Integer Programming

1
Integer Programming

• Integer programming is a solution method for many
  discrete optimization problems
• Programming Planning in this context
• Origins go back to military logistics in WWII
  (1940s).
• In a survey of Fortune 500 firms, 85 of those
  responding said that they had used linear or
  integer programming.
• Why is it so popular?
• Many different real-life situations can be
  modeled as integer programs (IPs).
• There are efficient algorithms to solve IPs.

2
Standard form of integer program (IP)

• maximize c1x1c2x2cnxn (objective function)
• subject to
• a11x1a12x2a1nxn ? b1 (functional
  constraints)
• a21x1a22x2a2nxn ? b2
• .
• am1x1am2x2amnxn ? bm
• x1, x2 , , xn ? Z (set constraints)
• Note Can also have equality or constraint
• in non-standard form.

3
Example of Integer Program(Production
Planning-Furniture Manufacturer)

• Technological data
• Production of 1 table requires 5 ft pine, 2 ft
  oak, 3 hrs labor
• 1 chair requires 1 ft pine, 3 ft oak, 2
  hrs labor
• 1 desk requires 9 ft pine, 4 ft oak, 5
  hrs labor
• Capacities for 1 week 1500 ft pine, 1000 ft
  oak,
• 20 employees (each works 40 hrs).
• Market data
• Goal Find a production schedule for 1 week
• that will maximize the profit.

profit demand
table 12/unit 40
chair 5/unit 130
desk 15/unit 30
4
Production Planning-Furniture Manufacturer
modeling the problem as integer program

• The goal can be achieved
• by making appropriate decisions.
• First define decision variables
• Let xt be the number of tables to be produced
• xc be the number of chairs to be produced
• xd be the number of desks to be produced.

5
Production Planning-Furniture Manufacturer
modeling the problem as integer program

• Objective is to maximize profit
• max 12xt 5xc 15xd
• Functional Constraints
• capacity constraints
• pine 5xt 1xc 9xd ? 1500
• oak 2xt 3xc 4xd ? 1000
• labor 3xt 2xc 5xd ? 800
• market demand constraints
• tables xt 40
• chairs xc 130
• desks xd 30
• Set Constraints
• xt , xc , xd ? Z

6
Solutions to integer programs

• A solution is an assignment of values to
  variables.
• A feasible solution is an assignment of values to
  variables such that all the constraints are
  satisfied.
• The objective function value of a solution is
  obtained by evaluating the objective function at
  the given point.
• An optimal solution (assuming maximization) is
  one whose objective function value is greater
  than or equal to that of all other feasible
  solutions.
• There are efficient algorithms for finding the
  optimal solutions of an integer program.

7
Next IP modeling techniques

• Modeling techniques
• Using binary variables
• Restrictions on number of options
• Contingent decisions
• Variables with k possible values
• Applications
• Facility Location Problem
• Knapsack Problem

8
Example of IP Facility Location

• A company is thinking about building new
  facilities in LA and SF.
• Relevant data
• Total capital available for investment 10M
• Question Which facilities should be built
• to maximize the total profit?

capital needed expected profit
1. factory in LA 6M 9M
2. factory in SF 3M 5M
3. warehouse in LA 5M 6M
4. warehouse in SF 2M 4M
9
Example of IP Facility Location

• Define decision variables (i 1, 2, 3, 4)
• Then the total expected benefit 9x15x26x34x4
• the total capital needed
  6x13x25x32x4
• Summarizing, the IP model is
• max 9x15x26x34x4
• s.t. 6x13x25x32x4 ? 10
• x1, x2, x3, x4 binary ( i.e., xi
  ?0,1 )

10
The Facility Location Problem adding new
requirements

• Extra requirement
• build at most one of the two warehouses.
• The corresponding constraint is
• x3 x4 ? 1
• Extra requirement
• build at least one of the two factories.
• The corresponding constraint is
• x1 x2 1

11
Modeling Technique Restrictions on the number
of options

• Suppose in a certain problem, n different options
  are considered. For i1,,n
• Restrictions At least p and at most q of the
  options can be chosen.
• The corresponding constraints are

12
Modeling Technique Contingent Decisions

• Back to the facility location problem.
• Requirement Cant build a warehouse unless
  there is a factory in the city.
• The corresponding constraints are
• x3 ? x1 (LA) x4 ? x2 (SF)
• Requirement Cant select option 3 unless
• at least one of options 1 and 2 is selected.
• The constraint x3 ? x1 x2
• Requirement Cant select option 4 unless
• at least two of options 1, 2 and 3 are
  selected.
• The constraint 2x4 ? x1 x2 x3

13

• More on Integer Programming and other discrete
  optimization problems and techniques
• Math 4620
• Linear and Nonlinear Programming
• Math 4630
• Discrete Modeling and Optimization