Managerial Decision Modeling with Spreadsheets Chapter 4 Linear Programming Sensitivity Analysis

Managerial Decision Modeling with Spreadsheets

Chapter 4

Linear Programming Sensitivity Analysis

Learning Objectives• Understand, using graphs, impact of changes in

objective function coefficients, right-hand-side values, and constraint coefficients on optimal solution of a linear programming problem.

• Generate answer and sensitivity reports using Excel's Solver.

• Interpret all parameters of reports for maximization and minimization problems.

• Analyze impact of simultaneous changes in input data values using 100% rule.

• Analyze impact of addition of new variable using pricing-out strategy.

4.1 Introduction

• Optimal solutions to LP problems have been examined under deterministic assumptions.

• Conditions in most real world situations are dynamic and changing.

• After an optimal solution to problem is found, input data values are varied to assess optimal solution sensitivity.

• This process is also referred to as sensitivity analysis or post-optimality analysis.

4.2 Sensitivity Analysis Using Graphs High Note Sound Company • Manufactures quality CD players and stereo receivers.

• Each product requires skilled craftsmanship.

• LP problem formulation:

Objective: maximize profit = $50C + $120Rsubject to

2C + 4R 80 (Hours of electricians' time available)

3C + R 60 (Hours of audio technicians' time available)

C, R 0 (Non-negativity constraints)

Where:

C = number of CD players to make.

R = number of receivers to make.

High Note Sound Company Problem Solution

Changes in Objective Function Coefficient

Impact of price change of ReceiversIf unit profit per stereo receiver (R) increased from $120 to $150, is corner point a still the optimal solution? YES ! But Profit is $3,000 = 0 ($50) + 20 ($150)

Changes in Objective Function Coefficient

Impact of price change of Receivers

If receiver’s profit coefficient changed from $120 to $80, slope of isoprofit line changes causing corner point (b) to become optimal. But Profit is $1,760 = 16 ($50) + 12 ($80).

4.3 Sensitivity Analysis Using Solver

• In Answer Report• Final Values: objective function, decision

variables.• Binding and nonbinding constraints• Slack: Unused resource.

• In Sensitivity Report• Adjustable Cells:

– Objective Function Coefficients– Reduced Cost– Allowable Changes

• Constraints: – Shadow Price – Allowable Changes

High Note Sound Company Answer Report

• Resources available:

– 80 hours of electricians’ time.

– 60 hours of audio technicians’ time.

• Final Values in table reveal optimal solution requires:

– all 80 hours of electricians’ time.

– Only 20 hours of audio technicians’ time.

• Binding and Non-binding Constraints:

– Electricians’ time constraint is binding.

– Audio technicians’ time constraint is non-binding.

• 40 unused hours of audio technicians’ time are referred to as slack.

Sensitivity Report

• Sensitivity report has two distinct components.

(1) Table titled Adjustable Cells

(2) Table titled Constraints.

• Tables permit one to answer several "what-if"

questions regarding problem solution.

• Consider a change to only a single input data value.

• Sensitivity information does not always apply to

simultaneous changes in several input data values.

High Note Sound Company Sensitivity Report

Changes in Constraint Right-hand-side (RHS)

• Primary information is provided by Shadow Price

• Shadow Price is change in optimal objective function value for one unit

increase in RHS.

• The shadow price is positive for binding constraints and is zero for nonbinding

constraints.

Changes in Right-hand-side (RHS)

RHS of Binding Constraint -

• If RHS of non-redundant constraint changes, size

of feasible region changes.

– If size of region increases, optimal objective

function improves.

– If size of region decreases, optimal objective

function worsens.

• Relationship expressed as Shadow Price.

• In case of electrician hours, shadow price is $30.

• For each additional hour of electrician time that firm can increase profits by $30.

• The range of RHS for electrician time with a shadow price of $30 is (0, 240).

• How to calculate shadow price and range? Excel.

High Note Sound Company

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$D$8 Electricians' Time 80.00 30.00 80.00 160.00 80.00

$D$9 Audio Technicians' Time 20.00 0.00 60.00 1E+30 40.00

Change in RHS of Nonbinding Constraint

• In case of audio technicians’ time, shadow price is zero.– Audio technicians’ time has 40 unused hours. – No interest in acquiring additional hours of resource. – Allowable increase for RHS value is infinity.

