We investigate traffic management in the mechanical transport system. A major factor in the management of the MTS is search of the optimal route. Therefore, this article is described the routing in the MTS taking into account the inaccuracies and uncertainties of transportation options. We have developed a routing algorithm that takes into account the temporal fuzzy nature of the variables. We have illustrated the example of the developed routing algorithm.
Mechanical transport system (MTS) is a class of transport systems using conveyors
for moving cargo [
Formulation of the problem We have following problem taking into account the changing reality and inaccuracies of incoming information. It is necessary to build a set of optimal routes for each of the MTS units. Information about received routes is stored in each node of MTS. And for determination of optimal route is used as a time parameter, and the parameter range. These parameters are presented in fuzzy form. The initial data are given on the MTS by expert to analyze the system. 3.
Routing Algorithm in MTS under fuzzy given distance and time
There are various ways to implement the routing of mechanical transport systems
taking into account only the distance parameter, such as Ford's algorithm, Floyd’
algorithm and etc [
Fuzzy temporal graph is a triple = (, !, ), where X is set of vertices of the
number of vertices c = n, T = {1,2, … , N}is the set of natural numbers,
determining (discrete) time; ! = {< !(!, !)| !, ! >} is fuzzy set of edges,
where !, ! ∈ , ! !, ! ∈ [
Step 1. To form the matrix D! (dimension N×N, where N is number of vertices in the graph). Each element i, j of the matrix d!!"() determines the length of the shortest arc leading from vertex i to vertex j. In the absence of such an arc put in d!!"() = ∞.
Step 2. Here ! denotes the dimension of the matrix × with d!!" (), = 1, − 1. To determine successively the elements of the matrix ! from elements of the matrix of !!! for , taking values 1, 2, … , : !!" = !!" =
min !!!,!!!
min !!!,!!! !!" + !!" !!
= 1, − 1 !!" !! + !!"
= 1, − 1 !!" = min !!" + !!" , !!" !! , = 1, − 1
As a result of the algorithm in the beginning is searched for the minimum distance, and then made to minimize the time.
Example of illustration the routing algorithm in MTS with fuzzy given parameters and temporal dependence You need to find the shortest routes to all nodes. Data are presented in fuzzy form. In round brackets is the distance, in square brackets indicates the time. Figure 1 shows example of MTS. ] ) 4 , 3 , 2 ( [ ) 6 , 5 , 4 (
(6,8,12)[(4,6,8)] The initial matrix of distance and time as follows (equation (6)): (6) (7) (3,4,7)[(1,3,4)] (5,6,7)[(2,3,5)]
So we get the matrix ! below ! = 1
1
Then the next phase for the elements use the formula (2) at the beginning, we obtain !!" = min{!!" + !!! } = min ∞ + 0,0,0 0,0,0 = ∞ Consequently, instead of element !!" in the matrix ! we have element “–“.
!!" = min{!!! + !!" } = min 0,0,0 0,0,0 + 3,4,6 [
Fill matrix !
!!!() = 0, !!!() = 0 ! = 1 2 0,0,0
0,0,0 1 −
2
We recalculate values of the matrix by the formula (4) with the new values obtained in the previous step.
We turn to the calculation of the matrix ! . Similarly, the use early in the formula (2): !!" = min{!!" + !!! ; !!" + !!" }
= min ∞ + 0,0,0 0,0,0 ; ∞ + ∞ = ∞
!!" = min{!!" + !!" ; !!" + !!! }
= min ∞ + 3,4,6 [
!!! = 0, !!! = 0, !!!() = 0
1 2 3
1 0,0,0 0,0,0 3,4,6 1,3,5 4,5,6 [
3 − − 0,0,0 [0,0,0]
We repeat similar iterations for the calculation of the matrix ! . Similarly, we
use early in the formula (2):
(17)
(18)
(19)
(20)
(21)
(22)
(23)
!!" = min{!!" + !!! ; !!" + !!" ; !!" + !!" }
= min 2,4,5 3,4,5 + 0,0,0 0,0,0 ; ∞ + ∞; ∞ + ∞
= 2,4,5 3,4,5 ( 4,1 )
!!" = min{!!" + !!" ; !!" + !!! ; !!" + !!" }
= min 2,4,5 3,4,5 + 3,4,6 [
Using equation (3), we obtain the following values of the elements of the matrix: !!" = min{!!! + !!" ; !!" + !!" ; !!" + !!" } = min 0,0,0 0,0,0 + ∞; 3,4,6 1,3,5 + ∞; ∞ + ∞ = ∞ !!" = min{!!" + !!" ; !!! + !!" ; !!" + !!" } = min ∞ + ∞; 0,0,0 0,0,0 + ∞; ∞ + ∞ = ∞ !!" = min{!!" + !!" ; !!" + !!" ; !!! + !!" } = min ∞ + ∞; ∞ + ∞; 0,0,0 0,0,0 + ∞ = ∞
We recalculate values of the matrix by the formula (4) with the new values obtained in the previous step.
!!" = min{!!" ; !!" + !!" }
= min 3,4,6 [
!!! = 0, !!! = 0, !!!() = 0, !!!() = 0 Fill the matrix ! (24) (25) (26) (27) (28) (29) (30) (31) 1 0,0,0 ! = 2 3 4 2,4,5 1 − − 0,0,0 As a result, similar calculations obtain intermediate matrix of routes ! . ! (32) (33) (34)
Using the ! intermediate matrix performs calculations and obtain the resulting matrix routes.
The specific management tool of MTS is routing. The proposed routing algorithm is applicable to any mechanical transport system. The use of the described routing method gives the best effect for MTS, which are operated in the unstable conditions for major changes of cargo flow intensity. This case of temporal dependence reflects the real environmental situation and its cost.
Acknowledgments. This work has been supported by the Russian Foundation for