# Optimizing Design of Smart Workplace through Multi-Objective Programming
## 4 Proposed Smart Workplace Design Optimization Procedure
### Promethee  
PROMETHEE [30; 31; 32; 33] stands for the Preference Ranking Organization Method for Enrichment of Evaluations. It is a multi-criterion decision aiding tool that utilizes pair-wise comparison of alternatives for each criterion to determine the strength of preference of alternative \(a\) over alternative \(b\). The implementation of PROMETHEE evaluation involves mainly quantitative data and it requires additional information, namely:  
* The weights of the evaluation criteria considered and
* The decision maker's preference function, used when comparing the performance of the alternatives with respect to each individual criterion.  
One distinction process of PROMETHEE is the use of the preference function (\(P_{j}\)) to translate the difference between the evaluations of two alternatives (\(a\) and \(b\)) for a particular criterion, into a preference degree ranging from 0 to 1. Let A be a finite set of alternatives, A = {\(a_{1},a_{2},\,\ldots,\,a_{n}\)}, C is a finite set of criteria, C = {\(c_{1},c_{2},\,\),\(c_{j}\,\ldots,\,c_{k}\)}, having a corresponding set of prespecified weights or importance's, W = { \(w_{1}\), \(w_{2}\),..., \(w_{j}\)..., \(w_{k}\) }. Then, for two alternatives \(a\),\(b\)\(\in\)\(A\):  
\[P_{j}(a,b)\ =\ G_{j}\big{[}\ c_{j}(a)\ -\ c_{j}(b)\ \big{]} \tag{1}\]  
Figure 2: Smart workplace model building blocks.  
\[0\leq P_{j}(a,b)\leq 1 \tag{2}\]  
\(P_{j}(a,b)\) is the preference function associated with the criterion performance \(c_{j}(i)\) for the \(i^{th}\) alternative, where \(G_{j}\) is a non-decreasing function of the observed deviation \(d\), between \(c_{j}(a)\) and \(c_{j}(b)\). The values of the criterion performance, \(c_{j}(i)\) of the alternative \(i\) can take crisp numerical, linguistic with corresponding numerical scale or fuzzy numbers as in [PERSON] et al. (2007) [33].  
In order to facilitate the selection of a specific preference function, six basic types have been proposed: usual function, U-shape function, V-shape function, level function, linear function and Gaussian function (as presented in Table 1) [PERSON] (1982) [30], [PERSON] and [PERSON] (1985) [32], [PERSON] and [PERSON] (1994) [33].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Shape** & **Definition** & **Threshold** \\ \hline \hline \multirow{2}{*}{Usual} & \multirow{2}{*}{\(P(d)=\begin{cases}0&d\leq 0\\ 1&d>0\end{cases}\)} & No threshold \\ \cline{1-1}  & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Preference functions of the PROMETHE technique.  
The values of the PROMETHEE compute the following quantities in the pair-wise evaluation of alternatives a and b, as follows:  
The overall preference index of alternative a over b, (_a,b_ \(\in A\)):  
\[\pi(a,b)=\sum_{j=1}^{k}P_{j}(a,b)w_{j} \tag{3}\]  
The average positive-outranking flow of alternative a:  
\[\varphi_{a}^{+}=\sum_{x\in A}\pi(a,x) \tag{4}\]  
The average negative-outranking flow of alternative a:  
\[\varphi_{a}^{-}=\sum_{x\in A}\pi(x,a) \tag{5}\]  
The net outranking flow of alternative a:  
\[\varphi_{a}^{net}=\varphi_{a}^{+}-\ \varphi_{a}^{-} \tag{6}\]  
where \(\varphi_{a}^{net}\) is called a net flow, and a higher value of it means a higher attractiveness of alternative a.  
In this research, we are going to employ the PROMETHEE technique in computing priorities of the workplace's operational objectives, with respect to two prespecified overall goals of the workplace. Detailed results of PROMETHEE computations will be illustrated in the provided example, in Section 5. The computation results are found in Appendix B (Tables A2-A4)  
Next section, we present a method for handling vague relations among the smart workplace optimization model building blocks.