Awkward Linear Programming Problem

Question

This is a linear programming problem I was given in my semester examinations. The question is attached as an image. Given its size , I couldn't type it out.

LPP problem

So basically we have to come up with all the inequalities for this problem given the conditions. Few of them are easy to note : Let no. of products for models I,II and III be x,y and z respectively. Then ,

$60x + 40y + 100z$ is the objective function

Given the minimum demand condition, we can say :

$x\ge 500$

$y\ge 500$

$z\ge 375$

Given the conditions on raw materials we can compute :

$2x + 3y + 5z \le 4000$ and

$4x + 2y + 7z \le 6000$

From the labour statements given , we can say that if $l$ is the labour required to produce a unit of model I , then for each unit of models II and III , $\frac{l}2$ and $\frac{l}3$ labour are required respectively.

The total labour capacity of the factory is $2500$ units of model I , so $2500l$ is the total labour capacity of the factory.

So we get :

$6x + 3y + 2z \le 15000$

Finally, from the statement that

$x:y:z = 3:2:5$ , we can deduce

$x = 0.6z \tag{1}$

$y = 0.4z \tag{2}$

and $z = z$

Now my doubt is :

From the last deduction, should we go back and plug in $(1)$ and $(2)$ in all the inequalities we derived earlier to make it a single variable problem ?

Or should we let it remain as it is ?

It is quite a herculean task to review the answer scripts in college , so I just want to verify if reducing the entire problem to a single variable is wrong from any frame of reference.

Answer 1

You are just asked to formulate a linear programming problem. You are not required to solve it by hand or reduce the number of variables though you are able to.

Besides the number of variables, another factor to consider is the interpretability. Sometimes it is good to have more variables, say in variables $x,y,z$ so that we can tell which mathematical constraints refer to which requirement.