THE APPLICATION OF THE DYNAMIC PROGRAMMING METHOD IN INVESTMENT OPTIMIZATION

This paper deals with the problem of investment in Measuring Transformers Factory in Zajecar and the application of the dynamic programming method as one of the methods used in business process optimization. Dynamic programming is a special case of nonlinear programming that is widely applicable to nonlinear systems in economics. Measuring Transformers Factory in Zajecar was founded in 1969. It manufactures electrical equipment, primarily low and medium voltage current measuring transformers, voltage transformers, bushings, etc. The company offers a wide range of products and for this paper’s needs the company’s management selected three products for each of which optimal investment costing was made. The purpose was to see which product would be the most profitable and thus proceed with the manufacturing and selling of that particular product or products.


Introduction
The principle of dynamic programming was, in a sense, known even before Second World War, but Professor Richard Bellman is considered to be the official creator of this method.In 1957, in his book 'Dynamic Programming' (Princeton University Press, Princeton), Bellman set the basic principles of this method.What he did was the following: he looked at a particular problem by making a hierarchy of sub-problems and then went on to solve the easiest one.That is how his principle of optimality came into being, which is the essence of dynamic programming.The main idea in this method's application is the division of a management process into several stages after which an optimal management model is chosen for each stage.The selected management model is the one which leads to optimal process functioning.At the same time one has to bear in mind that optimal management is characterized by the following: the future decisions have to lead to optimal management considering the present state and regardless of the previous state or previous management.
Since successful management means successful fulfillment of all business tasks of which a business process is composed, it is very important to deal with the issue of the optimal allocation of investment funds into specific activities or categories.The problem of revenue management has been dealt with by D. Zhang and D. Adelman 1 , 2 , while portfolio management has been the subject matter in the works of J. Han and B. Van Roy 3 .In the next sections, the mathematical model of Bellman's principle of optimality for additive objective functions will be presented.

Bellman's Principle of Optimality
Suppose that the functions f 1 (x 1 ), f 2 (x 2 ), ... f n (x n ), are such that the objective function is as follows: The problem pins down to finding the value of the variable x 1 , x 2 , ... x n when the objective function has the maximum value subject to the following constraints: we introduce a set of functions In the general case we get the following: The equations ( 2) and (3) are known as the Bellman Equation.
On the basis of these equations, the first task with n variable is broken down into n simpler problems each with one variable and in this way the task solving is done in stages.The function F 1 is determined indirectly while the functions F 2 , ...
In other words, objective function F , defined in (1), reaches its maximum value in the last iteration F n (b n ).
Finally, a set of found optimal values is to be defined, that is, one or more n values (x * 1 , ..., x * n ) whereby objective function F has the maximum value, in which case the optimal management strategy is finally defined as follows:

