Optimization of the Vehicle Movement Modes

Article describe process of optimization of vehicle movement modes. This task belongs to the class of combinatorial tasks. When in our task we take only six sector of the road and five transfers in each sector then number of the possible vehicle movement modes will be 18750. For decision of the task we use equation recurrence of Bellman, according to which the optimal choice has the property of independently from the initial state and solutions at the initial moment the subsequent solutions must be optimal. This allows to replace search of extremum of a function with many variables by search of extremum of a function with one variable. For optimization we use Mathcad system. The proposed procedures of optimization of vehicle movement modes allow to reduce time and costs of such calculations at least in several hundred times and effectively to carry out researches in any usage mode and exploitation of vehicles, providing good visual presentation of received results.


Introduction
Article describe process of optimization of vehicle movement modes. This task belongs to the class of combinatorial tasks. When in our task we take only six sector of the road and five transfers in each sector then number of the possible vehicle movement modes will be 18750. For decision of the task we use equation recurrence of Bellman, according to which the optimal choice has the property of independently from the initial state and solutions at the initial moment the subsequent solutions must be optimal. This allows to replace the extremum search of a function with many variables by extremum search of a function with one variable. For optimization we use Mathcad system.

Statement of the problem
It is known. when build roads, it is necessary to execute large amount of work. At the same time, on each road sector has different loading on vehicle: slopes of road, resistance to movement and so on. We know expense of fuel s(i,j,k) on each road sector i at transition from j th transfer to k th. The initial information for vehicle movement optimization is presented in table 1, the data of which will be presented later as a expenses matrix. For simplification we will to use transmission with four transfers.

Rout sector
i Vehicle movement on j-th transfer on (i-1)-th road sector Vehicle movement on k-th transfer on i-th road sector Expense of fuel s(i, j, k) on each rout sector i at transition from j th transfer to k th   when the vehicle transfer changes on the next road sector. In our task, for vehicle movement modes optimization we will take 5 sectors of road.

Method selection
Representation of all possible vehicle movement modes in the form of a marked graph of transfers provides clearness and ease of forming an acceptable set. This task belongs to the class of combinatorial tasks, in which the possible vehicle movement modes determine by the calculation.
where mi is the possible number of transfers of the vehicle on a on each road sector i.
So, if in our task we take only five sectors and in each sector five transfers, then the possible vehicle movement modes will be 3125.
We will use equation recurrence of Bellman, according to which the optimal choice has the property of independently from the initial state and solutions at the initial moment, the subsequent solutions must be optimal. This allows to replace the search of extremum of a function with many variables by search of extremum of a function with one variable.
where: ymin(i,j)minimum fuel expenses for the partial vehicle movement modes with the i -th sector and j-th transfer; ymin(i+1, k)-the same, after the (i+1)-th sector and with the k-th transfer of the vehicle. The algorithm of the dynamic programming method includes two stages. At the first stage, sequential optimization of vehicle movement modes is performed for partial route starting from the last sector, using equation recurrence of Bellman.
At the second stage, using the results of calculating the optimal vehicle movement modes, the optimal one is found, which provides the minimum fuel consumption.
The first stage include definition of the minimum total expenses: on a current sector of road i and the minimum expenses for the subsequent sectors of road (i -1) with allocation of expenses on the added sector of the road, provided the minimum expenses to all partial route of movement of a vehicle: 1. Determine the minimum expense for sectors of the route, after the 5th. In our example, the minimum fuel consumption after the 5th sector, as there are no existing sector, is 0. ymin (6.1) = 0.
2. Determine the minimum expense for partial vehicle movement mode starting with the 5th sector: Mark the arrows on the 5th sector: (1-1), (2-1), and (3-1). 3. Determine the minimum expense for partial vehicle movement mode starting with the 4th sector:  Mark the mark on the 4th sector of the arrow: (1-3), (2-1). 4. Determine the minimum expense for partial vehicle movement mode starting with the 3rd sector:    The second stage includes determination of the minimum expense for all rout starting with the 1th sector with allocation of expenses on each sector jf the rout -on the first sector allot the arrow (1-4); -on the second sector allot the arrow (4-2); -on the third sector allot the arrow (2-2); -on the fourth sector allot the arrow (2-1); -on the fifth sector of allot arrow (1-4). All these arrows are highlighted in bold on the graph of possible vehicle movement modes Fig. 2. Below is an algorithm and an example of optimization of the vehicle movement modes. by the Bellman method in Mathcad system.

Conclusion
The proposed procedures of optimization of vehicle movement modes allow to reduce time and costs of such calculations at least in several hundred times and effectively to carry out researches in any usage mode and exploitation of vehicles, providing good visual presentation of received results.