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Actual Question Statement:

In a factory there are 100 skilled and 125 unskilled workers. Two different goods A and B are produced. 2 Skilled and 5 unskilled workers can produce one unit of A per day using 10kg raw material and 175 units of electricity. 5 skilled and 3 unskilled workers can produce one unit of B using 13 kg raw materials and 130 units of electricity. The available quantities of raw materials and electricity per day are 500kg and 3000 units respectively. The profit per unit of A and B are 250 and 230 respectively. If total man power are to be used formulate a linear programming problem to find the maximum profit earned per day.

What I do not understand is, how to make use of the statement where it is said "Total man power are to be used"

I cannot come up with an equation for that, my other equations I've come up with are listed here:

Skilled Worker Availability: 2x + 5y ≤ 100

Unskilled Worker Availability: 5x + 3y ≤ 125

Raw Material Availability: 10x + 13y ≤ 500

Electricity Availability: 175x + 130y ≤ 3000

Non-negativity: x ≥ 0 and y ≥ 0

I think I'll need one more equation for the Total Manpower Utilization Part of it, but I can't figure it out. In the book the equations given are:

2x + 4y = 75

5x + 3y = 100

10x + 13y ≤ 500

175x + 130y ≤ 3000

x ≥ 0 and y ≥ 0

Are the first two equations for total manpower usage? I am very confused as to how they get it, because there's no explanation given for it in the book.

SubbSE
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  • The constraint means that you must use everybody. – lulu Dec 14 '23 at 12:14
  • Yes but I am unable to come up with an equation for it which is why I posted this question, because the equation that I came up with, adding Skilled Worker Availability and Unskilled Worker Availability, gives me a different equation as compared to one stated in the book, so I am confused. – SubbSE Dec 14 '23 at 12:22
  • All of the resources used by the manpower must be used. Those equations describe the total electricity and raw materials used (i.e. all of it). – Paul Dec 14 '23 at 12:23
  • Should say: I don't understand how "profit" is to be evaluated. I don't see any information regarding costs or revenue, just lists of resources What exactly tells you if one use of resources is more profitable than another? – lulu Dec 14 '23 at 12:24
  • Thanks for pointing out, I missed out that part of the question, I'll add it in right now! – SubbSE Dec 14 '23 at 12:30
  • I find it a bit odd, that it's mentioned that all workers should be used... So I think there should be an equation to define that as well? – SubbSE Dec 14 '23 at 12:32
  • This is all very vague and confusing. Using everybody would mean that $2x+5y=100, 5x+3y=125$ but that's impossible (no solution in integers). And even if it did have an integer solution, it would determine the allocation uniquely. So I doubt there is any real meaning behind "total man power are to be used" – lulu Dec 14 '23 at 12:33
  • Should add: the equations you say the official source supplies don't have an integer solution either, not that I can see where $75, 100$ come from in the problem. – lulu Dec 14 '23 at 12:36
  • Exactly, me neither, that's why I'm extra confused as to how and why they were derived in the first place.. What I had in mind was:

    7x + 8y = 225 to include all the workers, but that's just there as a part of the other two equations so I dunno what to make of it.

     – SubbSE Dec 14 '23 at 12:39
  • I'd ignore it. Solve the problem without it. You could modify the objective function by adding the goal that you want to use as many of the people as you can...but in that case, you'd have to be clear about how you are weighing the two goals. That is, how much profit would you give up to use one extra person, and such. – lulu Dec 14 '23 at 12:41
  • Thanks for the help, appreciate it! – SubbSE Dec 14 '23 at 12:45

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