More Linear Programming - help

[flyingpig]

Homework Statement

A small cargo ship leaving Hawaii for London has three holds for cargo. The forward hold has a capacity of 140 tons, and volume of 6,000 cubic feet; the central hold has a capacity of 800 tons, and a volume of 15,000 cubic feet; the aft hold has a capacity of 450tons and a volume of 9,000 cubic feet. Only the forward and aft holds have refrigerating systems. pa Assuming the papayas must be refrigerated. Set up, but do not solve, a linear programming model for selecting a cargo of sugar, papayas, and coconuts that maximizes profit. Use the data below

[PLAIN]http://img38.imageshack.us/img38/5041/unleddmj.jpg

The Attempt at a Solution

I don't have a clue...

 

[lanedance]

call each of the cargoes s,p and c.

clearly we have s,p,c>=0

can you write an equation for the profit, something like:
P(s,p,c) = e.s+f.p+g.c

you want to maximise the profit relative to the constraints - can you write down what the constraints are?

 

[flyingpig]

The dimensions and capacity is killing me. They are messing me up

 

[flyingpig]

I need to finish this (and my other problems) by Monday (tomorrow if I sleep now)

 

[Ray Vickson]

The dimensions and capacity is killing me. They are messing me up

Ask yourself: what would I need to know if I were planning the loading of the cargo? Would it be enough to know things like how many tons of papaya to put in the forward hold, etc.? If you did know all those things, could you check that the capacities were respected? Could you compute the profit?

RGV

 

[Mark44]

call each of the cargoes s,p and c.

clearly we have s,p,c>=0

More specifically, s, p, and c represent the weight, in tons of sugar, papaya, and coconuts, respectively.

The constraints are going to involve the amount (in tons) of each commodity, and the volume of each quantity. Note that the table gives the volume in units of cu. ft. per ton.

 

[flyingpig]

Here is my profit function

P = 50s + 120p + 65c

I don't know how to write the constraints!

 

[flyingpig]

I just tried writing my constraints

For forward hold

140 = 120s + 1200p + 600

Is that right? If so does that mean i have a lot of constraints?

 

[Ray Vickson]

I just tried writing my constraints

For forward hold

140 = 120s + 1200p + 600

Is that right? If so does that mean i have a lot of constraints?

Not quite right: you need 600c instead of 600. Also, you are computing volume, not weight, so why on Earth would you have 140? Why would you assume the hold is full? And you need a way to distinguish which hold the products are going into. So, yes: you will have a lot of variables and a lot of constraints. I get 9 variables and 9 constraints.

RGV

 

[flyingpig]

Not quite right: you need 600c instead of 600. Also, you are computing volume, not weight, so why on Earth would you have 140? Why would you assume the hold is full? And you need a way to distinguish which hold the products are going into. So, yes: you will have a lot of variables and a lot of constraints. I get 9 variables and 9 constraints.

RGV

Was my profit function correct though?

 

[flyingpig]

Volume:

forward - 6000 ≥ 120s + 1200p + 600c

central - 15000 ≥ 120s + 600c (I am assuming we don't even want to put papayas in the central one?)

aft - 9000 ≥120s + 1200p + 600c

Capacity

forward - 140 ≥ 1000s + 350p + 900c

central - 800 ≥ 1000s + 900c (same argument as before, no papayas)

aft = 450 ≥ 1000s + 350p + 900c

I have 6 constraints...I am missing three

≤ ≥

 

[flyingpig]

omg i forgot the constraints on how big and small can they hold! give me a minute!

 

[flyingpig]

Here is my new and fixed LOP

Max $$P = 50s + 120p + 65c$$

s.t.

forward

$$140 \geq 350p + 1000s + 900c$$
$$6000 \geq 1200p + 120s + 600c$$

central

$$800 \geq 1000s + 900c$$
$$15000 \geq 120s + 600c$$

aft

$$450 \geq 1000s + 900c + 350p$$
$$9000 \geq 120s + 600c + 1200p$$

$$p \leq 0$$
$$s \leq 0$$
$$c \leq 0$$

Again there is no p in the central hold because we don't even want any in there

 

[flyingpig]

Somebody please applaud me.

 

[Ray Vickson]

Somebody please applaud me.

So, you want to ship negative amounts? That's what your p <= 0, s <= 0, c <= 0 are saying. I replaced your p<= 0, etc., by p >= 0, etc., and solved your formulation _exactly_ as you wrote it. The solution was to ship p = 0.4 tons of papaya, and no sugar or coconuts, for a total profit of $48. On the other hand, my formulation gives a very different policy and earns a profit of $12,500. So, no, I don't think applause is in order.

I can't help you any more on this question without violating the rules of this forum

RGV

 

[flyingpig]

nono nonononon

$$p \geq 0$$
$$s \geq 0$$
$$c \geq 0$$

I was reading this all in tex and i took g for l for \leq instead of
\geq

 

[flyingpig]

No I just have to set up (but do not solve) the problem. However that be an bonus activity for tomorrow, but not today because I have three assignments due

 

[Ray Vickson]

nono nonononon

$$p \geq 0$$
$$s \geq 0$$
$$c \geq 0$$

I was reading this all in tex and i took g for l for \leq instead of
\geq

That's why this forum has a "preview post" button next to the "submit reply" button. You preview first, correct errors, then submit. You can preview again and again and again until you have it right.

RGV