# Linear Programming Dealing with Defects?

## Homework Statement

The problem goes like this:

A man is selling two types of lightbulbs. Lightbulb X costs $1.70, but 9% are defective, and so, cannot be sold. Lightbulb Y costs$0.90, but 12% are defective. The man wants to purchase at least 10 000 lightbulbs of each type, but he wants at least 10% of the total to be lightbulb X and at most 70% of total of lightbulb Y. Minimize the total cost.

Now my question is what do I do with the defect rates? I believe they are constraints, but have no clue how... And please, do not give me the answer, i would like to learn.

I like Pi

## The Attempt at a Solution

I have:

Let x = # of X lightbulbs
Let y = # of Y lightbulbs

Cmin = 1.7x + .9y (objective)

constraints:
x, y ≥ 10 000
x ≥ .10(x+y) ... x ≥ (.10/.90)y
y ≤ .70(x+y) ... y ≤ (.70/.30)x

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

The problem goes like this:

A man is selling two types of lightbulbs. Lightbulb X costs $1.70, but 9% are defective, and so, cannot be sold. Lightbulb Y costs$0.90, but 12% are defective. The man wants to purchase at least 10 000 lightbulbs of each type, but he wants at least 10% of the total to be lightbulb X and at most 70% of total of lightbulb Y. Minimize the total cost.

Now my question is what do I do with the defect rates? I believe they are constraints, but have no clue how... And please, do not give me the answer, i would like to learn.

I like Pi

## The Attempt at a Solution

I have:

Let x = # of X lightbulbs
Let y = # of Y lightbulbs

Cmin = 1.7x + .9y (objective)

constraints:
x, y ≥ 10 000
x ≥ .10(x+y) ... x ≥ (.10/.90)y
y ≤ .70(x+y) ... y ≤ (.70/.30)x

What you have so far is perfectly OK. The question says nothing at all about what to do with the defects. If your statement of the actual problem is accurate, the defect rates make no difference; they could be 100% without affecting anything. Are you sure you copied the problem correctly?

It would make more sense if they also specified a selling price for good type X and Y bulbs and then asked for a profit-maximizing solution, for example. Some other possibilities are: (i) require that we have no more that some specified percentage of defectives in the total bulb store; or (ii) have a disposal cost for each defective, and add that cost to the purchase cost, then minimize the total.

RGV

Last edited:
What you have so far is perfectly OK. The question says nothing at all about what to do with the defects. If your statement of the actual problem is accurate, the defect rates make no difference; they could be 100% without affecting anything. Are you sure you copied the problem correctly?

It would make more sense if they also specified a selling price for good type X and Y bulbs and then asked for a profit-maximizing solution, for example. Some other possibilities are: (i) require that we have no more that some specified percentage of defectives in the total bulb store; or (ii) have a disposal cost for each defective, and add that cost to the purchase cost, then minimize the total.

RGV

That is exactly how the question is written. However, there is a subquestion saying that if the demand for lightbulb X rises by 5% but total demand does not, how much type X lightbulbs and type Y lightbulbs should be ordered to minimize cost. What are the changes to the minimum solution?

Would that require the defective rates? My professor said that they are used in the constraints portion of the question...

Thanks,
I like Pi

Ray Vickson
Homework Helper
Dearly Missed
That is exactly how the question is written. However, there is a subquestion saying that if the demand for lightbulb X rises by 5% but total demand does not, how much type X lightbulbs and type Y lightbulbs should be ordered to minimize cost. What are the changes to the minimum solution?

Would that require the defective rates? My professor said that they are used in the constraints portion of the question...

Thanks,
I like Pi

I strongly disagree, but maybe some of the words used were meant differently; for example, maybe the problem really meant that he wants to sell at least 10,000 good bulbs of each type rather than purchase at least 10,000. The way it is worded now, he could purchase 100% defective bulbs and sell nothing at all.

RGV

I strongly disagree, but maybe some of the words used were meant differently; for example, maybe the problem really meant that he wants to sell at least 10,000 good bulbs of each type rather than purchase at least 10,000. The way it is worded now, he could purchase 100% defective bulbs and sell nothing at all.

RGV

That's exactly what I was thinking... So, in this case, because we are dealing with just cost and not a set amount needing to be sold, the defective rate doesn't apply?

Ray Vickson
Homework Helper
Dearly Missed
That's exactly what I was thinking... So, in this case, because we are dealing with just cost and not a set amount needing to be sold, the defective rate doesn't apply?

I thought that's what I said.

RGV

I thought that's what I said.

RGV

Yes, I am just clarifying.

Now what would i do if the demand for lightbulb X increases by 5%? How would i incorporate that into my linear programming model? Would that mean that I he wants to buy 5% more type X lightbulbs of what he's bought in the previous answers (10,000)?