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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 X 1 2 , X 1 /X 2 , e -X2 , X 1 , etc.

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linear programming

Linear Programming

Introduction

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

How much profit will be made each week?

  • How much profit will
  • be made weekly from
  • Space Rays?
  • How much profit will
  • be made weekly from
  • 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
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
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
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
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
Constraints -- Nonnegativity
  • Cannot produce a negative amount of Space Rays or Zappers

X1 0

X2 0

or

All X’s  0

the complete galaxy industries linear programming model
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
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.
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