Linear Programming

Joseph Mark

November 3, 2009

What is Linear Programming

Linear Programming – a management science technique that helps a business allocate the resources it has on hand to make a particular mix of products that will maximize profit.

Tool for maximizing or minimizing a quantity subject to constraints

Said to account for 50-90% of computing time used for management decisions

Real World Applications and Examples What crops to grow on limited farmland How to use resources in a bakery (eggs,

butter, sugar, eggs, flour) How to use labor force

Linear Programming

Uses an algorithm based on available data to produce an optimal solution

Can take on decimal values unlike integer programming However, sometimes the optimal solution is an

unfeasible solution It doesn’t make sense to produce 3.24 dolls to sell

Mixture Problem

In a mixture problem, limited resources are combined into products so that the profit from selling those products is a maximum Example: how different kinds of aviation fuel can be

manufactured using different kinds of crude oil An optimal production policy has two properties.

First it is possible; that is, it does not violate any of the limitations under which the manufacturer operates, such as availability of resources. Second, the optimal production policy gives the max profit

Mixture Problems with One Resource Suppose a toy manufacturer has 60 containers

of plastic and wants to make and sell skateboards. One skateboard requires 5 containers of plastic The profit on 1 skateboard is $1

Manufacturer can make 60/5 = 12 skateboards So the number of skateboards, x, that can be

made is between 0 and 12, or 0 ≤ x≤ 12 These values are the feasible set (set of all possible

solutions)

Notes on Feasible Region

Any point within the feasible region represents a possible production policy – any number within this region is possible to produce given the limited supplies.

For most problems, negative values are unfeasible, for example, how do you make negative skateboards?

Maximum profit always occurs at a corner point of a feasible region, in both simple and complex problems

Common Features of Mixture Problems Resources

Definite resources are available in limited, known quantities for the time period in question. Ex. plastic

Products Definite products can be made by combining, or mixing, the

resources. Ex. Skateboards Recipes

A recipe for each product specifies how many units of each resource are needed to make one unit of that product. Ex. 1 skateboard = 5 containers of plastic

Profits Each product earns a known profit per unit

Objective The objective in a mixture problem is to find out how much of each

product to make so to maximize profits without exceeding resource limitations

Two Products and One Resource Toy manufacturer makes skateboards and

dolls 1 doll requires 2 containers of plastic 1 skateboard requires 5 containers of plastic

Can make all dolls, all skateboards, or some combination of the two One doll makes $.55 profit One skateboard makes $1 profit

Total Profit will be $1x+$.55y, where x is the number of skateboards and y is the number of dolls manufactured

Mixture Charts

Answers 1. what are the resources 2. what quantity of each resource is available 3. what are the products 4. what are the recipes for the products 5. what are the unknown quantities 6. what is the profit formula

Mixture Chart for Skateboards and Dolls

Skateboards

(x units)

5 $1.00

Dolls

(y units)

2 $0.55

RESOURCES

Containers of Plastic

60 PROFIT

PR

OD

UC

TS

Resource Constraints

Tell how much of a resource you have You can’t use more than the amount available

For the skateboards and dolls problem The number of plastic containers used must be

less than 60 So 5x + 2y ≤ 60. Constraint inequalities are always associated

with equations for lines, thus linear programming.

Feasible Region for Skateboards and Dolls

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14

Skateboards

Do

lls

Finding the Optimal Production Policy Now need to find the point within the region

that gives the maximum profit Corner Point Principle

The corner point principle states that in a linear programming problem, the maximum value for the profit formula always corresponds to a corner point of a feasible region

Corner Point Principle

1. Determine the corner points of the feasible region

2. Evaluate the profit at each corner point of the feasible region

3. Choose the corner point with the highest profit as the production policy.

Optimal Production Policy for Skateboards and Dolls We have three corner points (0, 0) (12, 0) (0, 30) Evaluate profit formula at these points

$1x + $.55yMaximum profit at (0, 30) = $16.50

This point is called the optimal production policy

Feasible Region for Skateboards and Dolls

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14

Skateboards

Do

lls

Role of Profit Formula

The optimal production policy is dependent on the profit formula. For example, if the profit for skateboards were

$1.05 and profit for dolls were $.40, we would make all skateboards (12,0) instead of all dolls.

