LINEAR PROGRAMMING SENSITIVITY ANALYSIS

     

Learning Objectives

Learn sensitivity concepts 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 the impact of addition of new variable using pricing-out strategy.

Introduction (1 of 2)

 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 a problem is found, input data values are varied to assess optimal solution sensitivity.

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

Introduction (2 of 2)

 Sensitivity analysis determines the effect on optimal solutions of changes in parameter values of the objective function and constraint equations  Changes may be reactions to anticipated uncertainties in the parameters or the new or changed information concerning the model

The Role of Sensitivity Analysis of the Optimal Solution

 Is the optimal solution sensitive to changes in input parameters?

 Possible reasons for asking this question:  Parameter values used were only best estimates.

 Dynamic environment may cause changes.

 “What-if” analysis may provide economical and operational information.

1. Sensitivity Analysis of Objective Function Coefficients.

Ranges of Optimality

The value of the objective function will change if the coefficient multiplies a variable whose value is nonzero .

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.

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

Sensitivity Analysis Using Graphs

Example 1: Galaxy Industries (1 of 5)

Max 8X1 + 5X2 (Weekly profit) subject to 2X1 + 1X2 < = 1200 (Plastic) 3X1 + 4X2 < = 2400 (Production Time) X1 + X2 < = 800 (Total production) X1 - X2 < = 450 (Mix) X j > = 0, j = 1,2 (Nonnegativity)

Sensitivity Analysis Using Graphs

Example 1:(2 of 5)

We now demonstrate the search for an optimal solution X2 800 Then increase the profit, if possible...

...

and continue until it becomes infeasible

Profit =$5040

000 600 X1 400 600 800

Sensitivity Analysis Using Graphs

Example 1 (3 of 5)

MODEL SOLUTION Space Rays = 480 dozens Zappers = 240 dozens Profit = $5040 – This solution utilizes all the plastic and all the production hours.

– Total production is only 720 (not 800).

– Space Rays production exceeds Zapper by only 240 dozens (not 450).

Sensitivity Analysis Using Graphs

Example 1 (4 of 5)

X2 1200 800 The effects of changes in an objective function coefficient on the optimal solution 600 X1 400 600 800

Sensitivity Analysis Using Graphs

Example 1 (5 of 5)

X2 1200 Range of optimality 800 600 400 600 800 The effects of changes in an objective function coefficients on the optimal solution X1

Sensitivity Analysis Using Graphs

Example 2: Beaver Creek Pottery (1 of 4)

Maximize Z = $40x 1 subject to: 1x 4x 2 1 + $50x 2 + 2x 2  + 3x 2  x 1 , x 2  0 40 120

Sensitivity Analysis Using Graphs

Example 2 (2 of 4)

Maximize Z = $100x 1 subject to: 1x 4x 2 1 + $50x 2 + 2x 2  + 3x 2  x 1 , x 2  0 40 120

Sensitivity Analysis Using Graphs

Example 2 (3 of 4)

Maximize Z = $40x 1 subject to: 1x 4x 2 1 + $100x 2 + 2x 2  + 3x 2  x 1 , x 2  0 40 120

Sensitivity Analysis Using Graphs

(Objective Function Coefficient Sensitivity Range for c 1

Example 2 (4 of 4)

and c 2 )

objective function Z = $40x 1 sensitivity range for: x 1 : 25  c 1 x 2 : 30  c 2   66.67 80 + $50x 2

Objective Function Coefficient Ranges Excel “Solver” Results Screen (1 of 3)

Objective Function Coefficient Ranges Beaver Creek Example Sensitivity Report (2 of 3)

Objective Function Coefficient Ranges QM for Windows Sensitivity Range Screen

Beaver Creek Example (3 of 3)

Sensitivity Analysis Using Graphs

Example 3: High Note Sound Company(HNSC) (1 of 4)

 HNSC Manufactures quality CD players and stereo receivers.  Each product requires skilled craftsmanship.

 LP problem formulation: Objective: maximize profit = $50

C

+ $120

R

subject to 2

C

+ 4

R

3

C

+ R   80 60

C

,

R

 0 Where: (Hours of electricians' time available) (Hours of audio technicians' time available) (Non-negativity constraints)

C

= number of CD players to make.

R

= number of receivers to make.

Sensitivity Analysis Using Graphs

Example 3(2 of 4)

Sensitivity Analysis Using Graphs

Example 3 (3 of 4)

Impact of price change of Receivers If 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)

Sensitivity Analysis Using Graphs

Example 3 (4 of 4)

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).

Objective Function Coefficient Sensitivity Range (for a Cost Minimization Model)

Minimize Z = $6x 1 subject to: 2x 1 + 4x 2 4x x 1 1 + 3x 2 , x 2  0   + $3x 16 24 2 sensitivity ranges: 4  0  c c 1 2    4.5

Multiple changes

– The range of optimality is valid only when a single objective function coefficients changes.

