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Chapter 6

- Integer Linear Programming

Integer Linear Programming (ILP)Three Classes

- All-Integer Linear Program
- All variables must be integers
- 0-1 Integer Linear Program
- Integer variables must be 0 or 1, also known as

binary variables - Mixed-Integer Linear Program
- Some, but not all variables must be integers

ILP Solution Procedure

- Formulate the problem
- Solve the Relaxed LP Problem
- Relaxed means that we are temporarily

suspending the integer requirement - Round to find the nearest guaranteed integer

solution - Round down for a maximization problem
- Round up for a minimization problem
- Use Branch and Bound to seek an improved

solution.

Integer Linear Programming (ILP)

- All-Integer Linear Program
- All variables must be integers
- 0-1 Integer Linear Program
- Integer variables must be 0 or 1, also known as

binary variables - Mixed-Integer Linear Program
- Some, but not all variables must be integers

All Integer Linear Programming

- Northern Airlines is a small regional airline.

Management is now considering expanding the

company by buying additional aircraft. One of

the main decisions is whether to buy large or

small aircraft to use in the expansion. The

table below gives data on the large and small

aircraft that may be purchased. - As noted in the table, management does not want

to buy more than 2 small aircraft, while the

number of large aircraft to be purchased is not

limited. - How many aircraft of each type should be

purchased in order to maximize annual profit?

Define Variables - Northern Airlines

- Let
- S of Small Aircraft
- L of Large Aircraft

General Form - Northern Airlines

- Max 1S 5L
- s.t.
- 5S 50L ? 100
- 1S 0L ? 2
- and S, L ? 0 Integer

Northern Airlines Graph Solution

LP Relaxation (2, 1.8)

Budget

Small AC

Northern Airlines Graph Solution

Budget

Small AC

Rounded Solution (2, 1)

Northern Airlines Graph Solution

Optimal Solution (0, 2)

Budget

Small AC

Integer Linear Programming

- All-Integer Linear Program
- All variables must be integers
- 0-1 Integer Linear Program
- Integer variables must be 0 or 1, also known as

binary variables - Mixed-Integer Linear Program
- Some, but not all variables must be integers

0-1 ILP (Binary Integer Programming)

- Assists in selection process
- For example
- 1 corresponding to undertaking
- 0 corresponding to not undertaking
- Allows for modeling flexibility through
- Multiple choice constraints
- k out of n alternatives constraint
- Mutually exclusive constraints
- Conditional co-requisite constraint

ILP - Binary

- CAPEX Inc. is a high technology company that

faces some important capital budgeting decisions

over the next four years. The company must

decide among four opportunities - 1. Funding of a major RD project.
- 2. Acquisition of an existing company, RD Inc.

- 3. Building a new plant, and
- 4. Launching a new product.
- CAPEX does not have enough capital to fund all of

these projects. The table below gives the net

present value of each item together with the

schedule of outlays for each over the next four

years. All values are in millions of dollars.

General Form CAPEX Inc.

- Let
- X1 1 if RD Project funded, else 0
- X2 1 if acquire company, else 0
- X3 1 if build new plant, else 0
- X4 1 if launch new project, else 0

General Form CAPEX Inc

- Max 100X1 50X2 30X3 50X4
- s.t.
- 10X1 30X2 5X3 10X4 ? 40 Yr 1
- 15X1 0X2 5X3 10X4 ? 60 Yr 2
- 15X1 0X2 5X3 10X4 ? 80 Yr 3
- 20X1 0X2 5X3 10X4 ? 70 Yr 4
- and Xi ? 0, ? 1

Integer Linear Programming

- All-Integer Linear Program
- All variables must be integers
- 0-1 Integer Linear Program
- Integer variables must be 0 or 1, also known as

binary variables - Mixed-Integer Linear Program
- Some, but not all variables must be integers

Integer Programming Mixed Integer

- Hart Manufacturing makes three products. Each

product goes through three manufacturing

departments, A, B, and C. The required

production data are given in the table below.

(All data are for a monthly production schedule.)

General Form Hart Mfg.

- Let
- X1 units of product 1
- X2 units of product 2
- X3 units of product 3
- Y1 1 if production run, else 0
- Y2 1 if production run, else 0
- Y3 1 if production run, else 0

General Form Hart Manu.

- Max 25X1 28X2 30X3 400Y1 550Y2

600Y3 - s.t.
- 1.5X1 3X2 2X3 0Y1 0Y2

0Y3 ? 450 Dept. A - 2X1 X2 2.5X3 0Y1 0Y2

0Y3 ? 350 Dept. B - .25X1 .25X2 .25X3 0Y1 0Y2

0Y3 ? 50 Dept. C - 0X1 0X2 0X3 -175Y1 0Y2

0Y3 ? 0 Link P1 - 0X1 0X2 0X3 0Y1 -150Y2

0Y3 ? 0 Link P2 - 0X1 0X2 0X3 0Y1 0Y2

-140Y3 ? 0 Link P3 - and Xi ? 0 and Yi integer, ? 0, ? 1 (i.e.,

Binary)

Review Problems

- Ellis Electric
- Distribution Co.

