This paper provides mathematical indications that the size of the solution space is related to the probability of developing affordable systems. In particular, the bigger the size of the solution space, the more chances to find affordable solutions. It also provides mathematical indications of why adapting priorities to needs result in higher levels of system affordability. DOI: http://dx.doi.org/10.1016/j.procs.2014.03.067

1. Increasing the probability of developing
affordable systems by maximizing and
adapting the solution space
Alejandro Salado
Stevens Institute of Technology

2. Is system AFFORDABILITY important?

4. System affordabiltiy
𝐴 𝑡 =
𝑘1 𝐵 𝑡
1 + 𝑘2 𝐶 𝑡
𝑖𝑓 𝑆 𝑡 ≥ 𝐶 𝑡
0 𝑖𝑓 𝑆 𝑡 < 𝐶 𝑡
System affordability
Benefits
Investment
Budget

5. Requirements influence system affordabiltiy
EMPIRICAL EVIDENCE
THEORETICAL
UNDERSTANDING
 ?
Heuristics & rules of thumb Theorems & laws

6. Exploit benefits of a formal SYSTEMS THEORY
Requirements
Size solution space
Order solution space
System affordability

7. Some principles
MATHEMATICAL APPROACH
REQUIREMENTS
SHALL O=A+B

8. Hypotheses
↓ 𝐶𝑆 𝑜𝑟𝑑𝑒𝑟𝑒𝑟𝑟𝑜𝑟
→↑ 𝑝 𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡 = 𝑡 𝑛
↑ 𝐶𝑆𝑠𝑖𝑧𝑒 →↑ 𝑝 𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡 = 𝑡 𝑛
PROPOSITION 1
PROPOSITION 2
Compiant
space
Alignment
to stkh
needs
Real-life
limittion

9. Proof Proposition 1
𝑆𝑁𝑖 = 𝑆𝑁𝑖 𝑒 𝑖𝜃
𝑅𝑖 = 𝑅𝑖 𝑒 𝑖𝜃
𝑒𝑙𝑖𝑐𝑖𝑡 𝑆𝑁𝑖 = 𝑅𝑖 = 𝑅𝑖 + 𝑒𝑟𝑟𝑜𝑟
Relative
priorities
Need
Prioritized
needs
Minimize

10. Proof Proposition 1
𝑅𝑖 = 𝑅𝑖 𝑒 𝑖𝜃
Magnitude errors
Phase errors
Incorrect or incomplete
requirements
De-aligned priorities
with respect to stkh

11. Proof Proposition 1
Phase errors De-aligned priorities
with respect to stkh
𝑅𝑖(𝑡0 = 𝑅𝑖(𝑡0+𝑛
Requirements prioritization
BUT
Even in spiral!

12. Proof Proposition 1
𝐴 𝑡 =
𝑘1 𝐵 𝑡
1 + 𝑘2 𝐶 𝑡 𝑆 𝑡 ≥𝐶 𝑡
∆A
∆∅
≅
k1
∆B
∆∅
1+k2
∆C
∆∅
Time
dependency

13. Proof Proposition 1
∆A
∆∅
≅
k1
∆B
∆∅
1+k2
∆C
∆∅
∆𝐵
∆∅
∆𝐶
∆∅
∆𝐴
∆∅
≥ 0 N/A N/A
< 0 ≥ 0 < 0
< 0 < 0 ?

14. Hypotheses
↓ 𝐶𝑆 𝑜𝑟𝑑𝑒𝑟𝑒𝑟𝑟𝑜𝑟
→↑ 𝑝 𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡 = 𝑡 𝑛
↑ 𝐶𝑆𝑠𝑖𝑧𝑒 →↑ 𝑝 𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡 = 𝑡 𝑛
PROPOSITION 1
PROPOSITION 2
Compiant
space
Real-life
limittion

15. Proof Proposition 2
𝑝 𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝐾
𝑛 𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑙𝑒
𝑛 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑒
Effectiveness
design/exploration
method
Amount
of
affordable
solutions
in the CSAmount of
solutions in the
design spcae

16. Proof Proposition 2
𝑝 𝑎𝑓𝑓 𝐶𝑆1 = 𝐾1
𝑛 𝑎𝑓𝑓 𝐶𝑆1
𝑛 𝑢𝑛𝑖𝑣
𝑝 𝑎𝑓𝑓 𝐶𝑆2 = 𝐾2
𝑛 𝑎𝑓𝑓 𝐶𝑆2
𝑛 𝑢𝑛𝑖𝑣
𝑝 𝑎𝑓𝑓 𝐶𝑆1 = 𝑝 𝑎𝑓𝑓 𝐶𝑆2
𝐾1 𝑛 𝑎𝑓𝑓 𝐶𝑆1
𝐾2 𝑛 𝑎𝑓𝑓 𝐶𝑆2
Constant

