Automation of optimization of discrete technological processes

The article describes the procedure automation of optimization discrete technological processes with using of Bellman’s functional (recurrent) equation and system Mathcad. As rule the technological processes includes n of operations and each operation can be executed by various types of equipment. Expenses (cost, time, ...) on execution of i operation by k equipment after execution by j equipment (i-1) operation are known c (i, j, k). Expenses for execution by k equipment i operation can depend on the equipment j, which executed previous (i-1) operation. It is necessary to execute automation of optimization technological process with the minimum expenses. The algorithm of the decision of a problem by Bellman’s method includes two phases. The first phase is calculations of the minimum expenses for execution of all partial technological processes, from last operation of process to the first. The second phase is definition of the required optimum set of equipment which is carrying out all technological process with the minimum expenses. The proposed procedure of automation of optimization technological process using Bellman’s method and system Mathcad significantly decreases time and labour costs on execution of such calculations and efficiently to execute investigations related with change of equipment parameters.

The technological process includes n of operations.Each operation can be executed various types of equipment.Expenses (cost, time, …) on execution of i operation by k equipment after execution by j equipment (i -1) operation are known -c (i, j, k).Thus, the size of expenses for execution by k equipment of i operation can depend on the equipmentj, execution previous (i-1) operation.It is necessary to execute structurally-parametrical optimization of technological process with the minimum expenses.
The initial information is presented in the Table 1.

Fig. 1. Graph of discrete technological process
The circle (equipment) j executes (i-1) previous operation.The circle (equipment) k executes (i) current operation.
Arrow is possible tie of one type of the equipment with other.The circle (equipment) also means execution of operation by one type of the equipment and the beginning of execution of other operation by other type of the equipment.
The number j standing in the circle is type of the equipment with its parameters used on previous operation.The number k standing in the circle is type of the equipment with its parameters used on current operation.First and last circles of the graphfictitious equipment.
If at execution of any operation the subsequent equipment parameters mismatch previous equipment parameters then the given equipment cannot be used.In this case don't use an arrow between these equipment or use arrow with very big expenses so that this equipment could not be at the optimum set of the equipment.

Operation i
The equipment j executes previous (i-1) operation j = 1 j = 2 j = 3 The equipment k executes flowing i operation Expenses (cost, time, …) с(i,j,k) where: y (i+1, k) -the minimum expenses for the partial sets equipment, carrying out partial technological process, from i = (i+1) till 1 operation and k the equipment; y (i, j) -the same, from i till j=1 operation and j-th equipment.

Algorithm of the decision
The algorithm of the decision of a problem by means of Bellman's equation includes two stages.
The first stage includes calculations of the minimum expenses for execution of all partial technological processes, since the last operation.Results of calculation are putting above or under a circle fig. 2. The arrow is directed from the equipment which has ensured the minimum expenses for considered partial technological process is marking by a stroke.
The second stage includes definition of the required optimum set of equipment which is carrying out all technological process with the minimum expenses.
The first stage includes several steps.1. Calculation of the minimum expenses for execution of technological process including i = 5 operation.

MATEC Web of
Install stroke on an arrow (1,1) of the 3-th operation (see Figure 2) which has specified the equipment k = 1, ensured the minimum expenses.
-the minimum expenses for partial technological process y (3,2) Install stroke on an arrow (2,1) of the 3-th operation (see Figure 2) which has specified the equipment k = 1, ensured the minimum expenses.
-the minimum expenses for partial technological process y (3,3) Install stroke on an arrow (3,1) of the 3-th operation (see Figure 2) which has specified the equipment k = 1, ensured the minimum expenses.
4. Calculation of the minimum expenses for execution of 5-th, 4-th, 3-th and 2-nd operations: -the minimum expenses for partial technological process y (2,1) Install stroke on an arrow (1,3) of the 2-nd operation (see Figure 2) which has specified the equipment k = 3, ensured the minimum expenses.
-the minimum expenses for partial technological process y ( Install stroke on an arrow (2,3) of the 2-nd operation (see Figure 2) which has specified the equipment k = 3, ensured the minimum expenses.
-the minimum expenses for partial technological process y ( Install stroke on an arrow (1,1) of the first operation (see Figure 2) which has specified the equipment k = 1, ensured the minimum expenses.
Results of calculation are presented on Figure 2.

Fig. 2. Graph of the technological process with results of calculation
The second stage is carried out in reverse order, from search of the optimum equipment, carrying out last operation.Thus, in our example in the optimum set of equipment (see thick lines) enter on: -5-th (fictitious) operation the equipment under number k =1; 4-th operation the equipment under number k =1; 3-rd operation the equipment under number k =3; 2-nd operations the equipment under number k =1; 1-st operation the equipment under number k =1.

Automation of optimization in system Mathcad
The algorithm and programs automation of optimization of technological process by Bellman's method conducted in system Mathcad [2][3][4][5]

Conclusion
The proposed procedure of structurally-parametrical optimization of technological process using Bellman's method and system Mathcad significantly decreases time and labour costs on execution of such calculations and efficiently executes investigations related with change of equipment parameters.