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Increasing the probability of developing affordable systems by maximizing and
adapting the solution space
Alejandro SaladoStevens Institute of Technology
Is system AFFORDABILITY important?
System affordabiltiy
𝐴 (𝑡 )={¿ 𝑘1𝐵 (𝑡 )1+𝑘2𝐶 (𝑡 )
𝑖𝑓 𝑆 (𝑡 )≥𝐶 (𝑡 )
¿0 𝑖𝑓 𝑆 (𝑡 )<𝐶 (𝑡 )
System affordability
Benefits
Investment
Budget
Requirements influence system affordabiltiy
EMPIRICAL EVIDENCE THEORETICAL UNDERSTANDING
?Heuristics & rules of thumb Theorems & laws
Exploit benefits of a formal SYSTEMS THEORY
Requirements
Size solution spaceOrder solution space
System affordability
Some principles
MATHEMATICAL APPROACH
REQUIREMENTS
SHALL O=A+B
Hypotheses
↓𝐶𝑆𝑜𝑟𝑑𝑒𝑟 𝑒𝑟𝑟𝑜𝑟→↑𝑝𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 (𝑡=𝑡𝑛)
↑𝐶𝑆𝑠𝑖𝑧𝑒→↑𝑝𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 (𝑡=𝑡𝑛)
PROPOSITION 1
PROPOSITION 2
Compiant space
Alignment to stkh needs
Real-life limittion
Proof Proposition 1
𝑆𝑁 𝑖=𝑆𝑁 𝑖𝑒𝑖 𝜃
𝑅𝑖=𝑅 𝑖𝑒𝑖 𝜃
𝑒𝑙𝑖𝑐𝑖𝑡 (𝑆𝑁 𝑖 )=𝑅𝑖=𝑅𝑖+𝑒𝑟𝑟𝑜𝑟
Relative priorities
Need
Prioritized needs
Minimize
Proof Proposition 1
𝑅𝑖=𝑅 𝑖𝑒𝑖 𝜃
Magnitude errors
Phase errors
Incorrect or incomplete requirements
De-aligned priorities with respect to stkh
Proof Proposition 1
Phase errors De-aligned priorities with respect to stkh
¿Requirements prioritization
BUT
Even in spiral!
Proof Proposition 1
𝐴 (𝑡 )=𝑘1𝐵 (𝑡 )1+𝑘2𝐶 (𝑡 )|𝑆 (𝑡 )≥𝐶 ( 𝑡 )
∆A∆∅
≅k1∆B∆∅
1+k2∆C∆∅
Time dependency
Proof Proposition 1
∆A∆∅
≅k1∆B∆∅
1+k2∆C∆∅
≥ 0 N/A N/A< 0 ≥ 0 < 0< 0 < 0 ?
Hypotheses
↓𝐶𝑆𝑜𝑟𝑑𝑒𝑟 𝑒𝑟𝑟𝑜𝑟→↑𝑝𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 (𝑡=𝑡𝑛)
↑𝐶𝑆𝑠𝑖𝑧𝑒→↑𝑝𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 (𝑡=𝑡𝑛)
PROPOSITION 1
PROPOSITION 2
Compiant space
Real-life limittion
Proof Proposition 2
𝑝𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦=𝐾𝑛𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑙𝑒
𝑛𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑒
Effectiveness design/exploration
method
Amount of
affordable solutions in the CSAmount of
solutions in the design spcae
Proof Proposition 2
𝑝𝑎𝑓𝑓 (𝐶𝑆1 )=𝐾1
𝑛𝑎𝑓𝑓 (𝐶𝑆1 )𝑛𝑢𝑛𝑖𝑣
𝑝𝑎𝑓𝑓 (𝐶𝑆2 )=𝐾2
𝑛𝑎𝑓𝑓 (𝐶𝑆2 )𝑛𝑢𝑛𝑖𝑣
𝑝𝑎𝑓𝑓 (𝐶𝑆1 )=𝑝𝑎𝑓𝑓 (𝐶𝑆2 )𝐾1𝑛𝑎𝑓𝑓 (𝐶𝑆1 )𝐾2𝑛𝑎𝑓𝑓 (𝐶𝑆2 )
Constant
Proof Proposition 2
𝑝𝑎𝑓𝑓 (𝐶𝑆1 )=𝑝𝑎𝑓𝑓 (𝐶𝑆2 )𝐾1𝑛𝑎𝑓𝑓 (𝐶𝑆1 )𝐾2𝑛𝑎𝑓𝑓 (𝐶𝑆2 )
𝑎𝑓𝑓𝑜𝑟𝑑=𝒰 (𝑥 , 𝑦 )𝐶𝑆2⊂𝐶𝑆1𝐾1=𝐾 2
𝑝𝑎𝑓𝑓 (𝐶𝑆1 )≈𝑝𝑎𝑓𝑓 (𝐶𝑆2 )𝐶𝑆1𝑠𝑖𝑧𝑒
𝐶𝑆2𝑠𝑖𝑧𝑒
BUT THIS IS ONLY ONE TRY!!!
