Question

A firm produces three products. These products are processed on three different machines. The time required to manufacture one unit of each of the three products and the daily capacity of the three machines are given in the table below:

Machine	Time per unit (in minutes) of product-1	Time per unit (in minutes) of product-2	Time per unit (in minutes) of product-3	Machine Capacity (minutes per day)
${{M}_{1}}$	2	3	2	440
${{M}_{2}}$	4	-	3	470
${{M}_{3}}$	2	5	-	430

It is required to determine the daily number of units to be manufactured for each product. The profit per unit for product $1,2$ and $3$ is Rs. $4$, Rs. $3$ and Rs. $6$ respectively. It is assumed that all the amounts produced are consumed in the market. Formulate the mathematical (L.P) model that will maximize the daily profit.

Answer 1

Hint: First, we will define what linear programming is then we will draw the given table and then we will assume variables the number of units of products, 1, 2, and 3 manufactured daily and after that, we will make equation for-profit and maximize it then we will define constraints for it by using the condition on machine capacity.

Complete step by step answer:
Now, let’s understand what linear programming is. Now linear programming is a method to achieve the best outcome, such as maximum profit or lowest cost, in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization).
Now, we are given the following table:

Machine	Time per unit (in minutes) of product-1	Time per unit (in minutes) of product-2	Time per unit (in minutes) of product-3	Machine Capacity (minutes per day)
${{M}_{1}}$	2	3	2	440
${{M}_{2}}$	4	-	3	470
${{M}_{3}}$	2	5	-	430

First, we will decide the number of units of products 1, 2, and 3 to be produced daily. Now, let the number of units of products, 1, 2 and 3 manufactured daily be ${{x}_{1}},{{x}_{2}}\text{ and }{{x}_{3}}$ .
Now, we know that the production cannot be negative but it can be zero, therefore the negative production has no meaning and hence, ${{x}_{1}},{{x}_{2}},{{x}_{3}}\ge 0$
Now, it is given that the profit per unit for product $1,2$ and $3$ is Rs. $4$, Rs. $3$ and Rs. $6$ , therefore the profit for total production ${{x}_{1}},{{x}_{2}}\text{ and }{{x}_{3}}$ will be: $4{{x}_{1}}+3{{x}_{2}}+6{{x}_{3}}$ . Now this is our daily profit and we are asked to maximise it, therefore we will represent it as a linear function that we will be maximising:
$Z=4{{x}_{1}}+3{{x}_{2}}+6{{x}_{3}}$
Now, we have to write the constraints which will occur because of lack of availability of resources or demands. Now we are given the machine capacities therefore:
Machine ${{M}_{1}}$ can run $440$ minutes per day and given that for product 1 it takes 2 minutes, for product 2 it takes 3 minutes and for product 3 it takes 2 minutes, therefore:
$2{{x}_{1}}+3{{x}_{2}}+2{{x}_{3}}\le 440$
Similarly for other machines, we will have:
\[\begin{align}
  & 4{{x}_{1}}+0.{{x}_{2}}+3{{x}_{3}}\le 470\text{ for }{{\text{M}}_{2}} \\
 & 2{{x}_{1}}+5{{x}_{2}}+0.{{x}_{3}}\le 430\text{ for }{{\text{M}}_{3}} \\
\end{align}\]

Hence, the complete mathematical linear progression model for the problem can be written as:
Maximize: $Z=4{{x}_{1}}+3{{x}_{2}}+6{{x}_{3}}$
Subjected to the constraints:
$\begin{align}
  & ~2{{x}_{1}}+3{{x}_{2}}+2{{x}_{3}}\le 440 \\
 & 4{{x}_{1}}+3{{x}_{3}}\le 470\text{ } \\
 & 2{{x}_{1}}+5{{x}_{2}}\le 430\text{ } \\
\end{align}$

Note:
For the constraints we will have to consider every condition lets say if you write: $~2{{x}_{1}}+3{{x}_{2}}+2{{x}_{3}}<440$ instead of $~2{{x}_{1}}+3{{x}_{2}}+2{{x}_{3}}\le 440$ then it will not consider the case where $~2{{x}_{1}}+3{{x}_{2}}+2{{x}_{3}}\text{ is equal to }440$.