let defined multistep game Γ G on a graph tree Γ. The postman s problem on a complete graph with finite number of agents which responsible for the postman s route and with varying weights given to the graph edges is a stage game G in each vertex on graph tree Γ. The problem G will be called a competitive one-period postman problem with varying parameters. In this parer competitive many-period postman problem with with varying papametres is examined. A multistep game with the postman problem realized on each step having a finite number of agents and varying matrix of weights given to the graph edges depending on the step (period), will be called a competitive many-period postman problem with varying parameters. The payoffs of the postman problem participants (players) including the postman himself in the simultaneous games G are their profits minus costs for providing postman s movement on the routes. The routes are the players strategies. An algorithm for finding a compromise situation for solving simultaneous games is proposed. The theory is illustrated by the example of three-step game of three players.

2 7250 Xeniya Grigorieva and Oleg Malafeev Mathematics Subject Classification: 91 Axx Keywords: postman problem, multistep game, compromise solution 1 Introduction Remind that the postman s problem is to pass all streets on his way (edges of graph) and come back to the initial point by minimizing the total length of the way. It is clear that not only postman has such a task but also different couriers [1], [2]. The following generalization of this postman problem is examined in this paper. It is assumed that there is a finite number of agents (offices) that are responsible for the postman s route, and each of them wants to minimize its costs for providing postman s movement on his route, or equivalently maximize the money remained, that is its profit. This model have apply in working process office selling goods prolonged consumption. The movement of seller along route is provided different agents. Each agent wants to minimize its costs for providing postman s movement on his route, or equivalently maximize the money remained, that is its profit. Thus, a noncoalitional game of N individuals which provide a postman s movement on the given routes will be called a competitive one-period postman problem with a finite number of agents N. In reality postman s work takes a long time and model s parameters can change according to different conditions. This would be followed by a change in weights on some edges of the graph corresponding to the chosen route, that is why a new postman problem considering changes in the graph should be solved (in particular, can be change a graph itself: some edges may be disappear and some may be appear). A multistep game with the postman problem realized on each step having a finite number of agents and varying matrix of weights given to the graph edges depending on the step (period), will be called a competitive many-period postman problem with varying parameters. The payoffs of the postman problem participants (players) including the postman himself in the simultaneous games G are their profits minus costs for providing postman s movement on the routes. The routes are the players strategies. An algorithm for finding a compromise situation for solving simultaneous games is proposed. The theory is illustrated by the example of three-step game of three players.

3 A competitive many-period postman problem with varying parameters The postman problem Let G = (X, P ) - be a complete connected undirected graph, where X and P are the sets of vertices and edges of the graph G correspondingly, and A = {a(i, j)} (i,j) P is a matrix of weights a(i, j) given to the graph edges. Then the postman problem G(A) is the problem of finding the shortest route, which includes each edge at least one time and which ends in the vertex of starting the movement. Construct a cycle on the graph G, in which each edge (i, j) of the graph G is passed f(i, j) + 1 times, where f(i, j) is the number of additional postman s passings of the edge (i, j). But the total length of repeatedly passed edges must be minimal: L = (i,j) P a(i, j)f(i, j)) min (i,j). An algorithm for calculating Floyd s matrix would be used for making this cycle. Each matrix element (i, j) defines the length of the shortest way from vertex i to the vertex j. Intermediate route nodes from the node i to the node j are defined according to the matrix of nodes sequence [3]. Since any graph G has even number of vertices with an odd degree, then matches [3] or sets of non-adjacent edges can be made from the remained unpassed vertices with an odd degree. The constructed solution may not be unique. 3 One-period postman problem with finite number of agents. 3.1 The state of the postman problem with finite number of agents. Consider noncoalitional game with n players G = I = {1,..., n}, {R j } j=1, m, {H j i } i=1, n, j=1, m, {A j i } i=1, n, j=1, m, where I is a set of agents, {R j } is a set of ways in the route when each of them includes each edge at least once and ends in the vertex of starting the motion, H j i is a real function of costs of agent i on j-th way, A j i is a sum of agent i for providing postamn s movement on j-th way. Each agent has a problem of maximizing the functional F j i = A j i H j i max, j = 1, m, j

4 7252 Xeniya Grigorieva and Oleg Malafeev that is in fact a payoff function of i-th agent when postman goes on j-th way. Construct matrix F of agent s profits, in which rows correspond to the chosen ways, columns correspond to the agents: F = F F 1 n F m 1... F m n R R m By solving a problem of maximization we find ways that are suitable for each agent in seperate. But it is needed to find a way (one or more), that is suitable for all of them. For this purpose we use an algorithm for finding a compromise situation (thus, compromise solution is taken as optimal principle). 3.2 Algorithm for finding a compromise situation. Algorithm for finding a compromise solution for noncoalitional game G is described as follows [1]. Step 1. Construct ideal vector M = (M 1,..., M n ) where M i := F j i i = max F j i is a maximal profit value of agent i: j F Fn F1 m... Fn m... F j F j n n Step 2. For each way j find deviation from the maximum M i of the rest of the profit values, i. e. j i := M i F j i, i = 1, n: = M 1 F M n Fn M 1 F1 m... M n Fn m Step 3. From found deviations j i for each way j choose a maximal deviation from all agent i j i = max j i : j i M 1 F M n F 1 n = M 1 F1 m... M n Fn m n m 1... m n 1 i m i m Step 4. Choose minimal from all maximal deviations j i = min j j j i = j = min max j i. The way R j on which the minimum is reached is a compromise j i game solution for all agents.

