Linear Programming

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# Linear Programming - PowerPoint PPT Presentation

Linear Programming. Introduction. What is Linear Programming?. A Linear Programming model seeks to maximize or minimize a linear function , subject to a set of linear constraints . What are linear functions?. Linear function in 2 variables. No X 1 2 , X 1 /X 2 , e -X2 , X 1 , etc.

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### Linear Programming

Introduction

What is Linear Programming?
• A Linear Programming model seeks to maximize or minimize a linear function, subject to a set of linear constraints.
What are linear functions?

Linear function in 2 variables

No X12, X1/X2, e-X2,X1, etc.

• y = mx+b is the equation of a straight line
• e.g. y = -4/3 x +6
• Multiplying by 3 and rearranging: 4x + 3y = 18

Alinear function consists of the sum of positive, negative

or 0 constants times variables; e.g.

5X1 - 4X2 + 0X3 + 6X4 is a linear function in 4 variables.

What are Linear Constraints?
• Linear constraints have the form:

<Linear Function><has some relation to><a constant>

• The relation is one of the following:

, = ,  ---- they all contain the “equal to” part

Examples:

4X1 + 5X2 - 6X3 + 2X534

2X1 - 5X2 + 1X447

- 2X2 + 8X3 + 9X4 + 2X5=67

X1 0

X5 0

Example of a Linear Program

Subject to

X1 0, X2  0, X3  0, X4  0

MAX 4X1 + 7X3 - 6X4

s.t. 2X1 + 3X2 - 2X4 = 20

- 2X2 + 9X3 + 7X4 10

-2X1 + 3X2 + 4X3 + 8X4  35

X2  5

All X’s  0

Another Example

MIN 6X1 + 8X2 + 11X3 + 10X4 + 5X5 + 14X6

S.T. X1 + X2 + X3 20

X4 + X5 + X6  30

X1 + X4 = 12

X2 + X5 = 15

X3 + X6 = 22

All X’s  0

Components of a Linear Programming Model
• A linear programming model consists of:
• A set of decision variables
• A (linear) objective function
• A set of (linear) constraints
Why are Linear Programs Important?
• Many real world problems lend themselves to linear programming modeling.
• Other real world problems can be approximated by linear models.
• There are well-known successful applications in:
• Manufacturing, Marketing, Finance (investment), Advertising, Agriculture, Energy, etc.
• There are efficient solution techniques and software programs that solve linear programming models.
• The output generated from linear programming packages provides useful “what if” analysis.
Linear Programming Assumptions
• The parameter values are known with certainty.
• The objective function and constraints exhibit constant returns to scale.
• There areno interactionsbetween the decision variables (additivity assumption).
• Continuityof the decision variables means they can take on any value within a given feasible range.
• Integer programming models can only take on integer values within a given feasible range.
Example
• Galaxy Industries manufactures two toy gun models:
• Space Rays: Each dozen nets an \$8 profit and
• Requires 2 lbs. of plastic; 3 minutes of production time
• Zappers: Each dozen nets a \$5 profit and
• Requires 1 lb. of plastic; 4 minutes of production time
• Weekly resource limits
• 1000 pounds of plastic; 40 hours of production time
• Weekly production limits
• Maximum 700 dozen total units
• Space Rays cannot exceed Zappers by more than 350 dozen
Current Production
• Current reasoning calls for a production plan that:
• Produces as much as possible of the more profitable product, Space Ray (\$8 profit per dozen).
• Uses any left over resources to produce Zappers (\$5 profit per dozen), while remaining within the marketing guidelines of 700 total dozen produced and Space Rays – Zappers ≤ 350.
• Using a simple spreadsheet, letting the (cell for production of Zappers) = (cell for production of Space Rays – 350), trial and error gives the following good solution that uses all the available weekly plastic:

Space Rays = 450 dozen; Zappers = 100 dozen;

Profit = 8(450) + 5(100) = \$4100

This is a good solution – Can we do better?

The Mathematical Model
• Recall a mathematical model consists of:
• Set of decision variables
• Objective function
• Constraints
• Decision Variables

(Include both a measurement unit (dozens) and a time unit (week))

X1 = dozens of Space Rays produced weekly

X2 = dozens of Zappers produced weekly

2. OBJECTIVE FUNCTION
• Objective is to maximize the total weekly profit.

How much profit will be made each week?

• How much profit will
• Space Rays?
• How much profit will
• Zappers?

\$5 per dozen

\$8 per dozen

Make X2 dozen

Zappers per week

Make X1 dozen

Space Rays per week

5X2

8X1

+

MAX 8X1 + 5X2

3. Constraints -- PLASTIC
• At most 1000 pounds of plastic available weekly.

How much will be used?

• How much plastic will
• be used weekly making
• Space Rays?
• How much plastic will
• be used weekly making
• Zappers?

1 lb per dozen

2 lbs per dozen

Make X2 dozen

Zappers per week

Make X1 dozen

Space Rays per week

1X2

2X1

+

2X1 + 1X2 1000

Constraints -- Production Time
• At most 40 hours = 40x60 = 2400 minutes available weekly. How much will be used?
• How many minutes will
• be used weekly making
• Space Rays?
• How many minutes will
• be used weekly making
• Zappers?

4 min per dozen

3 min per dozen

Make X2 dozen

Zappers per week

Make X1 dozen

Space Rays per week

4X2

3X1

+

3X1 + 4X2 2400

Constraints -- Max Production
• At most 700 dozen total units can be produced weekly. How many will be produced?
• How many dozen
• Space Rays are
• produced weekly?
• How many dozen
• Zappers are
• Produced weekly?

Make X1 dozen

Space Rays per week

Make X2 dozen

Zappers per week

X1

+

X2

X1 + X2 700

Constraints -- Product Mix

Amount (in dozens) Space Rays exceed Zappers

• Space Rays can be at most 350 dozen units greater than Zappers each week. How many more dozen units of Space Rays will be produced weekly?
• How many dozen
• Space Rays are
• produced weekly?
• How many dozen
• Zappers are
• Produced weekly?

Make X2 dozen

Zappers per week

Make X1 dozen

Space Rays per week

X2

-

X1

X1 - X2 350

Constraints -- Nonnegativity
• Cannot produce a negative amount of Space Rays or Zappers

X1 0

X2 0

or

All X’s  0

The Complete Galaxy IndustriesLinear Programming Model

MAX 8X1 + 5X2

s.t. 2X1 + 1X2≤ 1000 (Plastic)

3X1 + 4X2 ≤ 2400 (Prod. Time)

X1 + X2 ≤ 700 (Total Prod.)

X1 - X2 ≤ 350 (Mix)

All X’s ≥ 0

Review
• A linear program seeks to maximize or minimize a linear objective subject to linear constraints.
• Many problems are or can be approximated by linear programming models.
• Linear programs possess the features of:
• Certainty, Constant Returns to Scale, Additivity and Continuity
• There exists efficient algorithms for solving linear programs that provide many sensitivity analyses as a by-product.