### Linear programming:

The technique of linear programming was formulated by a Russian mathematician L.V. Kantorovich. But the present version of simplex method was developed by Geoge B. Dentzig in 1947. Linear programming (LP) is an important technique of operations research developed for optimum utilization of resources.

According to famous Economist Robbins, the resources (land, labour, capital, materials, machines, etc.) are always limited. But each resource have various alternative uses. The problem before any manager is to select only those alternatives which can maximize the profit or minimize the cost of production. The linear programming technique is used for selecting the best possible strategy from a number of alternatives.

Linear programming consists of two words:

‘Linear and programming’. The world linear stand for indicating the rela­tionships between different variables of degree one whereas another word programming means planning and refers to the process of selecting best course of action from various alterna­tives.

Thus, linear programming is a mathematical technique for allocating limited resources is optimum manner. In the words of William M. Fox, “Linear programming is a planning technique that permits some objective function to be minimized or maximized within the framework of given situational restrictions.”

#### Chief characteristics:

All linear programming problems must have following five characteristics:

(a) Objective function:

There must be clearly defined objec­tive which can be stated in quantitative way. In business problems the objective is generally profit maximization or cost minimization.

(b) Constraints:

All constraints (limitations) regarding resources should be fully spelt out in mathematical form.

(c) Non-negativity:

The value of variables must be zero or positive and not negative. For example, in the case of production, the manager can decide about any particular product number in positive or minimum zero, not the negative.

(d) Linearity:

The relationships between variables must be linear. Linear means proportional relationship between two ‘or more variable, i.e., the degree of variables should be maximum one.

(e) Finiteness:

The number of inputs and outputs need to be finite. In the case of infinite factors, to compute feasible solution is not possible.

#### Assumptions:

(i) There are a number of constraints or restrictions- expressible in quantitative terms.

(ii) The prices of input and output both are constant.

(iii) The relationship between objective function and constraints are linear.

(iv) The objective function is to be optimized i.e., profit maximization or cost minimization.

LP has been considered an important tool due to following reasons:

1. LP makes logical thinking and provides better insight into business problems.

2. Manager can select the best solution with the help of LP by evaluating the cost and profit of various alternatives.

3. LP provides an information base for optimum alloca­tion of scarce resources.

4. LP assists in making adjustments according to changing conditions.

5. LP helps in solving multi-dimensional problems.

LP approach suffers from the following limitations also:

1. This technique could not solve the problems in which variables cannot be stated quantitatively.

2. In some cases, the results of LP give a confusing and misleading picture. For example, the result of this technique is for the purchase of 1.6 machines.

It is very difficult to decide whether to purchase one or two- machine because machine can be purchased in whole.

3. LP technique cannot solve the business problems of non-linear nature.

4. The factor of uncertainty is not considered in this technique.

5. This technique is highly mathematical and complicated.

6. If the numbers of variables or contrains involved in LP problems are quite large, then using costly electronic computers become essential, which can be operated, only by trained personel.

7. Under this technique to explain clearly the objective function is difficult.

#### Managerial uses and applications:

LP technique is applied to a wide variety of problems listed below:

(a) Optimizing the product mix when the production line works under certain specification;

(b) Securing least cost combination of inputs;

(c) Selecting the location of Plant;

(d) Deciding the transportation route;

(e) Utilizing the storage and distribution centres;

(f) Proper production scheduling and inventory control;

(g) Solving the blending problems;

(h) Minimizing the raw materials waste;

(i) Assigning job to specialized personnel.

The fundamental characteristic in all such cases is to find optimum combination of factors after evaluating known constraints. LP provides solution to business managers by understanding the complex problems in clear and sound way.

The basic problem before any manager is to decide the manner in which limited resources can be used for profit maximization and cost minimization. This needs best allocation of limited resources—for this purpose linear programming can be used advantageously.

#### Graphical method:

The business problems involving two variables can be easily solved by drawing the graph for various constraints. Following are the steps in graphical solution of linear programming problem (LPP):

1. Formulate LPP by writing the objective function (generally maximize profit) and the constraints.