Optimization posibilities for company's resources allocation.

Cociu, Nicolae ; Ignaton, Elemer ; Taroata, Anghel 等

1. INTRODUCTION

Resources represent all means involved in achieving a complex action and which to ensure effectiveness of action must be sized correctly and rationally, to have them at appropriate time and be used as optimal or rational.

Resources allocation is used in developing a program for rational allocation of necessary resources in terms of restrictions imposed by the temporal evolution of their availability and by an objective function optimization (maximize or minimize).

2. OPTIMIZATION POSSIBILITY (IN RESOURCES ALLOCATION)

To manufacture n products are used m resources of the company which are limited and must be allocated optimally. In a company to manufacture products [P.sub.1], [P.sub.2], ..., [P.sub.n] are used resources [R.sub.1], [R.sub.2], ..., [R.sub.m] which are available in limited quantities [b.sub.1], [b.sub.2], ..., [B.sub.m]. Resource [R.sub.i] join in the [P.sub.J] product in the [a.sub.ij] quantity, i = 1, 2, ..., m, j = 1, 2, ..., n, and estimated unit profit for the [P.sub.J] product is [c.sub.j], j = 1, 2, ..., n. We have the conditions: [a.sub.ij] [greater than or equal to] 0, [b.sub.j] [greater than or equal to] 0, [c.sub.j] [greater than or equal to] 0, i = 1, 2, ..., m; j = 1, 2, ..., n, (Andrasiu, M. et al. 1986).

These elements are summarized in Table 1. The optimal manufacturing plan must be determined, so we must know what quantity of products [P.sub.1], [P.sub.2], ..., [P.sub.n] have to be manufactured in the time period considered (day, week, month) and the conditions so that profits are maximal. It notes with [X.sub.J], the amount of product [P.sub.J] that will be manufactured, j = 1, 2, ..., n. The profit was noted by f. The mathematical pattern is obtained from relations (1) which represent a linear programming problem with an function--objective solvable by simplex algorithm primal or dual, or by a computer program: Lindo, QSB (Quantitative Systems for Business), DSSPOM (Decision Support Systems Production and Operations Management), QM etc., (Cociu, 1999; Filip, 2007).

If products [P.sub.J] are indivisible, integer, or a guiding mark or a piece, to the relations (1) must be added the restriction that variables [X.sub.J] be integer, therefore [X.sub.j] [member of] Z, resulting in a linear programming problem in integers, that can be solved with the algorithm of Gomory Branch and Bound algorithm or and with a program on computer.

Resource allocation efficiency through mathematical pattern (1) can be analyzed with the dual problem (2) attached to linear programming problem (1). If it solves the two linear programming problems, first and dual, given by relations (1) and (2), could be obtained information about the resource [R.sub.i]. consumption. If [Y.sub.i] > 0, then [R.sub.i] resource is used in full, is exhausted, the variant for optimal solution, so is the relation (3).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

If [Y.sub.i] = 0, then [R.sub.i] resource is not used entirely, and remain the quantity given by the relation (4).

[b.sub.j] - [n.summation over (j=1)] [a.sub.ij] x [X.sub.j] (4)

From Economic point of view, the dual variable [Y.sub.i] shows how much the optimal value for objective function f would increase, so the maximum profit if company would have an extra unit resource, in the resource [R.sub.i], i = 1, 2, ..., m. It can perform different manufacturing scenarios and choose the options that best matches the company's interests.

If from marketing study we know the maximum and minimize demand of each product, may be added to relations (1) the restriction (5).

[E.sub.j] [less than or equal to] [X.sub.j] [less than or equal to] [F.sub.j] (5)

where [E.sub.j] is the minimal market demand for [P.sub.j] product and [F.sub.j] the maximal market demand; [E.sub.j] [greater than or equal to] 0, [F.sub.j] [greater than or equal to] 0, j = 1, 2, ..., n,.

Optimal solution is obtained similarly for the pattern of relations (1) and (5) through the primal or dual simplex algorithm, respectively using some programs. Can be considered a generalization of the pattern given by relations (1), considering that all the consumptions of resources [a.sub.ij] belong to intervals, and considering the availability of resources to be allocated.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

The mathematical pattern given by the relations (7).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The pattern (7) is equivalent with pattern from relations (8).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Pattern (8) is a linear programming problem which is solved with the primal or dual simplex algorithm programs respectively using Qsb, Dsspom, Lindo, QM etc.

Resource allocation problem can be solved by multicriteria patterns by linear multi-objective programming. Resource allocation will be made so as to optimize multiple objective functions:

* [f.sub.1]--maximizing profits;

* [f.sub.2]--minimizing circulating funds;

* [f.sub.3]--minimize total costs of production. Is considered known:

[d.sub.j]--the amount of circulating funds per unit of product [P.sub.j];

[e.sub.j]--unit cost of product [P.sub.J], j = 1, 2, ..., n.

Multi criteria mathematical pattern will be given by relations (9).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

There are several possibilities for solving linear multi-objective programming problem. Generally, an optimum solution for the problem of linear programming with a single objective function of the three functions of the pattern above is not optimal also for other objective functions of the vectorial function f (X). Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Solving the problem of linear multi-objective programming is possible through STEP algorithm (Step method), POP algorithm (Progressive Orientation Procedure), maximizing the global utility etc.

3. CONCLUSIONS

By using linear programming, and the primal and dual problem in allocating company's resources, produces a better use of resources and various scenarios for optimal production plan are obtained.

4. REFERENCES

Andrasiu, M. et al. (1986). Methods for multi-criteria decisions, Technical Publishing House, Bucuresti

Andreica, M.; Stoica, M. & Luban, F. (1998). Quantitative methods in management, Economical Publishing House, ISBN 973-590-027-0, Bucuresti

Cociu, N. (1999). Decisions optimization in production systems, Eurobit Publishing House, ISBN 973-9441-65-3, Timisoara

Cociu, N. (1999). Optimizations in conceiving and exploitation of production systems, Eurobit Publishing House, ISBN 973-9441-65-15, Timisoara

Filip F. Gh. (2007). Support Systems for Decisions, Technical Publishing House, ISBN 978-973-31-2308-8, Bucuresti

Ionescu, Gh.; Cazan, E. & Negruta, A. (1999). Modelarea si optimizarea deciziilor managerial (Modelling and optimisation of managerial decisions), Editura Dacia, ISBN 973-35-0950-7, Cluj--Napoca Tab. 1. Resource allocation [R.sub.i] [P.sub.1] [P.sub.2] ... [R.sub.1] [a.sub.11] [a.sub.12] ... [R.sub.2] [a.sub.21] [a.sub.22] ... ... ... ... ... [R.sub.m] [a.sub.ml] [a.sub.m2] ... Unit Profit [c.sub.j] [c.sub.1] [c.sub.2] ... Avaialble resources [R.sub.i] [P.sub.n] [b.sub.i] [R.sub.1] [a.sub.1n] [b.sub.1] [R.sub.2] [a.sub.2n] [b.sub.2] ... ... ... [R.sub.m] [a.sub.mn] [b.sub.m] Unit Profit [c.sub.j] [c.sub.n]