# Maritime Supply Chain Optimization by Using Fuzzy Goal Programming
## 3 Proposed Model
### Assumptions  
1. There always has a deterministic dynamic demand at destination ports. 2. Objective functions and demands in destination ports are fuzzy. 3. All objective functions and constraints are linear. 4. Triangular fuzzy numbers are preferred for the fuzzy demand of the destination ports. 5. Linear membership functions are used to represent the objective goals. 6. The feeder ports are known. 7. Each feeder port has the capacity to meet the awaited cargo. 8. Hub-port operations are conducted in a single stage. 9. Operation capacity of hub-port is known. 10. Hub-port always provides storage cargo handling and transportation for all cargoes. 11. Unit stocking fees of the cargoes are fixed. 12. There is no cargo at the beginning of the first period. 13. Cargo handling cost is constant. 14. Arrival regime of cargoes varies by month. As given in Figure 1, this is a transshipment process under several parameters and decision variables. All possibilities have been taken into account as much as possible for the feeder port, hub-port and destination ports. The parameters deal with the demand, operation capacity, amount of cargo, situations of storage area, cost, safety stock and loss or damage of containers. There are six decision variables and the model tracks the changes at these decision variables. There are two goals, minimization of total cost and minimization of loss or damage of containers returned from the destination port. **Indices:**  
\(t\): Time (t in months)  
\(s\): Special cargo type (s)  
\(d\): Destination ports (d)  
**Parameters:**  
\(M_{dt}\): Fuzzy demand (M) of the destination ports (d) at time \(t\)  
\(o\): Operation capacity (o) of hub-port  
\(a_{s}\): The amount of special cargoes (s) for a 40 FT standard container  
\(SA_{s}\): Storage area (SA) for each special cargo (s) at the hub-port  
\(SA_{k}\): Storage area (SA) for containers (k) at the hub-port  
\(SA_{d}\): Storage area (SA) of destination ports (d)  
\(T_{s}\): Transportation cost (T) of special cargo (s)  
\(co\): Cost (c) of unit operation (o)  
\(T_{dt}\): Transportation cost (T) of unit container to the destination port (d) at time \(t\)  
\(Sc_{s}\): Unit storage cost (Sc) of special cargo (s) at hub-port  
\(Sc_{k}\): Unit storage cost (Sc) for a container (k) at hub-port  
\(Sc_{d}\): Storage cost (Sc) of a container at destination port (d)  
\(f_{s}\): Amount of safety stock (f) for special cargoes (s) at hub-port  
\(f_{s}\): Amount of safety stock (f) for containers (k) at hub-port  
\(f_{d}\): Amount of safety stock (f) at destination port (d)  
\(l_{d}\): loss or damage (l) of containers returned from destination port (d)  
**Decision Variables:**  
\(I_{st}\): The amount of special cargo (s) transported from the feeder ports at the time \(t\)  
\(K_{t}\): The amount of containerized cargo at time \(t\)  
\(L_{dt}\): The number of transported containers to the destination ports (d) at the time (t)  
\(R_{st}\): The amount of special cargo (s) at hub-port at the end of time \(t\)  
\(R_{t}\): The amount of containers at hub-port at the end of time \(t\)  
Figure 1: Structure of the proposed model for maritime supply chain. \(R_{dt}\): The amount of containers at destination port (d) at the end of time \(t\)  
Two goals are set for maritime supply chain optimization  
1. Minimization of total cost
2. Minimization of loss or damage of containers returned from destination port  
Min Total Cost (TC) \(=\) Total Supply Cost (TSC) \(+\) Total Operation Cost (TOC)  
+Total Distribution Cost (TDC)  
Total Returned Containers (TRC)=loss or damage of containers returned from destination port  
\[\text{TSC}\cong\sum_{t=1}^{T}\sum_{s=1}^{S}T_{s}\cdot I_{st}+\sum_{t=1}^{T} \sum_{s=1}^{S}Sc_{s}\cdot R_{st} \tag{3}\]  
\[\text{TOC}\cong\sum_{t=1}^{T}(co\cdot K_{t}+Sck\cdot R_{t}) \tag{4}\]  
\[\text{TDC}\cong\sum_{t=1}^{T}\sum_{d=1}^{D}T_{dt}\cdot L_{dt}+\sum_{t=1}^{T} \sum_{s=1}^{S}Sc_{d}\cdot R_{dt} \tag{5}\]  
\[\text{TC}=TSC+TOC+TDC \tag{6}\]  
\[\text{TRC}\cong\sum_{t=1}^{T}\sum_{d=1}^{D}L_{dt}\cdot l_{d} \tag{7}\]  
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