# Fuzzy Preference Programming Framework for Functional Assessment of Subway Networks
## 2 Background
### The Fuzzy Analytic Network Process  
In real-life decision-making situations, decision makers or experts could be uncertain about their own level of preference, due to incomplete information, insufficient knowledge, complexity, lack of appropriate measurement scale, or uncertainty within the decision environment. Decision makers also tend to specify preferences in the form of natural language expressions, which are most often vague and uncertain [17]. Fuzzy logic is a natural way to incorporate the uncertainty or the vagueness of the human judgment. When comparing two elements, the uncertain numerical ratio is expressed in a fuzzy manner rather than an exact one. Then, an appropriate prioritization procedure is applied to derive local priorities approximately satisfying the provided judgments. The concept of Fuzzy Analytical Hierarchy Process (FAHP) and Fuzzy Analytical Network Process (FANP) was extensively researched in literature and several fuzzy models were introduced. The common methods used for prioritization include, [PERSON]'s original Eigen Vector method, The Direct Least Squares Method proposed by [18], Logarithmic Least Squares Method also known as the geometric method [19], goal programming method proposed by [20], and Fuzzy Preference Programming Method proposed by [21]. Fuzzy Preference Programming (FPP) adopts the concept of \(\alpha\)-cuts to decompose fuzzy numbers into a number of intervals in order to represent the initial fuzzy sets. The method calculates priorities for each \(\alpha\)-level cut that are further aggregated into crisp local and global priorities [22]. The FPP method is based on two assumptions: The first assumption requires the existence of nonempty fuzzy feasible area \(\overline{P}\) on the simplex hyper plane \(Q^{n-1}\), as shown in Equation (1) \(Q^{n-1}\),  
\[Q^{n-1}=\ \Big{\{}(w_{1}\ldots\ w_{n})\Big{|}\ w_{i}\ >\ 0,\ \sum\
olimits_{i=1}^{n}w_{i}\ =1 \Big{\}}. \tag{1}\]  
The membership function of the fuzzy feasible area is expressed in terms of the intersection of all interval membership functions in \(\mu_{k}\) (\(R_{k}\,w\)), as shown in Equation (2):  
\[\mu_{\overline{P}}^{\ =\ }(w)\ =\ [\min\ \{\mu_{1}\ \ (R1\ w),\ \mu_{m}\ ( Rm\,w)\}\Big{|}\ w\in Q^{n-1}\Big{]} \tag{2}\]  
The second assumption: specifies a selection rule, which determines a priority vector having the highest degree of membership in the aggregated membership function \(\mu_{p}\) (\(w\)), since \(\overline{P}\) is a convex set, then there is always a priority vector (\(w\)) in \(Q^{n-1}\) that has a maximum degree of membership (\(\lambda\)); therefore, Equation (3) can be concluded:  
\[\mu_{\overline{P}}^{\ =\ }(w)\ =\ \lambda\ =\ \max\ [\min\ \{\mu_{1}\ \ (R_{1}\ w)\ \ldots\ \mu_{m}(R_{m}\,w)\}\big{|}\ w\ \in\ Q^{n-1}\Big{]} \tag{3}\]  
The max-min prioritization problem can be represented as the fuzzy programming problem shown in Equation (4):  
\[\text{Max }\lambda\] (4) \[\text{Subject to}\ \ \ d_{k}\,\lambda+\ R_{k}\,w\ \leq\ d_{K}\ \ \ \ k\ =\ 1,\,2\ldots\ m\ \ \ \ \ \ \ \ \ m=\ n\ (n\ -\ 1)/2.\] \[\ \Reference [22] also developed a non-linear fuzzy prioritization model to obtain directly crisp values of priorities from a set of comparison judgments, represented as triangular fuzzy numbers. If the optimal value of \(\lambda^{\star}\) is positive, this indicates all solution ratios satisfy the fuzzy judgment completely, which means that the initial set of fuzzy judgments is rather consistent. Alternatively, a negative value of (\(\lambda^{\star}\)) shows the solution ratios approximately satisfy all double-side inequalities. Therefore, the optimal value \(\lambda^{\star}\) can be used for measuring the consistency of the initial set of fuzzy judgments. FANP handles the uncertainty in quite a different manner than that of regular ANP with sensitivity analysis. Fuzzy ANP accommodates the subjectivity of human judgment as being expressed in natural language which entails 'fuzziness' in real-life problems.