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3QA3/2DA3

Management Science for Business

Lecture 04

C01: June 30, 2021

C02: July 6, 2021

Instructors:

Seyyed Hossein Alavi Zdravko Dimitrov

LP Sensitivity Analysis

(Post-Optimality Analysis)

Why Sensitivity Analysis?

 Certainty assumption for LP is not true for most real-world

problems

 Conditions are dynamic and changing

 What if one or more coefficients change?

 For example, profit changes from $9 to $11

➢What’s the effect on solution?

Why Sensitivity Analysis?

 Simply use computer to make necessary changes to model,

then quickly re-solve it!

If change is definite

 For example, we know with certainty: profit is decreased

from $10 to $9

➢Re-solve the model and obtain new optimal solution

Why Sensitivity Analysis?

 Hypothetical changes in input data values

 e.g. profit may be any value from $5 to $20

 It’s impractical to re-solve the model with respect to all possibilities!

 Shows how sensitive the solution is to change in each input data

 Determines a range for each input data in which the solution is optimal.

 Obtain this information from the current solution itself without re- solving

 Two main applications:

 Error in the estimation of input data/parameters

 Managers are interested in what-if analysis

Why Sensitivity Analysis?

 We examine:

 Changes in objective function coefficients (OFC)

 Changes in RHS

 First: changes in only one input data value at a time

 Either in one OFC or one RHS value

 Next: simultaneous changes in several input data values

Sensitivity Analysis for Flair Furniture Company

 Recall the Flair Company example from session 2:

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 7𝑇 + 5𝐶

Subject to

3𝑇 + 4𝐶 ≤ 2,400 (carpentry time)

2𝑇 + 1𝐶 ≤ 1,000 (painting time)

𝐶 ≤ 450 (maxchairs allowed)

𝑇 ≥ 100 (mintables required)

𝑇 ≥ 0 (non-negativity)

𝐶 ≥ 0 (non-negativity)

Optimal solution: 𝑇∗ = 320,𝐶∗ = 360

1- Changes to Objective Function Coefficients

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 7𝑇 + 5𝐶

 Change 7 to 3, 8, 11

 Keep other input data values the same

➢ Will the corner point 𝑇, 𝐶 = 320,360 remain the optimal solution?

1- Changes to Objective Function Coefficients

| | | | | | | | | |

0 200 400 600 800 1,000

800 –

–

600 –

–

400 –

–

200 –

–

0 –

N u

m b

e r

o f C

h a

ir s (

C )

Number of Tables (T)

2 3

Original - Optimal Level Profit Line

Original - Optimal Corner Point Solution

(T = 320, C = 360)

7T+5C = 4,040

1- Changes to Objective Function Coefficients

| | | | | | | | | |

0 200 400 600 800 1,000

800 –

–

600 –

–

400 –

–

200 –

–

0 –

N u

m b

e r

o f C

h a

ir s (

C )

Number of Tables (T)

2 3

7T+5C = 4,040

8T+5C = 4,360

Optimal Corner Point Solution

(T = 320, C = 360)

1- Changes to Objective Function Coefficients

| | | | | | | | | |

0 200 400 600 800 1,000

800 –

–

600 –

–

400 –

–

200 –

–

0 –

N u

m b

e r

o f C

h a

ir s (

C )

Number of Tables (T)

2 3

(T = 500, C = 0)

11T+5C = 5,500

3T+5C = 2,850(T = 200, C = 450)

7T+5C = 4,040

(T = 320, C = 360)

1- Changes to Objective Function Coefficients

Maximize 7𝑇 + 5𝐶

 Change 7 to 8: the optimal solution is still corner point 4

 Change 7 to 11: the optimal solution has changed to corner point 5

 Change 7 to 3: the optimal solution has changed to corner point 3

 Question:

Is there a range of possible values for the profit of a table within which the

current optimal corner solution remains optimal?

Allowable Increase/Decrease in Solver Sensitivity Report help us find the

range of optimality.

1- Changes to Objective Function Coefficients

Open “Class 04–Sensitivity Analysis.xlsx”

1- Changes to Objective Function Coefficients

Range of Optimality (𝑇∗ = 320,𝐶∗ = 360):

 For T with current coefficient 7: (7−3.25, 7+3) = (3.75, 10)

 Change 7 to 8: the optimal solution is still corner point 4

 Change 7 to 11: the optimal solution has changed to corner point 5

 Change 7 to 3: the optimal solution has changed to corner point 3

1- Changes to Objective Function Coefficients

Range of Optimality for (𝑇∗ = 320,𝐶∗ = 360):

 For T with current coefficient 7: (7−3.25, 7+3) = (3.75, 10)

 For C with current coefficient 5: (5−1.5, 5+4.33) = (3.5, 9.33)

❑ Note: we consider a change to only a single objective function

coefficient

➢ If they change simultaneously, the result may not apply.

