# Integration of Production Planning and Scheduling Based on RTN Representation under Uncertainties
## 2 Mathematical Model
### Planning Layer Model
#### 2.1.2 Chance Constrained Uncertain Model  
The uncertain factors in production planning have attracted more and more attention from scholars and researchers, as well as many methods and theories have been proposed to deal with the uncertain factors in production planning, including stochastic programming [26], fuzzy programming [27], robust optimization [28; 29], recoverable robustness [30], sensitivity analysis [31] etc. At present, many research results have been obtained by using stochastic programming to deal with the problem of demand uncertainty [32; 33]. Chance constrained programming is an important branch of stochastic programming. It allows decisions, to some extent, to not meet the constraints, but these decisions should make the probability of the constraints set up no less than a certain confidence level. The chance constrained method can more truly and effectively respond to demand uncertainty in the production planning. The deterministic planning model is replaced by the following uncertainty planning model:Objective function:(1)  
Constraints: (2), (3), (6)-(9)  
Constraint (10), deterministic constraint (4) is replaced by uncertainty constraint (10)  
Constraint (11), deterministic constraint (5) is replaced by uncertainty constraint (11). \[Td(i,k)=Td(i,k-1)+\tilde{O}rd_{ik}-D(i,k);i\in G_{p},k>1,k\in K. \tag{10}\]  
\[D(i,k)\leq\tilde{O}rd_{ik}+Td(i,k-1)+td_{i,k};i\in G_{p},k>1,k\in K. \tag{11}\]  
In Constraints (10) and (11), the parameter order demand quantity \(\tilde{O}rd_{ik}\) is a random variable, assuming that the parameter obeys normal distribution, namely \(N(E\big{[}\tilde{O}rd_{ik}\big{]},var\big{[}\tilde{O}rd_{ik}\big{]})\). Under a certain confidence level, Constraints (10) and (11) can be rewritten into Constraints (12) and (13). \[Pr\big{\{}\tilde{O}rd_{ik}\geq D(i,k)+Td(i,k)-Td(i,k-1)\big{\}}\geq\lambda;i \in G_{p}. \tag{12}\]  
\[Pr\big{\{}\tilde{O}rd_{ik}\geq D(i,k)-Td(i,k-1)-td_{i,k-1}\big{\}}\geq\beta;i \in G_{p}\;k\in K. \tag{13}\]  
By using the knowledge of Theorem 1 and Theorem 2 the probability density function can be used to further transform Constraints (12) and (13) into Constraints (14) and (15), and finally transform the model with demand uncertainty into a deterministic model with a confidence level. This transformation of the chance constraint has been solved in detail in many related studies [34; 35], and it will not be repeated here in this paper. \[\frac{[D(i,k)+Td(i,k)-Td(i,k-1)]-E[Ord_{ik}]}{\sqrt{var[Ord_{ik}]}}\leq\Phi^{- 1}(i-\lambda);i\in G_{p}. \tag{14}\]  
\[\frac{\big{[}D(i,k)-Td(i,k-1)-td_{i,k-1}\big{]}-E[Ord_{ik}]}{\sqrt{var[Ord_{ik }]}}\leq\Phi^{-1}(i-\beta);i\in G_{p}\;k\in K. \tag{15}\]  
**Theorem 1**. : _For the random linear programming model:_  
\[ \begin{array}{l}min\;Z=\sum\limits_{j=1}^{n}h_{j}\bullet x_{j}\;;\\ s.t.\left\{\begin{array}{l}\sum\limits_{j=1}^{n}g_{ij}\bullet x_{j}\leq\text {r}_{i}(\xi)\;(i=1,2,\cdots,m),\\ x_{j}\geq 0\;(j{=}1,2,\cdots,n).\end{array}\right.\end{array}  \tag{16}\]  
\(\text{r}_{i}(\xi)\;(i=1,2,\cdots,m)\) _is a single random variable, and the deterministic programming model with the formula at the confidence level \(\
u\) is as follows:_  
\[ \begin{array}{l}min\;Z=\sum\limits_{j=1}^{n}h_{j}\bullet x_{j}\;;\\ s.t.\left\{\begin{array}{l}\sum\limits_{j=1}^{n}g_{ij}\bullet x_{j}\leq\text {K}_{\
u}^{i}\;(i=1,2,\cdots,m),\\ x_{j}\geq 0\;(j{=}1,2,\cdots,n).\end{array}\right.\end{array}  \tag{17}\]  
_In the formula, \(\text{K}_{\
u}^{i}{=sup\Big{\{}K^{i}\big{|}K^{i}=F_{i}^{-1}(1-\
u)\Big{\}}}\),\(F_{i}(\bullet)\) is \(\text{r}_{i}(\xi)\;\) distribution function._  
**Theorem 2**. : _For the random linear programming model:_  
\[ \begin{array}{l}\text{min}\ Z=\sum\limits_{j=1}^{n}h_{j}\bullet x_{j}\ ;\\ s.t.\left\{\begin{array}{l}\sum\limits_{j=1}^{n}g_{ij}\bullet x_{j}\geq\mathbf{ r}_{i}(\xi)\ (i=1,2,\cdots,m),\\ x_{j}\geq 0\ (j=1,2,\cdots,n).\end{array}\right.\end{array}