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 Technologyasaladod@stevens.edu+49 176 321 31458