Binding Constraint; Slack vs Surplus LHS<=RHS vs LHS>=RHS

 A constraint is binding when LHS = RHS  Binding means the constraint is exactly satisfied

 For a <= constraint, this typically means that all the available amounts of that resource are fully used in the optimal solution

 For a non-binding ≤ constraint, 𝑅𝐻𝑆 − 𝐿𝐻𝑆 is called slack  Slack typically refers to the amount of unused resource in a ≤ constraint

 For a non-binding ≥ constraint, 𝐿𝐻𝑆 − 𝑅𝐻𝑆 is called surplus  Surplus typically refers to the amount of over-satisfaction of a constraint

 Therefore:

 If a constraint is binding, 𝑆𝑙𝑎𝑐𝑘 = 𝑆𝑢𝑟𝑝𝑙𝑢𝑠 = 0

 If a constraint is not binding, 𝑆𝑙𝑎𝑐𝑘 > 0 or 𝑆𝑢𝑟𝑝𝑙𝑢𝑠 > 0

Feasible

Region

N u

m b

e r

o f C

h a

ir s (

C )

Number of Tables (T)

1,000 –

–

800 –

–

600 –

–

400 –

–

200 –

–

0 –| | | | | | | | | | | | 0 200 400 600 800 1,000

Painting Time Constraint 2𝑇 + 1𝐶 ≤ 1,000

Carpentry Time Constraint 3𝑇 + 4𝐶 ≤ 2,400

Maximum Chairs Allowed Constraint

𝐶 ≤ 450

Minimum Tables Required Constraint

𝑇 ≥ 100

Optimal Corner Point (𝑇 = 320,𝐶 = 360) Binding (𝐿𝐻𝑆 = 𝑅𝐻𝑆)

Non-binding (𝑆𝑢𝑟𝑝𝑙𝑢𝑠 = 220 > 0)

Non-binding (𝑆𝑙𝑎𝑐𝑘 = 90 > 0)

Binding Constraint; Slack Vs. Surplus

2- Change in RHS LHS<=RHS vs LHS>=RHS

 For a ≤ constraint:

 Increasing the RHS → Bigger feasible region

 Decreasing the RHS → Smaller feasible region

 For a ≥ constraint:

 Increasing the RHS → Smaller feasible region

 Decreasing the RHS → Bigger feasible region

2- Change in RHS LHS<=RHS

 For a ≤ constraint:  Increasing the RHS → Bigger feasible region  Decreasing the RHS → Smaller feasible region

 When we make the feasible region bigger, the solution cannot be worse.  As the feasible region becomes larger, the original optimal solution can

always be attained, unless we find another point that yields a better objective function

 When we make the feasible region smaller, the solution cannot be better.

2- Increase in RHS LHS<=RHS

Optimal Level Profit Line for Revised Model

(Corner point 4A is optimal.

Maximum Profit = $4,820)

(T =560, C = 180)

Original

Feasible

Region

1,000 –

–

800 –

–

600 –

–

400 –

–

200 –

–

0 –| | | | | | | | | | | | 0 200 400 600 800 1,000

2 3

Original painting constraint: 2𝑇 + 1𝐶 ≤ 1,000;

change 1,000 to 1,300

Profit is increased by

$4,820-4,040=$780.

That is, a profit increase of

$780/300=$2.60 per

additional hour of painting

time.

2- Decrease in RHS LHS<=RHS

Original painting constraint: 2𝑇 + 1𝐶 ≤ 1,000;

change 1,000 to 900

Profit is decreased by

$4,040-3,780=$260. That

is, a profit decrease of

$260/100=$2.60 per

hour of painting time lost.

Optimal Level Profit Line for Revised Model

(Corner point 4B is optimal.

Maximum Profit = $3,780)

(T =240, C = 420)

Original

Feasible

Region

1,000 –

–

800 –

–

600 –

–

400 –

–

200 –

–

0 –| | | | | | | | | | | | 0 200 400 600 800 1,000

2 3

2- Change in RHS LHS>=RHS

 For a ≥ constraint:

 Increasing the RHS → Smaller feasible region

 Decreasing the RHS → Bigger feasible region

 When we make the feasible region bigger, the solution cannot

be worse.

