Branch and Bound Method

Example 1: Branch and Bound Method

A company is assembling a team to carry out a series of operations. There are four members of the team: A, B, C and D, and four operations to be carried out. Each team member can carry out exactly one operation. All four operations must be carried out successfully for the overall project to succeed, however the probability of a particular team member succeeding in a particular operation varies, as shown in the table below. For example, if the team members were assigned to operations in the order ABCD, then the overall probability of successful completion of the project is (0.9)(0.6)(0.85)(0.7) = 0.3213.

If there is any possible way that the team can be arranged such that the overall probability of success exceeds 45%, then the manager will approve the project. Will the manager approve the project? If yes, what is the arrangement of the team that gives the highest probability of success?

The formal branch and bound formulation for this problem follows:

Meaning of a node in the branch and bound tree: a person-operation assignment, full or partial.
Node selection policy: global best value of the bounding function
Variable selection policy: choose the next operation in natural order, 1 to 4.
Bounding function: for unassigned operations, choose the best unassigned person (the one with the highest probability of success) even if that person is chosen more than once.
Terminating Rule: when the incumbent solution objective function value is better than or equal to the bounding function values associated with all of the bud nodes.
Fathoming: when a bounding function gives a solution in which each operation is assigned to a different person.

The example below will step through the branch and bound method in order to find a solution to this problem.

Press the Start button to begin the example.

This animation was made using Alligator Flash Designer 7. More information about this program is available at Selteco Alligator.
You can view the source code for this animation using the trial verson of Alligator Flash Designer 7 and this file.

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