Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Sensitivity Analysis in
Sequencing Models
Yury Nikulin
Department of Mathematics and Statistics
University of Turku
Lappeenranta, August 20, 2013
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Outline
1
The World of Uncertainty
Reasons for Uncertainty
Consequences of Uncertainty
Modeling Uncertainty
2
Sensitivity Analysis
Problem formulation
Stability and Accuracy
3
Robust Optimization
4
Conclusion
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Reasons for Uncertainty
Consequences of Uncertainty
Modeling Uncertainty
Reasons for uncertainty
Uncertainty in optimization
inaccuracy of initial data
non-adequacy of models to real processes
errors of numerical methods
errors of rounding off
absence or lack of precise and reliable information etc.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Reasons for Uncertainty
Consequences of Uncertainty
Modeling Uncertainty
Reasons for uncertainty
Uncertainty in scheduling and time-tabling
equipment (machine) breakdowns or activity (job) disruption
earliness or tardiness, changes of job processing
changes of release dates, due dates, deadlines or resource availability
etc.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Reasons for Uncertainty
Consequences of Uncertainty
Modeling Uncertainty
Consequences of uncertainty
Self-propagated chain reactions
nuclear chain reaction
domino effect
butterfly effect
thermal runaway
cascading failure
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Reasons for Uncertainty
Consequences of Uncertainty
Modeling Uncertainty
Modeling uncertainty
Uncertainty can be modeled by means of:
possible scenarios (parameter realizations)
fuzzy numbers and fuzzy objective functions
various probabilistic measures
interval data
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Reasons for Uncertainty
Consequences of Uncertainty
Modeling Uncertainty
Interval uncertainty
Advantages
Lower and upper interval bounds are easy to specify
No statistical observations needed for the model
Easy implementation and practical use
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Reasons for Uncertainty
Consequences of Uncertainty
Modeling Uncertainty
Interval uncertainty
Advantages
Lower and upper interval bounds are easy to specify
No statistical observations needed for the model
Easy implementation and practical use
Disadvantages
Infinite number of problem input parameter realization - scenarios (can
be overcome for some models!)
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Reasons for Uncertainty
Consequences of Uncertainty
Modeling Uncertainty
Interval uncertainty
Approaches
Sensitivity analysis: qualitative and quantitative
Robust optimization techniques
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Reasons for Uncertainty
Consequences of Uncertainty
Modeling Uncertainty
Interval uncertainty
Approaches
Sensitivity analysis: qualitative and quantitative
Robust optimization techniques
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Optimizing on a set of substitutions
Problem description: input data
Let m ≥ 2 be a problem dimension. Let A = (a1 , a2 , . . . , am ) ∈ Rm
+ and
B = (b1 , b2 , . . . , bm ) ∈ Rm
+ be two row vectors where R+ = {u ∈ R : u > 0}.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Optimizing on a set of substitutions
Problem description: input data
Let m ≥ 2 be a problem dimension. Let A = (a1 , a2 , . . . , am ) ∈ Rm
+ and
B = (b1 , b2 , . . . , bm ) ∈ Rm
+ be two row vectors where R+ = {u ∈ R : u > 0}.
Problem description: feasible set and objective
Let Sm be a symmetric group of substitutions on a set Nm = {1, 2, . . . , m}.
The objective function is a linear function defined on a non-empty subset
T ⊂ Sm :
m
X
aj bt(j) ,
f (t, A, B) =
j=1
where t = (t(1), t(2), . . . , t(m)).
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Optimizing on a set of substitutions
Set of optima
The problem Z (T , A, B) consists in finding t ∗ = arg min f (t, A, B).
t∈T
Denote T (A, B) the set of all optimal substitutions in problem Z (T , A, B).
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Optimizing on a set of substitutions
Set of optima
The problem Z (T , A, B) consists in finding t ∗ = arg min f (t, A, B).
t∈T
Denote T (A, B) the set of all optimal substitutions in problem Z (T , A, B).
Degeneracy
For a given substitution t ∈ T ,
W (t, A, B) = {t ′ ∈ T :
m
X
| bt(j) − bt ′ (j) |> 0},
j=1
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Optimizing on a set of substitutions
Set of optima
The problem Z (T , A, B) consists in finding t ∗ = arg min f (t, A, B).
t∈T
Denote T (A, B) the set of all optimal substitutions in problem Z (T , A, B).
