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. PRESENTED BY
MUHAMMED JIYAD.K
1712012

2. • 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 = $100x1 + $50x2
subject to: 1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0

5. 2. Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
SENSITIVITY ANALYSIS USING GRAPH

6. 3. Maximize Z = $40x1 + $100x2
subject to: 1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
SENSITIVITY ANALYSIS USING GRAPH

7. 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. 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:
* An objective function coefficient lies within its range of
optimality
* There are no changes in any other input parameters.

9. • 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 c1
and c2 )

11. Objective Function Coefficient Sensitivity
Range (for a Cost Minimization Model)
Minimize Z = $6x1 + $3x2
subject to:
2x1 + 4x2  16
4x1 + 3x2  24
x1, x2  0
sensitivity ranges:
4  c1  
0  c2  4.5

12. • 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 = $40x1 + $50x2
subject to: 1x1 + 2x2  40
4x2 + 3x2 120
x1, x2  0

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 lower 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 = $40x1 + $50x2
subject to:
x1 + 2x2  40 hr of labor
4x1 + 3x2  120 lb of clay
x1, x2  0

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