Making expressions and inqquality constraints from "a story".

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November 12th 2006, 01:49 AM
anthmoo

Making expressions and inqquality constraints from "a story".

Ok this is the question.. (it requires use of the Simplex Algorithm) Please help! It's due tomorrow!

**Note: Only the thing i want to maximize and the constraints are needed! No need to trouble you with the whole simplex algorithm :p

Question:

Three processes, I, II and III and involved in the manufacture of three products, A, B and C. For each product, the manufacturing time (hours) and profit per item (£) are as shown.

Product---I---II---III---Profit
--A------1---2---3---120
--B------5---1---2---70
--C------4---4---1---100

The total manufacturing times on processes I, II, and III are 90, 35 and 60 hours respectively.

a) What mix of products yields the greatest profit?
b) What assumptions have you made in answering part (a)?
November 12th 2006, 05:34 AM
Soroban

Hello, anthmoo!

Quote:

Three processes, I, II and III and involved in the manufacture of three products, A, B and C.
For each product, the manufacturing time (hours) and profit per item (£) are as shown.

$\begin{array}{ccccc}\text{Product} & \text{I} & \text{II} & \text{III} & \text{Profit} \\ \hline \\<br /> A & 1 & 2 & 3 & 120 \\ B & 5 & 1 & 2 & 70 \\ C & 4 & 4 & 1 & 100 \\ \hline \\& 90 & 35 & 60 & \end{array}$

a) What mix of products yields the greatest profit?

The system of inequalities is: . $\begin{Bmatrix}A + 5B + 4C \:\leq \:90 \\ 2A + B + 4C\:\leq \:35 \\ 3A + 2B + C \:\leq \:60\end{Bmatrix}$

The profit function is: . $P \:=\:120A + 70B + 100C$

Quote:

b) What assumptions have you made in answering part (a)?

That $A,B,C\,\geq \,0$
. . Can't think of any others.
November 12th 2006, 05:39 AM
CaptainBlack

Quote:

Originally Posted by anthmoo

Ok this is the question.. (it requires use of the Simplex Algorithm) Please help! It's due tomorrow!

**Note: Only the thing i want to maximize and the constraints are needed! No need to trouble you with the whole simplex algorithm :p

Question:

Three processes, I, II and III and involved in the manufacture of three products, A, B and C. For each product, the manufacturing time (hours) and profit per item (£) are as shown.

Product---I---II---III---Profit
--A------1---2---3---120
--B------5---1---2---70
--C------4---4---1---100

The total manufacturing times on processes I, II, and III are 90, 35 and 60 hours respectively.

a) What mix of products yields the greatest profit?
b) What assumptions have you made in answering part (a)?

The constraints are the times required by the production mix on each
process, so if A, B and C denote the quanitties of the three products:

1A + 5B + 4C <= 90 ........ this is the time on process I and so has to be <=90
2A + 1B + 4C <= 35 ........ this is the time on process II and so has to be <=35
3A + 2B + 1C <= 60 ........ this is the time on process III and so has to be <=60

Also A>=0. B>=0, C>=0.

The objective to be maximised is profit:

P = 120A + 70B + 100C.
November 12th 2006, 06:16 AM
anthmoo

Thanks guys! Now it's time to go through the grueling simplex algorithm...

You're help is much appreciated and would an assumption (for part b) is that we assume that there are unlimited products?
November 12th 2006, 06:39 AM
CaptainBlack

Quote:

Originally Posted by anthmoo

Thanks guys! Now it's time to go through the grueling simplex algorithm...

You're help is much appreciated and would an assumption (for part b) is that we assume that there are unlimited products?

It looks to me as though you are to assume that the discrete number of
units of A, B and C can be treated as though they are continuous variables.

RonL