Very Basic Maths Assignment (sorry i'm a newbie) based on profit and graphs

  • Thread starter Thread starter adste89
  • Start date Start date
  • Tags Tags
    Assignment Graphs
Click For Summary
SUMMARY

This discussion centers on a linear programming problem involving the construction of luxury and standard houses on a plot of land. The builder must adhere to specific constraints: a minimum of 5 luxury houses and 10 standard houses, with a maximum of 30 houses total and a land area of 6000m2. The profit per luxury house is $12,000 and $8,000 per standard house. The optimal solution is determined to be 10 luxury houses and 20 standard houses, yielding a maximum profit of $280,000.

PREREQUISITES
  • Understanding of linear programming concepts
  • Familiarity with inequalities and constraints
  • Basic knowledge of profit maximization techniques
  • Experience using Excel Solver for optimization problems
NEXT STEPS
  • Learn how to formulate linear programming problems
  • Explore graphical methods for solving two-variable linear programming problems
  • Research the use of Excel Solver for complex optimization scenarios
  • Study operations research textbooks for advanced linear programming techniques
USEFUL FOR

This discussion is beneficial for engineering students, aspiring mathematicians, and anyone interested in optimization techniques for resource allocation and profit maximization in construction projects.

adste89
Messages
2
Reaction score
0

Homework Statement



Firstly I'm sorry to bother all of you that will find this question really basic but I haven't studied Maths since GCSE 7 years ago and I am now doing a degree in Engineering without A Level Maths so any help would be great...I need to work out the algebra and the main issue is how to plot it on a graph!? I'm guessing I need to use y=mx + c but I'm not sure how to do that on excel.

A builder has a plot of land available on which he can build either luxury of standard houses.
He decides to build at least 5 luxury and 10 standard houses. Planning restrictions limit him
to no more than 30 houses altogether. A luxury house requires 300m2
of land and a standard
house 150m2
. The plot is 6000m2
. Profit is $12; 000 per luxury house and $8000 per standard
house. How many of each type should he build to maximise his profit.


Thank you anyone that can help!

Adam
 
Physics news on Phys.org
ok so here is what I have worked out so far...

Luxury Houses = x
Basic Houses = y

(300x + 5y) + (150y + 10x) = 6000m2 i'm not too sure if this is correct

I also came to the conclusion for graph purposes the answer might be y= -0.5x + 10 ?

I also know the most profit is £280,000 with 10 luxury houses and 20 basic.
 
You need a change of variables. you know you need a certain number of lux and std minimum and you know the maximum houses so subtract the mins from the max to obtain something along the lines of:

30-10 -5 = x+y => y = a function of x

you can then replace all y with the function of x and can solve this 2 variable question as a single variable. Remember to subtract the area of the min requirements from your total area btw.
 
If the builder is going to build at least 5 luxury and 10 standard homes, then calculate the amount of profit obtained from their construction. From the total amount of land, subtract the area required by these 15 homes. For the remainder of the land area, find out the mix of luxury/standard homes which will fit that gives the most additional profit.
 
adste89 said:

Homework Statement



Firstly I'm sorry to bother all of you that will find this question really basic but I haven't studied Maths since GCSE 7 years ago and I am now doing a degree in Engineering without A Level Maths so any help would be great...I need to work out the algebra and the main issue is how to plot it on a graph!? I'm guessing I need to use y=mx + c but I'm not sure how to do that on excel.

A builder has a plot of land available on which he can build either luxury of standard houses.
He decides to build at least 5 luxury and 10 standard houses. Planning restrictions limit him
to no more than 30 houses altogether. A luxury house requires 300m2
of land and a standard
house 150m2
. The plot is 6000m2
. Profit is $12; 000 per luxury house and $8000 per standard
house. How many of each type should he build to maximise his profit.


Thank you anyone that can help!

Adam

This is a standard "linear programming" problem, and involves *inequalities*, not equations. Here is one formulation. Let L = number of luxury houses to build and S = number of standard houses to build. (No reason to use symbols x and y, when symbols L and S are so much more tied to their definitions.) Land used = 300L + 150S, and this must be <= 6000. Total number of houses = L + S, and this must be <= 30. Also: L >= 5 and S >= 10. Finally: profit = 12L + 8S (in $000,s). The final linear programming (LP) formulation is:
max 12L + 8S,
subject to
300L + 150S <= 6000
L + S <= 30
L >= 5, S >= 10

There are standard algorithms available for solving such problems manually, or you can submit it to a Solver tool in a spreadsheet (eg., the EXCEL Solver can deal with such problems having up to about 200-300 decision variables and about 200 inequalities and equations, together with simple bounds that are not part of the inequality count). However, simple 2-variable problems like this one can be tackled using a _graphical method_ because you have only two variables (L and S), so can plot the constraint region in a 2-dimensional plot and look where in the constraint region the 'objective function' 12L + 8 S is maximized. See, eg., http://math.tutorvista.com/algebra/graphical-method.html or any introductory Operations Research textbook.

RGV
 

Similar threads

Engineering I’m having second thoughts about my job position. How can I proceed with it?
  • · Replies 13 ·
Replies
13
Views
5K
Programs Double major in physics and math
  • · Replies 8 ·
Replies
8
Views
4K