A new simple algorithm for solution of optimization problems

This paper is devoted to developing a new sample algorithm after the solution of optimization problems subject to a budget constraint in firms’ production. In the calculation of the optimal composition of different items, there is one fundamental problem. In general, firm’s behaviors are under a budget constraint and they always try to optimally allocate their resources among production factors. It is possible that there are some production factors like quality of labor force which influence firm’s production after some period. In this case, when we want to solve optimization problems for these firms, we can encounter that the objective function (Revenue function, Production function or some other) of the optimization problem depends on the lags of the mentioned production factors. Therefore, the realization of the allocation of firm’s resources among the production factors doesn't seem plausible (because some of them are on the lags). In this context, we have tried to prepare a simple algorithm for optimal allocation of firm’s resources in the same period, using an optimal share which has been found for different lags of variables.

There are many samples for production, revenue, profit and cost function in microeconomic literatures.For example; Hu Z. H. examined the maximization problem of firm's production in the seller's market (Hu, 1998).Standard Cobb-Douglas production function is the objective function of this optimization problem while subject function has been constructed as the linear combination of labor and capital.Note that, the extension of Cobb-Douglas production function is applied in a lot of production practice.Another example which has been introduced by Grubbstrom R. W.He also used Cobb-Douglas production function (Q(X)) with n production factors but not standard version (Grubbstrom, 1995).In his work, sum of the coefficients of production function is equal to z (any value).Variation of parameter z leads to economic production analysis in three situations.Q(X) is called constant returns to scale, decreasing returns to scale and increasing returns to scale when z =1, 0 <  < 1 and  > 1 , respectively.
This research aims to analyze the optimization problems related to firm's production.In literature, there are many research works devoted to this topic.Markusen developed the identical production functions for firms in the X industry which are given by   =   (  ,  1 ,  2 , … ,   ), where Sj is referred to as "specialized inputs".He investigated profit maximization problem on the basis of CES type production function (Markusen, 1990).Another interesting work is Griliches's research on the production function (Griliches, 1967).He introduced the quality of labor which was measured by an occupational mix variable as the production factor.His next research work (Griliches, 1968) also has been devoted to the mentioned issue (firm's production).In the latter work, Cobb-Douglas production function has been used and estimated by means of OLS.The quality of labor force (measured by "occupation mix" index) and the quality of capital stock are independent variables of this study jointly with some other variables.We can also find a useful feedback about firm's production function and related optimization problems in some additional research works (Hillestad, 1975;Liou et al., 2006;Chen et al., 2003;Pujowidianto et al., 2009;Amoranto & Chun, 2011;and so on).
The question of our optimization problem is that firm's budget expenditures on some production factors, such as quality of labor force, rarely have a simultaneous impact on its production.For example, firms spend money in training of their employees, and the employees take time enough to acquire firmspecific skills (Naoki, 2011).Wagner note that "Usually, skill enhancement does not come to an end after passing the final exam in an apprenticeship program or a school…."(Wagner, 1997, pp. 421).He reported on the base of the IAB-Betriebspanel that share of employees who receive further training in small and medium sized firms less than share of the employees who receive further training in large firms.Note that, I encountered to mentioned evidence in some other research works (Abowd, Kramarz and Moreau, 1996;Fox and Smeets, 2011;Waldorf, 1973 etc.).So, these evidences let us to argue that, if a firm spends some money to improve its employees' skills in time t, then it is possible that these expenditures influence the firm's production after some period of time (t+k).This statement requests that both objective function (production function) and subject function (cost or constraint function) are on the lags of expenditures for improving the quality of the labor force.
So this research work tried to offer a special algorithm which consists of three steps to analyze the distribution of firm's factors spending (say b) in any quarter (say k-th quarter) among production factors (  divided into   ,   (suppose that x and y are the production factors)) was optimal or not.In applying of these steps, three cases can be appear.1) Maximum lag length greater than 3 (case 1) and 2) maximum lag length less than 3 (case 2) and 3) maximum lag length is equal to 3 (case 3).In Case 1, we can only analyze the distribution of the budget expenditures in past quarters.Actually, if we want to define optimal allocation of budget expenditures in Q1 of the next year we need value (limit) of the quarters after the next year (because maximum lag length greater than) which are unknown.In Case 2 and Case 3, we can both analyze the distribution of the budget expenditures in past quarters and define the optimal allocation at least for first quarter of the next year.Actually, if we want to define optimal allocation of budget expenditures in Q1 of the next year we need value (limit) of the quarters ( +1 ,  +2 ,  +3 ) of the next year (because maximum lag length less than and eaual to 3) which are known.As above mentioned, the optimal allocation at least for first quarter of the next year can be defined.It means that we can suggest the optimal distribution of budget expenditures at least for first quarter of the next year in Case 2 and Case 3.

