## 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, D) and four operations to be carried out (1, 2, 3, 4). 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 way to arrange the team so 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 steps through the branch and bound method to find the assignment of people to operations that gives the highest probability of success. We can then answer the questions faced by the manager.