Ellis Electric - Mixed-Integer Set-up Cost

- A problem faced by an electrical utility

each day is that of deciding which generators to

start up in order to minimize total cost. - The utility has three generators with the

characteristics shown in the table below. There

are two periods in a day, and the number of

megawatts needed in the first period is 2900.

The second period requires 3900 megawatts. - A generator may be started in either period but

if started in the first period, may be used in

the second period without incurring an additional

startup cost. All major generators (e.g. A, B,

and C) are turned off at the end of the day.

General Form Ellis Electric

- Let
- XA1 Power from Gen A in Period 1
- XB1 Power from Gen B in Period 1
- XC1 Power from Gen C in Period 1
- XA2 Power from Gen A in Period 2
- XB2 Power from Gen B in Period 2
- XC2 Power from Gen C in Period 2
- YA 1 if Generator A started else 0
- YB 1 if Generator A started else 0
- YC 1 if Generator A started else 0

General Form Ellis Electric

- Min 5XA1 5XA2 4XB1 4XB2 7XC1 4XC2

3000YA 2000YB 1000YC - s.t.
- 1XA1 0XA2 1XB1 0XB2 1XC 0XC2 1

0YA 0YB 0YC ? 2900 - 0XA1 1XA2 0XB1 1XB2 0XC 1XC2 1

0YA 0YB 0YC ? 3900 - 1XA1 0XA2 0XB1 0XB2 0XC 0XC2 1-

2100YA 0YB 0YC ? 0 - 0XA1 1XA2 0XB1 0XB2 0XC 0XC2 1-

2100YA 0YB 0YC ? 0 - 0XA1 0XA2 1XB1 0XB2 0XC 0XC2 1

0YA - 1800YB 0YC ? 0 - 0XA1 0XA2 0XB1 1XB2 0XC 0XC2 1

0YA - 1800YB 0YC ? 0 - 0XA1 0XA2 0XB1 0XB2 1XC 0XC2 1

0YA 0YB - 3000YC ? 0 - 0XA1 0XA2 0XB1 0XB2 0XC 1XC2 1

0YA 0YB - 3000YC ? 0 - and Xij ? 0 Yi INT, ? 0, ? 1 (Binary)

Distribution Company- Integer Transportation

- A distribution company wants to minimize the

cost of transporting goods from its warehouses A,

B, and C to the retail outlets 1, 2, and 3. - The fixed cost of operating a warehouse is

500 for A, 750 for B, and 600 for C, and at

least two of them have to be open. The

warehouses can be assumed to have adequate

storage capacity to store all units demanded,

i.e., assume each warehouse can store at least

525 units. - The costs for transporting one unit from

warehouse to retail outlet are given in the

following table

General Form Distribution Co.

- Let
- Xij units shipped from i to j
- YA 1 if warehouse A opens, else 0
- YB 1 if warehouse B opens, else 0
- YC 1 if warehouse C opens, else 0

General Form Distribution Co.

- Min
- 15XA1 32XA2 21XA3 9XB1 7XB2 6XB3

11XC1 18XC2 5XC3 500YA 750YB 600YC - s.t.
- 1XA1 0XA2 0XA3 1XB1 0XB2 0XB3

1XC1 0XC2 0XC3 0YA 0YB

0YC ? 200 - 0XA1 1XA2 0XA3 0XB1 1XB2 0XB3

0XC1 1XC2 0XC3 0YA 0YB

0YC ? 150 - 0XA1 0XA2 1XA3 0XB1 0XB2 1XB3

0XC1 0XC2 1XC3 0YA 0YB

0YC ? 175 - 1XA1 0XA2 0XA3 1XB1 0XB2 0XB3

1XC1 0XC2 0XC3 - 525YA 0YB 0YC

? 0 - 0XA1 1XA2 0XA3 0XB1 1XB2 0XB3

0XC1 1XC2 0XC3 0YA - 525YA 0YC

? 0 - 0XA1 0XA2 1XA3 0XB1 0XB2 1XB3

0XC1 0XC2 1XC3 0YA 0YB - 525YC

? 0 - 0XA1 0XA2 0XA3 0XB1 0XB2 0XB3

0XC1 0XC2 0XC3 1YA 1YB

1YC ? 2 - and Xij ? 0 Yi INT, ? 0, ? 1 (Binary)

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