Linear Programing and Management Decisions

Introduction:

Linear programming can be defined as the technique that can be applied to optimize various functions like cost and performance. The technique is applicable to variables that are related to linearly with each other. It can help to tackle very complicated issues and can improve the cost-effectiveness and performance of linear functions. There are few constraints or boundaries for each optimization. These limits are considered while optimizing the function using the approach of linear programming. There are some factors that are considered as standard and can be assumed in order to reduce the complexity of computation of optimization of various functions. The approach of Linear programming can be used for the optimization of various operations under some constraints. There are diverse applications of Linear programming in various fields like it has increasing applications in the field of engineering, transportation, and manufacturing, etc. It could be successfully used in order to maximize profit and sales of a business problem. 

Management decisions are those key decisions that are taken by the organization in order to maximize the profit of their business. Management decisions are performed by some managers that decide the goals of business and ways in which goals could be achieved. They decide the key points of any business, or they draft the policies that could play a part in achieving the specified objective of cost optimization in any business decision. There are many strategies that could be applied in order to achieve the objective of maximizing the profit of the business. 

Importance of Linear programming for Management:

The role of linear programming in improving managerial decisions is important. The technique can play a significant role in determining the factors that have a significant impact on the decision-making process of any business.

 It can play a vital role in reducing the function of the cost of the product and optimizing the profitability of any business. The strategy of linear programming can be used to compute the sales and prices of products. The profit of the company could be optimized by applying the technique of linear programming to it. The approach of linear programming can be very helpful for the managers for the decision making the process. The approach of linear programming could be very significant in determining the number of resources that are required for the production of any product. Such approximations could play an important role in analyzing the requirement of raw materials or resources required for different products to maximize the profit of the company. Effective managerial skills could play a significant role in maximizing the profit of the firm. The production level, as well as the profit of any business, could be maximized by applying the strategy of linear programming for effective management.

The practical example of linear programming is the graph that is formed by modeling the business problem in mathematical form and then by plotting it in the graphical form to determine the region of feasibility that can be used to get an optimized solution. Below is the graphical representation of this principle.

                                        Figure 1 The linear programming and its graph

There are different assumptions that are considered in order to implement the linear programming approach successfully. The pictorial representation of this assumption can be shown in the figure below.

                                                                    Figure 2 Assumptions of linear programming

There is a general algorithm that could be applied in order to determine the optimal solution of linear equations. The block diagram of the steps that are to be followed to perform linear programming is shown below.

                                            Figure 3 optimization of operation using linear programming

Advantages of Linear Programming on Decision making:

1. The technique of Linear programming can improve the quality of the decision that is taken by any manager. The decision-making approach becomes more technical and objective and produces better results.
2. It can help to use programming techniques to provide the solution of the approach that is more realistic, and the constraints are considered to obtain the exact solution. 
3. The realistic solutions are obtained by this approach.
4. It allows modification of the quantities and provides realistic results.
5. The process can help to tackle the bottlenecks of research and could be used to re-evaluate the techniques that are being applied to determine the optimal solution.

Example of Application of Linear Programming on Management Decisions 

There are different examples that could be considered while determining the practical application of linear programming in business. There can be different cases in which the function of cost or some other function relevant to the business. The examples in each case can be solved by mathematical modeling of the equations of the case. Consider an example of a company manager who is assigned with the task to optimize the profit of the production of the company either by minimizing the cost of maximizing the profit of the company. He can use the approach of linear programming to determine the maximum profit value. Let us consider a case in which there is a particular case in which the manager has to decide between different possibilities that are available to him for the sake of evaluating the best solution to tackle the situation. Consider the case that a manager is dealing with the situation in which a manufacturing company has two options of producing two chocolates that require raw material of chocolate powder and dried Milk. In order to manufacture the chocolate of type A or B the following things are required.

If the company has a total of 5 units of dried Milk and twelve units of chocolate powder, then the company needs to maximize profit. The amount of profit generated by selling chocolate of type A is the amount of 6 Rs per unit. The amount of profit generated by selling chocolate of type B is the amount of 5 Rs per unit. The manager is to decide the number of units that are to be produced by the company in order to get maximum profit.

Constraints:

Assumptions:

Assume the profit of the generation of the production of unit "A" is "X." Assume the profit of the generation of the production of unit "B" is "Y."               Assume the total profited that is generated is “P."

Objective Function:

The objective function that should be maximized is represented by the following equation:

Profit: Max P = 6X+5Y

The profit generated by this equation is represented by the equation above. The number of units that is multiplied by the type of product. The function that is to maximized is "P." The amount of dried Milk and chocolate powder availability are the factors that are limited. These things that restrict the result are called "Constraints."

The other equations that can be formed by considering the constraints are represented by the following equations:

The amount of Milk that is available limits the production and is represented as:

                                                             .X+Y ≤ 7

The amount of chocolate powder that is available limits the production and is represented as:

                                                            3X+2Y ≤ 15

 The only possible results are the integral values of X and Y

                                                         X ≥ 0  &  Y ≥ 0

The results of the mathematical modeling should fulfill all the constraints, and these all inequalities should be satisfied in order to make the best decision. The manager should ensure that all the conditions are fully satisfied in order to provide the best decision. In this way, the approach of linear programming could be used to maximize profit. The managerial decisions could be enhanced by the application of the above-mentioned constraints.

Conclusion:

Linear programming can play an important role in maximizing the profit or amount of sales or any other objective function under consideration. The management decisions could be significantly improved by applying these results in real-world problems. Mathematical modeling can be used to model real-world scenarios.

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