An approach to the sensitivity analysis for discrete optimization problems with perturbed objective function is presented. The problem is stated in the following general form: min {&Sgr;e∈Yc(e) : Y ∈ ℱ}, where c = (c(e), e ∈ E) is a vector of weights for some finite set E, and F ⊆ 2E is a given family of feasible subsets. It is assumed that the set of feasible solutions ℱ is fixed, but the coefficients of the vector c may vary. The main problem considered in this paper concerns the following question: How does the maximum relative error of a given feasible solution depend on inaccuracies of the problem data? Two particular perturbations of the vector c are considered: (i) it is assumed that c is in a closed ball with radius δ ≥ 0 and center c0 ≥ 0 or (ii) the relative deviation of c(e) from the value c0(e) is not greater than δ for any e ∈ E, i.e., |c(e) – c0(e) | ≤ δc0(e), e ∈ E. For a given feasible solution X ∈ ℱ, two functions of the parameter δ are introduced: the sensitivity function s(X, δ) and the accuracy function a(X, δ). The values of s(X, δ) and a(X, δ) are equal to the maximum relative error of the solution X when the perturbations of problem data are of the type (i) or (ii), respectively. Some general, but computationally inefficient formulae for calculating these functions are derived and their properties are studied. It is shown that s(X, δ) and a(X, δ) are nondecreasing, convex functions with a limited number of breakpoints. A practically efficient method of calculating upper and lower envelopes for the accuracy function is presented. This method is based on the notion of k-best solutions of the problem. It gives an interval to which the maximum relative error of the solution X must belong when the coefficients of vector c are given with accuracy δ. The approach is illustrated with an example of the symmetric traveling salesman problem.
