Integer Linear Programming

1
Chapter 6

• Integer Linear Programming

2
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

3
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.

4
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

5
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?

6
Define Variables - Northern Airlines

• Let
• S of Small Aircraft
• L of Large Aircraft

7
General Form - Northern Airlines

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

8
Northern Airlines Graph Solution
LP Relaxation (2, 1.8)
Budget
Small AC
9
Northern Airlines Graph Solution
Budget
Small AC
Rounded Solution (2, 1)
10
Northern Airlines Graph Solution
Optimal Solution (0, 2)
Budget
Small AC
11
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

12
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

13
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.

14
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

15
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

16
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

17
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.)

18
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

19
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)

20
Review Problems

• Ellis Electric
• Distribution Co.

21
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.

22
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

23
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)

24
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

25
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

26
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)