5 A competitive many-period postman problem with varying parameters A competitive many-period postman problem with varying parameters. Divide time segment [0, T ], during which a postman is going to move over the routes by periods [0, t 1 ], [t 1, t 2 ],..., [t k, T ], with 0 < t 1 < t 2 <... < T. Model parameters change under different conditions after each moment t k. It is assumed that there is a finite number of agents (offices) that provide postman movement on the route. They are interested in having minimal costs spent on the route during [0, T ]. Total costs are formed by adding costs on each time segment the segment [0, T ] is divided to. This many-period process can be illustrated in terms of graph tree Γ = (Z, L), where Z is the set of graph vertices, L is point-to-set mapping, defined on the set Z: L(z) Z, z Z. Finite graph tree with the initial vertex z 0 will be denoted by Γ(z 0 ). The root of Γ tree is the initial postman problem, vertices correspond to postman problems considering changes after each time-period, and branches are the chosen routes for corresponding problems, figure 1. Definition 4.1 A multistep game Γ(z 0 ) = I = {1,..., n}, Γ(z 0 ), {G(z, A z )} z Z, in which postman problem G(z, A z ) is realized on each step, with a finite number of agents and varying matrix A z of weights given to the graph edges depending on step or period, will be called a competitive many-period postman problem with varying parameters. Definition 4.2 Solving a competitive many-period postman problem with varying parameters means gaining such a sequence of ways for many-period process, that when summing up costs for each way forming this sequence we get minimal costs, that is equilibrium or compromise. Remark 4.1. There can be several ways for postman and it is not obligatory to choose an optimal one on each stage. The main thing is to have minimal final costs. An algorithm for finding a compromise situation would be used for solving this problem as well. Example 4.1. There is a complete graph G 0 with 4 vertices, figure 2. Matrix of weights assigned to the graph edges G 0 and matrix of ways with the shortest length between all pairs of vertices of the graph G F 0, calculated with the help of Floyd s algorithm, look as follows:

6 7254 Xeniya Grigorieva and Oleg Malafeev Figure 1: Figure 2:

7 A competitive many-period postman problem with varying parameters 7255 G 0 : G F 0 : It is necessary to choose edges that would be passed more than once because some graph vertices have odd degrees. For this purpose possible matches should be found. These are (1, 4), (2, 3), or (1, 2), (3, 4), or (2, 4), (1, 3) in this example. Therefore, possible graph passings are the following ways: R 1 = (1, 2), (2, 3), (3, 2), (2, 4), (4, 3), (3, 1), (1, 4), (4, 1); R 2 = (1, 2), (2, 3), (3, 4), (4, 2), (2, 4), (4, 1), (1, 3), (3, 1); R 3 = (1, 2), (2, 4), (4, 3), (3, 1), (1, 2), (2, 3), (3, 4), (4, 1). Other cycles passing graph edges in a different order are also possible. But these cycles would pass on same sets of two edges, which represent all possible matches on this graph, more than once. Thus, there are three different possible ways on this graph. Their lengths are as follows: L(R 1 ) = 24, L(R 2 ) = 25, L(R 3 ) = 26. After time t 1 graph changes. But when selecting different ways different changes arise. By selecting way R 1 graph G 0 has changed to graph G 1. Matrix of weights given to the graph edges G 1 and matrix of ways with the shortest length between all pairs of vertices of graph G F 1, calculated with the help of Floyd s algorithm, are as follows: R 1 G 1 : G F 1 : For passing graph G 1 there are the following ways: R 6 = R 1 = (1, 2), (2, 3), (3, 2), (2, 4), (4, 3), (3, 1), (1, 4), (4, 1); R 5 = R 2 = (1, 2), (2, 3), (3, 4), (4, 2), (2, 4), (4, 1), (1, 3), (3, 1); R 4 = R 3 = (1, 2), (2, 4), (4, 3), (3, 1), (1, 2), (2, 3), (3, 4), (4, 1). Their lengths: L(R 6 ) = 18, L(R 5 ) = 15, L(R 4 ) = 19.

