Linear Programming

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# Linear Programming - PowerPoint PPT Presentation

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

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## Linear Programming

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Presentation Transcript

### 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
• Binary variables
• Restricted to be 0 or 1
• Example – Is a plant built?
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
• 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
• 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

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

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

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

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?
• 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
• 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
• 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

Optimal

Build 4 Suburban Restaurants

Build 3 Downtown Restaurants

Average Daily Profit \$108,000

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