Linear ProgrammingLinear Programming

Integer Linear Models

When Variables Have To Be When Variables Have To Be IntegersIntegers

• 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

ExampleExample• 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/ObjectiveDecision Variables/Objective

• X1 = Number of restaurants built in suburbs

• X2 = Number of restaurants built downtown

MAX Expected Daily Profit

MAX Expected Daily Profit MAX 12X1 + 20X2 (in $1000’s)

ConstraintsConstraints• Cannot invest more than $27,000,000

• At least 2 downtown restaurants

• Number of managers used cannot exceed 19

Total Amount Invested CannotExceed 27

2X1 + 6X2 ≤ 27

# downtown restaurants Must beAt least 2

X2 ≥ 2

# Managers used CannotExceed 19

3X1 + 1X2 ≤ 19

In $1,000,000’s

The Complete ModelThe 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 RegionThe Linear Programming Feasible RegionX2

6

5

4

3

2

1

01 2 X13 4 5 6

X1, X2 ≥ 0

2X1 + 6X2 ≤ 27

2X1 + 6X

2 ≤ 27

X2 ≥ 2

X2 ≥ 2

3X1 + 1X2 ≤ 19

3X1 + 1X

2 ≤ 19

LPFeasibleRegion

Max 12X1 + 20X2

12X1 + 20X

2

Rounded down

FEASIBLEObjective Value = 100

(5,2)

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

Roundedup

(6,3)

Rounded off(5,3)

The Integer Programming Feasible RegionThe Integer Programming Feasible RegionX2

6

5

4

3

2

1

01 2 X13 4 5 6

X1, X2 ≥ 0

3X1 + 1X2 ≤ 19

2X1 + 6X2 ≤ 27

2X1 + 6X

2 ≤ 27

X2 ≥ 2

X2 ≥ 2

3X1 + 1X

2 ≤ 19

X1, X2 integer

Max 12X1 + 20X2

12X1 + 20X

2

ILP Optimum(4,3)

OBJ. VALUE = 108

Why Not Round To Get the Why Not Round To Get the Optimal Integer Solution?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 ModelsGeneral 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 SOLVERSolving 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 RestaurantsAverage Daily Profit $108,000

ReviewReview

• When to use integer models

• Why rounding will not always work

• Solution time

• No sensitivity analysis

• Objective function value cannot improve

• SOLVER solution approach