Optimization of the production structures.

Dobren, Flavius Andrei ; Dumitrescu, Constantin Dan ; Lazarescu, Cezara 等

1. INTRODUCTION

Optimization means to find, of the multitude of existing solutions, that precise solution that fulfills the maximum conditions, from one specific and established point of view. If that particular goal is oriented towards the economic domain, economic optimum represents that situation or state of economy that assures the highest efficiency. To choose from multiple solutions that particular one that assures the highest economic efficiency means to search for that solution which assures maximum results with minim effort and expenses, respecting certain restrictions that defined that particular domain.

Optimization of the processes in an industrial unit, considered as a complex system, is obtained based on three distinct levels (Nicolescu, 2001):

1. The level of the production system;

2. The level of technological production system;

3. The level of technical level;

The work presents the synthesis of the objectives through the optimization processes for each level of decision, the methods used for each level of optimization and the optimization criteria used for each level in part. During the optimization process, a specific objective, a certain optimization method, selected from a range of other methods, and a certain particular criteria of optimization correspond to each of these. A synthesis of these optimization processes pays attention to four factors: the level of optimization, the goals of the analysis processes, the maximization methods and the maximize criteria

Constantly, companies confront with the problem of assignation of the resources (Peel, 1993). For micro-units, small- and medium sized companies, the optimum solution of the problem can be reached through the usage of linear programming. A first phase is to transform the problem that conditions the existence of the restriction system into an equation and to establish the equation system; then the optimizing function is to be established, aiming the maximization/ minimization at the cost level of the whole of the distribution/ re-distribution process of the resources. (Andrei, 2005). The paper grants particular attention to the assessment of the human resources in companies because in small companies in Romania fluctuation employment is high, on the one hand and on the other hand the level of wages is negotiable from one person to another; these two elements compound a frequently used module.

To highlight this point, it is considered that an economic enterprise must produce [P.sub.1], [P.sub.2], Pn, benefiting at most of the [b.sub.1], [b.sub.2], bm quantities from the [R.sub.1], [R.sub.2], Rm resources. Knowing that the manufacturing of an unit from the Pj product needs the a quantity from the Ri resource, and that through the delivery of a quantity of the same product, the Cj, 1 [less than or equal to] i [less than or equal to] n profit is obtained, it is required to determine the x1, x2, ..., xn quantities from the P1, P2, ..., Pn products so that we obtain the highest benefit. So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The restrictive conditions are also called the linear programming restrictions. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Then PL--min is the matrix transcribed thus:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

and PL--max:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Obviously that the A * X=L restriction system may be: --incompatible

--exclusively compatible determined (only in the case m=n)

--undetermined compatible.

For our study, the last case is interesting because this is the situation where there is to choose from multiple solutions, the optimum one.

Let us assume that the range of A is m. If we mark the columns of the A matrix with aj, 1 [less than or equal to] j [less than or equal to] n, then the restrictions of the problem are transcribed:

a1x1 + a2x2 + ... + anxn = L (5)

A X [member of] Rn vector, whose components satisfy the restrictions of a linear problem, is called an admissible program (or a admissible solution, or a possible solution).An admissible [X.sup.o] program that minimizes (or maximizes, according to the problem) the linear function associated to that problem is called optimum program or optimum solution.

A X = (x1, x2, , xn)T program is called a base program if the vectors of the aj column, accordingly to the x components, non-null, are linear dependent. For the reason that A = m range a base program has at most m non-null components.

Optimum assignation of the resources

The moment when the resources allocation is solved, there is still the problem of their usage within the manufacturing process, so that the total costs would be minim, or the total profit would be maxim (Baron& Anghelache, 1996).

If it is considered that a plant/ department manufacturing system, manufactures Rj, j = 1,2,3, ..., n guiding marks with the Mi, i = 1,2,3, ..., m machines, with the operation sequence being random, the information about the process is:

the time standard for manufacturing a Rj guiding mark on the Mi machine is equal to tij (u.t./ piece)

the vacant time of a machine, for a time horizon (week, month, semester, year) is Ti .i = 1,2,3, ... m

the assessed time for manufacturing a guiding mark is estimated considered as being cj j= 1,2,3, ... n

it is thought that during manufacturing there is no vacant time, due to the fact that the operations are inter-changeable.

