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/day)
$M_{1}$ $2$ $3$ $2$ $440$
$M_{2}$ $4$ - $3$ $470$
$M_{2}$ $2$ $5$ - $430$
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.

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Solution

Formulation of Linear Programming Model:
Step-1:
From the study of the situation find the key decision to be made.In this connection,looking for variables helps considerably.In the given situation key decision is to decide the number of units of products $1, 2$ and $3$ to be produced daily.
Step-2:
Assume symbols for variable quantities noticed in step-1.Let the number of units of products, $1, 2,$ and $3$ manufactured daily be $x_{1}, x_{2}$ and $x_{3}$ .
Step-3:Express the feasible alternatives mathematically in terms of variables.Feasible alternatives are those which are physically,economically and financially possible.In the given situation feasible alternatives are sets of values of $x_{1}, x_{2}$ and $x_{3}$ .
where $x_{1}, x_{2}, x_{3} \geq 0$ .
since negative production has no meaning and is not feasible.
Step-4:Mention the objective quantitatively and express it as a linear function of variables.In the present situation,objective is to maximize and profit.
i.e.,maximize $Z = 4 x_{1} + 3 x_{2} + 6 x_{3}$
Step-5:Put into words the influencing factors or constraints.These occur generally because of constraints on availability(resources) or requirements(demands).Express these constraints also as linear equations/inequalities in terms of variables.
Here,constraints are on the machine capacities and can be mathematically expressed as
$2 x_{1} + 3 x_{2} + 2 x_{3} \leq 440$
$4 x_{1} + 0. x_{2} + 3 x_{3} \leq 470$
$2 x_{1} + 5 x_{2} + 0. x_{3} \leq 430$
$∴,$ the complete mathematical(L.P) model for the problem can be written as
Maximize $Z = 4 x_{1} + 3 x_{2} + 6 x_{3}$
subject to constraints, $2 x_{1} + 3 x_{2} + 2 x_{3} \leq 440$
$4 x_{1} + 3 x_{3} \leq 470$
$2 x_{1} + 5 x_{2} \leq 430$

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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/day)
$M_{1}$	$2$	$3$	$2$	$440$
$M_{2}$	$4$	-	$3$	$470$
$M_{2}$	$2$	$5$	-	$430$