FIA - Fuzzy Information Axiom (Axiomatic Design under fuzzy data)

Overview

FIA evaluates how well each alternative's achievable range (system range) matches the design target range under fuzzy uncertainty. Lower total information content I_i (in bits) means the alternative satisfies the functional requirements more reliably. An infinite I_ij means the alternative cannot satisfy FR j at all (zero intersection between system range and design range).

Output

utility, lower is better

Data

Fuzzy (TFN), uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy (Triangular) MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Assemble the TFN system range matrix R̃ = [r̃_ij] (rij = (l_ij, m_ij, u_ij)) and the TFN design range vector D̃ = (d̃_1, …, d̃_n) with d_j = (dl_j, dm_j, du_j). Linguistic terms (e.g. Low, Medium, High) must first be encoded into TFNs via the chapter's conversion scales (Fig.14-15 for intangibles, Fig.18-21 for tangible costs).

   $${\tilde{r}}_{ij}=(l_{ij},m_{ij},u_{ij}),{\tilde{d}}_j=(d{l}_j,d{m}_j,d{u}_j),0\le l\le m\le u$$

   Kahraman 2008 Ch.8 §2.2 p.213-215; Fig.14-15 p.223-224 linguistic encoding

2. 2

   For each (i, j) compute the System Area S_ij (planar area under the system-range TFN membership function) and the Common Area C_ij (planar area of the geometric intersection of system-range TFN and design-range TFN). For unit-height triangular fuzzy numbers, S_ij = (u_ij − l_ij) / 2. The Common Area is computed by integrating min(μ_sys(x), μ_des(x)) over the overlap support; for triangular membership functions this reduces to a piecewise polynomial of degree 2 with break-points at the membership crossovers. Both S_ij and C_ij are crisp scalars - the algorithm is NOT fuzzy arithmetic on TFNs; the fuzziness is captured only by the geometry of membership functions whose intersection area is taken (Kahraman 2008 Ch.8 §2.2 Fig.5 p.215).

   $$S_{ij}=\frac{1}{2}(u_{ij}-l_{ij});C_{ij}={\int }_{x\in \mathrm{supp}({\tilde{r}}_{ij})\cap \mathrm{supp}({\tilde{d}}_j)}\min ({\mu }_{{\tilde{r}}_{ij}}(x),{\mu }_{{\tilde{d}}_j}(x))dx$$

   Kahraman 2008 Ch.8 §2.2 Eq.(7) p.215; Fig.5 p.215; worked Eq.(16)-(17) p.223-224

3. 3

   Compute information content I_ij = log_2(S_ij / C_ij). If C_ij = 0 (empty intersection), then I_ij = ∞ and alternative i is infeasible on FR j (e.g. book Eq.(8) p.217 where FMS-I's ADMC system range does not overlap the design range, yielding I_{ADMC-1} = ∞).

   $$I_{ij}={\log }_2\left(\frac{S_{ij}}{C_{ij}}\right)(=\infty \text{ if }{C}_{ij}=0)$$

   Kahraman 2008 Ch.8 §2.2 Eq.(7) p.215; Eq.(8) p.217 infeasibility case; Suh 1990 Information Axiom

4. 4

   Aggregate per-alternative total information content I_i. The chapter presents two forms: (a) Unweighted sum I_i = Σ_j I_ij (Table 3 p.222, Table 5 p.225, Table 8 p.230 - all uses this form). (b) Weighted sum I_i = Σ_j w_j · I_ij (referenced for the weighted multi-attribute approach in §4.2). This implementation uses the weighted form (collapsing to unweighted when all w_j are equal); if w_j are not provided, default to uniform weights w_j = 1/n.

   $$I_i=\sum _{j=1}^n{w}_j\cdot I_{ij}\text{(reduces to }\sum _j{I}_{ij}\text{ when }{w}_j=1/n\cdot n=1)$$

   Kahraman 2008 Ch.8 §3 Table 3 p.222 (unweighted); §4.2 p.230 (weighted form)

5. 5

   Rank alternatives in ascending order of I_i (smaller information content = greater certainty of satisfying the FRs = better alternative). The alternative with minimum I_i is selected as the best design (Suh's Information Axiom).

   $$i^{\star }=\arg \underset{i\in \{1,\dots ,m\}}{\min }{I}_i$$

   Kahraman 2008 Ch.8 §3 Table 5 p.225 (FMS-II selected, min I = 15.275); Suh 1990 Information Axiom

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Fuzzy (Triangular) numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Edge cases and pitfalls

• •If C_ij = 0 (empty intersection), then I_ij = ∞ and alternative i is infeasible on FR j (e.g. book Eq.(8) p.217 where FMS-I's ADMC system range does not overlap the design range, yielding I_{ADMC-1} =
• •default to uniform weights w_j = 1/n.

• Zero common area: any FR with C_ij = 0 makes I_ij infinite and eliminates the alternative entirely. Decision makers must adjust design ranges or accept the elimination.

• System area S_ij uses the TFN triangular area formula (u-l)/2; non-triangular membership functions require a different area computation (Kahraman 2008 Ch.8 §3 footnote).

• FIA assumes independence among FRs (Suh's Independence Axiom). Strongly coupled criteria require axiom-1 (independence) reformulation before applying axiom-2 (information).

Works with

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