linear programming n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Linear Programming PowerPoint Presentation
Download Presentation
Linear Programming

Loading in 2 Seconds...

play fullscreen
1 / 15

Linear Programming - PowerPoint PPT Presentation


  • 84 Views
  • Uploaded on

Linear Programming. Integer Linear Models. When Variables Have To Be Integers. Example – one time production decisions Fractional values make no sense But if ongoing process, fractional values could represent work in progress Example -- building houses or planes, or scheduling crews

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Linear Programming' - lanza


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
linear programming

Linear Programming

Integer Linear Models

when variables have to be integers
When Variables Have To Be Integers
  • Example – one time production decisions
    • Fractional values make no sense
    • But if ongoing process, fractional values could represent work in progress
  • Example -- building houses or planes, or scheduling crews
  • Binary variables
      • Restricted to be 0 or 1
      • Example – Is a plant built?
types of integer programs ilp
Types of Integer Programs (ILP)
  • All Integer Linear Programs (AILP)
    • All the decision variables are required to be integers
  • Mixed Integer Linear Programs (MILP)
    • Only some of the variables are required to be integers
  • Binary Integer Linear Programs (BILP)
    • Variables are restricted to be 0 or 1
example
Example
  • Boxcar Burger will build restaurants in the suburbs and downtown
  • Suburbs
    • Profit $12000/day
    • $2,000,000 investment
    • Requires 3 managers
  • Downtown
    • Profit $20000/day
    • $6,000,000 investment
    • Requires 1 manager
  • Constraints
    • $27,000,000 budget
    • At least 2 downtown restaurants
    • 19 managers available
decision variables objective
Decision Variables/Objective
  • X1 = Number of restaurants built in suburbs
  • X2 = Number of restaurants built downtown

MAX Expected Daily Profit

MAX 12X1 + 20X2(in $1000’s)

MAX Expected Daily Profit

constraints
Constraints

In $1,000,000’s

  • Cannot invest more than $27,000,000
  • At least 2 downtown restaurants
  • Number of managers used cannot exceed 19

Total Amount Invested

Cannot

Exceed

27

27

2X1 + 6X2

# downtown restaurants

Must be

At least

2

2

X2

# Managers used

Cannot

Exceed

19

19

3X1 + 1X2

the complete model
The Complete Model

MAX 12X1 + 20X2 (in $1000’s)

s.t. 2X1 + 6X2 27 (Budget)

X2  2(Downtown)

3X1 + X2  19 (Managers)

Both X’s  0

Both X’s INTEGER!

the linear programming feasible region
The Linear Programming Feasible Region

X2

6

5

4

3

2

1

0

Max 12X1 + 20X2

LPFeasibleRegion

2X1 + 6X2≤ 27

3X1 + 1X2≤ 19

X2≥ 2

Rounded off

3X1 + 1X2≤ 19

2X1 + 6X2≤ 27

(5,3)

Roundedup

12X1 + 20X2

(6,3)

X2≥ 2

(5,2)

Rounded down

FEASIBLEObjective Value = 100

LP Optimum(5 7/16, 2 11/16)Obj. Value = 119

5

3

1

4

6

2

X1, X2≥ 0

X1

the integer programming feasible region
The Integer Programming Feasible Region

X2

6

5

4

3

2

1

0

Max 12X1 + 20X2

ILP Optimum(4,3)OBJ. VALUE = 108

2X1 + 6X2≤ 27

X2≥ 2

2X1 + 6X2≤ 27

12X1 + 20X2

X1, X2 integer

X2≥ 2

3

5

1

4

6

2

3X1 + 1X2≤ 19

3X1 + 1X2≤ 19

X1, X2≥ 0

X1

why not round to get the optimal integer solution
Why Not Round To Get the Optimal Integer Solution?
  • Rounding may yield the optimal integer solution
    • None did in this example
  • But it may yield an infeasible solution
    • Both (5,3) and (6,3) are infeasible solutions
  • Or a feasible solution that is not optimal
    • (5,2) is feasible but not optimal
    • Many times a feasible rounded point gives a “good” solution (giving close to the optimal value of the objective function) -- BUT NOT ALWAYS
general facts about integer models
General Facts About Integer Models
  • The solution time to solve integer models is longer than that of linear programs
    • Because many linear programs are solved en route to obtaining an optimal integer solution
  • For maximization models, the optimal value of the objective function will be less (or at least not greater than) the value for the equivalent linear model
    • Because constraints have been added – the integer constraints
  • There is no sensitivity analysis
    • Because the feasible region is not continuous
solving ilp s using solver
Solving ILP’s Using SOLVER
  • The only change in SOLVER is to add the integer constraints
    • In the Add Constraints dialogue box, highlight the cells required to be integer and choose “int” from the pull down menu for the sign
slide14

Optimal

Build 4 Suburban Restaurants

Build 3 Downtown Restaurants

Average Daily Profit $108,000

review
Review
  • When to use integer models
  • Why rounding will not always work
  • Solution time
  • No sensitivity analysis
  • Objective function value cannot improve
  • SOLVER solution approach