• Allowable decrease for RHS value is 40. – Once 40 hours is lost (current unused portion, or slack) of audio

technicians’ time, resource also becomes binding. – Any additional loss of time will clearly have adverse effect on

profit.

• The range of RHS for audio technicians’ time with a shadow price of $0 is (20, infinite).

Constraints

$D$8 Electricians' Time 80.00 30.00 80.00 160.00 80.00

$D$9 Audio Technicians' Time 20.00 0.00 60.00 1E+30 40.00

Change in Objective Function Coefficient (OFC)

Adjustable Cells• Reduced Cost value - shows the difference between the

marginal contribution of a decision variable and the

marginal worth of the resources it uses.

– Objective Function Coefficients

– Shadow Prices and Resources Used

• Allowable Increase and Allowable Decrease – the limits

to which the objective function coefficient of a decision

variable can be changed without affecting the optimality

of the current solution.

High Note Sound Company

Reduced Cost for each CD player

• The marginal contribution is the objective coefficient

• The marginal worth of the resources used:

– Resources Used: 2 hours of electrician time and 3 hours of

audio technician’s time.

– Shadow Prices: $30 for per hour of electrician time and $0 for

per hour of audio technician time.

– Marginal Cost: 2 x $30 + 3 x $0 = $60.

• Reduced Cost: $60 - $50 = $10

• Current value is 0. If one makes 1, firm will lose $10.

Allowable Increase and Decrease for the coefficient

of CD players

• Allowable Increase - indicates if the price of CD

players increases by $10, one will profit by making

additional CDs.

• Allowable Decrease – infinity (1E+30) indicates if $50

is not attractive enough to make CD – any price below

it will not make it attractive either!

Reduced Cost for each Stereo Receiver

• The marginal contribution is the objective coefficient

• The marginal worth of the resources used:

– Resources Used: 4 hours of electrician time and 1 hours of

audio technician’s time.

– Shadow Prices: $30 for per hour of electrician time and $0 for

per hour of audio technician time.

– Marginal Cost: 4 x $30 + 0 x $0 = $120.

• Reduced Cost: $120 - $120 = $0.

Allowable Increase and Decrease for each Stereo

Receiver

• Allowable Increase - infinity (1E+30) indicates if $120

is profitable enough to make receiver – any price above

it will also be profitable.

• Allowable Decrease – $20 indicates if the price of

receivers drops below than $100, it is not optimal to

produce 20 receivers and no CDs.

4.4 Sensitivity Analysis For A Larger Maximization Example

• Anderson Electronics Considering producing four potential products: VCRs, stereos, televisions (TVs), and DVD players:

Profit per unit:

VCR Stereo TV DVD

$29 $32 $72 $54

VCR Stereo TV DVD Supply Cost

Electronic Components 3 4 4 3 4,700 $7Non-electronic Components 2 2 4 3 4,500 $5Assembly time (hours) 1 1 3 2 2,500 $10Selling price (per unit) $70 $80 $150 $110

Anderson Electronics LP Formulation

Objective: maximize profit =

$29 V + $32 S + $72 T + $54 Dsubject to

3 V + 4 S + 4 T + 3 D 4700 (Electronic components)

2 V + 2 S + 4 T + 3 D 4500 (Non-electronic components)

1 V + 1 S + 3 T + 2 D 2500 (Assembly time in hours)

V, S, T, D 0

Where: V = number of VCRs to produce. S = number of Stereos to produce. T = number of TVs to produce. D = number of DVD players to produce.

Excel Solver Answer ReportTarget Cell (Max)

Cell Name Original Value Final Value

$F$8 Profit $0.00 $69,400.00

Adjustable Cells

$B$5 Solution value VCR 0.00 0.00

$C$5 Solution value Stereo 0.00 380.00

$D$5 Solution value TV 0.00 0.00

$E$5 Solution value DVD 0.00 1060.00

Constraints

Cell Name Cell Value Formula Status Slack

$F$10 Electronic comp 4700.00 $F$10<=$H$10 Binding 0.00

$F$11 Non-electronic comp 3940.00 $F$11<=$H$11 Not Binding 560.00

$F$12 Assembly time 2500.00 $F$12<=$H$12 Binding 0.00

Excel Solver Sensitivity Report

Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$5 Solution value VCR 0.00 -1.00 29.00 1.00 1E+30

