## 2 Modelling, Analysis and Optimization of a System
### Optimization: Artificial Bee Colony  
Investigation of processes involving intelligent behaviour present in nature has led researchers to develop new optimization methods. [PERSON] [51] developed the Artificial Bee Colony (ABC) algorithm by modelling the behaviour of bees searching for food. The basis of the ABC algorithm includes some assumptions for simplicity. The first of these is the nectar from each source can only be obtained by one worker bee. In other words, the number of duty bees is equal to the number of food sources. The number of worker bees is accepted as equal to the number of observer bees. The worker  
Figure 3: Flow diagram of the construct used for the most appropriate model of the Sinem GPP. bee assigned to a source with nectar fully consumed transforms into an explorer bee. Possible solutions of optimization problems for locations of food sources and nectar amounts of the sources are equivalent to the quality (fit) of the solutions related to those sources. Consequently, the ABC algorithm attempts to locate the source with most nectar and attempts to find a problem's minimum or maximum point within solution space. The process steps belonging to this model are given below [52; 53; 54; 55]:  
* At the start of the food search process, explorer bees randomly search the environment and begin to search for food. * After food sources have been found, explorer bees become worker bees and begin to carry the nectar found at the source into the hive. Each worker bee empties the nectar on return to the hive and at this point either returns to the food source or communicates information related to the food source with the observer bees waiting in the hive though dance displayed in the dance area. If the food source is consumed, the duty bee transforms into an explorer bee and begins to search for new food sources. * Observer bees waiting in the hive watch dances indicating rich food sources and choose a food source linked to the dance frequency which is proportional to the food quality. For initial production of food source areas, if we think of the food sources surrounding the hive as search space, the algorithm is based on randomly producing food sources equivalent to solutions in search space. The random location production process is completed by producing random values between the upper and lower limits for each parameter as seen in [54; 55]:  
\[x_{\bar{\mathrm{i}}}=x_{\bar{\mathrm{j}}}^{\min}+\mathrm{rad}(0,1)\left(x_{ \bar{\mathrm{j}}}^{\mathrm{maks}}-x_{\bar{\mathrm{j}}}^{\min}\right)\left\{ \begin{array}{l}\mathrm{i}=1,\ldots,\mathrm{SN}\\ \mathrm{j}=1,\ldots,\mathrm{D}\end{array} \right. \tag{10}\]  
where SN is the number of food sources and D represents the number of parameters to be optimized. After the initial stage, the worker bee, observer bee and explorer bee process for food sources begin, with an attempt made to find better sources. For stopping criteria of the ABC algorithm, the maximum cycle number (MCN), acceptable error value or other stopping criteria can be used. During this optimization process, the number of worker bees is equal to the number of food sources. Worker bees attempt to determine new food sources near food sources and assess their quality. If the new food source is better, it is remembered. To determine neighbouring food sources to the new source, the simulation is given in:  
\[v_{\bar{\mathrm{i}}}=x_{\bar{\mathrm{j}}}+\varphi_{\bar{\mathrm{i}}}\left(x_{ \bar{\mathrm{i}}}-x_{\bar{\mathrm{k}}j}\right) \tag{11}\]  
where, as the difference between the random values \(x_{\bar{\mathrm{i}}j}\) and \(x_{\bar{\mathrm{k}}j}\) reduces, in other words as the solutions equivalents, the variation amount in the \(x_{\bar{\mathrm{i}}j}\) parameter reduces. Thus, as the regional optimal solution is approached, the variation amount adaptively decreases. The \(v_{\bar{\mathrm{i}}}\) parameter produced within the limits represents a new source and its quality is assigned by calculating the fit value as given below:  
\[\mathrm{fitness}_{\bar{\mathrm{i}}}\left\{