17. Proof Proposition 2
𝑝 𝑎𝑓𝑓 𝐶𝑆1 = 𝑝 𝑎𝑓𝑓 𝐶𝑆2
𝐾1 𝑛 𝑎𝑓𝑓 𝐶𝑆1
𝐾2 𝑛 𝑎𝑓𝑓 𝐶𝑆2
𝑎𝑓𝑓𝑜𝑟𝑑 = 𝒰 𝑥, 𝑦
𝐶𝑆2 ⊂ 𝐶𝑆1
𝐾1 = 𝐾2
𝑝 𝑎𝑓𝑓 𝐶𝑆1 ≈ 𝑝 𝑎𝑓𝑓 𝐶𝑆2
𝐶𝑆1 𝑠𝑖𝑧𝑒
𝐶𝑆2 𝑠𝑖𝑧𝑒
BUT THIS IS ONLY ONE TRY!!!

18. Proof Proposition 2
𝑝 𝑎𝑓𝑓 𝑛
= 𝑝𝑠1
+ 𝑝 𝑓1
𝑝𝑠2
+ ⋯ + 𝑝 𝑓1
⋯ 𝑝 𝑓𝑛−1
𝑝𝑠 𝑛
𝐶𝑆𝑠𝑖𝑧𝑒 ≫ 𝑛 → 𝑝𝑠1
≈ 𝑝𝑠2
≈ ⋯ ≈ 𝑝𝑠 𝑛
No learning / No anchoring
𝑝 𝑎𝑓𝑓 𝑛
≈ 𝑝𝑠
𝑖=0
𝑛−1
1 − 𝑝𝑠
𝑖

19. Proof Proposition 2
𝑝 𝑎𝑓𝑓 𝑛
𝐶𝑆1
𝑝 𝑎𝑓𝑓 𝑛
𝐶𝑆2
=
𝐶𝑆1 𝑠𝑖𝑧𝑒
𝐶𝑆2 𝑠𝑖𝑧𝑒
𝑖=0
𝑛−1
1 − 𝑝𝑠
𝐶𝑆1 𝑠𝑖𝑧𝑒
𝐶𝑆2 𝑠𝑖𝑧𝑒
𝑖
𝑖=0
𝑛−1
1 − 𝑝𝑠
𝑖

20. Proof Proposition 2
Number of design iterations
Relativesizeofthesolutionspace
2 4 6 8 10
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Relativeincreasep(affordablesolutions)
10
15
20
25
30
35
40
45
ps = 0.10

21. Proof Proposition 2
Number of design iterations
Relativesizeofthesolutionspace
2 4 6 8 10
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Relativeincreasep(affordablesolutions)
10
15
20
25
30
35
40
45
ps = 0.10
Number of design iterations
Relativesizeofthesolutionspace
2 4 6 8 10
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Relativeincreaseofp(affordablesolutions)
10
15
20
25
30
35
40
45
ps = 0.01

22. Contributions
↓ 𝐶𝑆 𝑜𝑟𝑑𝑒𝑟𝑒𝑟𝑟𝑜𝑟
→↑ 𝑝 𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡 = 𝑡 𝑛
↑ 𝐶𝑆𝑠𝑖𝑧𝑒 →↑ 𝑝 𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡 = 𝑡 𝑛
THEOREM 1
THEOREM 2
Effective evolutionary
priroitization?
How to max CS with
requirements?

23. Limitations
Distribution of affordable solutions is considered uniform
CS contains many more solutions than rework cycles
Learning and anchoring effects self-cancel

24. Left for the future
Investigate SENSITIVITY of ps on paff
Investigate SENSITIVITY of uniformity assumptions on paff
Investigte SENSITIVITY of number of solutions on paff
Investigate effects of LEARNING and ANCHORING
Explore effects on PROJECT data

25. TOPIC TITLE:
INCREASING THE PROBABILITY OF DEVELOPING
AFFORDABLE SYSTEMS BY MAXIMIZING AND
ADAPTING THE SOLUTION SPACE
Alejandro Salado
Stevens Institute of Technology
asaladod@stevens.edu
+49 176 321 31458