Expert: Paul Klarreich - 8/28/2009

Question
Peach Inc. is producing very popular MP3 players. Currently, the two models produced are the xPod mini and the xPod maxi.

All xPods are highly profitable, with profit contributions of £20 for the xPod mini and £30 for the xPod maxi. Estimated demand for next month are 40,000 and 60,000 for xPod mini and xPod maxi, respectively. Both players are produced in the same factory. The production process consists of two main stages, both are required for both players. Stage 1 can produce a total of 100,000 per month xPods of any type (product mix is not relevant).

The output of Stage 2 depends on the type of  xPod produced. It can finish an xPod mini every 9 seconds, or an xPod maxi every 18 seconds, and can be operated for 350 hours a month (i.e., it can process up to 70,000 xPod maxi, or up to 140,000 xPod mini, or 35,000 maxi and 70,000 mini if capacity is distributed equally between products etc.). Management has furthermore decided, that production of the xPod maxi should never exceed twice the amount of the xPod mini.

Write down the constraints and objective function of the linear programming model.

This is what I have done but I am not quite sure.
Objective function: 20X + 30Y
Constraint: X =< 40000
Y =< 60000
X =< 100000
Y =< 100000
0.0025X =< 350 -> X =< 140000
0.005Y =< 350 -> Y =< 70000
0.0025X + 0.005Y =< 350
Y =< 2X
X, Y >= 0

Answer
Questioner: Thao
Country: Vietnam
Category: Advanced Math
Private: No
Subject: Linear programming
Question: Peach Inc. is producing very popular MP3 players. Currently, the two models produced are the xPod mini and the xPod maxi.

All xPods are highly profitable, with profit contributions of £20 for the xPod mini and £30 for the xPod maxi. Estimated demand for next month are 40,000 and 60,000 for xPod mini and xPod maxi, respectively. Both players are produced in the same factory. The production process consists of two main stages, both are required for both players. Stage 1 can produce a total of 100,000 per month xPods of any type (product mix is not relevant).

The output of Stage 2 depends on the type of  xPod produced. It can finish an xPod mini every 9 seconds, or an xPod maxi every 18 seconds, and can be operated for 350 hours a month (i.e., it can process up to 70,000 xPod maxi, or up to 140,000 xPod mini, or 35,000 maxi and 70,000 mini if capacity is distributed equally between products etc.). Management has furthermore decided, that production of the xPod maxi should never exceed twice the amount of the xPod mini.

Write down the constraints and objective function of the linear programming model.

This is what I have done but I am not quite sure.

>> For starters, let's call the xpod maxi the Ypod.  then:

x = number of xpods (mini) produced.
y = number of ypods (maxi) produced.



Objective function: 20X + 30Y    << Looks OKJ
Constraint: X =< 40000
Y =< 60000
X =< 100000   << I think you want to write X + Y <= 100000
Y =< 100000
0.0025X =< 350 -> X =< 140000
0.005Y =< 350 -> Y =< 70000
0.0025X + 0.005Y =< 350   << This is the key criteria, I think.

<< and you want to say something like this:

9 (seconds) times x + 18 (seconds) times y = 350 * 3600 seconds available.

>> Yes, that probably comes out to the same thing, but it isn't very obvious.

Y =< 2X
X, Y >= 0

I don't see anything really wrong with what you did.