8 7256 Xeniya Grigorieva and Oleg Malafeev By selecting way R 2 graph G 0 has changed to graph G 2. Matrix of weights given to the graph edges G 2 and matrix of ways with the shortest length between all pairs of vertices of graph G F 2, calculated with the help of Floyd s algorithm, are as follows: R 2 G 2 : G F 2 : For passing graph G 2 there are the following ways: R 7 = R 1 = (1, 2), (2, 3), (3, 2), (2, 4), (4, 3), (3, 1), (1, 4), (4, 1); R 8 = R 2 = (1, 2), (2, 3), (3, 4), (4, 2), (2, 4), (4, 1), (1, 3), (3, 1); R 9 = R 3 = (1, 2), (2, 4), (4, 3), (3, 1), (1, 2), (2, 3), (3, 4), (4, 1). Their lengths: L(R 7 ) = 14, L(R 8 ) = 13, L(R 9 ) = 12. By selecting way R 3 graph G 0 has changed to graph G 3. Matrix of weights given to the graph edges G 3 and matrix of ways with the shortest length between all pairs of vertices of graph G F 3, calculated with the help of Floyd s algorithm, are as follows: R 3 G 3 : G F 3 : For passing G 3 there are the following ways: R 10 = R 1 = (1, 2), (2, 3), (3, 2), (2, 4), (4, 3), (3, 1), (1, 4), (4, 1); R 11 = R 2 = (1, 2), (2, 3), (3, 4), (4, 2), (2, 4), (4, 1), (1, 3), (3, 1); R 12 = R 3 = (1, 2), (2, 4), (4, 3), (3, 1), (1, 2), (2, 3), (3, 4), (4, 1). Their lengths: L(R 10 ) = 20, L(R 11 ) = 21, L(R 12 ) = 19. Ways with total minimal length are the solution for many-period postman problem without firms that are competitors in terms of providing the movement. These are way R 2 on the time segment [0, t 1 ] and way R 9 on the time segment [t 1, T ]. Thus, by selecting these two ways for the route, we minimize the final costs. Costs will be equal to 37 units for moment T. Example 4.2. There is a postman that must move over its route during time T, parameters of the problem change after moment t 1. Thus, we get

9 A competitive many-period postman problem with varying parameters 7257 partition of the segment [0, T ] to segments [0, t 1 ], [t 1, T ]. Let there be three agents 1, 2 and 3, which provide the postman movement within his route. There are 3 possible ways of passing the route R 1, R 2, R 3 on time segment [0, t 1 ] and 9 possible ways R 4, R 5, R 6, R 7, R 8, R 9, R 10, R 11, R 12 on time segment [t 1, T ]. We need to find a compromise solution for time moment T, i. e. among set of ways R 1 +R 4, R 1 +R 5, R 1 +R 6, R 2 +R 7, R 2 +R 8, R 2 +R 9, R 3 +R 10, R 3 + R 11, R 3 + R 12 we need to find the minimal one. Costs functions: H R 1 1 H R 2 1 H R 3 1 H R 4 1 H R 5 1 H R 6 1 H R 7 1 H R 8 1 H R 9 1 H R 10 1 H R 11 1 H R 12 1 H R 1 2 H R 2 2 H R 3 2 H R 4 2 H R 5 2 H R 6 2 H R 7 2 H R 8 2 H R 9 2 H R 10 2 H R 11 2 H R 12 2 H R 1 3 H R 2 3 H R 3 3 H R 4 3 H R 5 3 H R 6 3 H R 7 3 H R 8 3 H R 9 3 H R 10 3 H R 11 3 H R 12 3 = Let the plan sum A j i be same for all agents for all possible ways and be equal to 30. Then matrix of profits of each way looks like follows: If recalculating for the sets of ways r 1 = R 1 + R 4, r 2 = R 1 + R 5, r 3 = R 1 + R 6, r 4 = R 2 + R 7, r 5 = R 2 + R 8, r 6 = R 2 + R 9, r 7 = R 3 + R 10, r 8 = R 3 + R 11, r 9 = R 3 + R 12, then matrix of profits for time T is: ) Ideal vector M = (24, 20, 19). 2) For each way j find deviation of the remained profit values from maximum M i from all agents i: ( ) =

10 7258 Xeniya Grigorieva and Oleg Malafeev max i 3) Among found deviations choose the maximal one for each agent (M i F j i ) = ( ). 4) Choose minimal from these maximal deviations min j max (M i F j i ) = 8. i Hence, we have compromise solution for game Γ G for all agents: way R 3 at the first stage [0, t 1 ] and way R 10 at the second stage [t 1, T ]. 5 Conclusion A competitive many-period postman problem with varying parameters and example of its numerical solution are examined in this paper. References [1] O.A. Malafeev, Managed conflict system, SPbSU, SPb., [2] T. Basar and G. Olsder, Dynamic Noncooperative Game Theory, 2-nd ed. Classics in Applied Mathematics, 23 New York, Academic Press, [3] Hamdy A. Taha, Operations Research: an introduction, Williams, Moscow, SPb, Kiev, Received: August 3, 2014