– When more than one variable changes we turn to the 1

00% rule.

The 100% Rule

1. For each increase (decrease) in an objective function coefficient, calculate (and express as a percentage) the ratio of the change in the coefficient to the maximum possible increase (decrease) as determined by the limits of the range of optimality.

2. Sum all these percent changes. If the total is less than 100 percent, the optimal solution

will not

change. If this total is greater than or equal to 100%, the optimal solution

may

change.

Reduced Costs

The reduced cost for a variable at its lower bound (usually zero) yields:  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.

2. Sensitivity Analysis of Right Hand Side Values

Changes in Right-Hand-Side Values of Constraints

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.

Sensitivity Analysis of Right-Hand SideValues

 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.

Changes in Constraint Quantity (RHS) Values Increasing the Labor Constraint (1 of 3)

Example 1: Beaver Creek

Maximize Z = $40x 1 subject to: 1x 4x 2 1 + $50x 2 + 2x 2  + 3x 2  x 1 , x 2  0 40 120

Changes in Constraint Quantity (RHS) Values Sensitivity Range for Labor Constraint (2 of 3)

Example 1: Beaver Creek

Sensitivity range for: 30  q 1  80 hr

Changes in Constraint Quantity (RHS) Values Sensitivity Range for Clay Constraint (3 of 3)

Example 1: Beaver Creek

Sensitivity range for: 60  q 2  160 lb

Constraint Quantity (RHS) Value Ranges by Computer Excel Sensitivity Range for Constraints (1 of 2 )

Example 1: Beaver Creek

Constraint Quantity (RHS) Value Ranges by Computer QM for Windows Sensitivity Range (2 of 2)

Example 1: Beaver Creek

Sensitivity Analysis of Right-Hand SideValues & Shadow Prices

In sensitivity analysis of right-hand sides of constraints we are interested in the following questions:  Keeping all other factors the same, how much would the optimal value of the objective function (for example, the profit) change if the right-hand side of a constraint is changed by one unit?

 For how many additional or fewer units this per unit change will be valid?

Changes in Right-hand-side (RHS) & Shadow Prices

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.

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

Shadow Prices (Dual Values)

 Defined as the marginal value of one additional unit of resource.

 The sensitivity range for a constraint quantity value is also the range over which the shadow price is valid.

Range of Feasibility

 The set of right - hand side values for which same set of constraints determines the optimal point.

 Within the range of feasibility, shadow prices remain constant; however, the optimal solution will change.

Excel Sensitivity Report for Shadow Prices (1 of 2)

Example 1:Beaver Creek Pottery

Maximize Z = $40x 1 to: x 1 + 2x 2 4x x 1 1 + 3x 2 , x 2  0   + $50x 2 subject 40 hr of labor 120 lb of clay

Solution Screen

(2 of 2)

Example 1: Beaver Creek Pottery

Example 2: High Note Sound Company Problem

(1 of 12)

 HNSC Manufactures quality CD players and stereo receivers.  Each product requires skilled craftsmanship.

 LP problem formulation: Objective: maximize profit = $50

C

+ $120

R

subject to 2

C

+ 4

R

3

C

+ R   80 60

C

,

R

 0 Where: (Hours of electricians' time available) (Hours of audio technicians' time available) (Non-negativity constraints)

C

= number of CD players to make.

R

= number of receivers to make.

Sensitivity Analysis of Right-hand-side (RHS) Values

Example 2: High Note Sound Company (2 of 12)

May change the feasible region size.

May change or move corner points.

Increase in Electricians’ Available Time

Example2: High Note Sound Company (3 of 12)

• As available electricians’ time increases, corner points

a

and

b

will move closer to one other. • Further increases in available electricians’ time may make this constraint redundant.

Decrease in Electricians’ Available Time

Example 2: High Note Sound Company (4 of 12)

• As available electricians’ time decreases, corner points

b

and

c

move closer to one another from their current locations. • Corner points

b

and

c

will no longer be feasible, and intersection of electricians’ time constraint with horizontal (

C

) axis will become a new feasible corner point.

Sensitivity Analysis Using Solver

Example 2: High Note Sound Company (5 of 12)

Solver Report

Example 2: High Note Sound Company (6 of 12)

Answer Report

Example 2: High Note Sound Company (7 of 12)

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.

Sensitivity Report

Example 2: High Note Sound Company (8 of 12)

 Primary information is provided by

Shadow Price

 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.

 Electricians’ time constraint is binding.

 Audio technicians’ time constraint is non-binding.  40 unused hours of audio technicians’ time are referred to as

slack

.

Changes in Right-Hand-Side (RHS)

Example 2: High Note Sound Company (9 of 12)

   In case of electrician hours Shadow Price is $30.

For each

additional

hour of electrician time that firm can increase profits by $30.

