# Integration of Production Planning and Scheduling Based on RTN Representation under Uncertainties
## 2 Mathematical Model
### Scheduling Layer Model
#### 2.2.2 Fuzzy Uncertain Model  
It is an effective method to describe the uncertainty of utility with fuzzy theory, because it is much easier to determine the membership function of a fuzzy number than the distribution function of a random variable. Different from the order demand, utilities are greatly influenced by the energy system, and the energy system is disturbed by many factors, so it is difficult to determine the specific distribution function to express the uncertainty of utilities. In this chapter, the fuzzy theory [36] is used to describe the uncertainty of utilities.  
The deterministic scheduling model is replaced by the following fuzzy scheduling model:  
Objective function: (20)  
Constraints: (21)-(26), (28)-(32), (34), (35)  
Constraint (36), deterministic constraint (33) is replaced by fuzzy constraint (36).  
\[\sum_{i\in I_{u}}\left(\lambda_{iu}\cdot w(i,n)\right)+\sum_{i\in I_{u}}\left( \varphi_{iu}\cdot b(i,n)\right)\leq\hat{U}_{u}^{\max};u\in U,n\in N. \tag{36}\]  
The utility supply \(\hat{U}_{u}^{\max}\) in Constraint (36) is an uncertain parameter. It is assumed that the triangular membership function of \(\hat{U}_{u}^{\max}\) is \(T_{ri}(u)\), which is expressed as follows:  
\[T_{ri}(u)=\left\{\begin{array}{ll}\frac{u-u_{1}}{u_{2}-u_{1}}\;,\;u_{1}\leq u \leq u  
Using the theory of fuzzy knowledge, Constraint (36) can be converted to Constraint (38). Finally, the uncertain model with utility constraints is transformed into a deterministic model with membership degree.  
\[\sum_{i\in l_{u}}\left(\lambda_{iu}\cdot w(i,n)\right)+\sum_{i\in l_{u}}\left( \varphi_{iu}\cdot b(i,n)\right)\leq w_{1}\cdot U_{1,\delta}^{\max}+w_{2}\cdot U _{2,\delta}^{\max}+w_{3}\cdot U_{3,\delta}^{\max};u\in U,n\in N. \tag{38}\]  
\(w_{1}\),\(w_{2}\),\(w_{3}\) are the weights, and \(w_{1}+w_{2}+w_{3}=1\); the fuzzy parameters of High-pressure Steam (HS) and Cooling Water (CW) include the most pessimistic value, the most possible value, and the most optimistic value. The three prominent values, i.e., the most pessimistic, the most possible, and the most optimistic values for each fuzzy number are usually estimated by decision maker [37; 38]. In this paper, the three prominent values are defined according to determining method of universe and the experience of engineering. The universe \(U\) of fuzzy number is defined as an equation according to [PERSON] and [PERSON] [39], where \(D_{min}\) is the minimum value in statistical data of fuzzy number, \(D_{max}\) is the maximum value in statistical data of fuzzy number. \(D_{1}\) and \(D_{2}\) are defined as an equation and according to the experience of engineering. \(D_{mf}\) is the value of which the frequency of occurrence is the highest in statistical data of fuzzy numbers.  
\[U=[D_{min}-D_{1},D_{max}+D_{2}], \tag{39}\]  
\[D_{1}=\ 0.2*\big{|}D_{min}-D_{mf}\big{|}, \tag{40}\]  
\[D_{2}=\ 0.2*\big{|}D_{max}-D_{mf}\big{|}. \tag{41}\]  
In order to describe fuzzy numbers more generally, in this paper we define \(u_{1}\), \(u_{2}\) and \(u_{3}\) as the most pessimistic value, the most possible value, and the most optimistic value. The most pessimistic value \(u_{1}\) is equal to \(D_{min}-D_{1}\) (the left boundary of universe). The most possible value \(u_{2}\) is equal to \(D_{mf}\). The most optimistic value \(u_{3}\) is equal to \(D_{max}+D