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

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).
$\tilde{r}_{ij} = (l_{ij}, m_{ij}, u_{ij}), \tilde{d}_{j} = (d l_{j}, d m_{j}, d u_{j}), 0 \leq l \leq m \leq u$
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).
$S_{ij} = \frac{1}{2} (u_{ij} - l_{ij}); C_{ij} = \int_{x \in supp (\tilde{r}_{ij}) \cap supp (\tilde{d}_{j})} min (μ_{\tilde{r}_{ij}} (x), μ_{\tilde{d}_{j}} (x)) d x$
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} = ∞).
$I_{ij} = lo g_{2} (\frac{S _{ij}}{C _{ij}}) (= \infty if C_{ij} = 0)$
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.
$I_{i} = j = 1 \sum n w_{j} \cdot I_{ij} (reduces to j \sum I_{ij} when w_{j} = 1/ n \cdot n = 1)$
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).
$i^{⋆} = ar g i \in {1, \dots, m} min I_{i}$

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 \leq l \leq m \leq u

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 supp (\tilde{r}_{ij}) \cap supp (\tilde{d}_{j})} min (μ_{\tilde{r}_{ij}} (x), μ_{\tilde{d}_{j}} (x)) d x

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} = lo g_{2} (\frac{S _{ij}}{C _{ij}}) (= \infty if C_{ij} = 0)

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} = j = 1 \sum n w_{j} \cdot I_{ij} (reduces to j \sum I_{ij} when w_{j} = 1/ n \cdot n = 1)

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^{⋆} = ar g i \in {1, \dots, m} min I_{i}