Limitations of Linear Programming

Although linear programming is a very useful technique for solving optimization problems there are certain important limitations in the application of linear programming. Some of these are discussed below:

1.      Firstly, the linear programming is a very useful technique for solving optimization where the constraints and the objective function can be stated in terms of linear expression. In real/world business problems many objective functions and constraints cannot be expressed linearly. Suppose for example, that a firm producing a wide variety of consumer products has some unutilized capacity in the first shift. The management is considering the production of two new products - x and Y to utilize the capacity. It wants to determine what quantities of each product should be produced monthly to maximize the profits as summing that all products manufactured can be sold. The market research has indicated that the contribution per unit of product X remains constant at a level of Rs. 80 irrespective of the volume of sales. The contribution per unit of product y however falls as the sales volume increases and it can be represented as being Rs (100-0.1) where y is the number of units of product y sold. The objective function can be expressed as:

Maximize Z = 80 x + (100-0.01y)y = 80x + 100y - 0.01y2

It may be noticed that this is a quadratic objective function and therefore, linear programming cannot be applied here.

2.      In linear programming problems coefficients in the objective function and the constraint   

Equations must be completely known and they should not change during the period of study, i.e., they should be known constants. In practical situations it may not be possible to state all coefficients in the objective function and constrains with certainty. Furthermore, these coefficients may actually be random variables each with certainty. Furthermore these coefficients may actually be random variables each with an underlying probability distraction for the values. Such problems cannot be solved using linear programming.

3.      Yet another important limitation of linear programming is that it may give fractional valued answers. For example, a solution may call for production of 96-7 units of product A and 63-3 units of product B. while rounding off to the nearest integer values may give reasonable accurate results in some cases. But the results may be poor in many other cases. For example assume due to limited space, a firm has to decide as to which of the ten sales depots each with its own requirement of space and estimated annual profit should be opened in order to maximize the annual profit. It is obvious that a fractional valued answer would mean nothing in this case where the decision can take only integer values say 0 and 1, the former signifying the decision to open and the later signifying decision not to open a particular sale depot.

4.      Linear programming will fail to give a solution if management has confecting multiple goals. In L.P model there is only one goal which is expressed in the objective function e.g., maximizing the value of the profit function or minimizing the cost function. One has to resort to goal programming (G.P) in solutions involving multiple goals.

5.      Other limitations of L.P includes:

-            Does not take into consideration the effect of time and uncertainty.

-          Parameters appearing in the model are assumed to be constants but in real-life situations they are frequently neither known nor constants.

6.      Linear programming problem requires that total measure of effectiveness and total resource usage resulting from the joint performance of the activities must equal the respective sums of these quantities resulting from each activity being performed individually.

In some situations it may not be true. For example consider a situation when a byproduct is produced with the scrap material from the primary product this material would still have to be procured if only one of the two products were produced. However the total material requirements if both products are produced are less than the sum of requirements if each were produced individually. It may not be possible to handle such situations with linear programming problems.

7.      Many real-world problems are so complex in terms of the number of variables and relationships constrained in them that they tax the capacity of even the largest computer. The approximations which must be made to reduce the problem to meaningful dimensions frequently place the final results in some doubt.

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