However, there may be reasons for wanting to produce both products

Adding Minimums to Mixture Chart

Skateboards

(x units)

5 4 (1)$1.00

(2)$1.05

Dolls

(y units)

2 10 (1)$0.55

(2)$0.40

RESOURCES

Containers of Plastic

60 PROFIT

PR

OD

UC

TS

MINIMUMS

Adding Minimums

Feasible Region for Skateboards and Dolls

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14

Skateboards

Do

lls

Find New Optimal Solution

Have new corner points at (4,10) (4,20) (8,10)

Can find these points algebraically (4,10) easy to see Upper left point has 4 as x-coordinate

Substitute x=4 into second line 5x+2y=60 5(4)+2y=60 -> y = 20

Same for lower right, substitute y=10 instead 5x+2(10)=60 -> x=8

Finding New Optimal Solution Substitute new corner points into profit

formula Using profit formula 1:

(4, 10) = 1(4) + .55(10) = $9.50 (4, 20) = 1(4) + .55(20) = $15.00 (8, 10) = 1(8) + .55(10) = $13.50

Using profit formula 2: (4, 10) = 1.05(4) + .40(10) = $8.20 (4, 20) = 1.05(4) + .40(20) = $12.20 (8, 10) = 1.05(8) + .40(10) = $12.40

Summary of Pictorial Method 1. Identify resources and products 2. Make a Mixture Chart showing resources,

products, recipes for creating products, profit of each product, and the amount of each resource on hand. If problem has minimums, include that as well

3. Assign unknowns, x or y, to each product. Use the mixture chart to write down the resource constraints, the minimum constraints, and the profit formula

4. Graph the line corresponding to each resource constraint and determine which side of the line is in the feasible solution (≥ or ≤ )

5. Find the corner points and evaluate the profit formula at these points

Mixture Problems with 2 Resources Consider the toy manufacturer and now has a

second constraint of time. Suppose there are 360 minutes of available

labor. One skateboard requires 15 minutes One doll requires 18 minutes Still maintain zero minimum constraints

New Mixture Chart

Skateboards

(x units)

5 15 (1)$1.00

Dolls

(y units)

2 18 (1)$0.55

Containers of Plastic

60 PROFIT

PR

OD

UC

TS

Minutes

360

RESOURCES

Resource Constraints

5x + 2y ≤ 60 containers of plastic 15x + 18y ≤ 360 minutes available

Profit Formula $1x + $0.55y

Graph two lines and find the intersection

Feasible Region for Skateboards and Dolls

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30

Skateboards

Do

lls Plastics

Minutes

Find New Corner Points

(0,0) (12,0) (0,20) Find the intersection point

5x+2y=60 15x+18y=360

Solve for one variable 18(5x+2y=60) -> 90x+36y=1080 -2(15x+18y=360) -> -30x-36y=-720 60x = 360 x = 6 5(6)+2y=60 y = 15(6,15) is last point

Evaluate

Evaluate your new points in your profit formula $1x + $.55y

(0,0) = $0 (12,0) = 12 + 0 = $12 (0,20) = 0 + .55(20) = $11 (6,15) = 6 + .55(15) = $14.25

So the optimal production policy is to make 6 skateboards and 15 dolls for a profit of $14.25

Mixing Two Juices

A juice manufacturer produces and sells two fruit beverages: Cranapple and Appleberry

1 gal of Cranapple is 3qts Cranberry Juice and 1qt Apple Juice

1 gal of Appleberry is 2qts Apple Juice and 2qts Cranberry Juice

Cranapple makes a profit of 2cents/gallon Appleberry makes a profit of 5cents/gallon

Constraints

You have 200 quarts of Cranberry Juice available and 100 quarts of Apple Juice available, how many gallons of each drink mixture should we make? Cranberry constraint 3x+2y ≤ 200 Apple Juice constraint 2x+2y ≤ 100

X = gallons of cranapple juice Y = gallons of appleberry juice

With Minimum Constraints

Want x (gallons of cranapple juice) ≥ 20 Want y (gallons of appleberry juice) ≥ 10

Get corner points of (20, 10) (20, 40) (50, 25) (60,10)

Evaluate in profit formula 2x+5y (20,40) is optimal point

But substituting (20,40) into our cranberry juice constraint, we find we only use 3(20)+2(40)=140 of our allotted 200 quarts available. So we have 60 quarts of slack. So perhaps sell cranberry juice separately or purchase more

apple juice if it’s profitable.

Complex Regions

Sometimes there are so many corner points of a feasible region that multiple calculations are needed to determine the coordinates and profits for each one. Computing the profit for every corner point for even the fastest computer could be impossible

Also, it is not possible to visualize the feasible region as a part of two-dimensional space where there are more than two products. Each product is represented by an unknown, and each unknown is represented by a dimension of space. If we have 50 products, we would need 50 dimensions and we couldn’t visualize such a region.

EXCEL

Linear Programming in Excel

LINDO

Courses for Additional Information

BUAD 361 – Intro to Operations Technology

BUAD 467 – Adv. Data Management & Modeling

MATH 323 – Intro to Operations Research