تجزیه و تحلیل حساسیت در بهینه سازی و مسائل قابلیت اطمینان

The paper starts giving the main results that allow a sensitivity analysis to be performed in a general optimization problem, including sensitivities of the objective function, the primal and the dual variables with respect to data. In particular, general results are given for non-linear programming, and closed formulas for linear programming problems are supplied. Next, the methods are applied to a collection of civil engineering reliability problems, which includes a bridge crane, a retaining wall and a composite breakwater. Finally, the sensitivity analysis formulas are extended to calculus of variations problems and a slope stability problem is used to illustrate the methods.

This paper deals with sensitivity analysis. Sensitivity analysis discusses “how” and “how much” changes in the parameters of an optimization problem modify the optimal objective function value and the point where the optimum is attained (see [1]). Today, it is not enough to give users the solutions to their problems. In addition, they require knowledge of how these solutions depend on data and/or assumptions. Therefore, data analysts must be able to supply the sensitivity of their conclusions to model and data. Sensitivity analysis allows the analyst to assess the effects of changes in the data values, to detect outliers or wrong data, to define testing strategies, to increase the reliability, to optimize resources, reduce costs, etc. Sensitivity analysis increases the confidence in the model and its predictions, by providing an understanding of how the model responds to changes in the inputs. Adding a sensitivity analysis to a study means adding extra quality to it. Sensitivity analysis is not a standard procedure, however, it is very useful to (a) the designer, who can know which data values are the most influential on the design, (b) to the builder, who can know how changes in the material properties or the prices influence the total reliability or the cost of the work being designed, and (c) to the code maker, who can know the costs and reliability implications associated with changes in the safety factors or failure probabilities. The methodology proposed below is very simple, efficient and allows all the sensitivities to be calculated simultaneously. At the same time it is the natural way of evaluating sensitivities when optimization procedures are present. The paper is structured as follows. In Section 2 the statement of optimization problems and the conditions to be satisfied are presented. Section 3 gives the formula to get sentivities with respect to the objective function. In Section 4, a general method for deriving all possible sensitivities is given. Section 5 deals with some examples and the interpretation of the sensitivity results. In Section 6 the methodology is extended to calculus of variations, and finally, Section 7 provides some relevant conclusions.

The main conclusions from this paper are: 1. There exist very simple and closed formulas for sensitivity analysis of the objective function and the primal and dual variables with respect to data in regular cases of linear programming problems (see [25]) 2. In the case of regular cases of non-linear programming a methodology has been given to perform a sensitivity analysis of the objective function and the primal and dual variables with respect to data. This implies solving the optimization problem and then constructing and solving a linear system of equations. 3. In non-regular cases a more involved methodology is required, which is given in another paper. 4. Sensitivity analysis can be done for calculus of variations problems in a similar way as it is done in optimization problems (linear and non-linear) and in optimal control problems, obtaining results for calculus of variations that are the parallel versions of those for the other problems. 5. Theorem 2 and Theorem 3 are the counterparts of Theorem 1 for calculus of variations in the finite and infinite cases, respectively. They allow obtaining closed formulas for the objective function sensitivities with respect to the data. 6. For calculus of variations, the sensitivities of the primal and dual variables and functions with respect to variable and function type data require perturbation analysis, as indicated in the paper. 7. The practical applications presented in this paper have illustrated and clarified the theory, and demonstrated the goodness of the proposed technique, together with the importance of the practical applications that can benefit from the proposed methods.