Changes in RHS of a Non-binding Constraint

Example 2: High Note Sound Company (10 of 12)

 Audio technicians’ time has 40 unused hours.   No interest in acquiring additional hours of resource. Shadow price for audio technicians’ time is zero.

 Allowable increase for RHS value is infinity (shown as 1E+30 by Solver).  Once 40 hours is lost (current unused portion, or slack) of audio technicians’ time, resource also becomes binding.  Any

additional

profit. loss of time will clearly have adverse effect on

Changes in the Objective Function Coefficient (OFC)

Example 2: High Note Sound Company (11 of 12) Adjustable Cells



Reduced Cost

value - shows amount one will ‘lose’ if  solution is forced to make an additional unit.  Current value is 0. If one makes 1, firm will lose $10. 

Allowable Increase

- indicates if price 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!

Changes in Objective Function Coefficient (OFC)

Example 2: High Note Sound Company Example (12 of 12)

Sensitivity Analysis For A Larger Maximization Example

Example 3: Anderson Electronics (1 of 13)

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

Profit per unit: VCR

$41

Stereo TV DVD

$32 $72 $54

LP Formulation

Example 3: Anderson Electronics (2 of 13)

Objective: maximize profit = $29

V

+ $32

S

+ $72

T

+ $54

D

subject to

3 2

V V

+ 4 + 2

S S

+ 4 + 4

T T

+ 3 + 3

D D

1

V

,

V S

, + 1

T

,

S D

+ 3  0

T

+ 2

D

   4700 4500 2500 (Electronic components) (Non-electronic components) (Assembly time in hours) 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 Set-up and Solution

Example 3: Anderson Electronics Example (3 of 13)

Excel Solver Answer Report

Example 3: Anderson Electronics (4 of 13)

Excel Solver Sensitivity Report

Example 3: Anderson Electronics (5 of 13)

Excel Solver Sensitivity Report

Example 3: Anderson Electronics (6 of 13)

Adjustable 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) and

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

Excel Solver Sensitivity Report

Example 3: Anderson Electronics (7 of 13)

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 and $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 and $80:

Objective Coefficient – Allocable Decrease

($72 - infinity) and

Objective Coefficient – Allocable Increase

($72 + $8).

Constraints on the Sensitivity Report

Example 3: Anderson Electronics (8 of 13)

Constraints on the Sensitivity Report

Example 3: Anderson Electronics (9 of 13)

Simultaneous Changes In Parameter Values

Example 3: Anderson Electronics (10 of 13)

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, 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.

Simultaneous Changes In Parameter Values

Example 3:Anderson Electronics (11 of 13)

• 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).

Simultaneous Changes In Parameter Values

Example 3: Anderson Electronics (12 of 13)

Simultaneous Changes in OFC Values

• 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.

Checking Validity of the 100% Rule

•

Example 3: Anderson Electronics Example (13 of 13)

• 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 Required Profit on 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

• 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.

Sensitivity Analysis - Minimization Example

Example 4: Burn-Off Diet Drink Example (1 of 5)

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

LP Formulation

Example 4: Burn-Off Diet Drink Example (2 of 5)

Objective: minimize daily dose cost in cents.

4A + 7B + 6C + 3D Subject to

A

+

B

+

C

+

D

 36 (Daily dose requirement) 3

A

+ 4

B

+ 8

C

+ 10

D

 280 (Chemical X requirement) 5

A

+ 3

B

+ 6

C

+ 6

D

 200 (Chemical Y requirement) 10

A

+ 25

B

+ 20

C

+ 40

D

 1050 (Chemical Z max limit)

A

,

B

,

C

,

D

 0

Excel Solution

Example 4: Burn-Off Diet Drink (3 of 5)

Solver Answer Report

Example 4: Burn-Off Diet Drink (4 of 5)

Solver Sensitivity Report

Example 4: Burn-Off Diet Drink (5 of 5)

Problem Statement (1 of 3)

Example Problem 5

• Two airplane parts: no.1 and no. 2.

• Three manufacturing stages: stamping, drilling, milling.

• Decision variables: x 1 x 2 (number of part no.1 to produce) (number of part no.2 to produce) • Model: Maximize Z = $650x 1 subject to: 4x 6.2x

9.1x

1 1 1 + 910x 2 + 7.5x

+ 4.9x

+ 4.1x

2 2 2 x 1 , x 2     105 (stamping,hr) 90 (drilling, hr) 110 (finishing, hr) 0

Graphical Solution (2 of 3)

Example Problem 5

Maximize Z = $650x 1 subject to: 4x 6.2x

9.1x

1 1 1 + 7.5x

+ 4.9x

+ 4.1x

2 2 2 x 1 , x 2     105 90 110 0 + $910x 2 s1 = 0, s2 = 0, s3 = 11.35 hr 485.33  137.76  c 1 q 1   1,151.43

89.10

Graphical Solution

Excel Solution (3 of 3)

Example Problem 5

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.