Linear programming prob: planning parking lots for mall

[alliebrak]

A new mall with 2 major department stores and 55 specialty shops is being built. You are in charge of planning and building the parking lots for the mall. The planners provide you with the following information:

- The total number of parking spaces must range from 2,000 to 2,400.
- For every employee parking space, there must be at least 9 public parking spaces.
- There must be at least 20 employee parking spaces per department store and 2 employee parking spaces per specialty shop.

You anticipate that building costs will be $580 per public parking space and $600 per employee parking space.

The mall planners excpect that revenue from each public parking space will be at least $1000 per week and each employee pakring space will be at least $100 per week.


Design a proposal to the planners showing the feasible numbers of public and empoloyee parking spaces. How many parking spaces of each type should be built to minimize the cost of building the lot? How many parking spaces of each type should be built to maximize weekly revenue?

 

[Mrspi]

A new mall with 2 major department stores and 55 specialty shops is being built. You are in charge of planning and building the parking lots for the mall. The planners provide you with the following information:

- The total number of parking spaces must range from 2,000 to 2,400.
- For every employee parking space, there must be at least 9 public parking spaces.
- There must be at least 20 employee parking spaces per department store and 2 employee parking spaces per specialty shop.

You anticipate that building costs will be $580 per public parking space and $600 per employee parking space.

The mall planners excpect that revenue from each public parking space will be at least $1000 per week and each employee pakring space will be at least $100 per week.


Design a proposal to the planners showing the feasible numbers of public and empoloyee parking spaces. How many parking spaces of each type should be built to minimize the cost of building the lot? How many parking spaces of each type should be built to maximize weekly revenue?

Please show us what you've done so far. Did you define variables?

Once you have defined variables, you can write inequalities to represent each of the constraints (conditions that must be met).

Until we see some effort on your part, I'm not willing to go any further.

 

[pitoten]

It looks like you have to assign variables to 2 things to solve the problem: (employee spaces will be x) and (custo spaces will be y) and extract equations about them from the information.

You can extract a rule about the number of employee spaces (x) like,

x must be greater than or equal to 150, from the third statement. Statements like this show what the upper and lower domains of x and y are, and therefore, narrow down what values will need to be considered....

You can also extract an equation on how the two variables relate to each other .

(x = 9*y) from the second statement. There are also other statements about how much $ each kind costs and how much $ each kind makes that have to be considered.

After you get all the equations, you can combine them into a single equation that relates the number each kind of space to the money that is made...

That is,

f(x) + g(y) = m where

f(x) = the function f needed to express the number of employee spaces (x) and
g(y) = the function g needed to express the number of custo space (y).
m = money made



You build the functions by following the requirements listed in the problem.

You also know x in terms of y, so you can sub in y=x/9 in g(y), and get m as a function of x only: m(x). This is the equation will solve the problem (and remember the allowable domain for x) You now find the maxima and minima of m(x), if any, which is found by finding where dm/dx=0 + taking the second derivative of m(x) to see if it is a maxima or minima, which answers both questions.