Give, if possible, an upper bound for the optimum value that is as tight as possible. Explain why this bound holds or why it is not possible to give an upper bound.

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Exercise 12

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Assume that we are solving a pure integer program with three decision variables. The goal is to minimize the objective function and the branch-and-bound method is used to solve the problem.

At an intermediate stage we have the branching tree as shown below. Each circle represents the LP relaxation of a sub-problem. The additional constraint of each sub-problem is shown within each circle, the optimum values and optimum solutions of the LP relaxations are shown under each circle.

Infeasible

(a)    Give, if possible, an upper bound for the optimum value that is as tight as possible. Explain why this bound holds or why it is not possible to give an upper bound.

(b)   Give, if possible, a lower bound for the optimum value that is as tight as possible. Explain why this bound holds or why it is not possible to give a lower bound.

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