$C$5 Solution value Stereo 380.00 0.00 32.00 40.00 1.67

$D$5 Solution value TV 0.00 -8.00 72.00 8.00 1E+30

$E$5 Solution value DVD 1060.00 0.00 54.00 10.00 5.00

Constrains

$F$10 Electronic comp 4700.00 2.00 4700.00 2800.00 950.00

$F$11 Non-electronic comp 3940.00 0.00 4500.00 1E+30 560.00

$F$12 Assembly time 2500.00 24.00 2500.00 466.67 1325.00

Excel Solver Sensitivity ReportAdjustable Cells

Non Zero value decision variables, Stereos and DVDs:Produce 380 Stereos with unit profit of $32. • Decision should not change as profit is between $31.33 and

$72: Objective Coefficient – Allocable Decrease ($32 - $1.67)

and Objective Coefficient – Allocable Increase ($32+$40)

Produce 1060 DVDs with unit profit of $54. • Decision should not change as profit is between $49 and

$64: Objective Coefficient – Allocable Decrease ($54 - $5)

Objective Coefficient – Allocable Increase ($54+$10)

Zero value decision variables, VCRs and TVs:

Produce 0 VCRs with unit cost of $1.00 (Reduced Cost).

• Decision to make 0 should not change as profit is below $29 –

but should change over $30:

Objective Coefficient – Allocable Decrease ($29 - infinity) and

Objective Coefficient – Allocable Increase ($29 + $1).

Produce 0 TVs with unit cost of $8.00 (Reduced Cost).

• Decision to make 0 should not change as profit is below $72 –

but should change over $80:

Objective Coefficient – Allocable Decrease ($72 - infinity) and

Objective Coefficient – Allocable Increase ($72 + $8).

Constraints

Nonzero Shadow Prices:

• Electronic Components, Shadow price $2

– Each additional unit of electronic components will allow

Anderson to increase its profit by $2.

– The shadow price is $2 for RHS between (3750, 7500).

RHS - Allocable Increase (4700 + 2800) and

RHS - Allocable Decrease (4700 - 950).

Constraints

Nonzero Shadow Prices:

• Assembly Time, Shadow price $24

– Each additional hour of assembly time will allow Anderson

to increase its profit by $24.

– The shadow price is $24 for RHS between (1175, 2966.67).

RHS - Allocable Increase (2500 + 466.67) and

Constraints

Zero Shadow Price:

• Non-electronic components, Shadow price $0

– Nonbinding constraint, 560 units of unused resources

– The shadow price is $0 for RHS between (3940, infinite).

RHS - Allocable Increase (infinite) and

4.5 Simultaneous Changes Using the 100% Rule Possible to analyze impact of simultaneous changes on optimal

solution only under specific condition: (Change / Allowable change) 1

• If decrease RHS from 4,700 to 4,200 units in electronic component, allowable decrease is 950.

The ratio is: 500 / 950 = 0.5263• If increase 200 hours (from 2,500 to 2,700) in assembly

time, allowable increase is 466.67.

The ratio is: 200 / 466.67 = 0.4285• The sum of these ratios is:

Sum of ratios = 0.5263 + 0.4285 = 0.9548 < 1

Since sum does not exceed 1, information provided in sensitivity report is valid to analyze impact of changes.

4.5 Simultaneous Changes In Parameter Values

Anderson Electronics

• Decrease of 500 units in electronic component availability

reduces size of feasible region and causes profit to

decrease.

– Magnitude of decrease is $1,000 (500 units x $2 per unit).

• Increase of 200 hours of assembly time results in larger

feasible region and net increase in profit.

– Magnitude of increase is $4,800 (200 hours x $24 per hour).

• Net impact of both changes simultaneously is an increase

in profit by $3,800 ( $4,800 - $1,000).

4.5 Simultaneous Changes In OFC ValuesAnderson Electronics• What is impact if selling price of DVDs drops by $3

per unit and at same time selling price of stereos increases by $8 per unit?

• For current solution to remain optimal, allowable decrease in DVD players is $5, while allowable increase in OFC for stereos is $40. – Sum of ratios is:

Sum of ratios = $3 / $5 + $8 / $40 = 0.80 < 1 – $3 decrease in profit per DVD player causes total profit to

decrease by $3,180 (i.e., $3 x 1,060). – $8 increase in unit profit of each stereo results in total

profit of $3,040 (i.e., $8 x 380). • Net impact is a decrease in profit of only $140 to a

new value of $69,260.

4.6 Pricing-out New Variables

• Information given in sensitivity report can be used to study impact of introduction of new decision variables (products).

