Sensitivity analysis in linear programming problem ( Muhammed Jiyad)

Topic includes : *Sensitivity Analysis *Objective function *Right Hand Side(RHS) *Sensitivity analysis using graph *Objective function coefficient *Reduced cost *Shadow pricing *Shadow pricing Microsoft Excel sensitivity report and solution.

1. SENSITIVITY ANALYSIS IN LINEAR PROGRAMMING PROBLEM PRESENTED BY MUHAMMED JIYAD.K 1712012

2. SENSITIVITY ANALYSIS Sensitivity analysis serves as an integral part of solving linear programming model & is normally carried out after the optimal solution is obtained. It determines how sensitive the optimal solution is to making changes in the original model. Sensitivity analysis allows us to determine how “sensitive” the optimal solution is to changes in data values. Sensitivity analysis is important to the manager who must operate in a dynamic environment with imprecise estimates of the coefficients. Sensitivity analysis allows him to ask certain what-if questions about the problem. Sensitivity analysis is used to determine how the optimal solution is affected by changes, within specified ranges, in: • the objective function coefficients • the right-hand side (RHS) values

3. OBJECTIVE FUNCTION The feasible region does not change. Since constraints are not affected, decision variable values remain the same. Objective function value will change. RIGHT HAND SIDE Feasible region changes. If a nonbinding constraint is changed, the solution is not affected. If a binding constraint is changed, the same corner point remains optimal but the variable values will change.

4. SENSITIVITY ANALYSIS USING GRAPH 1. Maximize Z = $100x 1 + $50x 2 subject to: 1x 1 + 2x 2  40 4x 2 + 3x 2  120 x 1 , x 2 

5. 2. Maximize Z = $40x 1 + $50x 2 subject to: 1x 1 + 2x 2  40 4x 2 + 3x 2  120 x 1 , x 2  SENSITIVITY ANALYSIS USING GRAPH

6. 3. Maximize Z = $40x 1 + $100x 2 subject to: 1x 1 + 2x 2  40 4x 2 + 3x 2  120 x 1 , x 2  SENSITIVITY ANALYSIS USING GRAPH

7. OBJECTIVE FUNCTION COEFFICIENT SENSITIVITY RANGE The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point will remain optimal. The sensitivity range for the x1 coefficient is designed as c1 .

8. SENSITIVITY ANALYSIS OF OBJECTIVE FUNCTION COEFFICIENTS RANGES OF OPTIMALITY The value of the objective function will change if the coefficient multiplies a variable whose value is non - zero . The optimal solution will remain unchanged as long as: * A n objective function coefficient lies within its range of optimality * T here are no changes in any other input parameters.

9. SENSITIVITY ANALYSIS OF OBJECTIVE FUNCTION COEFFICIENTS The optimality range for an objective coefficient is the range of values over which the current optimal solution point will remain optimal . For two variable LP problems the optimality ranges of objective function coefficients can be found by setting the slope of the objective function equal to the slopes of each of the binding constraints .

10. Sensitivity Analysis Using Graphs ( Objective Function Coefficient Sensitivity Range for c 1 and c 2 )

11. Objective Function Coefficient Sensitivity Range (for a Cost Minimization Model) Minimize Z = $6x 1 + $3x 2 subject to: 2x 1 + 4x 2  16 4x 1 + 3x 2  24 x 1 , x 2  sensitivity ranges: 4  c 1    c 2  4.5

12. SENSITIVITY ANALYSIS FOR RHS VALUES The sensitivity range for a RHS value is the range of values over which the quantity (RHS) values can change without changing the solution variable mix, including slack variables Any change in the right hand side of a binding constraint will change the optimal solution Any change in the right-hand side of a nonbinding constraint that is less than its slack or surplus, will cause no change in the optimal solution .

13. Changes in Constraint Quantity (RHS) Values Increasing the Labor Constraint Maximize Z = $40x 1 + $50x 2 subject to: 1x 1 + 2x 2  40 4x 2 + 3x 2  120 x 1 , x 2 

14. Changes in Constraint Quantity (RHS) Values Increasing the Labor Constraint

15. Changes in Constraint Quantity (RHS) Values Sensitivity Range for Clay Constraint

16. REDUCED COST The reduced cost for a variable at its lowe r bound yields: The minimum amount by which the OFC of a variable should change to cause that variable to become non-zero. The amount the profit coefficient must change before the variable can take on a value above its lower bound . The amount the optimal profit will change per unit increase in the variable from its lower bound . The amount by which the objective function value would change if the variable were forced to change from 0 to 1.

17. SHADOW PRICING Defined as the marginal value of one additional unit of resource Shadow Price is change in optimal objective function value for one unit increase in RHS. In linear programming problems the shadow price of a constraint is the difference between the optimized value of the objective function and the value of the objective function, evaluated at the optional basis. The sensitivity range for a constraint quantity value is also the range over which the shadow price is valid.

18. Shadow Prices (Excel Sensitivity Report ) * Maximize Z = $40x 1 + $50x 2 subject to: x 1 + 2x 2  40 hr of labor 4x 1 + 3x 2  120 lb of clay x 1 , x 2 

19. Solution (Excel Screen)

20. CONCLUSION In this topic we include sensitivity analysis, objective function, Right Hand Side (RHS), sensitivity analysis using graphs, objective function coefficient for cost minimization model, sensitivity analysis for RHS value changes, changes in constraint quantity, reduced cost and shadow pricing. So far, we discussed all those things In this presentation. So I conclude with a note. “Sensitivity analysis serves as an integral part of solving linear programming model “ Thank you