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