The Concept of Problem Complexity
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The concept of problem complexity
Alejandro SaladoStevens Institute of Technology
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A) Anyone born in Hoboken?
B) Anyone born elsewhere?
C) Anyone unborn?
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Choose your preferred concept
Constraints: fixed budget and schedule
A very SIMPLE system
Performance = 1.00
A very COMPLEX system
Performance = 1.00
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What is a COMPLEX system?
Emergence
Dynamic loops
Interconnectivity
Number of parts
InteractionDisorder Unexpected
Diversity
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Exercise 1:
Draw the most COMPLEX figure you can imagine
Time: 2 s
Hint: loops, crossings, corners, randomness...
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Exercise 2:
Draw the SIMPLEST figure you can imagine
Time: 2 s
Hint: a straight line
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What happened?
A SIMPLE system was DIFFICULT to develop
A COMPLEX system was EASY to develop
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What is a COMPLEX system?
ACADEMIAProperty of a model
Based on system elements
INDUSTRYDifficulty to developBased on system / project properties
Needs a system architecture Does NOT help in mitigating/reducing complexity
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Why MEASURING complexity?
1. Understand system behavior
2. Design for some -ilities
3. Estimate development effort
Science drive
Design drive
Decision drive
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System complexity
FUNCTIONAL PHYSICAL ORGANIZAT.
Interdependence between system
functions
Interdependence between system
components
Interdependence between
organizations
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System complexity
SYSTEM PROJECT ENVIRON.
System of interest
The system developing the system
Where the system
operates
COGNITION
Understanding of people
interacting with system
*Sheard and Mostashari, extracting 39 factors from more than 300 definitions
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One more thought
Which one is more complex?
A standard car batteryA standard laptop
battery, but with 100 h autonomy
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Does the problem definition induce complexity?
Do requirements influence system complexity?
Can we anticipate a complexity bound?
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Perhaps a complexity SPECTRUM?
System complexity
Problem complexity
Organiz. complexity
Functional complexity
Structural complexity
Some correlations / overlaps already measured
Need a common unit of measurement
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How to MEASURE system complexity?
SCIENCE DESIGN ESTIMATION
DisorderBehavior
InterconnectednessParametric cost
estimators
π»=ββπ=1
π
ππ β πππ2 (ππ) πΆ=πΆ1+πΆ2 βπΆ3N parts, N I/Fs, N reqs,
materials...
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πΉ :πΆπβπΈ
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The power of joint ENTROPY
πΆ (πΆ1β―πΆπ)=ββπ1
β―βπππ (π1β―ππ )β πππ π [π (π1β―ππ ) ]
πΆ (πΆ1β―πΆπ)β₯πππ₯ (πΆπ)
πΆ (πΆ1β―πΆπ)β€βππΆπ
Property 1.
Property 2.
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... And therefore
Effort to reduce FUNCTIONAL/STRUCTURAL complexity may be limited/jeopardized by
how the PROJECT is organized or the REQUIREMENTS to be fulfilled!
MATHEMATICAL justification if joint entropy can be applied
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Problem Complexity
A function of the SIZE of the solution space
Design space
CS1 CS2
*CS: compliant space
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Problem Complexity
A function of AMOUNT of requirements and CONFLICTS between them
DSM?Flawed
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Problem Complexity
πΆπ=πΎ β(βπ=1π
ππ βπ π π)πΈ
ββπ=1
π
π» ππ π
Inspired on COSYSMO (Valerdi, 2008)
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Problem Complexity
πΆπ=πΎ β(βπ=1π
ππ βπ π π)πΈ
ββπ=1
π
π» ππ π
Calibration factor
Size of requirement set Conflicting requirements
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Problem Complexity
πΆπ=πΎ β(βπ=1π
ππ βπ π π)πΈ
ββπ=1
π
π» ππ π
Functional requirementRelative weight
Diseconomies of scale*
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Problem Complexity
πΆπ=πΎ β(βπ=1π
ππ βπ π π)πΈ
ββπ=1
π
π» ππ π
Amount of conflicting requirements *
Diseconomies of scale*
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Problem Complexity
πΆπ=πΎ β(βπ=1π
ππ βπ π π)πΈ
ββπ=1
π
π ππ» π
NOT IN THIS PAPER!
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Heuristics to identify conflicting requirements
H1β₯ 2 phases of matter
H4Competing for
resources
H3Opposing directions
laws of physics
H2Opposing directions
laws of society
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Case StudyID Requirement (fuzzy)
R1 Standard driving functionality
R2 4x wheel traction
R3 Big trunk
R4 Airbag
R5 Auto parking
R6 Auto breaking
R7 High speed & acceleration
R8 High autonomy
Requirement de-scoping
Industry benchmarkVs.
Conflict-based
Problem complexityVs.
Expert judgment
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Case Study: BenchmarkID Requirement (fuzzy)
R1 Standard driving functionality
R2 4x wheel traction
R3 Big trunk
R4 Airbag
R5 Auto parking
R6 Auto breaking
R7 High speed & acceleration
R8 High autonomy
COSYSMO assessment based on industry experts
Dinh
1
2
1
1
3
3
2
2
De-scoped
Yes
Yes
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Case Study: Conflict basedID Requirement (fuzzy)
R1 Standard driving functionality
R2 4x wheel traction
R3 Big trunk
R4 Airbag
R5 Auto parking
R6 Auto breaking
R7 High speed & acceleration
R8 High autonomy
Sensitivity based on (notional) problem complexity metric
Dinh
1
2
1
1
3
3
2
2
De-scoped
Yes
rf
X
X
X
X
X
H3
+mass
-mass/+energy
-mass/-energy
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Case Study: Comparative analysis
Element
Problem complexity
Resulting functionality
Resulting performance
Relative complexity subject matter expert
Dinh de-scoped requirements
Based on industry experts
Benchmark
58.61
3/5
3/3
β
3
Conflict-based
33.22
5/5
2/3
β
1
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Contributions
System complexity
Problem complexity
Organiz. complexity
Functional complexity
Structural complexity
πΆ (πΆ1β―πΆπ)=ββπ1
β―βπππ (π1β―ππ )β πππ π [π (π1β―ππ ) ]
πΆπ=πΎ β(βπ=1π
ππ βπ π π)πΈ
ββπ=1
π
π» ππ π
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Left for the future
VALIDATE heuristics based on subject matter expert
Perform RELATIVE calibration of problem complexity
Perform ABSOLUTE calibration of problem complexity
Further investigate IMPLICATIONS of joint entropy
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TOPIC TITLE:THE CONCEPT OF PROBLEM COMPLEXITY
Alejandro SaladoStevens Institute of [email protected]+49 176 321 31458