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Linear Programming
What is Linear Programming?
What is Linear Programming?Say you own a 500 square acre farm. On
this farm you can grow wheat, barley, corn or some combination of the 3. You have a limited supply of fertilizer and pesticide, both of which are needed (in different quantities) for each crop grown. Let’s say wheat sells at $7 a bushel, barley is $3, and corn is $3.50.
So, how many of each crop should you grow to maximize your profit?
What is Linear Programming?
A mathematical tool for maximizing or minimizing a quantity (usually profit or cost of production), subject to certain constraints.
Of all computations and decisions made by management in business, 50-90% of those involve linear programming.
Background on Linear Programming
• As a field of mathematics, LP is still a small child (in math years)
• Developed by Leonid Kantorovich around the time of WWII
• Further developed over followingdecades
• Today, easily the most commonly used field for optimization
• Economics, business management,transportation, technology, planning, production, …
the list goes on…
Maximizing ProfitProblem where a limited
number of resources are used to produce a combination of products to maximize profit from the sale
Production of…
• Pretty much anything
• Maximizing Problems consist of…1. Resources
2. Products
3. Recipes
4. Profit
5. Objective
Setting Up A toy manufacturer can produce skateboards
and dolls. Both require the precious resource of plastic, of which there are 60 units available. Skateboards take five units of plastic and make $1 profit. Dolls take two units of plastic and make $0.55 profit. The company wants to make at least 1 doll and at least 1 skateboard.
What is the number of dolls and skateboards the company can produce to maximize profit?
Setting Up Mixture ProblemsFirst identify components of the problem:
1. Variables (Products)
2. Constraints (Resources and Recipes)
3. Profits
4. Objective – Maximize profit
Make Mixture Chart or Formulas
2 Groups of Equations:
- Profit Equation (profit equation)
- Constraint Inequalities
With these, create Feasible Region
ResourcesPlastic (60)
Profit
Products Skateboards(x units)
5 $1.00
Dolls(y units)
2 $0.55
Feasible Region – region which consists of all possible solution choices for a particular problem
Using the constraint equation we get the following graph:
Constraints:
5s + 2d ≤ 60
s > 0
d > 0
Corner Point Principle
Which point is optimal?
Corner Point PrincipleThe maximal value always
corresponds to a corner point
Corner Point Principle
Plug in corner points to profit formula:
Quick PracticeA clothing company has 100 yards of cloth and produces shirts (x units) and
vests (y units). Shirts require 10 units and have profit value of $5, while vests require 4 units and have profit value of $4.
What is the optimal production solution?Steps 3 & 4: Feasible Region & Corner Points
( 0, 25 )
( 10, 0 )
( 0, 0 )
Point Calculation of Profit Formula$5.00x + $4.00y = P
(0, 0) $5.00 (0) + $4.00 (0) = $0.00
(0, 25) $5.00 (0) + $4.00 (25) = $100.00
(10, 0) $5.00 (10) + $4.00(0) = $50.00
Quick PracticeWhat if the company decides to also put a
“non-zero constraint” on all production?
Must produce at least 3 shirts and 10 vests.
Constraints become: 10x + 4y ≤ 100 …
x ≥ 3
y ≥ 10
Feasible Region becomes:
( 3, 17.5 )
( 6, 10 )
( 3, 10 )
Corner Points:
Point Calculation of Profit Formula$5.00x + $4.00y = P
(3, 10) $5.00 (3) + $4.00 (10) = $55.00
(3, 17.5) $5.00 (3) + $4.00 (17) = $83.00
(6, 10) $5.00 (6) + $4.00(10) = $70.00
Great Job!
Linear Programming
• Find the minimum and maximum values by graphing the inequalities and finding the vertices of the polygon formed.
• Substitute the vertices into the function and find the largest and smallest values.
6
4
2
2 3 4
3
1
1
5
5
7
8
y ≤ x + 3
y ≥ 2
1 ≤ x ≤5
Linear Programming
• The vertices of the quadrilateral formed are:
(1, 2) (1, 4) (5, 2) (5, 8)
• Plug these points into the function P = 3x - 2y
Linear Programming
P = 3x - 2y
• P(1, 2) = 3(1) - 2(2) = 3 - 4 = -1
• P(1, 4) = 3(1) - 2(4) = 3 - 8 = -5
• P(5, 2) = 3(5) - 2(2) = 15 - 4 = 11
• P(5, 8) = 3(5) - 2(8) = 15 - 16 = -1
Linear Programming
• f(1, 4) = -5 minimum
• f(5, 2) = 11 maximum