Table AI Indices, parameters, decision...

Table AI

Indices, parameters, decision variables and constrains for CSP₁

Symbol	Description
Values calculated
ML_p,s	If the product p can be delivered by means of transport s, then ML_p,s=1 otherwise ML_p,s=0
MC_p,c	If the product p can be operated by distribution center c, then MC_p,c=1 otherwise MC_p,c=0
ST	The large fixed value, for example, the total number of ordered products
Decision variables
X_k,p,c,o,s	The size of product p delivery from the manufacturer (factory) k to the distribution center c using means of transport s due to the customer order o
Y_k,c,s	If the delivery is carried out from manufacturer (factory) k to the distribution center c using means of transport s, then Y_k,c,s=1 otherwise Y_k,c,s=0
Constraints
The volume of supply to distribution centers from producers resulting from customer orders
$\sum_{k \in K} \sum_{c \in C} \sum_{s \in S} B_{c, o} \cdot M L_{p, s} \cdot M C_{p, c} \cdot X_{k, p, c, o, s} ⩽ Z_{o, p} \forall o \in O, p \in P$	(C1)
The manufacturer can not dispatch more products than its production capacity
$\sum_{c \in C} \sum_{o \in O} \sum_{s \in S} X_{k, p, c, o, s} ⩽ C K_{k, p} \forall k \in K, p \in P$	(C2)
If the delivery is carried out from the factory to the distribution center, there must be a course specific means of transport (link decision variables X_k,p,c,o,s and Y_k,c,s)
$\begin{array}{l} X_{k, p, c, o, s} ⩽ ST \cdot Y_{k, c, s} \forall k \in K, p \in P, c \in C, o \in O, s \in S \\ \sum_{o \in O} \sum_{p \in P} X_{k, p, c, o, s} ⩾ Y_{k, c, s} \forall k \in K, c \in C, s \in S \end{array}$	(C3)
At most one course in a given period for a given means of transport
$\sum_{k \in K} \sum_{c \in C} Y_{k, c, s} ⩽ 1 \forall s \in S$	(C4)
The volume (tonnage) loaded on the means of transport can not be greater than its capacity (tonnage) – note that this version of the model assumes only one gauge if it were to be considered that more similar equations arise only with other factors
$\sum_{o \in O} \sum_{p \in P} V P_{p} \cdot X_{k, p, c, o, s} ⩾ V F_{s} \cdot Y_{k, c, s} \forall k \in K, c \in C, s \in S$	(C5)
Binarity and integrity
$\begin{array}{l} X_{k, p, c, o, s} \in Z^{+} \forall k \in K, p \in P, c \in C, o \in O, s \in S \\ Y_{k, c, s} \in {0, 1} \forall k \in K, c \in C, s \in S \end{array}$	(C6)

Symbol	Description
Values calculated
ML_p,s	If the product p can be delivered by means of transport s, then ML_p,s=1 otherwise ML_p,s=0
MC_p,c	If the product p can be operated by distribution center c, then MC_p,c=1 otherwise MC_p,c=0
ST	The large fixed value, for example, the total number of ordered products
Decision variables
X_k,p,c,o,s	The size of product p delivery from the manufacturer (factory) k to the distribution center c using means of transport s due to the customer order o
Y_k,c,s	If the delivery is carried out from manufacturer (factory) k to the distribution center c using means of transport s, then Y_k,c,s=1 otherwise Y_k,c,s=0
Constraints
The volume of supply to distribution centers from producers resulting from customer orders
$\sum_{k \in K} \sum_{c \in C} \sum_{s \in S} B_{c, o} \cdot M L_{p, s} \cdot M C_{p, c} \cdot X_{k, p, c, o, s} ⩽ Z_{o, p} \forall o \in O, p \in P$	(C1)
The manufacturer can not dispatch more products than its production capacity
$\sum_{c \in C} \sum_{o \in O} \sum_{s \in S} X_{k, p, c, o, s} ⩽ C K_{k, p} \forall k \in K, p \in P$	(C2)
If the delivery is carried out from the factory to the distribution center, there must be a course specific means of transport (link decision variables X_k,p,c,o,s and Y_k,c,s)
$\begin{array}{l} X_{k, p, c, o, s} ⩽ ST \cdot Y_{k, c, s} \forall k \in K, p \in P, c \in C, o \in O, s \in S \\ \sum_{o \in O} \sum_{p \in P} X_{k, p, c, o, s} ⩾ Y_{k, c, s} \forall k \in K, c \in C, s \in S \end{array}$	(C3)
At most one course in a given period for a given means of transport
$\sum_{k \in K} \sum_{c \in C} Y_{k, c, s} ⩽ 1 \forall s \in S$	(C4)
The volume (tonnage) loaded on the means of transport can not be greater than its capacity (tonnage) – note that this version of the model assumes only one gauge if it were to be considered that more similar equations arise only with other factors
$\sum_{o \in O} \sum_{p \in P} V P_{p} \cdot X_{k, p, c, o, s} ⩾ V F_{s} \cdot Y_{k, c, s} \forall k \in K, c \in C, s \in S$	(C5)
Binarity and integrity
$\begin{array}{l} X_{k, p, c, o, s} \in Z^{+} \forall k \in K, p \in P, c \in C, o \in O, s \in S \\ Y_{k, c, s} \in {0, 1} \forall k \in K, c \in C, s \in S \end{array}$	(C6)

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