Degeneracy
For a given substitution t ∈ T ,
W (t, A, B) = {t ′ ∈ T :
m
X
| bt(j) − bt ′ (j) |> 0},
j=1
Non-trivial optima
Any optimum t ∗ ∈ T (A, B) with W (t ∗ , A, B) = ∅ we call trivial, otherwise
nontrivial. The set of all nontrivial optima t ∗ denote as T̂ (A, B).
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Sequencing model
Job processing on a single machine
Let A encode job processing times, and
let any particular t = (t(1), t(2), . . . , t(m)) ∈ T specify for each job j ∈ Nm the
place t(j) at which it has to be processed.
Sequencing structuring and objectives
Then, given B = (m, m − 1, . . . , 1), for any particular t ∈ T , the objective
value is calculates as
f (t, A, B) =
m
X
(m − j + 1)at −1 (j) =
j=1
m
X
Cj ,
j=1
where t −1 is inverse to t, so at −1 (j) is a processing time of the job placed at
j-th position in the ordering.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Solutions quality measure
Absolute error
For given t ∗ ∈ T (A, B) and A ∈ Rm
+ the so-called absolute error is defined:
ε(t ∗ , A, B) := f (t ∗ , A, B) − min f (t, A, B) = max q(t ∗, t, A, B) =
t∈T
n
max 0,
max
t∈W (t ∗ ,A,B)
t∈T
o
q(t ∗, t, A, B) =
q(t ∗, t, A, B),
max
t∈W̄ (t ∗ ,A,B)
where
q(t, t ′ , A, B) = f (t, A, B) − f (t ′ , A, B) =
m
X
aj (bt(j) − bt ′ (j) ),
j=1
W̄ (t ∗ , A, B) := W (t ∗ , A, B) ∪ {t ∗ }.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Example
Example
Input data
Let m = 3, A = (1, 2, 5), B = (3, 2, 1). T = {t1 , t2 , . . . , t6 }, t1 = (1, 2, 3),
t2 = (1, 3, 2), t3 = (2, 1, 3), t4 = (2, 3, 1), t5 = (3, 1, 2), t6 = (3, 2, 1).
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Example
Example
Input data
Let m = 3, A = (1, 2, 5), B = (3, 2, 1). T = {t1 , t2 , . . . , t6 }, t1 = (1, 2, 3),
t2 = (1, 3, 2), t3 = (2, 1, 3), t4 = (2, 3, 1), t5 = (3, 1, 2), t6 = (3, 2, 1).
Objectives
Then we calculate f (t1 , A, B) = 12, f (t2 , A, B) = 15, f (t3 , A, B) = 13,
f (t4 , A, B) = 19, f (t5 , A, B) = 17, f (t6 , A, B) = 20.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Example
Example
Input data
Let m = 3, A = (1, 2, 5), B = (3, 2, 1). T = {t1 , t2 , . . . , t6 }, t1 = (1, 2, 3),
t2 = (1, 3, 2), t3 = (2, 1, 3), t4 = (2, 3, 1), t5 = (3, 1, 2), t6 = (3, 2, 1).
Objectives
Then we calculate f (t1 , A, B) = 12, f (t2 , A, B) = 15, f (t3 , A, B) = 13,
f (t4 , A, B) = 19, f (t5 , A, B) = 17, f (t6 , A, B) = 20.
Optimality
Then since t1 ∈ T̂ (A, b), we have ε(t ∗ , A, B) = 0.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Example (cont.)
Example
Perturbed data
à = (1 + δ, 2, 5), where δ > 0. Then f (t1 , A, B) = 12 + 3δ,
f (t2 , A, B) = 15 + 3δ, f (t3 , A, B) = 13 + 2δ, f (t4 , A, B) = 19 + 2δ,
f (t5 , A, B) = 17 + δ, f (t6 , A, B) = 20 + δ.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Example (cont.)
Example
Perturbed data
à = (1 + δ, 2, 5), where δ > 0. Then f (t1 , A, B) = 12 + 3δ,
f (t2 , A, B) = 15 + 3δ, f (t3 , A, B) = 13 + 2δ, f (t4 , A, B) = 19 + 2δ,
f (t5 , A, B) = 17 + δ, f (t6 , A, B) = 20 + δ.
Optimality in perturbed problem
When δ ≤ 1, optimality of t1 is preserved, i.e. t1 ∈ T (Ã, B).
