Solving Max Profit w/ Simplex Method: Arthur Sanderson Ltd.

[sara_87]

Homework Statement

Arthur Sanderson Ltd. is a small paint factory which produces interior and exterior house paints for wholesale distribution. Two raw materials are used in the manufacture of the paints i.e. neutral base and pigment. The maximum availability of neutral base is 6 tons per day. The maximum availability of pigment is 8 tons per day. The raw material requirements for the paints are shown in Table 1.

Tons of Raw Material per Ton of Paint
Exterior Interior
Neutral Base 1 2
Pigment 2 1

Table 1

A market survey has established that the daily demand for interior cannot exceed that of exterior paint by more than 1 ton. The survey also shows that the maximum demand for interior paint is limited to 2 tons daily. The profits made by the company are £3000 on each ton of exterior paint and £2000 on each ton of interior paint. The company wishes to determine the number of tons of each paint it must produce each day in order to maximise its profit and satisfy the constraints of demand and raw materials usage.

Homework Equations

The Attempt at a Solution

ok this is kind of dum but i can't do the question ( i know it's an easy question don't make me feel worse!) because I'm not sure if i have the right linear programming problem or whatever

this is what i did:

3000x + 2000y = z
x + 2y =< 6
2x + y =< 8
y =< 2
x =< y+1

x, y>= o

tell me it's right! :(

 

[Mark44]

Your setup looks OK to me, but you still need to find a solution. You should mention that you wish to maximize the objective function z = 3000x + 2000y.

The simplest way to solve this is by graphing the inequalities, the boundaries of which will define your feasible region. This region has a number of corner points. Evaluate your objective function at each of the corner points. One of these will produce the maximum value of your objective function.

 

[sara_87]

thank you! i know how to do the rest :D

 

[sara_87]

i have another question and I'm just going to add it on to this one because I'm stuck on the same thing

Question:

The Crystal Glass Works produces glass high quality glass ornaments. The company owns 10 glass cutting machines, 9 glass polishing machines, one packing machine and operates a standard 40-hour working week. The company manufactures two products, ‘Sparkle Fountain’ and ‘Rainbow of Lights’. Each ‘Sparkle Fountain’ requires 0.4 hours on the glasscutter, 0.5 hours on the glass polisher and 2.4 minutes on the packing machine. It can either be sold for £10 or be further processed to produce a ‘Rainbow of Lights’. The additional processing times for each ‘Rainbow of Lights’ are 5 hours on the glasscutter, 2 hours on the glass polisher and 48 minutes on the packing machine. Each ‘Rainbow of Lights’ can be sold for £90.

firstly i have to find the programming model thing to find maximum profits...

ok this is what i did:

maximise:

and then i get stuck!

i'm going to pick sparkle fountain to be x
and rainbow of lights to be y

so we need to maximise: 10x + 90y = z

then... i know that on the machines they can't take longer than 40 hours (i think) so:

lets just say

10a =< 40
9b =< 40
c =< 40

(where a is the glass cutting machine, b the polishing, and c the paking)

and also:

0.4 x + 5.4 y >= i dunno
0.5 x + 2.5 y >= i dunno
2.4 x + 50.4 >= i dunno

i don't know what to do! :(
i know that what I'm doing couldn't be more wrong! :yuck: and i know the rest just the begining, which is the main bit, that i keep dying on

 

[Mark44]

i have another question and I'm just going to add it on to this one because I'm stuck on the same thing

Question:

The Crystal Glass Works produces glass high quality glass ornaments. The company owns 10 glass cutting machines, 9 glass polishing machines, one packing machine and operates a standard 40-hour working week. The company manufactures two products, ‘Sparkle Fountain’ and ‘Rainbow of Lights’. Each ‘Sparkle Fountain’ requires 0.4 hours on the glasscutter, 0.5 hours on the glass polisher and 2.4 minutes on the packing machine. It can either be sold for £10 or be further processed to produce a ‘Rainbow of Lights’. The additional processing times for each ‘Rainbow of Lights’ are 5 hours on the glasscutter, 2 hours on the glass polisher and 48 minutes on the packing machine. Each ‘Rainbow of Lights’ can be sold for £90.

