## 2 Modeling and Problem Formulation
### Process-Planning Problem Formulation for Sustainable CPS
To propose a general framework to support the planning of processes in sustainable CPSs, a general form of the objective function is first presented. The general form of the objective function will be tailored for the case of sequential processes to illustrate its usage next. The general form of the objective function is a function of the decision variable, \(x\). Therefore, we use \(G(x)\) to denote the objective function. As the cost due to the total processing time and energy consumption factors are considered in this study, the objective function \(G(x)\) is a function of the total processing time of the configuration calculated by \(\Gamma(x,B)\) and the energy consumption of the configuration calculated by \(Eng(x,B)\). Therefore, \(G(x)\) is described by \(G(\Gamma(x,B),Eng(x,B))\). Note that \(\Gamma(x,B)\) is related to time, whereas \(Eng(x,B)\) is related to energy consumption. There are several different ways to combine \(\Gamma(x,B)\) and \(Eng(x,B)\) to define the objective function \(G\). For example, one may introduce weighting coefficients to combine \(\Gamma(x,B)\) and \(Eng(x,B)\) to define the objective function \(G\). Alternatively, one may define \(G(\Gamma(x,B),Eng(x,B))\) as the weighted sum of functions of \(\Gamma(x,B)\) and \(Eng(x,B)\), e.g., \(G(\Gamma(x,B),Eng(x,B))=w_{1}G_{1}(\Gamma(x,B))+w_{2}G_{2}(Eng(x,B))\), where \(G_{1}\) is a monotonic decreasing function and \(G_{2}\) is a monotonic increasing function. The objective function should be tailored properly according to the characteristics of the type of production processes under consideration and the goal of the production process. We will elaborate \(G(\Gamma(x,B),Eng(x,B))\) for the special case of sequential processes later to illustrate how the objective function is defined. In our problem formulation, we use \(\Re(x,B,\Re eq_{n})\) to denote the constraints to satisfy the requirements of the operations specified in \(\Re eq_{n}\). We use \(\mathrm{T}(x,B)\) to denote the constraints to satisfy time requirements specified in \(\Re eq_{n}\). We use \(\Pi(x,B)\) to denote the other types of constraints. We formulate the following problem to maximize the objective function. \[\max(G(x)=G(\Gamma(x,B),Eng(x,B)) \tag{1}\]  
s.t. \[\Re(x,B,\Re eq_{n}) \tag{2}\]  
\[\mathrm{T}(x,B) \tag{3}\]  
\[\Pi(x,B) \tag{4}\]  
\[x_{aj}\in\{0,1\}\ \forall a\in\{1,2,3,\ldots,|RA|\},\forall j\in\{1,2,3,\ldots, J_{a}\} \tag{5}\]  
The problem formulation defined by (1) through (5) should be tailored properly according to the characteristics of the type of production processes under consideration and the goal of the production process. To illustrate how to tailor the problem formulation defined by (1) through (5) to create a process with desirable properties, let us consider the problem of composing a process \(\Psi_{n}\) for a process agent \(n\) with a sequential Cyber World model \(\Omega_{n}\) in a CPS. Suppose the composed process must satisfy three requirements: (i) each operation in the process \(\Omega_{n}\) must be performed by an agent, (ii) the total processing time of the process must be no greater than a given upper bound \(\omega_{n}\) and (iii) the number of times an operation can be performed by each agent cannot exceed a pre-specified upper bound \(\omega_{n}\). In this case, we may tailor the problem defined by (1) through (5) for composing a sequential process as follows. If the \(\Omega_{n}\) is a sequential process, the constraints \(\Re(x,B,\Re eq_{n})\) to satisfy the requirements of the operations can be represented by \(\sum\limits_{a\in RA}\sum\limits_{j=1}^{I_{a}}x_{aj}a_{aj}\geq d_{nk}\forall k \in\{1,\ldots,K_{n}\}\). If \(\Omega_{n}\) is a sequential process, \(\Gamma(x,B)=\sum\limits_{a\in RA}\sum\limits_{j=1}^{I_{a}}x_{aj}\tau_{aj}\). Let \(\omega_{n}\) denote the upper bound of the total processing time of the process \(\Psi_{n}\). In this case, the constraints to satisfy the time requirements \(\mathrm{T}(x,B)\) can be represented by \(\omega_{n}\geq\Gamma(x,B)\).