Numerical techniques-Linear programming(graphical method)

[Monsterboy]

Homework Statement

Two grades of paper,grade A and grade B,are produced by a paper mill.It takes 12 minutes and 24 minutes to produce a ton of these two grades of papers respectively,with corresponding profits of $20 and $50. The mill works for 160 hours in a week and has the capacity of producing 400 tonnes of grade A and 300 tonnes of grade B paper.Find the quantity of each of these grades of paper to be produced so that the profit is maximum.

Homework Equations

See below

The Attempt at a Solution

well ,we need to find the max profit
assume 'Z' as profit and the equation for Z is Z = 20A + 50B (Objective function)
where A,B are number of grade A and grade B papers
we have to maximize Z right??

i am not sure about how to put (inequality/equality) constraints.
(400x10^3)A +(300x10^3)B <=160 ??

i don't how to relate the mass(tonnes) of these papers with time(hrs/min) in the constraints

 

[Mark44]

Homework Statement

Two grades of paper,grade A and grade B,are produced by a paper mill.It takes 12 minutes and 24 minutes to produce a ton of these two grades of papers respectively,with corresponding profits of $20 and $50. The mill works for 160 hours in a week and has the capacity of producing 400 tonnes of grade A and 300 tonnes of grade B paper.Find the quantity of each of these grades of paper to be produced so that the profit is maximum.

Homework Equations

See below

The Attempt at a Solution

well ,we need to find the max profit
assume 'Z' as profit and the equation for Z is Z = 20A + 50B (Objective function)
where A,B are number of grade A and grade B papers
we have to maximize Z right??

Right.

i am not sure about how to put (inequality/equality) constraints.
(400x10^3)A +(300x10^3)B <=160 ??

Not right.
How long, in hours, does it take to make a tonne of grade A paper? How long to make a tonne of grade B paper? The mill operates only 160 hours a week, so that is one constraint on how much paper can be made.

The other constraints are that the mill can make at most 400 tonnes of grade A paper and 300 tonnes of grade B paper. These are two other constraints.

Also, the amount of paper of either grade must be nonnegative.

i don't how to relate the mass(tonnes) of these papers with time(hrs/min) in the constraints

As an aside, "ton" and "tonne" are not the same, and you have used both. The first is 2000 lb. while the second is 1000 kg, or 2200 lb.

 

[Joffan]

As an aside on your aside Mark44, "ton" is even more complicated. A ton can be either 2000lb or 2240lb, depending on where you are - possibly other options that I don't know about too. A tonne is exactly 1000kg.

 

[Ray Vickson]

Homework Statement

Two grades of paper,grade A and grade B,are produced by a paper mill.It takes 12 minutes and 24 minutes to produce a ton of these two grades of papers respectively,with corresponding profits of $20 and $50. The mill works for 160 hours in a week and has the capacity of producing 400 tonnes of grade A and 300 tonnes of grade B paper.Find the quantity of each of these grades of paper to be produced so that the profit is maximum.

Homework Equations

See below

The Attempt at a Solution

well ,we need to find the max profit
assume 'Z' as profit and the equation for Z is Z = 20A + 50B (Objective function)
where A,B are number of grade A and grade B papers
we have to maximize Z right??

i am not sure about how to put (inequality/equality) constraints.
(400x10^3)A +(300x10^3)B <=160 ??

i don't how to relate the mass(tonnes) of these papers with time(hrs/min) in the constraints

Always try to approach such problems systematically: write down the definitions of your decision variables (including proper units); write out the constraints carefully---don't write down random formulas, think things through systematically; and write out the objective to be maximized or minimized. Always try to avoid (if you can) mixtures of numbers of very different magnitudes in left-hand-side coefficients when modelling (especially in large models), because the problem is going to be solved on a computer using finite precision arithmetic, and you want to reduce the effects of roundoff effects (which can be VERY serious in large linear models if you are not careful).

So, first of all: choose sensible units, such as TA = tons of grade A paper to produce per week and TB = tons of grade B per week. That makes your constraint nicer: weekly time used = 12TA + 24TB (minutes). How many minutes per week are available? (You need to be careful: if you use 'minutes' on the left you need to use 'minutes' on the right.) What are the limits on TA and TB? What is the profit ($/week) for given values of TA and TB?

RGV

 

[Joffan]

It might be simpler to express the time for production in hours: 0.2 hours for a tonne of A and 0.4 hours for a tonne of B.

The problem doesn't state explicitly that the mill can only produce one product at a time but I'm assuming that must be so, otherwise it is not an interesting problem.