Transcript Linear Programming - William & Mary

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



Can make all dolls, all skateboards, or some
combination of the two



1 doll requires 2 containers of plastic
1 skateboard requires 5 containers of plastic
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
RESOURCES
Containers of Plastic
PRODUCTS
60
PROFIT
Skateboards
(x units)
5
$1.00
Dolls
(y units)
2
$0.55
Resource Constraints

Tell how much of a resource you have


For the skateboards and dolls problem



You can’t use more than the amount available
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
Dolls
35
30
25
20
15
10
5
0
0
2
4
6
8
Skateboards
10
12
14
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 + $.55y
Maximum profit
at (0, 30) = $16.50
Feasible Region for Skateboards and Dolls
Dolls
35
30
This point is called
the optimal
production policy
25
20
15
10
5
0
0
2
4
6
8
Skateboards
10
12
14
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
RESOURCES
PRODUCTS
Containers of Plastic
60
MINIMUMS
PROFIT
Skateboards
(x units)
5
4
(1)$1.00
(2)$1.05
Dolls
(y units)
2
10
(1)$0.55
(2)$0.40
Adding Minimums
Feasible Region for Skateboards and Dolls
Dolls
35
30
25
20
15
10
5
0
0
2
4
6
8
Skateboards
10
12
14
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
PRODUCTS
RESOURCES
Containers of Plastic
Minutes
60
360
PROFIT
Skateboards
(x units)
5
15
(1)$1.00
Dolls
(y units)
2
18
(1)$0.55
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
Dolls
35
30
25
20
15
10
Plastics
Minutes
5
0
0
5
10
15
Skateboards
20
25
30
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)
-2(15x+18y=360)
(6,15) is last point
-> 90x+36y=1080
-> -30x-36y=-720
60x = 360
x=6
5(6)+2y=60
y = 15
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