Proof Proposition 2
𝑝𝑎𝑓𝑓 𝑛=𝑝𝑠1+𝑝 𝑓 1𝑝 𝑠2+⋯+𝑝 𝑓 1⋯𝑝 𝑓 𝑛−1𝑝𝑠𝑛
No learning / No anchoring
𝑝𝑎𝑓𝑓 𝑛≈𝑝𝑠∑𝑖=0
𝑛−1
(1−𝑝𝑠 )𝑖
Proof Proposition 2
𝑝𝑎𝑓𝑓 𝑛 (𝐶𝑆1 )𝑝𝑎𝑓𝑓 𝑛(𝐶𝑆2 )
=𝐶𝑆1 𝑠𝑖𝑧𝑒𝐶𝑆2𝑠𝑖𝑧𝑒
∑𝑖=0
𝑛−1
(1−𝑝𝑠𝐶𝑆1𝑠𝑖𝑧𝑒
𝐶𝑆2𝑠𝑖𝑧𝑒)𝑖
∑𝑖=0
𝑛−1
(1−𝑝𝑠 )𝑖
Proof Proposition 2
Number of design iterations
Rel
ativ
e si
ze o
f the
sol
utio
n sp
ace
2 4 6 8 101.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Rel
ativ
e in
crea
se p
(affo
rdab
le s
olut
ions
)
10
15
20
25
30
35
40
45
ps = 0.10
Proof Proposition 2
Number of design iterations
Rel
ativ
e si
ze o
f the
sol
utio
n sp
ace
2 4 6 8 101.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Rel
ativ
e in
crea
se p
(affo
rdab
le s
olut
ions
)
10
15
20
25
30
35
40
45ps = 0.10
Number of design iterations
Rel
ativ
e si
ze o
f the
sol
utio
n sp
ace
2 4 6 8 101.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Rel
ativ
e in
crea
se o
f p(a
fford
able
sol
utio
ns)
10
15
20
25
30
35
40
45ps = 0.01
Contributions
↓𝐶𝑆𝑜𝑟𝑑𝑒𝑟 𝑒𝑟𝑟𝑜𝑟→↑𝑝𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 (𝑡=𝑡𝑛)
↑𝐶𝑆𝑠𝑖𝑧𝑒→↑𝑝𝑎𝑓𝑓𝑜𝑟𝑑𝑎𝑏𝑖𝑙𝑖𝑡𝑦 (𝑡=𝑡𝑛)
THEOREM 1
THEOREM 2
Effective evolutionary priroitization?
How to max CS with requirements?
Limitations
Distribution of affordable solutions is considered uniform
CS contains many more solutions than rework cycles
Learning and anchoring effects self-cancel
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
TOPIC TITLE:INCREASING THE PROBABILITY OF DEVELOPING AFFORDABLE SYSTEMS BY MAXIMIZING AND ADAPTING THE SOLUTION SPACE
Alejandro SaladoStevens Institute of [email protected]+49 176 321 31458
