Optimization problem - defining domain

Mar 2017
358
3
Massachusetts
Hi,

Can someone help me formally define what the domain would be for this question?

Screen Shot 2017-08-12 at 3.22.54 PM.png

I was fortunately able to reach the correct solution for this question without a well defined range, but for future sake, I would like to be able to know how to derive it.

I know that the domain would consider the variables dealing with the maximum passenger capacity of 15,000 people and also the minimum fare sales of $130,000 ... but I'm struggling to construct a domain in the context of these restrictions.

- Olivia
 
Feb 2014
1,748
651
United States
How did you set up the problem? To do optimization you must have defined a function. The argument of that function is what will determine the domain. I see what your problem is here. You are thinking about constraints on two different variables. That can be translated into the language of range and domain, but may not be the most intuitive way to think about the problem of these constraints. So let's start with what you have done.
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
You say, at one point "range" and, at another, "domain". I presume you are thinking of the number of passengers per day as a function of the fare so I would say that, since they have "space to serve up to 15,000 passengers per day" the range will be 0 to 15,000. At the same time, you are told that, at $20 fare, they serve 10,000 passengers per day and "if the fare increases by $0.50 200 fewer people will ride the bus". (This problem does not say "per day" but I assume that is intended.) That is, taking "P" to be the number of passengers and "f" the fare, P= 10000- 200((f- 20)/0.50= 10000- 400(f- 20).

There cannot be less than 0 passengers so the fare cannot be more than f such that 10000- 400(f- 20)= 0. That is, 400(f- 20)= 10000, f- 20= 25 or f= $45.00. On the other hand, since there cannot be more than 15000 passengers, the fare cannot be les than f such that 10000- 400(f- 20)= 15000. 400(f- 20)= -5000, f- 20= -12.5, f= $7.50. The domain is $7.50 to $45.00.

But you really don't need to know either domain or range to answer this question as you appear to have discovered.
 
Mar 2017
358
3
Massachusetts
Yea it seems slightly more tedious than usual to find the domain and range just by briefly looking at your response. Thanks for letting me know
 
Mar 2017
358
3
Massachusetts
How did you set up the problem? To do optimization you must have defined a function. The argument of that function is what will determine the domain. I see what your problem is here. You are thinking about constraints on two different variables. That can be translated into the language of range and domain, but may not be the most intuitive way to think about the problem of these constraints. So let's start with what you have done.
Basically I just created a revenue function... r(x) = (10,000 - 200x)(20 + 0.5x)

"x" represents the number of times the fare increased by $0.5

I'm curious what the domain restriction would be for this.
 
Mar 2017
358
3
Massachusetts
How did you set up the problem? To do optimization you must have defined a function. The argument of that function is what will determine the domain. I see what your problem is here. You are thinking about constraints on two different variables. That can be translated into the language of range and domain, but may not be the most intuitive way to think about the problem of these constraints. So let's start with what you have done.
Basically I just created a revenue function... r(x) = (10,000 - 200x)(20 + 0.5x)

"x" represents the number of times the fare increased by $0.5

I'm curious what the domain restriction would be for this.
 
Feb 2014
1,748
651
United States
Basically I just created a revenue function... r(x) = (10,000 - 200x)(20 + 0.5x)

"x" represents the number of times the fare increased by $0.5

I'm curious what the domain restriction would be for this.
Well x must clearly be an integer so the domain is the set of integers. Now you could also put some reasonableness bounds on x, which would further restrict the domain.

I reiterate that I think putting constraints in as explicit functions is a more straightforward way to go and leads to understanding LaGrangian multipliers.
 
Last edited:
Mar 2017
358
3
Massachusetts
Okay, sounds good. Thanks for your help!