Integer Programming: A Popular Method for Optimizing Real-Life Situations, Slides of Discrete Structures and Graph Theory

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Integer Programming

Integer Programming

• 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.

Standard form of integer program (IP)

• In vector form:

maximize cx (objective function) subject to A x ≤ b (functional constraints) x ∈ (set constraints)

Input for IP: 1×n vector c , m×n matrice A , m×1 vector b.

Output of IP: n×1 integer vector x.

• Note : More often, we will consider

mixed integer programs (MIP), that is, some variables are integer, the others are continuous.

n Z +

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

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+ Docsity.com

Solutions to integer programs

• A solution is an assignment of values to variables.
• A solution can hence be thought of

as an n -dimensional vector.

• 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.

Next time :

IP modeling techniques