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Sensitivity Analysis Sensitivity Analysis

Introduction to Sensitivity AnalysisIntroduction to Sensitivity Analysis Graphical Sensitivity AnalysisGraphical Sensitivity Analysis Sensitivity Analysis: Computer Sensitivity Analysis: Computer

SolutionSolution Simultaneous ChangesSimultaneous Changes

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Introduction to Introduction to Sensitivity AnalysisSensitivity Analysis

Sensitivity analysisSensitivity analysis (or post-optimality (or post-optimality analysis) is used to determine how the optimal analysis) is used to determine how the optimal solution is affected by changes, within solution is affected by changes, within specified ranges, in:specified ranges, in: the objective function coefficientsthe objective function coefficients the right-hand side (RHS) valuesthe right-hand side (RHS) values

Sensitivity analysis is important to a manager Sensitivity analysis is important to a manager who must operate in a dynamic environment who must operate in a dynamic environment with imprecise estimates of the coefficients. with imprecise estimates of the coefficients.

Sensitivity analysis allows a manager to ask Sensitivity analysis allows a manager to ask certain certain what-if questionswhat-if questions about the problem. about the problem.

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Example 1Example 1

LP FormulationLP Formulation

Max 5Max 5xx11 + 7 + 7xx22

s.t. s.t. xx11 << 6 6

22xx11 + 3 + 3xx22 << 19 19

xx11 + + xx22 << 8 8

xx11, , xx22 >> 0 0

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Example 1Example 1 Graphical SolutionGraphical Solution

22xx11 + 3 + 3xx22 << 19 19

xx22

xx11

xx11 + + xx22 << 8 8Max 5x1 + 7x2Max 5x1 + 7x2

xx11 << 6 6

Optimal Solution:Optimal Solution: xx11 = 5, = 5, xx22 = 3 = 3

88

77

66

55

44

33

22

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1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 1010

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Objective Function Objective Function CoefficientsCoefficients

Let us consider how changes in the objective Let us consider how changes in the objective function coefficients might affect the optimal function coefficients might affect the optimal solution.solution.

The The range of optimalityrange of optimality for each coefficient for each coefficient provides the range of values over which the provides the range of values over which the current solution will remain optimal.current solution will remain optimal.

Managers should focus on those objective Managers should focus on those objective coefficients that have a narrow range of optimality coefficients that have a narrow range of optimality and coefficients near the endpoints of the range.and coefficients near the endpoints of the range.

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Example 1Example 1

Changing Slope of Objective FunctionChanging Slope of Objective Function

xx11

FeasibleFeasibleRegionRegion

1111 2222

33334444

5555

xx22 Coincides withCoincides withxx11 + + xx22 << 8 8

constraint lineconstraint line

88

77

66

55

44

33

22

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1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 1010

Coincides withCoincides with22xx11 + 3 + 3xx22 << 19 19

constraint lineconstraint line

Objective functionObjective functionline for 5line for 5xx11 + 7 + 7xx22

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Range of OptimalityRange of Optimality

Graphically, the limits of a range of Graphically, the limits of a range of optimality are found by changing the slope of optimality are found by changing the slope of the objective function line within the limits of the objective function line within the limits of the slopes of the binding constraint lines.the slopes of the binding constraint lines.

Slope of an objective function line, Max Slope of an objective function line, Max cc11xx11 + + cc22xx22, is -, is -cc11//cc22, and the slope of a , and the slope of a constraint, constraint, aa11xx11 + + aa22xx22 = = bb, is -, is -aa11//aa22..

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Example 1Example 1 Range of Optimality for Range of Optimality for cc11

The slope of the objective function line is The slope of the objective function line is --cc11//cc22. The slope of the first binding constraint, . The slope of the first binding constraint, xx11 + + xx22 = 8, is -1 and the slope of the second binding = 8, is -1 and the slope of the second binding constraint, constraint, xx11 + 3 + 3xx22 = 19, is -2/3. = 19, is -2/3.

