Sensitivity Analysis

1
Sensitivity Analysis

• How will a change in a coefficient of the
  objective function affect the optimal solution?
• How will a change in the right-hand side value
  for a constraint affect the optimal solution?

2
Pet Food Co. Linear Equations
3
Pet Food Co. Graph Solution
Line 3
Line 2
4
Pet Food Co. Optimal Solution

• Extreme Point is optimal if
• Slope of Line 3 lt Slope of objective function lt
  Slope of Line 2

5
Pet Food Co. Calculate Slope of Line 3

• 1P1 1P2 gt 500
• 1P2 gt -1P1 500
• P2 gt -P1 500
•  
• Slope of Intercept of
• Line 3 Line 3 on P2 axis

6
Pet Food Co. Calculate Slope of Line 2

• 0P1 1P2 gt 200
• 1P2 gt -0P1 200
•  
• Slope of Intercept of
• Line 2 Line 2 on P2 axis

7
Pet Food Co. Optimal Solution

• Extreme Point 4 is optimal if
• -1 lt Slope of objective function lt 0

8
Calculating Slope-Intercept

• General form of objective function
• Z CP1P1 CP2P2
• Slope-intercept for objective function
• P2 -(CP1/CP2) P1 Z/CP2
• Slope of Intercept of
• Obj. Function Obj. Function on x2 axis

9
Pet Food Co. Optimal Solution

• Extreme Point is optimal if
• -1 lt -(CP1/CP2) lt 0
• Or
• 0 lt (CP1/CP2) lt 1

10
Pet Food Co. Compute the Range of Optimality

• Extreme Point is optimal if
• 0 lt (CP1/CP2) lt 1
• Compute range for CP1, hold CP2 constant
• 0 lt (CP1/8) lt 1

11
Pet Food Co. Compute the Range of Optimality

• From the left-hand inequality, we have
• 0 lt (CP1/8)
• Thus,
• 0 lt CP1

12
Pet Food Co. Compute the Range of Optimality

• From the right-hand inequality, we have
• (CP1/8) lt 1
• Thus,
• CP1 lt 8

13
Pet Food Co. Compute the Range of Optimality

• Summarizing these limits
• 0 lt CP1 lt 8

14
Pet Food Co. Compute the Range of Optimality

• Extreme Point is optimal if
• 0 lt (CP1/CP2) lt 1
• Compute range for CP2, hold CP1 constant
• 0 lt (5/CP2) lt 1

15
Pet Food Co. Compute the Range of Optimality

• From the left-hand inequality, we have
• 0 lt (5/CP2)
• Thus,
• (1/5) 0 lt (1/CP2)
• Invert
• 5/0 gt CP2
• Division by zero is undefined (infinite).
• This means the cost of P2 can increase to
  infinity without changing the optimal solution

16
Pet Food Co. Compute the Range of Optimality

• From the right-hand inequality, we have
• (5/CP2) lt 1
• Thus,
• CP2 gt 5

17
Pet Food Co. Compute the Range of Optimality

• Summarizing these limits
• 0 lt CP1 lt 8
• 5 lt CP2 lt Infinite

18
Sensitivity Analysis

• How will a change in a coefficient of the
  objective function affect the optimal solution?
• How will a change in the right-hand side value
  for a constraint affect the optimal solution?

19
Pet Food Company Graph Solution
Line 3
Line 2
20
Pet Food Co. Range of Feasibility

• Constraint 1 is not binding
• Therefore, the shadow price is zero
• Slack is 100
• Range of Feasibility
• 300 lt Constraint 1 RHS lt Infinite

21
Pet Food Co. Change in the Right-hand Side

• Constraint 2 add 1 to right-hand side
• 0P1 1P2 gt 201
• 1P1 1P2 gt 500
• Solve for P1
• -1(0P1 1P2 201)
• 1P1 1P2 500
• P1 299
• Solve for P2
• 1(299) 1P2 gt 500
• P2 201

22
Pet Food Co. Change in the Right-hand Side

• Solve objective function
• z 5(299) 8(201)
• z 3103
• Shadow Price
• 3103 3100 3
• Thus cost increases at 3.00 per lb. added of P2
  per batch
• Conversely, if we decrease lbs. of P2 per batch
  by 1 the objective function will decrease by 3.00

23
Pet Food Co. Range of Feasibility

• Constraint 2 RHS 200
• Allowable Increase 300
• Allowable Decrease 100
• Range of Feasibility
• 100 lt Constraint 2 RHS lt 500

24
Pet Food Co. Change in the Right-hand Side

• Constraint 3 add 1 to right-hand side
• 0P1 1P2 gt 200
• 1P1 1P2 gt 501
• Solve for P1
• -1(0P1 1P2 200)
• 1P1 1P2 501
• P1 301
• Solve for P2
• 1(301) 1P2 gt 501
• P2 200

25
Pet Food Co. Change in the Right-hand Side

• Solve objective function
• z 5(301) 8(199)
• z 3097
• Shadow Price
• 3105 3100 5
• Thus cost increases at 5.00 per lb. added of P2
  per batch
• Conversely, if we decrease lbs. of P2 per batch
  by 1 the objective function will decrease by 5.00

26
Pet Food Co. Range of Feasibility

• Constraint 3 RHS 500
• Allowable Increase 100
• Allowable Decrease 300
• Range of Feasibility
• 200 lt Constraint 3 RHS lt 600

27
Pet Food Co. Linear Equations Slack/ Surplus
Variables

• Min
• 5P1 8P2 0S1 0S2 0S3
• s.t.
• 1P1 1S1
  400
• 1P2 - 1S2
  200
• 1P1 1P2 - 1S3
  500
• P1, P2, S1 ,S2 ,S3 gt 0

28
Pet Food Co. Slack Variables

• For each constraint the difference between the
  RHS and LHS (RHS-LHS). It is the amount of
  resource left over.
• Constraint 1 S1 100 lbs.

29
Pet Food Co. Surplus Variables

• For each constraint the difference between the
  LHS and RHS (LHS-RHS). It is the amount bt which
  a minimum requirement is exceeded.
• Constraint 2 S2 0 lbs.
• Constraint 3 S3 0 lbs.

30
Pet Food Co. Constraint Limits

• Range of Feasibility
• 300 lt Constraint 1 lt Infinite
• 100 lt Constraint 2 lt 500
• 200 lt Constraint 3 lt 600