# Data Reduction of Digital Twin Simulation Experiments Using Different Optimisation Methods
## 2 Literature Background
### Simulation Optimisation  
The data used for the simulation optimisation to mimic the possible scenarios of the process are obtained from the enterprise information system, e.g., enterprise resource planning (ERP). These data are very good for strategic or tactical planning (not so often used for operative planning) or control. Using the internet of things (IoT), it is possible to read or download the data from the different objects in the company, e.g., the production lines, AGVs, etc. These data are especially suitable for operative planning or control. The current data processing is beyond the computing ability of traditional computational models due to the immense volume of constantly or exponentially generated data from the objects in the company. Researchers in data analytics use and design new efficient algorithms to handle massive data analytics problems, e.g., population-based algorithms including swarm intelligence and evolutionary algorithms--see Figure 3. These methods are used for extracting useful information from a big volume of raw data. Figure 2: Steps required to provide semantically valid data to the simulation model [18]. Some of our department's projects are also focused on the simulation of processes in industrial companies. We create DTs (discrete event simulation models) which represent real physical processes in different parts of a company (mainly production lines, warehouses, etc.). The main problem of these simulation studies (mainly focusing on the optimisation of the modelled processes) is the big search space (many possible solutions to the modelled problem). The volume of data generated by the optimisation processes is also affected by the number of the simulation model input parameters. The search space is usually boundary constrained as follows:  
\[\widetilde{X}=\prod_{j=1}^{n}\widetilde{X}_{j}=\prod_{j=1}^{n}[a_{j},b_{j}],\ a_{j}\leq b_{j},a_{j},b_{j}\in\mathbb{R} \tag{1}\]  
where  
* \(\widetilde{X}\) denotes the search space--the domain of the input parameters of the discrete event simulation model;
* \(j\) denotes the index of the \(j\)-th decision variable (simulation model input parameter) of the simulation model;
* \(n\) denotes the dimension of the search space;
* \(a_{j}\) denotes the lower bound of the interval of the \(j\)-th decision variable;
* \(b_{j}\) denotes the upper bound of the interval of the \(j\)-th decision variable. Many of the modelled problems are NP-hard problems in which we often cannot evaluate all the possible solution candidates,  
\[\mathbf{X}[j]=x_{j}\forall j:j=\{1,2,\dots,n\} \tag{2}\]  
where  
* \(\mathbf{X}[j]\) denotes a possible solution candidate;
* \(x_{j}\) denotes the value of the \(j\)-th decision variable. The basic problem of the simulation optimisation is to find the optimum of the objective function representing the aim of the simulation optimisation as follows:  
\[\tilde{\mathbf{X}}=\operatorname*{argmin}_{\mathbf{X}\in\widetilde{X}}F( \mathbf{X})=\left\{\tilde{\mathbf{X}}\in\widetilde{X}:F\left(\tilde{\mathbf{X }}\right)\leq\ F(\mathbf{X})\forall\mathbf{X}\in\widetilde{X}\right\} \tag{3}\]  
where  
* \(\tilde{\mathbf{X}}\) denotes the global minimum of the objective function \(F(\mathbf{X})\);  
Figure 3: Relationship between data science with evolutionary algorithm and swarm intelligence [19]. \(F(\textbf{X})\) denotes the objective function value of the solution candidate. The objective function represents the goal of the simulation study. Each solution candidate (possible solution of the modelled problem representing the vector of the values for each decision variable \(\textbf{X}=[x_{1},x_{2},\ldots,x_{n}\;]\) in search space) is evaluated by the objective function value--the range includes real numbers \(F(\textbf{X})\subseteq\mathbb{R}\). The objective function maximisation can be converted to function minimisation. Testing all possible solutions to a problem is a very inefficient way (and mostly impossible) of finding a suitable solution candidate or the global optimum.