# Computing Idle Times in Fuzzy Flexible Job Shop Scheduling
## 2 The Fuzzy Flexible Job Shop Problem
### Fuzzy Processing Times  
It is fairly common in the literature to model uncertain processing times as triangular fuzzy numbers (TFNs), defined by an interval, \([a^{1},a^{3}]\), of possible values (its support) and a modal value, \(a^{2}\)[16]. A TFN can be represented as a triplet, \(\widehat{a}=(a^{1},a^{2},a^{3})\), so its membership function is given by the following:  
\[\mu_{\widehat{a}}(x)= \begin{cases}\frac{x-a^{1}}{a^{2}-a^{3}}&a^{1}\leq x\leq a ^{2}\\ \frac{x-a^{3}}{a^{2}-a^{3}}&a^{2}\leq x\leq a^{3}\\ 0&\text{otherwise}\end{cases}  \tag{1}\]  
Arithmetic operations on TFNs are defined based on the Extension Principle [46], so for two TFNs, \(\widehat{a}\) and \(\widehat{b}\), the sum is a TFN defined as follows:  
\[\widehat{a}+\widehat{b}=(a^{1}+b^{1},a^{2}+b^{2},a^{3}+b^{3}), \tag{2}\]  
And their difference is a TFN defined as follows:  
\[\widehat{a}-\widehat{b}=(a^{1}-b^{3},a^{2}-b^{2},a^{3}-b^{1}). \tag{3}\]  
At first, the resulting expression for the subtraction may seem counter-intuitive. However, notice that the smallest possible value that the uncertain variable, \(a\), described by \(\widehat{a}\), may take is \(a^{1}\), while the greatest possible value for the uncertain variable, \(b\), described by \(\widehat{b}\), is \(b^{3}\) and, hence, the smallest possible value for their difference is \(a^{1}-b^{3}\). Analogously, the greatest possible value for the difference will be \(a^{3}-b^{1}\). This results in support for the difference where the extreme values of the two involved TFNs can be said to be \"crossed\". Interestingly enough, the same expressions for the sum and the difference are obtained if we reconstruct them via interval computation from the alpha-cuts [16; 46]. A consequence of these definitions is that, in the set of TFNs, the difference is not necessarily the inverse of the sum, that is, \(\widehat{a}\
eq(\widehat{a}+\widehat{b})-\widehat{b}\). This is because underlying the definition (3) that results from the Extension Principle is the assumption that the uncertain variables \(a\) and \(b\) associated with \(\widehat{a}\) and \(\widehat{b}\) are non-interactive; i.e., they can take values in their support independently of each other. Therefore, when the TFNs \(\widehat{a}\) and \(\widehat{b}\) are linked, the resulting difference may be an \"over-uncertain\" or \"pessimistic\" quantity. This may prove to be an issue in certain applications; it is, for instance, the main difficulty in critical-path analysis and backpropagation in activity networks with fuzzy durations [16]. \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(\square^{C}\) & refers to a measure computed using the Coarse approach \\ \(I_{\phi}(o)\) & idle time of an operation in \(\phi\) \\ \(CIT_{\phi}(r)\) & core idle time of a resource in \(\phi\) \\ \(TCIT_{\phi}\) & total core idle time of \(\phi\) \\ \(\rho\) & a crisp scenario \\ \(f^{\rho}_{\phi}\) & value of the objective function \(f\) under the crisp scenario \(\rho\) in solution \(\phi\) \\ \(\mathbb{E}[\widehat{a}]\) & expected value of \(\widehat{a}\) \\ \(S(\widehat{a})\) & spread of \(\widehat{a}\) \\ \(MVP(\widehat{a})\) & modal value position of \(\widehat{a}\) \\ \(RDEV(\phi,\rho)\) & relative distance to the expected value of \(\phi\) with respect to the crisp scenario \(\rho\) \\ \(UI(\phi,\mathbf{S})\) & used uncertainty of solution \(\phi\) in the set of possible scenarios \(\mathbf{S}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._Unlike the cases of addition and subtraction, the set of TFNs is not closed under the extended maximum. For this reason, in fuzzy scheduling, it is usual to approximate the maximum of two TFNs via interpolation as follows:  
\[\max(\widehat{a},\widehat{b})\approx(\max(a^{1},b^{1}),\max(a^{2},b^{2}),\max( a^{3},b^{3})). \tag{4}\]  
The approximation always preserves the support and the modal value, and, if the result of the extended maximum is a TFN, it coincides with the approximated value (cf. [47]).