firstly i have to find the programming model thing to find maximum profits...

ok this is what i did:

maximise:

and then i get stuck!

i'm going to pick sparkle fountain to be x
and rainbow of lights to be y

so we need to maximise: 10x + 90y = z

then... i know that on the machines they can't take longer than 40 hours (i think) so:

For the stuff below, you should be more specific about what a, b, and c represent. I am very leery of these three inequalities. Since your objective function involves only x and y, your constraints likewise should involve only x and y.

lets just say

10a =< 40
9b =< 40
c =< 40

(where a is the glass cutting machine, b the polishing, and c the paking)

and also:

The inequalities below should probably go the other way. Before doing that, see if you can match up a phrase or sentence in the problem statement to each inequality, and convince yourself that you are translating correctly.

0.4 x + 5.4 y >= i dunno
0.5 x + 2.5 y >= i dunno
2.4 x + 50.4 >= i dunno

i don't know what to do! :(
i know that what I'm doing couldn't be more wrong! :yuck: and i know the rest just the begining, which is the main bit, that i keep dying on

 

[sara_87]

ok thanks
then this (i have a huge urge to say 'i think' but i won't):

maximise 10x + 90y = z
0.4 x + 5.4 y =< 40
0.5 x + 2.5 y =< 40
2.4 x + 50.4 =< 40


then they say that there are only 10 glass cutting machines how would i add this constraint it can't be that x + y <= 10 because that wouldn't make sense
i'm not sure how i would add this...

 

[Mark44]

Look at the number of hours each product takes on the glass cutter, the glass polisher, and the packing machine.
If x is the number of sparkle fountains produced in a week and y is the number of rainbow thingies produced per week, then you know that

.4x <= 40 and 5y <= 40 (cutting)

Do the same for polishing and packing.

These inequalities should replace the ones you have in post #6. There are also the nonnegativity constraints - you can't make a negative number of either product.

 

[sara_87]

ok ok thanx

 

[sara_87]

i lied i don't get it!
it was really late the other day and my brain decided to sleep so i told myself i'd think about it later on, and so now is later on and my brain is awake... but it's not functioning

this is what I've got:
maximise:
10x + 90y = z
0.4x<= 40
5y <=40
0.5x <= 40
2y <= 40
2.4x <=40
48y <= 40

x, y >= 0

how would i include the fact that there are only 10 glass cutting machines, 9 polishing ones, and 1 packing one?

 

[Mark44]

OK, I've thought about it a bit more, also. Since there are 10 glass cutting machines, there are 40*10 hours available on them. And similar logic for the other two machines.

Maximize z = 10x + 90y where x = no. of Sparkle Fountains per wk, and y = no. of Rainbow of Lights per wk
Subject to:
.4x + 5.4y <= 400 (40 hr. * 10 glass cutting machines)
.5x + 2.5y <= 360 (40 hr. * 9 glass polishing machines)
.04x + .84y <= 40 (40 hr. * 1 packing machine)
x, y >= 0
The units are hours in the first three inequalities.
I'm assuming that when the problem says "The additional processing times for each ‘Rainbow of Lights’ are ...", you need to add on the times for producing a Sparkle thingie.

You can do a sanity check on what I have by checking two extreme cases: all sparkle fountains (set y = 0) and all rainbow of lights (set x = 0). For each of these cases find the largest value of x or y that satisfies all three inequalities, and evaluate your objective function. It's likely that some pair of values for x and y with neither being zero maximizes the objective function.

 

[sara_87]

YES! i see! this is what i was thinking! except i didn't know how to come up with it like u did and i didn't know how to say it...

anyway thank u so much!

 

[Mark44]

Here's a tip: Instead of writing this
maximise:
10x + 90y = z
0.4x<= 40
5y <=40
0.5x <= 40
2y <= 40
2.4x <=40
48y <= 40
add some explanatory text the way I did. I.e., text that explains what the variables represent and text that explains what each inequality represents. It'll really help you understand what you're trying to represent in symbols.

If your first attempt is incorrect (as mine was) having some explanatory text can help you understand why it is incorrect and maybe guide you toward a new equation/inequality.

 

[sara_87]

thanx i will do :)