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Few studies have been done on techniques to solve multiple objective nonlinear integer problems. This paper formulates an algorithm for nonlinear integer goal programming using a branch-and-bound method and Hwang & Masud's nonlinear goal programming method. The application of this algorithm is demonstrated by solving reliability problems with single and multiple objectives. The single objective nonlinear integer problem is solved by the nonlinear integer goal programming taking the constraints at priority level one and the objective at priority level two. One interesting feature in this algorithm is that the problem is solved by traditional nonlinear search techniques, such as Hooke and Jeeves pattern search, that are originally intended for solving the ``unconstrained'' problem. However, there is no way to guarantee finding the global optimum for a given problem. This means that the investigator must usually be satisfied with a local optimum or a set of local optima.