WOOD 492
MODELLING FOR DECISION SUPPORT
Lecture 10
Introduction to Sensitivity Analysis
Why Sensitivity Analysis?
• LPs are used for finding an optimal plan
– Solution of key decision variables that generates the best value
for the objective function, based on given parameters
• LP solutions are based on the “certainty” assumption
– What if we are not 100% sure about the parameter values?
– How can we determine the impact of parameter changes on the
optimal solution?
– Which parameters are the most important one to estimate
correctly based on the sensitivity of the objective function?
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Sensitivity Analysis
• Is extremely useful when making decisions based on the
results of an optimization model
• Helps us answer questions like:
– What is the value of additional capacity/resources?
– How much would the prices have to change before …..
– How sensitive is the objective value to the estimates of
parameter values?
– What is the range of parameter values for which the optimal
solution stays the same?
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Example for Sensitivity Anlaysis
•
•
•
•
Producing 5 products using 3 moulders, 2 sander, and labour
Factory works 2 shifts, 8 hours per shift, 6 days / week
Each product takes 20 hrs of labour
8 workers working 48 hours / week
Price
Moulder Hours
Sander Hours
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Prod 1
550
12
10
Prod 2
600
20
8
Prod 3
350
Prod 4
400
25
Prod 5
200
15
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LP formulation
Price
Moulder Hours
Sander Hours
Prod 1
Prod 2
Prod 3
Prod 1
550
12
10
Prod 2
600
20
8
Prod 4
Prod 3
350
Prod 4
400
25
Prod 5
200
15
16
Prod 5
Z
Answer
Upper
Lower
Obj
Moulder
Sander
Labor
12.00
7.20
0.00
0.00
0.00
$550
12
10
20
$600
20
8
20
$350
$400
25
$200
15
20
20
16
20
$10,920
<=
<=
<=
RHS
288
192
384
1. What is the value of an extra hour of moulding, sanding or labour?
2. How much more expensive should product 3 be before we start
producing it?
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Shadow price (Answering Q1) by Trial & Error
Answer
Upper
Lower
Obj
Moulder
Sander
Labor
Prod 1
Prod 2
Prod 3
Prod 4
Prod 5
12.00
7.20
0.00
0.00
0.00
$550
12
10
20
$600
20
8
20
$350
$400
25
$200
15
20
20
Z new = $10,926.25
Z old = $10,920
16
20
$10,920
<=
<=
<=
RHS
288
192
384
Change to 289
Difference (shadow price)= $6.25 = value of an additional hour of moulder
time
Shadow price: marginal value of a resource/constraint. Can be calculated by
adding 1 to the RHS of a constraint and calculating the difference in the
objective function.
Reduced Cost (Answering Q2) by Trial & Error
Answer
Upper
Lower
Obj
Moulder
Sander
Labor
Prod 1
Prod 2
Prod 3
Prod 4
Prod 5
12.00
7.20
0.00
0.00
0.00
$550
12
10
20
$600
20
8
20
$350
$400
25
$200
15
20
20
16
20
$10,920
<=
<=
<=
RHS
288
192
384
Increase gradually
Price of product 3 has to increase by $125 before it would be produced.
Reduced cost = -125
Reduced Cost: If a variable = 0 in the optimal solution, then its reduced cost
is the amount its objective function coefficient (price in this example) needs to
change before it will come into the solution (>0).
New value of Obj. Coefficient = Old value of Obj. Coeff – Reduced Cost
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Sensitivity Analysis with Simplex
• When LP problems are large with many variables and
constraints
– Re-solving LPs may require a large computational effort
– Simplex algorithm eliminates the need to resolve the LP for
every change in parameters
– While we won’t get into the details of sensitivity analysis with
Simplex method, we can view the results in Excel Solver’s
sensitivity report
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Example of the sensitivity report in Excel
• Furniture Manufacturer (refer to Example_3 files on the
course website)
How sensitive is the
solution to this
estimated price?
Variable
Tables
Desks
Dressers
Answer
22.00
33.33
20.00
$3,160
Obj Ftn
$50.00
$45.00
$28
used
sign
RHS
Saw-hrs
9
3
3
358
<=
500
Assemble-hrs
5
6
2
350
<=
350
Finish-hrs
5
3
2
250
<=
250
1
20
<=
20
Sales Demand
How would the
solution change if
we had more
assembly hours?
Sensitivity report
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Lab 3 Preview
CB 1
4000 m3
Sawdust (m3)
Log deck
CB 2
5000 m3
Logs sorted by
diameter (m3)
Head saw
Planks and boards
(MBF)
Trimmer
Lumber products
(MBF)
Market
Chips (m3)
CB 3
2500 m3
Lab 3 Preview
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Lab 3 Preview
• Objective: maximize profits ($)
• Decision variables:
– how much to cut from each cut block (m3)
– How much logs to process with each sawing pattern(m3)
– Which lumber products to produce (mbf)
• Constraints
–
–
–
–
–
Available timber (volume in the cut blocks)
Machine hours available, downtime
Material balance related to conversion factors
No demand constraints
Sawing patterns are not mutually exclusive (we can use a
combination of SP1 and SP2)
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Some unit conversions
• 1 MBF = volume of a block of 1” by 12” by 12”
– 1BF= 144 cubic inch
– 1 BF=0.0254*0.3048*0.3048 cubic meters ≈ 0.002360 m3
– 1000 BF=1 MBF= 2.36 m3
• Calculating chip conversion factors (m3/m3*, unitless):
*Note: In this lab, the chip conversion factors are given in relation to the
total log volume, not the lumber volume
Total log vol = lumber vol+ sawdust + chips (all units should be in m3)
= lumber vol(MBF)*2.360 (m3/MBF)+ 0.1 * total log vol+ chips
chips= total log vol - 0.1 total log vol- lumber vol * 2.360
= 0.9 total log volume – (lumber recovery factor* total log vol)*2.360
=total log volume * (0.9 – lumber recovery factor * 2.360)
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Lab 3 preview
• Speed of Head saw = 250 ft/minute = 15,000 ft/hr
– Accounting for 10% downtime= 15000 *0.9= 13,500 ft/hr
– Each log is 8 ft tall and there’s a 4 ft gap between logs so for every
12 ft of conveyor, there is one log passing
– 13,500/12= 1125 logs/hour (passing through head saw)
– Combination of this log/hr rate and log volumes is used to calculate
processing time (hr/m3) for the head saw, as shown in the Excel file
• Speed of Trimmer= 95 lugs/minute = 5,700 lugs/hr
– Accounting for 90% coverage= 5,700 *0.9=5,130 boards/hr
– Accounting for 10% downtime= 5,130 *0.9= 4,617 boards/hr
– Combination of this board/hr rate and lumber volumes is used to
calculate processing time (hr/MBF) for the trimmer, as shown in the
Excel file
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Next Class
• Quiz on Friday, Sept 28
• More sensitivity analysis examples
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