The quantity from the Rj guiding mark is named with Xj j=1,2,3, ..., n, which is produced on the Mi machine.

With these elements, the mathematical model of the problem is:

[summation] tij x Xj [less than or equal to] Ti (6)

in which: i = 1,2,3, ..., m, j = 1,2,3, ..., n, Xj [greater than or equal to] 0, si Xj # N The function for optimizing is:

[summation] cj x Xj [right arrow] min (7)

The problem's solution is obtained by using the Gomory solving algorithm for the case of whole numbers, or by using the specific programs for linear programming (QSB. LINDO). Solving this problem leads to linear programming in whole numbers with real numbers coefficients.

2. OPTIMIZATION OF THE MANUFACTURING CAPACITY BY USING A SPECIFIC CALCULUS PROGRAM

In order to dimension the production structure according to the agenda enforced by the market conditions (from Romania, and also from the European Community market), the user may access the condition (Dumitrescu et al., 1998).

Cpyear = [summation]Qyearj [right arrow] (pcs/year) (8)

If we restrict the area of the manufacturing capacity notion to the company level, from the point of view of outputs of manufacturing systems, the production capacity is actually a transfer capacity of that system (which can be a work place, work team, department or organization). For this case, the production capacity actually represents the maximum quantity of a certain array and quality obtainable along a specified period, in fixed technical--organizing conditions, without considering the narrow places.

The Calculus Program for the Production Capacity and Dimensioning of the Humane Resources / Investments (The PcDHr/I program) has the following distinctive sections:

* Section 1--The calculus of the production capacity of a production system, paying attention to the four cases he may be found in.

* Section 2--The optimum dimensioning of the productions system (the calculus of the production areas needed to the production process)

* Section 3--Human resources/ payment fund/ calculus for the necessary investment in order to realize the production system.

The program also offers the possibility to establish the human resources essentials, in order to draw up the flow chart (in the case of small/medium enterprises). Furthermore, the value of the necessary investment, the flow of furnishing process of equipment, tools and the endowment of the work place with specialized equipments can be established. The program detains the calculus possibility of the production capacity, considering the characteristics of the manufacturing processes (homogeneous/non-homogeneous) and the arrayal characteristics (homogeneous/ non-homogeneous arrays); in these four cases, the manufacturing capacity is evaluated distinctively.

3. CONCLUSIONS

1. Optimization means to determine the solution that in the conditions of pre-established restrictions assures maximum of results with minimum effort or expenses. Within a company the process optimization may be achieved on three different levels: the level of production system, the technological level, and the technical level; each level is characterized through its goals, optimization methods and proper methods.

2. Applying the linear programming method in order to establish the optimum allocation of resources in the case of SME, we can generate a general mathematical model:

[summation] Xj[[alpha]i, [beta]ij]# [[gamma]i] , [delta]]i = 1,2,3, ... m, (9)

where: Xj [greater than or equal to] 0, si Xj # N j = 1,2,3, ..., n

3. The production capacity of a SME may be optimized by enforcing a specific calculus program, starting from the condition Cpyearj = [summation]Qyearj [right arrow] pcs/ year, generally accepted within Romania and the European economic area. The program includes three distinct sections mentioned before: the calculus of the production capacity of a production system, the optimum dimensioning of the productions system and the human resources/ payment fund / calculus for the necessary investment in order to realize the production system.

This system may be implemented in homogeneous / non-homogeneous conditions, as much as in the production process and in the arrayal characteristics.

4. Research has aimed several specific aspects:

* the program is for small and medium sized companies' use; the information is to be introduced fast & will be used immediately;

* the cost of this program is low; it permits to use a high data volume, relevant to the company;

* the program may be used to dimension a manufacturing structure and the necessary staff to activate the developed structure and also for a first evaluation of the necessary investment to enable the dimensioned structure.

4. REFERENCES

Andrei, S. Statistics, Theory and Applications (2005) All Publishing, Bucharest

Baron, T. & Anghelache, C. The Statistics (1996), Economical Publishing, Bucharest

Dumitrescu, C.; Fantana, N. & Militaru, C., General Management Factors (1998), EUROBIT Publishing, Timisoara

Nicolescu, O. Small and Medium Entreprises' Management (2001 ), Economical Publishing, Bucharest

Peel, M. Introduction to Management (1993), Alternativ Publishing, Bucharest