• For example:

– If problem is re-solved with a new product in model, will it be recommend that a new product be made?

– Or, will it be recommend that a new product not be made, and continue making same products (that is, stereos and DVD players)?

Could Anderson Electronics Propose a New Product?

• Anderson Electronics considers a new product, home-theater system (HTS). Could the company propose this new product?

• Answer to such question involves a procedure called pricing-out.

Pricing-Out Procedure

Home-Theater System (HTS)

• Requires:– 5 units of electronic components

– 4 units of non-electronic components

– 4 hours of assembly time.

• Selling price: $175 per unit.

• The actual cost is 5 x $7 + 4 x $5 + 4 x 10 = $95.

• The net profit is $175 - $95 = $80.

Pricing-out procedure

Home-Theater System (HTS)

• Resources required to make this player:– No longer available to meet existing production plan (380 stereos and

1060 DVD players) for $69,400 total profit.

• Checking validity of the 100% Rule:

Calculate ratio of reduction in each resource’s availability to allowable decrease for that resource.

Sum of ratios = 5/950 + 4/560 + 4/1325 = 0.015 < 1

• Profit loss if the resources are used for each HTS:5 x shadow price of electronic components +

4 x shadow price of non-electronic components +

4 x shadow price of assembly time

or 5 x $2 + 4 x $0 + 4 x $24 = $106.

Pricing-out procedure

Home-Theater System (HTS) • Profit contribution of each HTS has to at least make

up shortfall in profit. • OFC for HTS must be at least $106 in order for

optimal solution to have non-zero value. • The unit profit of HTS is $80. Therefore, Anderson

Electronics should not propose this new product.

Revised Excel Layout

V S T T H

VCR Stereo TV DVD HTS

Solution value 0.00 380 0.00 1060 0.00

Selling price per unit $70 $80 $150 $110 $175 $147,000 <-- Revenue

Cost price per unit $41 $48 $78 $56 $95 $77,600 <-- Cost

Profit $29 $32 $72 $54 $80 $69,400 <-- Objective

Constraints Cost

Electronic comp 3 4 4 3 5 4700.00 <= 4700 $7

Non-electronic comp 2 2 4 3 4 3940.00 <= 4500 $5

Assembly time 1 1 3 2 4 2500.00 <= 2500 $10

LHS Sign RHS

Revised Excel Solver Sensitivity Report

Adjustable Cells

$B$5 Solution value VCR 0.00 -1.00 29.00 1.00 1E+30

$C$5 Solution value Stereo 380.00 0.00 32.00 40.00 1.67

$D$5 Solution value TV 0.00 -8.00 72.00 8.00 1E+30

$E$5 Solution value DVD 1060.00 0.00 54.00 10.00 5.00

$F$5 Solution value HTS 0.00 -26.00 80.00 26.00 1E+30

Constraints

$G$10 Electronic comp 4700.00 2.00 4700.00 2800.00 950.00

$G$11 Non-electronic comp 3940.00 0.00 4500.00 1E+30 560.00

$G$12 Assembly time 2500.00 24.00 2500.00 466.67 1325.00

4.7 Sensitivity Analysis - Minimization Example

Burn-Off Diet Drink

• Plans to introduce miracle drink that will magically burn fat away.

Ingredient A Ingredient B Ingredient C Ingredient D Requirement

Chemical X 3 4 8 10 At least 280 units

Chemical Y 5 3 6 6 At least 200 units

Chemical Z 10 25 20 40 At most 1,050 units

Cost per ounce

4 cents 7 cents 6 cents 3 cents

Burn-Off Diet Drink LP Formulation

Objective: minimize daily dose cost in cents.

4A + 7B + 6C + 3D

Subject to

A + B + C + D 36 (Daily dose requirement)

3A + 4B + 8C + 10D 280 (Chemical X requirement)

5A + 3B + 6C + 6D 200 (Chemical Y requirement)

10A + 25B + 20C + 40D 1050 (Chemical Z max limit)

