Wyndor with variations Every LP problem will fall into one of the following categories: 1. The problem has one (or more) optimal solution(s) 2. The problem has no optimal solution because a) The problem is unbounded b) The problem is inconsistent (infeasible) Example (No Feasible Solution) Maximize Z = 3x1 + 5x2 subject to x1 ≥ 5 x2 ≥ 4 3x1 + 2x2 ≤ 18 and x1 ≥ 0, x2 ≥ 0. x2 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x1 Example (Unbounded Solution) Maximize Z = 5x1 + 12x2 subject to x1 ≤ 5 2x1 –x2 ≤ 2 and x1 ≥ 0, x2 ≥ 0. x2 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x1 If we find an optimal solution we either have a • Normal solution • Multiple optima or • Degenerate solution Wyndor Glass normal solution Wyndor Glass Co. Product-Mix Problem Unit Profit Doors $300 Windows $500 Hours Hours Used Per Unit Produced Used Plant 1 1 0 2 Plant 2 0 2 0 Plant 3 3 2 0 Units Produced Doors 2 Windows 6 <= <= <= Hours Available 4 12 18 Total Profit $3 600 GLP optimal solution GLP sensitivity report SYNOPSIS OF THE SOLUTION OUTPUT • Optimal values for the - decision variables - slack/surplus variables - objective function • Shadow prices and allowable range of RHS changes • Allowable increase/decrease of objective coefficients • Reduced costs Multiple optima – graphical solution Wyndor – Multiple optima Wyndor, Degenerate solution – graphical representation Wyndor – degenerate solution • The meaning of "Adjustable cells" (upper part of the sensitivity report) • Nondegenerate solution • Allowable increase/decrease tells us how much we can increase/reduce a given coefficient in the objective function witkout changing the optimal solution (everything else fixed) • Whenever a coefficient is changing less than allowable the optimal solution will not change • If the coefficient of a variable is increased with the exact upper allowable amount there will be an alternative optimal solution with, for a max(min) model, higher (lower) value of the variable • If the coeff. is reduced with the exact maximum allowable amount there will be an alternative optimal solution with, for a max (min) model, lower (higher) value of the variable • Signs we can use to point out multiple optima: Allowable increase/decrease for the objective function coeff.. = 0 (i.e. any change of the coefficient will lead to another unique solution) • Degenerate solution • The signs of multiple optima are not valid • The objective funtion coefficient must change with at least, and possibly a lot more than "allowable increase/decrease" to give a new optimal solution Sensitivity analysis contd. • Meaning of ”Constraints” (lower part) • Shadow price: The change in OV (objective function value) when the RHS (right hand side) of that constraint is increased by one unit, everyhing else held constant • Allowable increase/decrease ranges give us the region for which the shadow prices are valid • Signs of a degenerate solution: Some of the shadow prices will have a 0 allowable increase or decrease (we only get effects on OV of one-sided changes in RHS for these) Shortcomings of Solver’s Sensitivity Report 1. The report gives sensitivity information for perturbations of parameters only in the immediate neighborhood of the solution, and only for changing one parameter at a time. 2. The report gives sensitivity only for effects upon the OV. 3. The report gives no sensitivity information for changes in the model’s technical coefficients.