Is there a more accurate way to estimate sales quantities?

In summary, the conversation discusses a homework problem involving estimating the number of matchboxes sold based on price and a maximum error tolerance of 1,000 units. The solution involves finding the price range for a given quantity sold and calculating the percentage error in the estimate. The quality of the estimate depends on its purpose and alternatives.
  • #1
help
14
0
Homework Statement
homework. Am I thinking right? modular inequality
Relevant Equations
The number of matchboxes that will be sold in a month can be estimated according to
the price p of each box. The estimated quantity of boxes sold is related
with the price p according to the equation

q = 16000 - 2500p
when the price is between 2 and 6.

The company cannot sell with a price below 2 and greater than or equal to 6.
Note that q is just an estimate of the quantity sold. In normal scenarios, this
estimate has a maximum error of 1,000 units in relation to the actual quantity determined
r at the end of the month.
Determine the price range that should be used for the number of boxes sold
is equal to 7,250 units, within the maximum error margin of the survey.



So, I started this problem and I would like to be sure of what I did.

attached resolution attempt
attached resolution attempt
 

Attachments

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  • #2
help said:
Homework Statement:: homework. Am I thinking right? modular inequality
Relevant Equations:: The number of matchboxes that will be sold in a month can be estimated according to
the price p of each box. The estimated quantity of boxes sold is related with the price p according to the equation

q = 16000 - 2500p
when the price is between 2 and 6.

The company cannot sell with a price below 2 and greater than or equal to 6.
Note that q is just an estimate of the quantity sold. In normal scenarios, this estimate has a maximum error of 1,000 units in relation to the actual quantity determined r at the end of the month. Determine the price range that should be used for the number of boxes sold is equal to 7,250 units, within the maximum error margin of the survey.

So, I started this problem and I would like to be sure of what I did.

attached resolution attempt

attached resolution attempt
Hello @help . (Maybe use a better name)
:welcome:

Your solution:
Optimization_01-PF987897.jpg


It looks correct to me.
 
  • #3
SammyS said:
It looks correct to me.

Hi, thanks for replying.
Letter Bb)
Over 6 months, several prices were tried. The following actual sales figures are observed, according to
with each price charged:
1587736308426.png

The company believes that a good estimate q of the quantity sold can be given by

Give the lowest value you can for E so that https://mathhelpforum.com/attachments/1587578280478-png.40117/ in all observations made

So, I would like to be sure of what I did.

pqrE
335003250250
3,234003570170
3,532503100150
430002890110
4,527502550200
5250025000
 
  • #4
help said:
The company believes that a good estimate q of the quantity sold can be given by
Something missing here, like a new formula for q?
help said:
Give the lowest value you can for E so that
So that what?
 
  • #5
Sorry!
The company believes that a good estimate q of the quantity sold can be given by
q = 5000 - 500p.

Give the lowest value you can for E so that |q - r | ≤ E in all observations made.
 
  • #6
help said:
Sorry!
The company believes that a good estimate q of the quantity sold can be given by
q = 5000 - 500p.

Give the lowest value you can for E so that |q - r | ≤ E in all observations made.
Then your work in post #3 looks correct, but what is your answer for the lowest value of E meeting the condition?
 
  • #7
the lowest possible value for E would be zero correct?

to end the question has an item c...

c)
Calculate the percentage of error observed |q-r| in relation to an actual quantity sold each
month.
From there, your opinion on the quality of this estimate.

So, I would like to be sure of what I did.

percentage error = E/r

what does your opinion mean about the quality of this estimate?

pqrE
335003250250
7,69%​
3,234003570170
4,76%​
3,532503100150
4,84%​
430002890110
3,81%​
4,527502550200
7,84%​
5250025000
0,00%​
 
  • #8
help said:
the lowest possible value for E would be zero correct?
No.
Your confusion comes from having used E in your table in a different way from the way it is used in the question.
Your table lists six values, so let's call these Ei, i=1..6.
The question is asking for the least value of another variable, E, such that |Ei|≤E for all i.
 
  • #9
haruspex said:
No.
Your confusion comes from having used E in your table in a different way from the way it is used in the question.
Your table lists six values, so let's call these Ei, i=1..6.
The question is asking for the least value of another variable, E, such that |Ei|≤E for all i.
it's clearing up a little bit. I can leave | q-3000 | ≤500 because so 2500≤q≤3500. That is E = 500 ?

pqrr-q|r-q|
335003250-250250
3,234003570170170
3,532503100-150150
430002890-110110
4,527502550-200200
52500250000
 
  • #10
help said:
it's clearing up a little bit. I can leave | q-3000 | ≤500 because so 2500≤q≤3500. That is E = 500 ?

pqrr-q|r-q|
335003250-250250
3,234003570170170
3,532503100-150150
430002890-110110
4,527502550-200200
52500250000
I don't understand how you get 500.
You have six values for |r-q|. What is the smallest number E such that none of the six values exceed E?
 
  • #11
Oh ok I got it. It is a maximum error. then E = 250
 
  • #12
help said:
Oh ok I got it. It is a maximum error. then E = 250
Right.
 
  • #13
Thanks!
to end the question has an item c...

c)
Calculate the percentage of error observed |q-r| in relation to an actual quantity sold each
month.
From there, your opinion on the quality of this estimate.

So, I would like to be sure of what I did.

percentage error = E/r

what does your opinion mean about the quality of this estimate?
pqrr-q|r-q|
335003250-250250
7,69%​
3,234003570170170
4,76%​
3,532503100-150150
4,84%​
430002890-110110
3,81%​
4,527502550-200200
7,84%​
52500250000
0,00%​
 
  • #14
help said:
Thanks!
to end the question has an item c...

c)
Calculate the percentage of error observed |q-r| in relation to an actual quantity sold each
month.
From there, your opinion on the quality of this estimate.

So, I would like to be sure of what I did.

percentage error = E/r

what does your opinion mean about the quality of this estimate?
pqrr-q|r-q|
335003250-250250
7,69%​
3,234003570170170
4,76%​
3,532503100-150150
4,84%​
430002890-110110
3,81%​
4,527502550-200200
7,84%​
52500250000
0,00%​
You have calculated the percentages correctly.
How good the estimate is depends on what it is to be used for and what the alternatives are.
Plot q and r against p. Does it suggest a better formula for q?
 

What is a linear optimization problem?

A linear optimization problem is a mathematical model used to find the optimal solution to a problem with linear constraints. It involves maximizing or minimizing an objective function while satisfying a set of linear constraints.

What are the components of a linear optimization problem?

The components of a linear optimization problem include the decision variables, the objective function, and the constraints. The decision variables represent the quantities that need to be determined, the objective function is the function to be maximized or minimized, and the constraints are the conditions that must be satisfied.

What is the difference between linear and nonlinear optimization problems?

A linear optimization problem has a linear objective function and linear constraints, while a nonlinear optimization problem has a nonlinear objective function and/or nonlinear constraints. Linear optimization problems are generally easier to solve and have well-defined solutions, while nonlinear problems can be more complex and may have multiple solutions.

What are some real-world applications of linear optimization?

Linear optimization is used in a variety of fields, including business, engineering, and economics. Some common applications include production planning, resource allocation, portfolio optimization, and transportation planning.

What are some common methods for solving linear optimization problems?

Some common methods for solving linear optimization problems include the simplex method, the interior point method, and the branch and bound method. These methods use different algorithms to find the optimal solution to the problem.

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