Business process optimization in MTF
Measuring Transformers Factory (MTF) in Zajecar was founded in 1969.Its scope of business is manufacturing electrical equipment and it has a wide range of products including various types of current, voltage protective and measuring transformers.It also supplies the market with different kinds of medium voltage supporting and bushing types of insulators and inductive reactors.The company's product order quantity differs depending on the demand; it can supply large quantities (more than 100 items or more than 50 items), medium quantities (from 50-100, or from 20-50 items) or small quantities (fewer than 50, or fewer than 20 items).The price depends on both on investment capital and on the quantity of the sold products; this further means that the profit the company makes over a period of time depends on the selling price of its products in the market.
The company's management does its best to successfully perform all its business activities.The main business goal, of course, is meeting the demand in the market; however, the decision on optimal allocation of investment regarding particular products which will ensure maximum profit is not less important.
After the company's management had agreed, we conducted the following research: three important products manufactured by MTF were singled out and we tried to define the most effective way of investing in these products which would ultimately prove to be the most profitable for the company's business.
These products are the following: The low voltage current transformer STEM -081 up to 0.72kV, to which we will further refer as T 1, is ordered in medium quantities (50-100 items), large quantities (more than 100 items) and in small quantities (fewer than 50 items).The price is EUR 17, 01 per unit.
The medium voltage current transformer STEM -N 3821, up to 35 kV, to which we will further refer as T 2 , is ordered in medium quantities (20-50 items), large quantities (more than 50 items), or small quantities (fewer than 20 items).The price is EUR 331,42 per unit.
The transformer JNT sm-24/12 of 24 kV, to which we will further refer as T 3, is ordered in medium quantities (from 20-50 items), in large quantities (more than 50 items) or small quantities (fewer than 20 items).The price is EUR 355, 53 per unit.
According to the data provided by the Company's Finance Department the percentage of sales for each of the products in relation to the quantity sold is presented in the following table:  The company has the amount of money of EUR 80 000 at its disposal.It is to invest this money in the production of the three transformers with the aim of making maximum profit.
In order to solve this investment problem the above-mentioned mathematical model of Bellman's principle will be used.In this case, it will have the following form: f i (x i ) stands for the expected profit after investing the sum x i in product i, whereby i=1,2,3.Objective function F(x 1 ,x 2 ,x 3 ) stands for the overall profit achieved by investing the amount of money of EUR 80 000 in products T 1, T 2, and T 3. On the basis of the introduced conditions in this problem, the mathematical model of this problem will have the following form: subject to the following constraints: After this the application of Bellman's principle of optimality comes, whose algorithm has already been described.As already mentioned, the problem is solved in stages but the result of one stage depends on the result gained in the previous one.Functions F k (b k ) are determined successively for k=1,2,3.The following is also to be taken into account: The value of function F 1 (b 1 ) is found in the first stage in the following way: ( ) ( ) Further, the value of the function is as follows: The next stage determines the optimal allocation of investment in products T 1 and T 2, so objective function ( ) 2 2 b F will have the following form: Using the gained values for function F 1 (b 1 ) and the data on investment efficiency in Product T 2 , and using the formula given at (7), the function's values are as follows F 2 (b 2 ) : the following is true: , we have the following : , we have the following:  At the end of this stage, for 80000 2 = b we will have the following:   On the basis of these results, we can conclude that in this investment phase it is necessary to invest x 1 =60000 in the production of transformer T 1 , while the amount of money (EUR) needed for the production of T 2 is x 2 =20000.
Finally, in the third stage, the allocation of funds among the three products is carried out, and in that case, objective function ( ) will be as follows: Considering (8), the calculated values of function 3 F for expected investment capital are as follows: , and , the values are as follows: , the values of F 3 are as follows: , the following is true: And if the invested amount of money b 3 =80000, whose allocation is crucial for the solution of this problem, the values of function F 3 will be as follows: Finally we come up with the following values: x 3 =0, x 2 =20000 and x 1 =60000, which means that the amount of EUR 80 000 is to be allocated in the following manner: EUR 60 000 in the production of T 1 and EUR 20 000 in the production of T 2, while no investment in T 3 is to be made.In this way the company would make maximum profit, that is EUR 46 200.
All the results of the dynamic analysis carried out in Measuring Transformers Factory in Zajecar are presented in the following table.The results shown in Table 3 will be the same as those acquired by the application of Bellman software application that will be dealt with in the next section.In that way we will practically check and prove the previously used procedures in Bellman's principle of optimality which is, as described so far, applied in the optimization of investment allocation in Measuring Transformers Factory in Zajecar.

The software implementation
As there was a need for the dynamic analysis of the dynamic programming problem, we created the original software application of Bellman's principle of optimality by making a database application in Access.This application, called Bellman, is used for checking the data acquired in the previous analysis.
The main principle this application relies on is a known algorithm based on Bellman's principle of optimality which ensures optimal planning of the socalled multi-stage process of management.The software, whose main components are shown in Picture 1, is such that it first requires a user to enter the total investment capital, that is, the number of projects which are to be analyzed.After that, the investment value is entered as well as the expected value of profit for each level of investment and for each project respectively.
After the values have been entered, the software finds the optimal investment plan which ensures maximum profit.All the calculated parameters and the main characteristics of the DP problem in question are presented in the form of the MS Access Report as shown in Picture 2. It is clear that the data obtained by the dynamic analysis of Bellman's principle of optimality completely match the results gained by the software analysis application.
Picture 1. Process for solving DP problems Picture 2. Software implementation of investment optimization in MTF in Zajecar It goes without saying that this software application is of a general character and can be used in solving other problems when it is necessary to apply dynamic programming.It may be interesting to mention a problem many banks have when it comes to financing business plans -they are to select the most optimal business plans and finance them and not the others given the limited resources.

Conclusion
After conducting the dynamic analysis by applying Bellman's principle of optimality and after the results had been proven by Bellman software application, we can conclude that we solved the problem of investment optimization in Measuring Transformers Factory in Zajecar regarding the three types of transforms marked as T 1 , T 2 and T 3 in this paper.

F
n are determined by the recurrence formula (relation) (3), whereby ( )

Table 1 .
The percentage of sales

Table 2 ,
below presents the expected profit in case of investing EUR 20000 that is, EUR 40000 that is, EUR 60000 and EUR 80000 in each of the products T 1 , T 2 , and T 3.

Table 2 .
The expected profit

Table 3 .
The results of the dynamic analysis