Suppose that a linear programming problem has 20 functional constraints in inequality form such that their right-hand sides (the bi) are uncertain parameters, so chance constraints with some a are introduced in place of these constraints. After next substituting the deterministic equivalents of these chance constraints and solving the resulting new linear programming model, its optimal solution is found to satisfy 10 of these deterministic equivalents with equality whereas there is some slack in the other 10 deterministic equivalents. Answer the following questions under the assumption that the 20 uncertain bi have mutually independent normal distributions.
(a) When choosing a = 0.95, what are the lower bound and upper bound on the probability that all of these 20 original constraints will turn out to be satisfied by the optimal solution for the new linear programming problem so this solution actually will be feasible for the original problem.
(b) Now repeat part (a) with a = 0.99.
(c) Suppose that all 20 of these functional constraints are considered to be hard constraints, i.e., constraints that must be satisfied if at all possible. Therefore, the decision maker desires to use a value of a that will guarantee a probability of at least 0.95 that the optimal solution for the new linear programming problem actually will turn out to be feasible for the original problem. Use trial and error to find the smallest value of a (to three significant digits) that will provide the decision maker with the desired guarantee.