Find the range of values for Find the range of values for cc11 (with (with cc22 staying staying 7) such that the objective function line slope lies 7) such that the objective function line slope lies between that of the two binding constraints:between that of the two binding constraints:

-1 -1 << - -cc11/7 /7 << -2/3 -2/3

Multiplying through by -7 (and reversing the Multiplying through by -7 (and reversing the inequalities):inequalities):

14/3 14/3 << cc11 << 7 7

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Example 1Example 1 Range of Optimality for Range of Optimality for cc22

Find the range of values for Find the range of values for cc22 ( with ( with cc11 staying 5) staying 5) such that the objective function line slope lies between such that the objective function line slope lies between that of the two binding constraints:that of the two binding constraints:

-1 -1 << -5/ -5/cc22 << -2/3 -2/3

Multiplying by -1: 1 Multiplying by -1: 1 >> 5/ 5/cc22 >> 2/3 2/3

Inverting, Inverting, 1 1 << cc22/5 /5 << 3/2 3/2

Multiplying by 5: Multiplying by 5: 5 5 << cc22 << 15/2 15/2

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Software packages such as Software packages such as The Management The Management ScientistScientist and and

Microsoft ExcelMicrosoft Excel provide the following LP provide the following LP information:information:

Information about the objective function:Information about the objective function:• its optimal valueits optimal value• coefficient ranges (ranges of optimality)coefficient ranges (ranges of optimality)

Information about the decision variables:Information about the decision variables:• their optimal valuestheir optimal values• their reduced coststheir reduced costs

Information about the constraints:Information about the constraints:• the amount of slack or surplusthe amount of slack or surplus• the dual pricesthe dual prices• right-hand side ranges (ranges of feasibility)right-hand side ranges (ranges of feasibility)

Sensitivity Analysis: Sensitivity Analysis: Computer SolutionComputer Solution

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Example 1Example 1

Range of Optimality for Range of Optimality for cc11 and and cc22

Adjustable CellsAdjustable Cells

FinalFinal ReducedReduced ObjectiveObjective AllowableAllowable AllowableAllowableCellCell NameName ValueValue CostCost CoefficientCoefficient IncreaseIncrease DecreaseDecrease

$B$8$B$8 X1X1 5.05.0 0.00.0 55 22 0.333333330.33333333$C$8$C$8 X2X2 3.03.0 0.00.0 77 0.50.5 22

ConstraintsConstraints

FinalFinal ShadowShadow ConstraintConstraint AllowableAllowable AllowableAllowableCellCell NameName ValueValue PricePrice R.H. SideR.H. Side IncreaseIncrease DecreaseDecrease

$B$13$B$13 #1#1 55 00 66 1E+301E+30 11$B$14$B$14 #2#2 1919 22 1919 55 11$B$15$B$15 #3#3 88 11 88 0.333333330.33333333 1.666666671.66666667

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Right-Hand SidesRight-Hand Sides Let us consider how a change in the right-hand Let us consider how a change in the right-hand

side for a constraint might affect the feasible side for a constraint might affect the feasible region and perhaps cause a change in the region and perhaps cause a change in the optimal solution.optimal solution.

The improvement in the value of the optimal The improvement in the value of the optimal solution per unit increase in the right-hand side solution per unit increase in the right-hand side is called the is called the dual pricedual price..

The The range of feasibilityrange of feasibility is the range over which is the range over which the dual price is applicable.the dual price is applicable.

As the RHS increases, other constraints will As the RHS increases, other constraints will become binding and limit the change in the become binding and limit the change in the value of the objective function.value of the objective function.

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Dual PriceDual Price

Graphically, a dual price is determined by adding Graphically, a dual price is determined by adding +1 to the right hand side value in question and +1 to the right hand side value in question and then resolving for the optimal solution in terms of then resolving for the optimal solution in terms of the same two binding constraints. the same two binding constraints.

The dual price is equal to the difference in the The dual price is equal to the difference in the values of the objective functions between the new values of the objective functions between the new and original problems.and original problems.

The dual price for a nonbinding constraint is 0.The dual price for a nonbinding constraint is 0. A A negative dual pricenegative dual price indicates that the objective indicates that the objective

function will function will notnot improve if the RHS is increased. improve if the RHS is increased.

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Relevant Cost and Sunk Relevant Cost and Sunk CostCost

A resource cost is a A resource cost is a relevant costrelevant cost if the amount if the amount paid for it is dependent upon the amount of the paid for it is dependent upon the amount of the resource used by the decision variables. resource used by the decision variables.

Relevant costs Relevant costs areare reflected in the objective reflected in the objective function coefficients. function coefficients.

A resource cost is a A resource cost is a sunk costsunk cost if it must be paid if it must be paid regardless of the amount of the resource regardless of the amount of the resource actually used by the decision variables. actually used by the decision variables.

Sunk resource costs areSunk resource costs are not not reflected in the reflected in the objective function coefficients.objective function coefficients.

