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پاورپوینت کامل Sensitivity Analysis: An Applied Approach 37 اسلاید در PowerPoint

اسلاید ۴: ۵.۱ – A Graphical Approach to Sensitivity AnalysisGraphical analysis of the effect of a change in an objective function value for the Giapetto LP shows:By inspection, we can see that making the slope of the isoprofit line more negative than the finishing constraint (slope = -2) will cause the optimal point to switch from point B to point C. Likewise, making the slope of the isoprofit line less negative than the carpentry constraint (slope = -1) will cause the optimal point to switch from point B to point A.Clearly, the slope of the isoprofit line must be between -2 and -1 for the current basis to remain optimal.

اسلاید ۵: ۵.۱ – A Graphical Approach to Sensitivity AnalysisThe values of the contribution to profit for soldiers for which the current optimal basis (x1,x2,s3) will remain optimal can be determined as follows:Let c1 be the contribution ($3 per soldier) to the profit. For what values of c1 does the current basis remain optimal3x1 + 2×2 = constantRearranging:At present c1 = 3 and each isoprofit line has the form:Since -2 < slope < -1:Solving for c1 yields:Note: the profit will change in this range of c1

اسلاید ۶: ۵.۱ – A Graphical Approach to Sensitivity AnalysisGraphical Analysis of the Effect of a Change in RHS on the LP’s Optimal Solution (using the Giapetto problem).A graphical analysis can also be used to determine whether a change in the rhs of a constraint will make the current basis no longer optimal. For example, let b1 = number of available finishing hours.The current optimal solution (point B) is where the carpentry and finishing constraints are binding. If the value of b1 is changed, then as long as where the carpentry and finishing constraints are binding, the optimal solution will still occur where the carpentry and finishing constraints intersect.

اسلاید ۷: ۵.۱ – A Graphical Approach to Sensitivity AnalysisIn the Giapetto problem to the right, we see that if b1 > 120, x1 will be greater than 40 and will violate the demand constraint. Also, if b1 < 80, x1 will be less than 0 and the nonnegativity constraint for x1 will be violated.Therefore: 80 b1 120The current basis remains optimal for 80 b1 120, but the decision variable values and z-value will change.

اسلاید ۸: ۵.۱ – A Graphical Approach to Sensitivity AnalysisShadow Prices (using the Giapetto problem) It is often important to determine how a change in a constraint’s rhs changes the LP’s optimal z-value. We define:The shadow price for the i th constraint of an LP is the amount by which the optimal z-value is improved (increased in a max problem or decreased in a min problem) if the rhs of the i th constraint is increased by one. This definition applies only if the change in the rhs of constraint i leaves the current basis optimal.For the finishing constraint, 100 + D finishing hours are available (assuming the current basis remains optimal). The LP’s optimal solution is then x1 = 20 + D and x2 = 60 – D with z = 3×1 + 2×2 = 3(20 + D) + 2(60 – D) = 180 + D. Thus, as long as the current basis remains optimal, a one-unit increase in the number of finishing hours will increase the optimal z-value by $1. So, the shadow price for the first (finishing hours) constraint is $1.

اسلاید ۹: ۵.۱ – A Graphical Approach to Sensitivity AnalysisImportance of Sensitivity AnalysisSensitivity analysis is important for several reasons:Values of LP parameters might change. If a parameter changes, sensitivity analysis shows it is unnecessary to solve the problem again. For example in the Giapetto problem, if the profit contribution of a soldier changes to $3.50, sensitivity analysis shows the current solution remains optimal.Uncertainty about LP parameters. In the Giapetto problem for example, if the weekly demand for soldiers is at least 20, the optimal solution remains 20 soldiers and 60 trains. Thus, even if demand for soldiers is uncertain, the company can be fairly confident that it is still optimal to produce 20 soldiers and 60 trains.

اسلاید ۱۰: ۵.۲ – The Computer and Sensitivity AnalysisIf an LP has more than two decision variables, the range of values for a rhs (or objective function coefficient) for which the basis remains optimal cannot be determined graphically. These ranges can be computed by hand but this is often tedious, so they are usually determined by a packaged computer program. LINDO will be used and the interpretation of its sensitivity analysis discussed.

اسلاید ۱۱: ۵.۲ – The Computer and Sensitivity AnalysisConsider the following maximization problem. Winco sells four types of products. The resources needed to produce one unit of each are: Product 1Product 2Product 3Product 4AvailableRaw material23474600Hours of labor34565000Sales price$4$6$7$8To meet customer demand, exactly 950 total units must be produced. Customers demand that at least 400 units of product 4 be produced. Formulate an LP to maximize profit.Let xi = number of units of product i produced by Winco.

اسلاید ۱۲: ۵.۲ – The Computer and Sensitivity AnalysisThe Winco LP formulation:max z = 4×1 + 6×2 +7×3 + 8x4s.t. x1 + x2 + x3 + x4 = 950 x4 400 2×1 + 3×2 + 4×3 + 7×4 4600 3×1 + 4×2 + 5×3 + 6×4 5000 x1,x2,x3,x4 0

اسلاید ۱۳: ۵.۲ – The Computer and Sensitivity AnalysisMAX 4 X1 + 6 X2 + 7 X3 + 8 X4 SUBJECT TO 2) X1 + X2 + X3 + X4 = 950 3) X4 >= 400 4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= 4600 5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000 END LP OPTIMUM FOUND AT STEP 4 OBJECTIVE FUNCTION VALUE 1) 6650.000 VARIABLE VALUE REDUCED COST X1 0.000000 1.000000 X2 400.000000 0.000000 X3 150.000000 0.000000 X4 400.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 3.000000 3) 0.000000 -2.000000 4) 0.000000 1.000000 5) 250.000000 0.000000 NO. ITERATIONS= 4LINDO output and sensitivity analysis example(s).Reduced cost is the amount the objective function coefficient for variable i would have to be increased for there to be an alternative optimal solution.

اسلاید ۱۴: ۵.۲ – The Computer and Sensitivity Analysis RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X1 4.000000 1.000000 INFINITY X2 6.000000 0.666667 0.500000 X3 7.000000 1.000000 0.500000 X4 8.000000 2.000000 INFINITY RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 950.000000 50.000000 100.000000 3 400.000000 37.500000 125.000000 4 4600.000000 250.000000 150.000000 5 5000.000000 INFINITY 250.000000LINDO sensitivity analysis example(s).Allowable range (w/o changing basis) for the x2 coefficient (c2) is:5.