Chapter 6 Optimization Models with Integer Variables Introduction

Chapter 6 Optimization Models with Integer Variables

Introduction • Binary variable: – A decision variable that is permitted to take only two possible values, 0 or 1 – Usually a 0– 1 variable corresponds to an activity that either is or is not undertaken. – If it equals 1, the activity is undertaken; if it equals 0, the activity is not undertaken.

Solving models with binary variables • Complete enumeration – look at all possible solutions and select the best – impractical because as the number of variables increases the number of possible solutions to enumerate will increase exponentially – a model with 100 binary changing cells will have 2100 possible solutions to enumerate -- 2100 is an extremely large number, so it would take even a very fast computer a long time to check each one of them.

Solving models with binary variables • implicit enumeration – Branch and Bound method – used by Solver in IP models – Branching – systematically creating two problems (branches) setting a given binary variable to 0 or 1 – incumbent solution – current best feasible solution; is a lower bound for a Max problem – Upper bound – the maximum possible objective function for a given branch for a Max problem – Initially LP relaxation serves as the upper bound – If Upper bound < incumbent solution abandon the branch

Solver Tolerance setting A tolerance setting of 5% means that Solver stops as soon as it finds a feasible (integer) solution to the IP model that is within 5% of the current upper bound.

Solver messages