The Concept of Problem Complexity
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The document discusses the concept of problem complexity, highlighting the challenges of developing complex systems compared to simple ones, and the effects of requirements on complexity. It emphasizes the need for a common measurement unit for complexity and presents mathematical formulations to quantify it. Additionally, it includes case studies on problem complexity and suggests further research on heuristics and calibration methods.
The Concept of Problem Complexity
- 1. The concept of problem complexity Alejandro Salado Stevens Institute of Technology
- 2. A) Anyone born in Hoboken? B) Anyone born elsewhere? C) Anyone unborn?
- 3. Choose your preferred concept Constraints: fixed budget and schedule A very SIMPLE system Performance = 1.00 A very COMPLEX system Performance = 1.00
- 4. What is a COMPLEX system? Emergence Dynamic loops Interconnectivity Number of parts Interaction Disorder Unexpected Diversity
- 5. Exercise 1: Draw the most COMPLEX figure you can imagine Time: 2 s Hint: loops, crossings, corners, randomness...
- 7. Exercise 2: Draw the SIMPLEST figure you can imagine Time: 2 s Hint: a straight line
- 9. What happened? A SIMPLE system was DIFFICULT to develop A COMPLEX system was EASY to develop
- 10. What is a COMPLEX system? ACADEMIA Property of a model Based on system elements INDUSTRY Difficulty to develop Based on system / project properties Needs a system architecture Does NOT help in mitigating/reducing complexity
- 11. Why MEASURING complexity? 1. Understand system behavior 2. Design for some -ilities 3. Estimate development effort Science drive Design drive Decision drive
- 12. System complexity FUNCTIONAL PHYSICAL ORGANIZAT. Interdependence between system functions Interdependence between system components Interdependence between organizations
- 13. 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
- 14. One more thought Which one is more complex? A standard car battery A standard laptop battery, but with 100 h autonomy
- 15. Does the problem definition induce complexity? Do requirements influence system complexity? Can we anticipate a complexity bound?
- 16. 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
- 17. How to MEASURE system complexity? SCIENCE DESIGN ESTIMATION Disorder Behavior Interconnectedness Parametric cost estimators 𝐻 = − 𝑖=1 𝑛 𝑝𝑖 ∙ 𝑙𝑜𝑔2 𝑝𝑖 𝐶 = 𝐶1 + 𝐶2 ∙ 𝐶3 N parts, N I/Fs, N reqs, materials...
- 18. 𝐹: 𝐶𝑀 → 𝐸
- 19. The power of joint ENTROPY 𝐶 𝐶1 ⋯ 𝐶 𝑛 = − 𝑐1 ⋯ 𝑐 𝑛 𝑃 𝑐1 ⋯ 𝑐 𝑛 ∙ 𝑙𝑜𝑔𝑗 𝑃 𝑐1 ⋯ 𝑐 𝑛 𝐶 𝐶1 ⋯ 𝐶 𝑛 ≥ 𝑚𝑎𝑥 𝐶𝑖 𝐶 𝐶1 ⋯ 𝐶 𝑛 ≤ 𝑖 𝐶𝑖 Property 1. Property 2.
- 20. ... 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
- 21. Problem Complexity A function of the SIZE of the solution space Design space CS1 CS2 *CS: compliant space
- 22. Problem Complexity A function of AMOUNT of requirements and CONFLICTS between them DSM? Flawed
- 23. Problem Complexity 𝐶 𝑝 = 𝐾 ∙ 𝑖=1 𝑛 𝑎𝑖 ∙ 𝑟𝑓 𝑖 𝐸 ∙ 𝑗=1 𝑚 𝐻𝑗 𝑏 𝑗 Inspired on COSYSMO (Valerdi, 2008)
- 24. Problem Complexity 𝐶 𝑝 = 𝐾 ∙ 𝑖=1 𝑛 𝑎𝑖 ∙ 𝑟𝑓 𝑖 𝐸 ∙ 𝑗=1 𝑚 𝐻𝑗 𝑏 𝑗 Calibration factor Size of requirement set Conflicting requirements
- 25. Problem Complexity 𝐶 𝑝 = 𝐾 ∙ 𝑖=1 𝑛 𝑎𝑖 ∙ 𝑟𝑓 𝑖 𝐸 ∙ 𝑗=1 𝑚 𝐻𝑗 𝑏 𝑗 Functional requirementRelative weight Diseconomies of scale*
- 26. Problem Complexity 𝐶 𝑝 = 𝐾 ∙ 𝑖=1 𝑛 𝑎𝑖 ∙ 𝑟𝑓 𝑖 𝐸 ∙ 𝑗=1 𝑚 𝐻𝑗 𝑏 𝑗 Amount of conflicting requirements * Diseconomies of scale*
- 27. Problem Complexity 𝐶 𝑝 = 𝐾 ∙ 𝑖=1 𝑛 𝑎𝑖 ∙ 𝑟𝑓 𝑖 𝐸 ∙ 𝑗=1 𝑚 𝑏𝑗 𝐻 𝑗 NOT IN THIS PAPER!
- 28. Heuristics to identify conflicting requirements H1 ≥ 2 phases of matter H4 Competing for resources H3 Opposing directions laws of physics H2 Opposing directions laws of society
- 29. Case Study ID 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 benchmark Vs. Conflict-based Problem complexity Vs. Expert judgment
- 30. Case Study: Benchmark ID 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
- 31. Case Study: Conflict based ID 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
- 32. 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
- 33. Contributions System complexity Problem complexity Organiz. complexity Functional complexity Structural complexity 𝐶 𝐶1 ⋯ 𝐶 𝑛 = − 𝑐1 ⋯ 𝑐 𝑛 𝑃 𝑐1 ⋯ 𝑐 𝑛 ∙ 𝑙𝑜𝑔𝑗 𝑃 𝑐1 ⋯ 𝑐 𝑛 𝐶 𝑝 = 𝐾 ∙ 𝑖=1 𝑛 𝑎𝑖 ∙ 𝑟𝑓 𝑖 𝐸 ∙ 𝑗=1 𝑚 𝐻𝑗 𝑏 𝑗
- 34. 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
- 35. TOPIC TITLE: THE CONCEPT OF PROBLEM COMPLEXITY Alejandro Salado Stevens Institute of Technology asaladod@stevens.edu +49 176 321 31458