A, B, C, D 0

Excel Solution

A B C D

Ingr A Ingr B Ingr C Ingr D

Number of ounces 10.250 0.000 4.125 21.625

Cost (cents) 4 7 6 3 130.625 <-- Objective

Constraints

Daily dosage 1 1 1 1 36.00 >= 36

Chemical X 3 4 8 10 280.00 >= 280

Chemical Y 5 3 6 6 205.75 >= 200

Chemical Z 10 25 20 40 1050.00 <= 1050

LHS Sign RHS

Solver Answer Report Burn-Off Diet Drink

Target Cells

$F$6 Cost (cents) 0.000 130.625

Adjustable Cells

$B$5 Number of ounces Ingr A 0.000 10.250

$C$5 Number of ounces Ingr B 0.000 0.000

$D$5 Number of ounces Ingr C 0.000 4.125

$E$5 Number of ounces Ingr D 0.000 21.625

Constraints

Cell Name Cell Value Formula Status Slack

$F$11 Chemical Z 1050.000 $F$11<=$H$11 Binding 0.000

$F$8 Daily dosage 36.000 $F$8>=$H$8 Binding 0.000

$F$9 Chemical X 280.000 $F$9>=$H$9 Binding 0.000

$F$10 Chemical Y 205.750 $F$10>=$H$10 Not Binding 5.750

Solver Sensitivity Report

Adjustable Cells

$B$5 Number of ounces Ingr A 10.250 0.000 4.000 3.500 2.500

$C$5 Number of ounces Ingr B 0.000 5.688 7.000 1E+30 5.688

$D$5 Number of ounces Ingr C 4.125 0.000 6.000 15.000 2.333

$E$5 Number of ounces Ingr D 21.625 0.000 3.000 3.800 1E+30

Constraints

$F$11 Chemical Z 1050.000 -0.238 1050.000 47.143 346.000

$F$8 Daily dosage 36.000 3.750 36.000 16.500 1.278

$F$9 Chemical X 280.000 0.875 280.000 41.000 11.000

$F$10 Chemical Y 205.750 0.000 200.000 5.750 1E+30

Nonzero Reduced Cost: Ingredient B

• The reduced cost of ingredient B is $5.688.

– Each ounce of ingredient B used to make the drink will

cause the total cost per daily dosage to increase by 5.688

cents.

– The current cost of ingredient B is 7 cents. If the cost of

ingredient B is lower by 5.688 cents, then it becomes cost-

effective to use this ingredient.

– When the cost of ingredient B is above 1.312 cents (=7-

5.688), the current corner point solution remains optimal.

Zero Reduced Cost: Ingredient C

• The reduced cost of ingredient C is $0.

– The current cost of ingredient C is 6 cents per

ounce. The range for the cost coefficient of this

ingredient is between 3.667 cents (=6-2.333) and

21 cents (=6+15).

– When the cost of ingredient C is between this

range, the current corner point solution remains

optimal.

Nonzero Shadow Price: Chemical X

• The shadow price of chemical X is 0.875.

• For each additional unit of chemical X required to

be present in the drink, the total cost will increase

by 0.875 cents.

• The shadow price remains to be 0.875 if the

requirement for chemical X is between 269 units

(=280-11) and 321 units (=280+41).

Nonzero Shadow Price: Chemical Z

• The shadow price of chemical Z is -0.238.

• Each unit increase in the maximum limit allowed

for chemical Z will reduce the total cost by 0.238

cents.

• The shadow price remains to be -0.238 if the

maximum limit is between 704 units (=1050-346)

and 1097.143 units (=1050+47.143).

Simultaneous Changes In Parameter ValuesBurn-Off can decrease the minimum requirement for chemical

X by 5 units provided the maximum limit allowed for chemical Z is reduced by 50 units.

• The sum of each proportion of change to allowable change is

5/11 + 50/346 = 0.399 < 1• Since sum does not exceed 1, information provided in

sensitivity report is valid to analyze impact of changes. • The reduced cost from the change in chemical X is

0.875 x 5 = 4.375 cents.• The reduced cost from the change in chemical Z is

0.238 x 50 = 11.9 cents.• The net impact is an increase in total cost of 7.525 cents

(=11.9-4.375).

Summary• Sensitivity analysis used by management to answer

series of “ what-if ” questions about LP model inputs. • Tests sensitivity of optimal solution to changes:

– Profit or cost coefficients, and

– Constraint RHS values.

• Explored sensitivity analysis graphically (with two decision variables).

• Discussed interpretation of information:– In answer and sensitivity reports generated by Solver.

– In reports used to analyze simultaneous changes in model parameter values.

– Determine potential impact of new variable in model.

Managerial Decision Modeling with Spreadsheets Chapter 4 Linear Programming Sensitivity Analysis

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