Integer Programming

1
Integer Programming

• Introduction to Integer Programming (IP)
• Difficulties of LP relaxation
• IP Formulations
• Branch and Bound Algorithms
• Reference Chapter 9 in W. L. Winstons book.

2
Integer Programming Model

• An Integer Programming model is a linear
  programming problem where some or all of the
  variables are required to be non-negative
  integers.
• These models are in general substantially harder
  than solving linear programming models.
• Network models are special cases of integer
  programming models and are very efficiently
  solvable.
• We will discuss several applications of integer
  programming models.
• We will study the branch and bound technique, one
  of the most popular algorithm to solve integer
  programming models.

3
Classifications of IP Models

• Pure IP Model Where all variables must take
  integer values.
• Maximize z 3x1 2x2
• subject to x1 x2 6 x1, x2 ³ 0, x1
  and x2 integer
• Mixed IP Model Where some variables must be
  integer while others can take real values.
• Maximize z 3x1 2x2
• subject to x1 x2 6 x1, x2 ³ 0, x1
  integer
• 0-1 IP Model Where all variables must take
  values 0 or 1 .
• Maximize z x1 - x2
• subject to x1 2x2 2 2x1 - x2
  1, x1, x2 0 or 1

4
Classifications of IP Models (contd.)

• LP Relaxation The LP obtained by omitting all
  integer or 0-1 constraints on variables is called
  the LP relaxation of IP.
• IP
• Maximize z 21x1 11x2 subject to
  7x1 4x2 13 x1, x2 ³ 0, x1 and x2
  integer
• LP Relaxation
• Maximize z 21x1 11x2 subject to
  7x1 4x2 13 x1, x2 ³ 0
• Result
• Optimal objective function value of IP
  Optimal obj. function value of LP relaxation

5
IP and LP Relaxation
3
2
7x1 4x2 13
x2
1
x
x
x
x
3
1
2
x1
6
Simple Approaches for Solving IP

• Approach 1
• Enumerate all possible solutions
• Determine their objective function values
• Select the solution with the maximum (or,
  minimum) value.
• Any potential difficulty with this approach?
• Approach 2
• Solve the LP relaxation
• Round-off the solution to the nearest feasible
  integer solution
• Any potential difficulty with this approach?

7
Capital Budgeting Problem

• Stockco Co. is considering four investments
• It has 14,000 available for investment
• Formulate an IP model to maximize the NPV
  obtained from the investments
• IP
• Maximize z 16x1 22x2 12x3 8x4
• subject to
• 5x1 7x2 4x3 3x4 14
• x1, x2,,x3, x4 0, 1

8
Fixed Charge Problem

• Gandhi cloth company manufactures three types of
  clothing shirts, shorts, and pants
• Machinery must be rented on a weekly basis to
  make each type of clothing. Rental Cost
• 200 per week for shirt machinery
• 150 per week for shorts machinery
• 100 per week for pants machinery
• There are 150 hours of labor available per week
  and 160 square yards of cloth
• Find a solution to maximize the weekly profit

9
Fixed Charge Problem (contd.)

• Decision Variables
• x1 number of shirts produced each week
• x2 number of shorts produced each week
• x3 number of pants produced each week
• y1 1 if shirts are produced and 0 otherwise
• y2 1 if shorts are produced and 0 otherwise
• y3 1 if pants are produced and 0 otherwise
• Formulation
• Max. z 6x1 4x2 7x3 - 200y1 - 150 y2 - 100y3
• subject to
• 3x1 2x2 6x3 150
• 4x1 3x2 4x3 160 x1 M y1, x2 M
  y2, x3 M y3
• x1, x2,,x3 ³ 0, and integer y1,
  y2,,y3 ³ 0 or 1

10
Either-Or Constraints

• Dorian Auto is considering manufacturing three
  types of auto compact, midsize, large.
• Resources required and profits obtained from
  these cars are given below.
• We have 6,000 tons of steel and 60,000 hours of
  labor available.
• If any car is produced, we must produce at least
  1,000 units of that car.
• Find a production plan to maximize the profit.

11
Either-Or Constraints (contd.)

• Decision Variables
• x1, x2, x3 number of compact, midsize and large
  cars produced
• y1, y2, y3 1 if compact , midsize and large
  cars are produced or not
• Formulation
• Maximize z 2x1 3x2 4x3
• subject to
• x1 My1 x2 My2 x3 My3
• 1000 - x1 M(1-y1)
• 1000 - x2 M(1-y2)
• 1000 - x3 M(1-y3)
• 1.5 x1 3x2 5x3 6000
• 30 x1 25x2 40 x3 60000
• x1, x2, x3 ³ 0 and integer y1, y2, y3 0 or 1

12
Set Covering Problems

• Western Airlines has decided to have hubs in USA.
  
