# Optimizing Design of Smart Workplace through Multi-Objective Programming
## 4 Proposed Smart Workplace Design Optimization Procedure
### Fuzzy Relations  
Fuzzy relations have been applied in many areas, such as in fuzzy control, project evaluation, clustering, and inexact modeling. They generalize the concept of relations in the same way the fuzzy sets generalize the concept of sets. The concept of fuzzy set theory was introduced in 1965 by [PERSON] [34]. The theory was a means of representing and manipulating data that are not exact or precise, but rather ambiguous or fuzzy. In fuzzy set theory, an element x may partially belong to a set A, expressed by a degree of membership \(\mu_{A}(x)\). A fuzzy set A in X, where X is a non-empty set, is characterized by its membership function degree:  
\[\mu_{A}\text{: X}\rightarrow\text{[0,1]} \tag{7}\]  
\(\mu_{A}(x)\) is the membership degree of the element x in the fuzzy set A for each x \(\in\) X. Where, the value zero represents complete non-membership, the value one represents complete membership and any other values in between are used to represent the intermediate degree of membership. Now, a relation mathematically describes a situation where certain elements of sets are linked to one another in some defined aspect. It is important to distinguish between crisp and fuzzy relations:  
- Crisp Relation:  
A crisp relation is defined on the Cartesian product of two universal sets. Let \(X\), \(Y\subseteq R\) be two universal sets then, the crisp relation \(R\) is defined by its membership function as follows:  
\[R=\ X\ \times\ Y=\{(x,y)|\ x\in X\,\ y\ \in Y\} \tag{8}\]  
\[\mu_{R}(x,y)=\left\{ \begin{array}{ll}1,\ (x,y)\in R\\ 0,\ (x,y)\
otin R\end{array} \right. \tag{9}\]where, the value 1 means complete membership for the pair in relation and value 0 means no relation at all. - Fuzzy relation:  
A fuzzy relation is defined also on the Cartesian product of two universal sets. Let, \(X\), \(Y\subseteq R\) be universal sets, then;  
\[R=\ X\ \times\ Y=\{(x,y),\ \mu_{R}(x,y)|x\in X\,\ y\ \in Y\} \tag{10}\]  
\[R:\ X\ \times\ Y\ \rightarrow\ [0,1] \tag{11}\]  
\(R\) is called a fuzzy relation in \(X\times Y\subseteq R\). The value 1 means complete membership of pair (\(x\),\(y\)) in relation \(R\), whereas, the value 0 means complete non-membership, and any value in between means the intermediate degree of membership in \(R\). A fuzzy relation \(R\) between vectors \(x_{m}\) and \(y_{n}\) can be presented in the form of a two-dimensional \(m\times n\) matrix:  
\[R= \begin{array}{c}x_{1}\\ x_{2}\\ \cdots\\ x_{m}\\ \end{array} \left[ \begin{array}{cccc}\mu_{R}(x_{1},y_{1})&\cdots&\cdots&\mu_{ R}(x_{1},y_{n})\\ \cdots&\cdots&\cdots&\cdots\\ \mu_{R}(x_{m},y_{1})&\cdots&\cdots&\mu_{R}(x_{m},y_{n})\\ \end{array} \right] \tag{12}\]  
where \(\mu_{R}\big{(}x_{i},y_{j}\big{)}\) are membership functions for pair \((x_{i},\ y_{j})\) in the relation \(R\). Fuzzy relation between elements of two sets can be expressed also by linguistic terms, for instance: \"\(x\) is considerable smaller than \(y\)\", \"\(x\) is very close to \(y\)\", and \"Approximately equal\", etc. In this research, the fuzzy relation \"Positively impact\" will be established among the three basic model building blocks: SETs, SFs, and OOs of the considered workplace. The construction of fuzzy relation matrices that quantify relations among the mentioned three sets of the model building blocks. This requires a unified numerical scale to quantify the relations using the expert's assigned linguistic value with a corresponding numerical value representing a membership for each element or relation pair. Therefore, to simplify the impact assessment process, a set of experts or specializing decision analysts can assign a linguistic term with a corresponding membership for each relation pair (see Table 2). The membership values of the fuzzy relation \"Positively impact\" are graded from 0 up to one. Negative and neutral impacts are considered as zero up to 0.1 and expressed linguistically as \"Never\". Using the above linguistic scale, the decision analyst can quantify the relation membership value between every two elements of any two related sets.