Linear Optimization, or Linear Programming, solution process.
Prof. Peter Schoch, Sussex County Community College
So, this was introduced into my son's high school Precalculus I class, and he really didn't understand it. After reading his textbook, I could understand why, the explanation was terrible. So, I decided to do a few examples for him, to show him how it would work. While I used Maple to do the computational pieces, I didn't use any builtin functions, I kept it as simple as possible.

Carpenter problem


The question asks how many large bookcases and how many small bookcases to make each week to maximize profit.
So, we assign x = large bookcases and y = small bookcases
Objective
Each large yields $35 and each small yields $20. Want to maximize the profit 35x+20y. However, we can only make 32 of them combined, and they combine in a specific amount.
Constraints
x≥2; we must make at lest 2 large
y≥4; we must make at least 4 small
4x+2y≤32 ; this is a resource limitation statement.
To compute the values that might maximize the profit/income, you must substitute in the values of the "corners"/vertices and select the one that does that
"Corners" or Vertices

Maximization value

(2,12)


(1.1) 

(2,4)


(1.2) 

(6,4)


(1.3) 


Table 1: Maximization Condition 
Make 2 large and 12 small for $310 if he sells it all.


Small town transport


The town can purchase large and small vans and wants to maximize the number of passengers they carry.
We assign x=large vans and y=small vans.
Objective
Large can carry 15 and small can carry 7: 15x+7y must be maximized.
Constraints

x

y


Money to purchase


(2.1) 


(2.2) 

≤100000

Money to service


(2.3) 


(2.4) 

≤500


Table 2: Constraints 
To compute the values that might maximize the profit/income, you must substitute in the values of the "corners"/vertices and select the one that does that

(2.5) 

(2.6) 
This solves the system of linear equations to obtain results of x=4 and y=2, if we treated the equations as equalities..
"Corners" or Vertices

Maximization value

(0,5)


(2.7) 

(5,0)


(2.8) 

(4,2)


(2.9) 

(0,0)


(2.10) 


Table 3: Maximization Condition 
Buy 5 large vans to maximize passengers. Note that the mix of 4 large and 2 small is nearly equal.


Skis problem


The question stated how many pairs of downhill skis and crosscountry skis we need to make to maximize the income/profit.
So, we assign x= # downhill skis and y= # crosscountry skis.
Objective
Each pair of downhill skis yields $40 and each crosscountry skis yields $30. We want to maximize profit/income: 40x+30y.
Constraints

x

y


Hours to Fabricate


(3.1) 


(3.2) 

≤108

Hours to Finish


(3.3) 


(3.4) 

≤24


Table 4: Constraints 
To compute the values that might maximize the profit/income, you must substitute in the values of the "corners"/vertices and select the one that does that

(3.5) 

(3.6) 
"Corners" or Vertices

Maximization value

(0,0)


(3.7) 

(0,24)


(3.8) 

(18,0)


(3.9) 

(6,18)


(3.10) 


Table 5: Maximization Condition 
So, make 6 downhill skis and 18 crosscountry for a profit of $780.


Maximize storage space


How many of Cabinet A and how many of Cabinet B should they purchase to maximize the storage capacity (not space).
Assign
x=Cabinet A
y=Cabinet B
Objective
Cabinet A is 12 cu. ft. and Cabinet B has 18 cu. ft., maximize 12x+18y
Constraints

x

y


Floor space


(4.1) 


(4.2) 

≤60

Budget


(4.3) 


(4.4) 

≤600


Table 6: Constraints 
This means that there is no solution for the storage VOLUME problem!
Recasting the problem for square footage...
Assign
x=Cabinet A
y=Cabinet B
Objective
Cabinet A is 3 sq. ft. and Cabinet B has 6 sq. ft. maximize 3x+6y, but remain under 60.
Constraints

x

y


Budget


(4.5) 


(4.6) 

≤600


Table 7: Constraints 
To compute the values that might maximize the square footage, you must substitute in the values of the "corners" and vertices and select the one that does that
"Corners" or Vertices

Maximization value

(0,0)


(4.7) 

(0,8)


(4.8) 

(8,0)


(4.9) 


Table 8: Maximization Condition 
So, to maximize the floor space used, buy 0 Cabinet A and 6 Cabinet B. But any moron who maximizes floor space usage and not actual storage capacity should be fired.


Stenciling problem


The question is how many small and large boxes should stencil to sell at least 12 boxes AND maximize profit.
Assignment
x=# small boxes
y= # large boxes
Objective
10x+20y must be maximized.
Constraints

x

y


Hours to Fabricate


(5.1) 


(5.2) 

≤30

Number to sell


(5.3) 


(5.4) 

≥12


Table 9: Constraints 
To compute the values that might maximize the profit/income, you must substitute in the values of the "corners" and vertices and select the one that does that
"Corners" or Vertices

Maximization value

(6,6)


(5.5) 

(12,0)


(5.6) 

(15,0)


(5.7) 


Table 10: Maximization Condition 
So, make 6 small and 6 large for a profit of $180.