Description of the problem
We noted in the introduction that, the main purpose of this research is to determine a new simple algorithm after the solution of optimization problems in which there are some variables on the lags in objective and subject functions.Suppose that, firm's production depends on n production factors.Let's denote these factors by Xi (i = 1, 2, … , n).Assume that, some of them currently have an impact while some of them have the impact with the lags on the firm's production.Let, the firm wants to maximize its production in the condition of budget constraint.Now suppose that this production process has been described by any production function f and budget constraint has been described by constraint function C. Then we can write mathematical shape of this problem as follows (Simon and Blume, 1994;Chiang, 1984;Just et. al., 1983): Subject to: Where, i = 1, 2, …, n and t = 1, 2, …, T. D is a nonnegative subset of R,  ,− = [ 1,− 1 ,  2,− 2 , … ,  ,−  ] is a vector of production factors, p = 0,1, 2, … , P and :  →  is an objective function or production function, :  →  is a constraint function or cost function, and bt is a budget constraints in time t.
Note that (1)-( 2) is constraint optimization problem on the X (production factors) and it can be constructed as a linear and non-linear optimization problem.There are various ways of solution to such problems in the optimization course.Note that it doesn't matter us.Because this paper describes an algorithm after solution and we will assume that the solution of ( 1)-( 2) optimization problem have already been found.Suppose that this solution is as follows: Where,   are optimal allocation of firm's resources bt (budget constraints) among  ,− , respectively and i = 1, 2, …, n; p = 0,1, 2, … , P.
There is a problem that the constraint condition (2) is on the lags of the variables because objective function (1) relies on the function of lags of these variables.Now suppose that, is an optimal solution of (1)-( 2) optimization problem.Where, i = 1, 2, …, n ; p = 0,1, 2, … , P ; t = 1, 2, … ,T.Let, subject function ( 2) is the linear budget constraint condition which is following: Then, we can define optimal shares of firm's budget among production factors as follows: Where,  −, * are optimal value of production factors,   is firm's budget constraints in time t and i = 1, 2, …, n ; p = 0,1, 2, … , P ; t = 1, 2, … , T.
In Table 1, we have tried to describe this situation.Now suppose that   is the limit in first quarter of current year in Table 1.For solution ( 7) and ( 8), firm must spend  1   share of value   for X1, must spend  2   share of value   for X2 ,…….. and must spend     shares of value   for Xs , k1, k2,….,kslags before, respectively for maximizing its production in Q1 of current year.In same way,  1  −1 ,  2  −1 , ………. and    −1 shares of value  −1 must be spent for X1, k1 lags before, for X2, k2 lags before, ……….and for Xs, ks lags before, respectively for maximizing its production in Q4 of previous year (see Table 1).But it seems impossible.This phenomena causes the problem to use the results of the solutions of ( 1)-( 2 ] (p = 0,1, 2, … , P) of the ( 1)-( 2) optimization problem under condition of

A Solution of the problem and result
First of all, assume that there are three properties; [A] The value of firm's budget expenditures for the next year is known in advance (in the current year), [B] Expenditures in the current quarter are proportionate to the expenditures in the previous quarter.Mathematically, it can be written as follows: Now for simplicity, suppose that firm has only three production factors: X1 , X2, and X3.Assume that each factor influences firm's production with the lags and suppose that, each factor has only one stage lag effect1 .Suppose that, the optimal solution is as following: Now let's accept that  1 = 1,  2 = 2,  3 = 3 and use Table 1 for only two years: current and next.Then, on the base of optimal solution (10), to maximize, the firm must spend  1 share of value   ( 1   ) 1 lag before for X1,  2 share of value   ( 2   ) 2 lag before for X2 and  3 share of value   ( 3   ) 3 lag before for X3 (see Table 2).If we continue the process with the same pattern, analogously,  1 share of value  −1 ( 1  −1 ) 1 lag before for X1,  2 share of value  −1 ( 2  −1 ) 2 lag before for X2 and  3 share of value  −1 ( 3  −1 ) 3 lag before for X3 (see Table 2) must be spent for maximizing of the influence of budget spending.We can see this process more clearly in the Table 2. Now, let's begin with the Q2 of the current year.We can see that,  3 = 3 and  3 >  1 ,  2 has been provided.The question is how we can share the value  −3 among X1 , X2 and X3 in Q2 of current year, so that spending can have maximum influence to firm's production?So, we can write the mentioned optimization principle in Table 2 as following: Let,  −3 =  −3  −3 ( −3 is any positive number) Now we can get a new optimal allocation of budget expenditures among X1 , X2, and X3 in Q2 of the current year as following: Then, we can say that ( 12) is the optimal distribution of budget expenditures among X 1 , X 2, and X 3 in Q2 of the current year.It means that this distribution can have the maximum influence on firm's production in the next quarters.But there is one important point in this statement:  −3 must be assigned by Now we can apply this statement to the general case.So, let's define the optimal distribution of budget limit for any time (period) among  1 ,  2 , … ,   .For this, below simple algorithm can be used: Step 1: The longest lag is defined.Suppose that, this lag is   and the condition   ≥  1 ,  2 , … ,  −1 is available.
Step 2: The budget constraint ( −  ) and distribution of this value among the production factors are reconstructed as following: 1  −(  − 1 ) +  2  −(  − 2 ) + ⋯ +    −(  −  ) +  +1  −(  −0) + ⋯ +    −(  −0) =  −  (13) quarters after the next year (because lag length greater than 3) which are unknown.In Case 2 and Case 3, we can both analyze the distribution of the budget expenditures in past quarters and define the optimal allocation at least for first quarter of the next year.Actually, if we want to define optimal allocation of budget expenditures in first quarter of the next year we need value (limit) of the quarters of the next year (because maximum lag length less than and eaual to 3) which are known.As above mentioned, the optimal allocation at least for first quarter of the next year can be defined.It means that we can suggest the optimal distribution of budget expenditures at least for first quarter of the next year in Case 2.
this table considers the case of   ≥  1 ,  2 , … ,  −1 .But it is not verdict.This table can be constructed for other cases.