Definition of LInear Programing || steps of Linear Programing || Linear Programing in Quantitative techniques || linear Programing Example

 

Linear programming

Linear programming is a powerful quantitative techniques which design for solving the allocation problem. Linear programming indicates the planning of decision variables which are directly proportional to achieve the optimal solution by using limited resources and try to solved the problem.

 

Decision variables

Decision variables refers to the economic variables or physical quantities which are competing with one another for sharing limited resources. Decision variables indicate that what problem we solve like want to produce the a product, but problem is that what quantity we want to produce. Company need to take decision that what product they will produce with their limited resources.

Objective function

every company has their own object for the decision variables. They will get the answer that  what they want to accomplish from the decision variables. Some company try to increase the profit that means maximization of profit otherwise the try to decrease the cost that means minimization of cost and sometime want to decrease the time amount for performing a function. The objective function are express by Capital letter Z.

constraint

In the world there is a limitation of resources,  so company can’t use huge resources for their decision problem. Company have so many resources that are time, cost, manpower, production capacity, limited raw material. The resources are expressed by the equalities (=) and inequalities ( ≥ or ≤)

Non-negativity restriction

We can produce some amount or may be zero production that means no production but a company can’t produce negative production.

Steps for formulation the LP problem.

step 1 – identify the decision variables of interest to the decision maker and express them by x, y, z.

x= number of x product to be produce

y=number of y product to be produce.

Step 2- ascertain the objective of the decision maker, that they try to maximize the profit or minimize the cost or time.

Step3-  a company want to maximization the profit or minimization the cost or time. Every company have their own objective function that they want to achieve. If a company sell a product then their objective will be increase the profit margin. Then the function are express by the Z = x+y.

 Suppose, x company want to maximize profit where they get 2 dollar for  x and 3 for the y. so their objective function be z= 2x+3y

Step4 -  introduce constraint , constraint are the limited resources of a company. Which are express by  ≥ or ≤ sign. Limitation of manpower that a company have. So constraint, suppose  x and y will get 10 people for selling product. Then the constant will be x+y ≤ 10

Step5-     we already know the  non-negativity restriction which means that negative production can not happen.

 

 

Problem solution

Product mix problem

An animal feed company must produce 200 kg of mixture consisting of  ingredient  x and y daily. X cost Rs. 3 per kg and y cost  Rs. 8 per kg. Not more than 80kg of x can be used. And at  least 60 kg of y must be used. Find how much of each ingredient should be used if the  company want to minimize cost. 

 

 

Solution

 

Let x= kg of ingredient x can be produced

 Y= kg of ingredient y can be produced

Since the objective is to minimize the cost, the objective function is given by

Minimize Z= 3x+8y

Subject to constraint

x+y = 200  ( total mixture to be produced)

x ≤ 80 ( maximum use of x ingredient ) x can be used as equal 80 or less than 80

x ≥ 60 (minimum use of ingredient y) we can use  for the at least used ingredient

x ≥ 0   (non- negativity restriction)

In case of Purchasing problem solution we have to find out the cost of product to be produced and cost for purchasing that particular product .