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The study deals with the multi-choice mathematical programming problem, where the right hand side of the constraints is multi-choice in nature. However, the problem of multi-choice linear programming cannot be solved directly by standard linear or nonlinear programming techniques. The aim of this paper is to transform such problems to a standard mathematical linear programming problem. For each constraint, exactly one parameter value is selected out of a multiple number of parameter values. This process of selection can be established in different ways. In this paper, we present a new simple technique enabling us to handle such problem as a mixed integer linear programming problem and consequently solve them by using standard linear programming software. Our main aim depends on inserting a specific number of binary variables and using them to construct a linear combination which gives just one parameter among the multiple choice values for each choice of the values of the binary variables. A numerical example is presented to illustrate our analysis.

Most of real life and industrial problems involve a process called optimization, which means finding the maximum or minimum value of some quantity called objective function, subject to a system of linear inequa- lities or equalities called constraints. The graph of the system of constraints is called the feasible region and the optimal values of the objective function occur at vertices of the feasible region. Many research papers handle this problem from more than one point of view and gave very helpful and considerable results.

Linear programming is frequently applied in real-life problems and therefore it is very important to introduce new tools in the approach that allow the model to fit into the real-life problems. Linear programming was developed by Dantzig [

Recently, a considerable work and many attractive results have appeared in this research area, see [

Chang [

This paper is organized as follows: The first section is devoted for introduction and historical survey of the subject, in the second section, we give the mathematical model of the general multi-choice linear programming problem. While in Section 3 we present our technique and explain how it works to transform the multi-choice linear programming problem to a standard linear programming one, in addition, we give our main results in a compact mathematical form. In Section 4, we offer a numerical example to show the procedure of our technique. We conclude the paper by a brief discussion to compare our result with other results given by Biswal and Acharya [

The mathematical model of a multi-choice linear programming problem is presented as:

Find

subject to:

The right hand side of each constraint (2) has a set of

Let

Setting:

In order to achieve our main aim of selecting just one value among the

inserting a set of K binary variables, namely:

the following form:

Finally, we replace the right hand side of the

Now, we can rewrite the mathematical model (1)-(3) in more convenient form as:

Find

subject to:

where

A factory produce three different types of washing machines A, B, and C. Producing of one washing machine of type A requires 10 hours general labor, 5 technical hours, and 1 packing hour. While the washing machine of type B requires 4 hours general labor, 7 technical hours, and 2 packing hours. and the washing machine of type C requires 5 hours general labor, 2 technical hours, and 3 packing hours. The factory can afford up to 900 or 1000 or 1100 hours of general labor, up to 1000 or 1450 or 1600 or 2000 hours of technical labor, and up to 300 or 400 packing hours per week. A washing machine of type A, type B, and type C earns a profit of £130, £100, and £80 respectively. The factory decided to produce at least 20 washing machines of type C per week. How many washing machines per week should the factory produce from each type to maximize profit?

Let

The following table summarize the information above:

Since the manufacturer wants to maximize the profit, so, we have the following mathematical multi-choice problem:

subject to the constrains:

Applying the above technique in (5) and (6), we can transform the given multi-choice programming problem to a mixed integer linear programming problem. Referring to the data above, we have

subject to:

The above mathematical model is treated by either LINGO or MATLAB software.

The corresponding Matlab code is:

The solution is obtained as:

Biswal and Acharya [

The aim of the article was to describe a new efficient technique for solving mathematical multi-choice problems the process depend on obtaining a number of linear mathematical expressions equal to the number of the multi-choice constrains and contain a specified number of binary variables. We tried to set the models as general as possible and that way make them applicable to any given mathematical multi-choice linear pro- gramming task. In practice it is possible to describe the majority of such problems. We can state that these models can serve as a simple and easy method for solving this type of mathematical problems using available softwares as Matlab and Lingo. Transforming the multi-choice problem to a linear problem is considered the main advantage of this method. The present method can be extended to a multi-objective multi-choice linear programming problem.

The authors are thankful to the editor in chief and the learned referees for their valuable suggestions regarding the improvement of this paper.

Tarek A.Khalil,Yashpal SinghRaghav,N.Badra, (2016) Optimal Solution of Multi-Choice Mathematical Programming Problem Using a New Technique. American Journal of Operations Research,06,167-172. doi: 10.4236/ajor.2016.62019