# A New Modified Firefly Algorithm for Optimizing a Supply Chain Network Problem
## 3 Results and Discussion  
### Problem Description  
We consider a supply chain network problem to demonstrate the efficiency of the proposed modified FA. The investigated supply chain network problem in this study includes the decision associated with the distribution of multiple products that are produced by multiple manufacturers and distributed through multiple Distribution Centres (DCs) and Retailers. The schematic description of this problem is depicted in Figure 4.  
In the following, the problem's assumptions and model formulation are presented:  
_i. Indices_  
_i_: Index of manufacturing plants, \(i\) = 1, 2,..., \(I\)  
_j_: Index of DCs, \(j\) = 1, 2,..., \(J\)  
_k_: Index of retailers, \(k\) = 1, 2,..., \(K\)  
_p_: Index of different kinds of items, \(p\) = 1, 2,..., \(P\)  
_t_: Index of the planning period, \(t\) = 1, 2,..., \(T\)  
_I_: Number of manufacturing plants  
_J_: Number of DCs  
_K_: Number of retailers  
_P_: Number of items types  
_T_: Number of periods  
Figure 4: The considered supply chain network. _. Parameters_  
\(D_{\text{Jpt}}\) Demand in retailer \(k\) for item \(p\) at the end of period \(t\)  
\(C_{\text{Jpt}}\) Supply capacity of manufacturing plant \(i\) for item \(p\) in period \(t\)  
\(\eta\)\(\eta\)Holding cost per unit of item \(p\) at DC \(j\) in each period  
\(L_{\bar{y}}\)Distance between manufacturer \(i\) and DC \(j\)  
\(v_{p}\)Shipping cost per unit of item \(p\) along unit distance  
\(L^{\prime}_{\bar{y}}\)Distance (in kilometres) between DC \(j\) and retailer \(k\)  
\(C^{\prime}_{\bar{\mu}}\)Distribution capacity of DC \(j\) in period \(t\)  
\(V_{\bar{\mu}}\)Total storage (holding capacity) capacity of DC \(j\) during period \(t\)  
\(\eta^{\prime}\)\(v_{p}\)Holding cost per unit of item \(p\) at retailer \(k\) in each period  
\(S_{\text{Jpt}}\)Backlog cost per unit of item \(p\) at retailer \(k\)  
\(U_{\bar{\mu}t}\)Total holding capacity of retailer \(k\) during period \(t\)  
\(\text{\emph{ii}}\)Decision variables_  
\(X_{\text{ijpt}}\)Quantity of items \(p\) shipped from manufacturing plant \(i\) to DC \(j\) in period \(t\)  
\(I_{\text{Jpt}}\)Inventory level of item \(p\) at DC \(j\) at the end of period \(t\)  
\(Y_{\bar{\lambda}pt}\)Quantity of item \(p\) shipped from DC \(j\) to retailer \(k\) during period \(t\)  
\(B_{\text{ppt}}\)Backlog amount of item \(p\) at retailer \(k\) in period \(t\)  
\(I_{\text{Jpt}}\)Inventory of item \(p\) at retailer \(k\) at the end of period \(t\)  
\(\bullet\)Objective Functions  
\[ \begin{array}{ll}MinZ_{1}&=\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{P}\sum \limits_{p=1}^{P}\sum\limits_{t=1}^{T}(v_{p}L_{ij})X_{\text{ijpt}}+\sum \limits_{j=1}^{I}\sum\limits_{k=1}^{K}\sum\limits_{p=1}^{P}\sum\limits_{t=1}^ {T}(v_{p}L^{\prime}_{\bar{\mu}k})\ Y_{\bar{\lambda}pt}\\ &+\sum\limits_{j=1}^{J}\sum\limits_{p=1}^{P}\sum\limits_{t^{\prime}=1}^{t} \eta_{jp}I_{\text{jpt^{\prime}}}+\sum\limits_{k=1}^{K}\sum\limits_{p=1}^{P} \sum\limits_{t^{\prime}=1}^{t}\eta^{\prime}_{kp}I_{\text{kpt^{\prime}}}+\sum \limits_{k=1}^{K}\sum\limits_{p=1}^{P}\sum\limits_{t^{\prime}=1}^{t}S_{kp}B_ {\text{kpt^{\prime}}}\end{array}  \tag{5}\]  
Equation (5) represents the objective function that minimizes the total costs, consist of shipping costs from the plants to DCs (\(\sum\limits_{i=1}^{I}\sum\limits_{j=1}^{J}\sum\limits_{p=1}^{P}\sum\limits_{t= 1}^{T}(v_{p}L_{ij})X_{\text{ijpt}}\)) and from DCs to retailers (\(\sum\limits_{j=1}^{J}\sum\limits_{k=1}^{K}\sum\limits_{p=1}^{P}\sum\limits_{t =1}^{T}(v_{p}L^{\prime}_{\bar{\mu}k})\ Y_{\bar{\lambda}pt}\)), costs of holding in both DCs (\(\sum\limits_{j=1}^{J}\sum\limits_{p=1}^{P}\sum\limits_{t^{\prime}=1}^{t}\eta _{jp}I_{\text{jpt^{\prime}}}\)) and retailers (\(\sum\limits_{k=1}^{K}\sum\limits_{p=1}^{P}\sum\limits_{t^{\prime}=1}^{t}\eta^{ \prime}_{kp}I_{\text{kpt^{\prime}}}\)) and penalty charges regarding the quantity of backorders at retailers (\(\sum\limits_{k=1}^{K}\sum\limits_{p=1}^{P}\sum\limits_{t^{\prime}=1}^{t}S_{kp}B _{\text{kpt^{\prime}}}\)).