The concept of problem complexity

Alejandro SaladoStevens Institute of Technology

A) Anyone born in Hoboken?

B) Anyone born elsewhere?

C) Anyone unborn?

Choose your preferred concept

Constraints: fixed budget and schedule

A very SIMPLE system

Performance = 1.00

A very COMPLEX system

Performance = 1.00

What is a COMPLEX system?

Emergence

Dynamic loops

Interconnectivity

Number of parts

InteractionDisorder Unexpected

Diversity

Exercise 1:

Draw the most COMPLEX figure you can imagine

Time: 2 s

Hint: loops, crossings, corners, randomness...

Exercise 2:

Draw the SIMPLEST figure you can imagine

Time: 2 s

Hint: a straight line

What happened?

A SIMPLE system was DIFFICULT to develop

A COMPLEX system was EASY to develop

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

Why MEASURING complexity?

1. Understand system behavior

2. Design for some -ilities

3. Estimate development effort

Science drive

Design drive

Decision drive

System complexity

FUNCTIONAL PHYSICAL ORGANIZAT.

Interdependence between system

functions

Interdependence between system

components

Interdependence between

organizations

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

One more thought

Which one is more complex?

A standard car batteryA standard laptop

battery, but with 100 h autonomy

Does the problem definition induce complexity?

Do requirements influence system complexity?

Can we anticipate a complexity bound?

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

How to MEASURE system complexity?

SCIENCE DESIGN ESTIMATION

DisorderBehavior

InterconnectednessParametric cost

estimators

𝐻=−∑𝑖=1

𝑛

𝑝𝑖 ∙ 𝑙𝑜𝑔2 (𝑝𝑖) 𝐶=𝐶1+𝐶2 ∙𝐶3N parts, N I/Fs, N reqs,

materials...

𝐹 :𝐶𝑀→𝐸

The power of joint ENTROPY

𝐶 (𝐶1⋯𝐶𝑛)=−∑𝑐1

⋯∑𝑐𝑛𝑃 (𝑐1⋯𝑐𝑛 )∙ 𝑙𝑜𝑔 𝑗 [𝑃 (𝑐1⋯𝑐𝑛 ) ]

𝐶 (𝐶1⋯𝐶𝑛)≥𝑚𝑎𝑥 (𝐶𝑖)

𝐶 (𝐶1⋯𝐶𝑛)≤∑𝑖𝐶𝑖

Property 1.

Property 2.

... 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

Problem Complexity

A function of the SIZE of the solution space

Design space

CS1 CS2

*CS: compliant space

Problem Complexity

A function of AMOUNT of requirements and CONFLICTS between them

DSM?Flawed

Problem Complexity

𝐶𝑝=𝐾 ∙(∑𝑖=1𝑛

𝑎𝑖 ∙𝑟 𝑓 𝑖)𝐸

∙∏𝑗=1

𝑚

𝐻 𝑗𝑏 𝑗

Inspired on COSYSMO (Valerdi, 2008)

Problem Complexity

𝐶𝑝=𝐾 ∙(∑𝑖=1𝑛

𝑎𝑖 ∙𝑟 𝑓 𝑖)𝐸

∙∏𝑗=1

𝑚

𝐻 𝑗𝑏 𝑗

Calibration factor

Size of requirement set Conflicting requirements

Problem Complexity

𝐶𝑝=𝐾 ∙(∑𝑖=1𝑛

𝑎𝑖 ∙𝑟 𝑓 𝑖)𝐸

∙∏𝑗=1

𝑚

𝐻 𝑗𝑏 𝑗

Functional requirementRelative weight

Diseconomies of scale*

Problem Complexity

𝐶𝑝=𝐾 ∙(∑𝑖=1𝑛

𝑎𝑖 ∙𝑟 𝑓 𝑖)𝐸

∙∏𝑗=1

𝑚

𝐻 𝑗𝑏 𝑗

Amount of conflicting requirements *

Diseconomies of scale*

Problem Complexity

𝐶𝑝=𝐾 ∙(∑𝑖=1𝑛

𝑎𝑖 ∙𝑟 𝑓 𝑖)𝐸

∙∏𝑗=1

𝑚

𝑏 𝑗𝐻 𝑗

NOT IN THIS PAPER!

Heuristics to identify conflicting requirements

H1≥ 2 phases of matter

H4Competing for

resources

H3Opposing directions

laws of physics

H2Opposing directions

laws of society

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

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

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

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

↑

3

Conflict-based

33.22

5/5

2/3

↓

1

Contributions

System complexity

Problem complexity

Organiz. complexity

Functional complexity

Structural complexity

𝐶 (𝐶1⋯𝐶𝑛)=−∑𝑐1

⋯∑𝑐𝑛𝑃 (𝑐1⋯𝑐𝑛 )∙ 𝑙𝑜𝑔 𝑗 [𝑃 (𝑐1⋯𝑐𝑛 ) ]

𝐶𝑝=𝐾 ∙(∑𝑖=1𝑛

𝑎𝑖 ∙𝑟 𝑓 𝑖)𝐸

∙∏𝑗=1

𝑚

𝐻 𝑗𝑏 𝑗

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

TOPIC TITLE:THE CONCEPT OF PROBLEM COMPLEXITY

Alejandro SaladoStevens Institute of Technologyasaladod@stevens.edu+49 176 321 31458