• Western runs flights between the following
  cities Atlanta, Boston, Chicago, Denver,
  Houston, Los Angeles, New Orleans, New York,
  Pittsburgh, Salt Lake City, San Francisco, and
  Seattle.
• Western needs to have a hub within 1000 miles of
  each of these cities.
• Determine the minimum number of hubs

13
Formulation of Set Covering Problems

• Decision Variables
• xi 1 if a hub is located in city i
• xi 0 if a hub is not located in city i
• Minimize xAT xBO xCH xDE xHO xLA xNO
  xNY xPI xSL xSF xSE
• subject to

14
Additional Applications

• Location of fire stations needed to cover all
  cities
• Location of fire stations to cover all regions
• Truck despatching problem
• Political redistricting
• Capital investments

15
Branch and Bound Algorithm

• Branch and bound algorithms are the most popular
  methods for solving integer programming problems
• They enumerate the entire solution space but only
  implicity hence they are called implicit
  enumeration algorithms.
• A general-purpose solution technique which must
  be specialized for individual IP's.
• Running time grows exponentially with the problem
  size, but small to moderate size problems can be
  solved in reasonable time.

16
An Example

• Telfa Corporation makes tables and chairs
• A table requires one hour of labor and 9 square
  board feet of wood
• A chair requires one hour of labor and 5 square
  board feet of wood
• Each table contributes 8 to profit, and each
• chair contributes 5 to profit.
• 6 hours of labor and 45 square board feet is
• available
• Find a product mix to maximize the profit
• Maximize z 8x1 5x2
• subject to x1 x2 6 9x1 5x2 45 x1,
  x2 ³ 0 x1, x2 integer

17
Feasible Region for Telfas Problem

• Subproblem 1 The LP relaxation of original
• Optimal LP Solution x1 3.75 and x2 2.25 and
  z 41.25
• Subproblem 2 Subproblem 1 Constraint x1 ³ 4
• Subproblem 3 Subproblem 1 Constraint x1 3

18
Feasible Region for Subproblems

• Branching The process of decomposing a
  subproblem into two or more subproblems is called
  branching.
• Optimal solution of Subproblem 2
• z 41, x1 4, x2 9/5 1.8
• Subproblem 4 Subproblem 2 Constraint x2 ³ 2
• Subproblem 5 Subproblem 2 Constraint x2 1

19
Feasible Region for Subproblems 4 5
20
The Branch and Bound Tree
3
Optimal solution of Subproblem 5 z 40.05,
x1 4.44, x2 1 Subproblem 6 Subproblem
5 Constraint x1 ³ 5 Subproblem 5 Subproblem 5
Constraint x1 4
21
Feasible Region for Subproblems 6 7
Optimal solution of Subproblem 7 z 37,
x1 4, x2 1 Optimal solution of
Subproblem 6 z 40, x1 5, x2 0
22
The Branch and Bound Tree
1
x1 ³ 4
x1 3
Subproblem 3 z 3 x1 3 x2 1, LB 39
7
2
x2 1
x2 ³ 2
3
4
5
6
23
Solving Knapsack Problems

• Max z 16x1 22x2 12x3 8x4
• subject to
• 5x1 7x2 4x3 3x4 14
• xi 0 or 1 for all i 1, 2, 3, 4
• LP Relaxation
• Max z 16x1 22x2 12x3 8x4
• subject to
• 5x1 7x2 4x3 3x4 14
• 0 xi 1 for all i 1, 2, 3, 4
• Soving the LP Relaxation
• Order xis in the decreasing order of ci/ai where
  ci are the cost coefficients and ais are the
  coefficients in the constraint
• Select items in this order until the constraint
  is satisfied with equality

24
The Branch and Bound Tree
25
Strategies of Branch and Bound

• The branch and bound algorithm is a divide and
  conquer algorithm, where a problem is divided
  into smaller and smaller subproblems. Each
  subproblem is solved separately, and the best
  solution is taken.
• Lower Bound (LB) Objective function value of the
  best solution found so far.
• Branching Strategy The process of decomposing a
  subproblem into two or more subproblems is called
  branching.

26
Strategies of Branch and Bound (contd.)

• Upper Bounding Strategy The process of obtaining
  an upper bound (UB) for each subproblem is called
  an upper bounding strategy.
• Pruning Strategy If for a subproblem, UB LB,
  then the subproblem need not be explored further.
• Searching Strategy The order in which
  subproblems are examined. Popular search
  strategies LIFO and FIFO.