 When we make the feasible region smaller, the solution cannot

be better.

Shadow Price of a Constraint

 Shadow Price is the change in the objective function value when

the RHS value of the constraint is relaxed by one unit.

 Flair Company example:

 Painting hours:

 If we change 1,000 to 1,001 and 999, the objective function value will be

4,040 + 2.6 = 4,042.6 and 4,040 − 2.6 = 4,037.4

 Carpentry hours:

 If we change 2,400 to 2,401 and 2,399, the objective function value will be

4,040 + 0.6 = 4,040.6 and 4,040 − 0.6 = 4,039.4

 The shadow price for a non-binding constraint is = 0

 The shadow price for a binding constraint can be ≠ 0

Shadow Price of a Constraint

 Validity of shadow price for a constraint:

 The range of feasibility (for a constraint) is the interval over

which the RHS value of a constraint can vary while its shadow price

remains constant.

 is determined by Allowable Decrease and Allowable Increase

interval in Solver.

 For non-binding constraints, the shadow price is zero and one of

Allowable Decrease or Allowable Increase is infinity, shown as

1E+30 in Solver.

Use of Shadow Price

1. Shows us the value of extra capacity:

 The value of one extra hour in carpentry department is $0.6

 The value of one extra hour in painting department is $2.6

2. New product Evaluation:

 Shows the opportunity cost of producing a new product that uses

the same resources available for the current products

Reduced Cost

Reduced cost is the shadow price of the non-negativity constraints

 A non-negativity constraint: 𝑥 ≥ 0

 e.g. increase the RHS of 𝑥 ≥ 0 to 𝑥 ≥ 1

 If a non-negativity constraint is non-binding (𝑥 > 0), then the reduced cost (shadow price) is = 0.

 If a non-negativity constraint is binding (𝑥 = 0), then the reduced cost (shadow price) is ≥ 0.

 Therefore, we always have: 𝑥 × 𝑟𝑒𝑑𝑢𝑐𝑒𝑑 𝑐𝑜𝑠𝑡 = 0

(complementary slackness)

 Can be interpreted in two ways …

Reduced Cost – Interpretation 1

The magnitude (absolute value) of the reduced cost is the minimum amount by which the Objective Function Coefficient (OFC) of a variable should change in order for it to affect the optimal solution.

➢ That decision variable will then have a non-zero value in the new optimal solution

Maximize 7𝑇 + 5𝐶

 In Flair Company example, change OFC of 𝑇 from 7 to 15:  The production of 𝐶 will change to 0

 Reduced cost of 𝐶 will change to -0.5

 Improvement to make 𝐶 to enter the optimal solution:

 5 + | − 2.5| = 7.5

 Re-solve the model with a new OFC of > 7.5 for 𝐶.

Reduced Cost – Interpretation 2

Reduced cost is the difference between the marginal contribution

of a decision variable (i.e. objective function coefficient) and the

marginal worth of the resources it uses (i.e. sum of constraint

coefficients times the shadow prices).

Maximize 7𝑇 + 5𝐶

• In Flair Company example, the marginal contributions of the

decision variables T and C are $7 and $5, respectively.

Reduced Cost – Interpretation 2

The marginal worth (MW) of the resources a decision variable uses:

𝑀𝑊 = ෍ 𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 × 𝑆ℎ𝑎𝑑𝑜𝑤 𝑃𝑟𝑖𝑐𝑒

 In Flair’s example

 3𝑇 + 4𝐶 ≤ 2,400 (carpentry time) shadow price: 0.6

 2𝑇 + C ≤ 1,000 (painting time) shadow price: 2.6

 𝐶 ≤ 450 (max chairs allowed) shadow price: 0

 𝑇 ≥ 100 (min retable required) shadow price: 0

➢ MW for T = 3 × 0.6 + 2 × 2.6 + 1 × 0 = $7

➢ MW for C = 4 × 0.6 + 1 × 2.6 + 1 × 0 = $5

Reduced Cost – Interpretation 2

 Flair Company example:

Maximize 7T+5C

MW of T = $7

MW of C = $5

❖ If a decision variable has a non-zero value at optimality, its

marginal contribution to the objective function value will equal

the marginal worth of the resources it consumes (reduced cost =

0).

Practice Problems

 CHAPTER 4

 Discussion Questions 6, 8

 Problems 13, 15, 21, 22, 23