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Stability function
Perturbed data set
For a given ρ ∈ [0, q(A)), where q(A) = min{aj : j ∈ Nm }, we consider a set
n
o
′
Ω(ρ, A) := A′ = (a′1 , a′2 , . . . , a′m ) ∈ Rm
+ : | a j − aj |≤ ρ, j ∈ Nm .
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Stability function
Perturbed data set
For a given ρ ∈ [0, q(A)), where q(A) = min{aj : j ∈ Nm }, we consider a set
n
o
′
Ω(ρ, A) := A′ = (a′1 , a′2 , . . . , a′m ) ∈ Rm
+ : | a j − aj |≤ ρ, j ∈ Nm .
Stability function
For an optimal solution t ∗ ∈ T̂ (A, B) and ρ ∈ [0, q(A)), the value of the
stability function is defined as follows:
S(t ∗ , ρ) :=
max ε(t ∗ , A′ , B).
A′ ∈Ω(ρ,A)
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Accuracy function
Perturbed data set
For a given δ ∈ [0, 1), we consider a set
n
o
′
Θ(δ, A) := A′ = (a′1 , a′2 , . . . , a′m ) ∈ Rm
+ : | a j − aj |≤ δaj , j ∈ Nm .
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Accuracy function
Perturbed data set
For a given δ ∈ [0, 1), we consider a set
n
o
′
Θ(δ, A) := A′ = (a′1 , a′2 , . . . , a′m ) ∈ Rm
+ : | a j − aj |≤ δaj , j ∈ Nm .
Accuracy function
For an optimal solution t ∗ ∈ T̂ (A, B) and δ ∈ [0, 1), the value of the accuracy
function is defined as follows:
A(t ∗ , δ) :=
max ε(t ∗ , A′ , B).
A′ ∈Θ(δ,A)
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Analytical expression of stability function
Theorem A
For an optimal solution t ∗ ∈ T̂ (A, B) and ρ ∈ [0, q(A)), the stability function
can be expressed by the formula:
S(t ∗ , ρ) =
max
t∈W̄ (t ∗ ,A,B)
m
o
n
X
| bt ∗ (j) − bt(j) | .
q(t ∗ , t, A, B) + ρ
Yury Nikulin
j=1
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Analytical expression of accuracy function
Theorem B
For an optimal solution t ∗ ∈ T̂ (A, B) and δ ∈ [0, 1), the accuracy function can
be expressed by the formula:
A(t ∗ , δ) =
max
t∈W̄ (t ∗ ,A,B)
m
o
n
X
aj | bt ∗ (j) − bt(j) | .
q(t ∗ , t, A, B) + δ
Yury Nikulin
j=1
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Stability and Accuracy Radii
Stability Radius
n
o
RS(t ∗ , A) = sup ρ ∈ [0, q(A)) : S(t ∗ , ρ) = 0 ,
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Stability and Accuracy Radii
Stability Radius
n
o
RS(t ∗ , A) = sup ρ ∈ [0, q(A)) : S(t ∗ , ρ) = 0 ,
Accuracy Radius
n
o
RA(t ∗ , A) = sup δ ∈ [0, 1) : A(t ∗ , δ) = 0 .
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Analytical expression of stability radius
Theorem C
For an optimal solution t ∗ ∈ T̂ (A, B) its stability radius can be expressed by
the formula:
n
q(t, t ∗ , A, B) o
.
RS(t ∗ , A) = min q(A), min
m
t∈W (t ∗ ,A,B) P
| bt ∗ (j) − bt(j) |
j=1
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Analytical expression of accuracy radius
Theorem D
For an optimal solution t ∗ ∈ T̂ (A, B) its accuracy radius can be expressed by
the formula:
o
n
q(t, t ∗ , A, B)
.
RA(t ∗ , A) = min 1, min
m
t∈W (t ∗ ,A,B) P
aj | bt ∗ (j) − bt(j) |
j=1
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Example
Example
Input data
Let m = 3, A = (1, 2, 5), B = (3, 2, 1). T = {t1 , t2 , . . . , t6 }, t1 = (1, 2, 3),
t2 = (1, 3, 2), t3 = (2, 1, 3), t4 = (2, 3, 1), t5 = (3, 1, 2), t6 = (3, 2, 1).