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Cautionary Note onCautionary Note onthe Interpretation of Dual the Interpretation of Dual

PricesPrices Resource cost is sunkResource cost is sunk

The dual price is the maximum amount you The dual price is the maximum amount you should be willing to pay for one additional should be willing to pay for one additional unit of the resource.unit of the resource.

Resource cost is relevantResource cost is relevant

The dual price is the maximum premium The dual price is the maximum premium over the normal cost that you should be over the normal cost that you should be willing to pay for one unit of the resource.willing to pay for one unit of the resource.

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Example 1Example 1 Dual PricesDual Prices

Constraint 1Constraint 1: Since : Since xx11 << 6 is not a binding 6 is not a binding constraint, its dual price is 0.constraint, its dual price is 0.

Constraint 2Constraint 2: Change the RHS value of the second : Change the RHS value of the second constraint to 20 and resolve for the optimal constraint to 20 and resolve for the optimal

point determined by the last two constraints: point determined by the last two constraints: 22xx11 + 3 + 3xx22 = 20 and = 20 and xx11 + + xx22 = 8. = 8.

The solution is The solution is xx11 = 4, = 4, xx22 = 4, = 4, zz = 48. = 48.

Hence, Hence,

the dual price = the dual price = zznewnew - - zzoldold = 48 - 46 = 2. = 48 - 46 = 2.

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Example 1Example 1 Dual PricesDual Prices

Constraint 3Constraint 3: Change the RHS value of the : Change the RHS value of the third constraint to 9 and resolve for the optimal third constraint to 9 and resolve for the optimal point determined by the last two constraints: point determined by the last two constraints: 22xx11 + 3 + 3xx22 = = 19 and 19 and xx11 + + xx22 = 9. = 9.

The solution is: The solution is: xx11 = 8, = 8, xx22 = 1, = 1, zz = 47. = 47.

The dual price is The dual price is zznewnew - - zzoldold = 47 - 46 = 1. = 47 - 46 = 1.

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Example 1Example 1

Dual PricesDual Prices

Adjustable CellsAdjustable Cells

FinalFinal ReducedReduced ObjectiveObjective AllowableAllowable AllowableAllowableCellCell NameName ValueValue CostCost CoefficientCoefficient IncreaseIncrease DecreaseDecrease

$B$8$B$8 X1X1 5.05.0 0.00.0 55 22 0.333333330.33333333$C$8$C$8 X2X2 3.03.0 0.00.0 77 0.50.5 22

ConstraintsConstraints

FinalFinal ShadowShadow ConstraintConstraint AllowableAllowable AllowableAllowableCellCell NameName ValueValue PricePrice R.H. SideR.H. Side IncreaseIncrease DecreaseDecrease

$B$13$B$13 #1#1 55 00 66 1E+301E+30 11$B$14$B$14 #2#2 1919 22 1919 55 11$B$15$B$15 #3#3 88 11 88 0.333333330.33333333 1.666666671.66666667

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Range of FeasibilityRange of Feasibility The The range of feasibilityrange of feasibility for a change in the right for a change in the right

hand side value is the range of values for this hand side value is the range of values for this coefficient in which the original dual price coefficient in which the original dual price remains constant.remains constant.

Graphically, the range of feasibility is determined Graphically, the range of feasibility is determined by finding the values of a right hand side by finding the values of a right hand side coefficient such that the same two lines that coefficient such that the same two lines that determined the original optimal solution continue determined the original optimal solution continue to determine the optimal solution for the problem.to determine the optimal solution for the problem.

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Example 1Example 1 Range of FeasibilityRange of Feasibility

Adjustable CellsAdjustable Cells

FinalFinal ReducedReduced ObjectiveObjective AllowableAllowable AllowableAllowableCellCell NameName ValueValue CostCost CoefficientCoefficient IncreaseIncrease DecreaseDecrease

$B$8$B$8 X1X1 5.05.0 0.00.0 55 22 0.333333330.33333333$C$8$C$8 X2X2 3.03.0 0.00.0 77 0.50.5 22

ConstraintsConstraints

FinalFinal ShadowShadow ConstraintConstraint AllowableAllowable AllowableAllowableCellCell NameName ValueValue PricePrice R.H. SideR.H. Side IncreaseIncrease DecreaseDecrease

$B$13$B$13 #1#1 55 00 66 1E+301E+30 11$B$14$B$14 #2#2 1919 22 1919 55 11$B$15$B$15 #3#3 88 11 88 0.333333330.33333333 1.666666671.66666667