## 4 Developed Model
### Integrating Total Cost functions  
The parameters and variables of problem in objective function affect significantly on the manufacturing system in optimizing the model and reducing fact the total cost. The researchers can consider more parameters in the related sections and operating parts of supply chain system to analyze the problems and obtain the best solution for solving the supply chain cost problem. However, the manufacturing supply chain is seeking to increase competitive performance in global market by providing efficient and effective tools than can be gain advantages for their industry as a supply chain processing system [39]. It is demonstrated how firms are using technology for the integration of avenues accessible to achieve the advantages in satisfying the needs of market sections profitability [40]. Some hypotheses can be considered in this part for cost in integrated manufacturing functions:  
* There is just one supplier used for providing the raw material in this model. * The holding and maintenance costs in all models of supply, production, and distribution are considered and can be supposed to be unchanged. * The quantity for the suppliers and the plant is defined. * The demands for the plant manufacture and production is based on customer's order while the quantity is obvious and would be modified per period of time. * The periods are determined as t = 1, 2, 3,..., T.
* The materials produce one product
* One center for distributing is considered. Each one of the above hypotheses would be expandable. The obtained Model by integrating costs is as follow;  
\[ \begin{split} Min=\sum\limits_{t=1}^{T}A^{Q}(t)Z^{Q}(t)+A^{P}(t)Z ^{P}(t)+A^{P}(t)Z^{D}(t)+C^{Q}(t)Q(t)+C^{P}(t)P(t)\\ +C^{D}(t)D(t)+h^{Q}(t)I^{Q}(t)+h^{Q}(t)I^{P}(t)\\ +h^{D}(t)I^{D}(t)\end{split}  \tag{19}\]  
\(s.t.\)  
\[X^{Q}(t)\leq A^{Q}(t) \tag{20}\]  
\[X^{P}(t)\leq A^{P}(t) \tag{21}\]  
\[X^{D}(t)\leq A^{D}(t) \tag{22}\]  
\[\sum\limits_{t=1}^{T}Q(t)+I^{Q}(t-1)\geq\sum\limits_{t=1}^{T}P(t)\qquad t=1,2,3,\ldots,T \tag{23}\]  
\[\sum\limits_{t=1}^{T}P(t)+I^{P}(t-1)\geq\sum\limits_{t=1}^{T}D(t)\qquad t=1,2,3,\ldots,T \tag{24}\]  
\[\sum\limits_{t=1}^{T}D(t)+I^{D}(t-1)\geq d(t)\qquad t=1,2,3,\ldots,T \tag{25}\]  
\[I^{Q}(t)=[(Q(t)-Q_{dw}(t))-d(t)]+I^{Q}(t-1) \tag{26}\]  
\[I^{P}(t)=[(P(t)-P_{dw}(t))-d(t)]+I^{P}(t-1) \tag{27}\]  
\[I^{D}(t)=[(D(t)-D_{dw}(t))-d(t)]+I^{D}(t-1) \tag{28}\]  
\[X^{Q}(t)-MZ^{Q}(t)<0 \tag{29}\]\[X^{P}(t)\leq A^{P}(t) \tag{21}\]0  
\[X^{P}(t)\leq A^{P}(t) \tag{21}\]1  
\[X^{P}(t)\leq A^{P}(t) \tag{21}\]2  
The analytical discrete-time chart model is formulated to minimize the total operational costs in the whole system of the supply chain. The system is minimizing the total cost through integrating the supply, production, and distribution models and functions into an aggregate objective function including all holding costs for the inventory in each with achieving the main goal in current supply chain system. As constraint one shows, the cost for providing raw materials is less than or equal to target unit price. In each period of time, the cost of manufactured products as second part of supply chain process is less than or equal to considered target unit price in constraint two. Constraint three presents the cost of distributed product as less than or equal to the considered target price in supply chain. Constraints four to six determined the product's operational volume with last inventory being greater than or equal to the amount or volume of distributing products in the period of time, while the amount of distributing products to warehouses must be large enough to respond to demands in conventional time for delivery. In constraints seven, eight, and nine, the volume of inventory in each level of supply, production, and distribution depends on the remaining products or materials from the last time and having planned to provide or produce the required amount for the next step of operation process in supply chain manufacturing system.