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Example
Example
Input data
Let m = 3, A = (1, 2, 5), B = (3, 2, 1). T = {t1 , t2 , . . . , t6 }, t1 = (1, 2, 3),
t2 = (1, 3, 2), t3 = (2, 1, 3), t4 = (2, 3, 1), t5 = (3, 1, 2), t6 = (3, 2, 1).
Objectives
Then we calculate f (t1 , A, B) = 12, f (t2 , A, B) = 15, f (t3 , A, B) = 13,
f (t4 , A, B) = 19, f (t5 , A, B) = 17, f (t6 , A, B) = 20.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Problem formulation
Stability and Accuracy
Example (cont.)
Example
Stability and accuracy
t1 ∈ T̂ (A, b), we calculate
S(t1 , ρ) = max{−3 + 2ρ, −1 + 2ρ, −7 + 4ρ, −5 + 4ρ, −8 + 4ρ} = −1 + 2ρ,
RS(t1 , A) = 0.5;
A(t1 , δ) = max{−3 + 7δ, −1 + 3δ, −7 + 13δ, −5 + 9δ, −8 + 12δ},
RA(t1 , A) =
Yury Nikulin
1
.
3
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
What is robust optimization?
Playing against the worst-case
Instead of producing an optimal solution for a normal situation, which is
described by deterministic models but rarely occurs in practice, and where
recovery to optimality can be complicated, the aim of robust optimization is to
produce solutions that optimize additionally constructed objectives assuring
the optimal solution to remain feasible after worst case realization of
uncertain problem input parameters.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
What is a robust solution?
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
What is a robust solution?
Straightforward Robustness
An optimal solution is robust if it remains optimal under any realization of the
input data.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
What is a robust solution?
Straightforward Robustness
An optimal solution is robust if it remains optimal under any realization of the
input data.
Absolute Robustness
A solution which minimizes the maximum among all scenarios objective
value.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
What is a robust solution?
Straightforward Robustness
An optimal solution is robust if it remains optimal under any realization of the
input data.
Absolute Robustness
A solution which minimizes the maximum among all scenarios objective
value.
Relative Robustness
A solution which minimizes the maximum regret or, among all scenarios, the
maximum deviation from optimal solution.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
SA vs. RO
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
SA vs. RO
Fixing margins
ρ ∈ [0, q(A)) or δ ∈ [0, 1).
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
SA vs. RO
Fixing margins
ρ ∈ [0, q(A)) or δ ∈ [0, 1).
Inducing intervals
[aj − ρ, aj + ρ] or [aj (1 − δ), aj (1 + δ)]
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
SA vs. RO
Fixing margins
ρ ∈ [0, q(A)) or δ ∈ [0, 1).
Inducing intervals
[aj − ρ, aj + ρ] or [aj (1 − δ), aj (1 + δ)]
Optimizing stability and accuracy
min ′ max ε(t, A′ , B),
t∈T A ∈Ω(ρ,A)
min max ε(t, A′ , B).
t∈T A′ ∈Θ(δ,A)
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
SA vs. RO
Analytical expressions vs. worst-case scenario
Analytical expressions of stability and accuracy functions explicitly contain
information about the worst case scenario for every fixed t ∈ T .
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
SA vs. RO
Analytical expressions vs. worst-case scenario
Analytical expressions of stability and accuracy functions explicitly contain
information about the worst case scenario for every fixed t ∈ T .
Two sides of the same coin
Developed independently, numerical measures of SA and RO are pretty
much similar with same complexity challenge.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Eager to know more?
Brand new book, to appear in 2014
Accuracy and stability functions for a problem of minimization a linear form on
a set of substitutions
Chapter in: Sequencing and Scheduling with Inaccurate Data,
Nova Publishers,
Editors: Yuri N. Sotskov and Frank Werner.
Yury Nikulin
Sensitivity Analysis in Sequencing Models
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The World of Uncertainty
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Robust Optimization
Conclusion
Reliable Modeling
Take care about initial parameters in your model to make it more realistic and
reliable!
... Initial data is the shaky bridge above the precipice separating the model
and the reality. It is necessary to take care of its stability and robustness in
order to not stumble on the way to the target...
Yury Nikulin
Sensitivity Analysis in Sequencing Models
Outline
The World of Uncertainty
Sensitivity Analysis
Robust Optimization
Conclusion
Questions and Answers
Thank you for your time and interest!
Yury Nikulin
Sensitivity Analysis in Sequencing Models