Table AII Indices, parameters, decision...

Table AII

Indices, parameters, decision variables and constrains for CSP₂

Symbol	Description
Indices
H	Set of pallets/grouped shipments
h	Shipment/pallet (h=1, …, H)
Values calculated
U_h	The size of the shipment/pallet for aggregate index h (based on UO_c,o,g)
DOC_h,c	If the shipment/pallet h is dispatched from the center c, then DOC_h,c=1 otherwise DOC_h,c=0
DOG_h,g	If the shipment/pallet h contains product type g, then DOG_h,g=1 otherwise DOG_h,g=0
DOF_h,f	If the means of transport f can carry shipment h, then DOF_h,f=1 w otherwise DOF_h,f=0
Decision variables
X_k,p,c,o,s	The same meaning as for the CSP₁
UO_c,o,g	The size of the shipment/pallet for product type g delivered to the customer o from the distribution center c
Constraints
Determination of the coefficient ML_p,s
$M L_{p, s} = \sum_{g \in G} \sum_{u \in U} A_{p, g} \cdot D_{u, g} \cdot H S_{s, u} \forall p \in P, s \in S$	(C7)
Determination of the coefficient MC_p,c
$M C_{p, c} = \sum_{g \in G} A_{p, g} \cdot L_{c, g} \forall p \in P, c \in C$	(C8)
Determination of the size of the shipment/pallet for product type g delivered to the customer o from the distribution center c
$\sum_{k \in K} \sum_{p \in P} \sum_{s \in S} V P_{p} \cdot A_{p, g} \cdot D_{u, g} \cdot H S_{s, u} X_{k, p, c, o, s} = U O_{c, o, g} \forall c \in C, o \in O, g \in G$	(C9)
Aggregation and unification attributes the shipment/pallet and its renumbering
$h = [c, o, g] \forall c \in C, o \in O, g \in G : U O_{c, o, g} > 0 U_{h} = U O_{c, o, g} \forall c \in C, o \in O, g \in G, u \in U : h = [c, o, g]$	(C10)
Determination of the coefficient DOS_h,o
$D O S_{h, o} = 1 \forall c \in C, o \in O, g \in G, h \in H : h = [c, o, g]$	(C11)
Determination of the coefficient DOC_h,c
$D O C_{h, c} = 1 \forall c \in C, o \in O, g \in G, h \in H : h = [c, o, g]$	(C12)
Determination of the coefficient DOG_h,g
$D O G_{h, g} = 1 \forall c \in C, o \in O, g \in G, h \in H, p \in P : h = [c, o, g], A_{p, g} = 1$	(C13)
Determination of the coefficient DOF_h,f
$D O F_{h, f} = 1 \forall h \in H, f \in F, g \in G, u \in U : D O G_{h, g} = 1, D_{u, g} = 1, H F_{f, u} = 1$	(C14)

Symbol	Description
Indices
H	Set of pallets/grouped shipments
h	Shipment/pallet (h=1, …, H)
Values calculated
U_h	The size of the shipment/pallet for aggregate index h (based on UO_c,o,g)
DOC_h,c	If the shipment/pallet h is dispatched from the center c, then DOC_h,c=1 otherwise DOC_h,c=0
DOG_h,g	If the shipment/pallet h contains product type g, then DOG_h,g=1 otherwise DOG_h,g=0
DOF_h,f	If the means of transport f can carry shipment h, then DOF_h,f=1 w otherwise DOF_h,f=0
Decision variables
X_k,p,c,o,s	The same meaning as for the CSP₁
UO_c,o,g	The size of the shipment/pallet for product type g delivered to the customer o from the distribution center c
Constraints
Determination of the coefficient ML_p,s
$M L_{p, s} = \sum_{g \in G} \sum_{u \in U} A_{p, g} \cdot D_{u, g} \cdot H S_{s, u} \forall p \in P, s \in S$	(C7)
Determination of the coefficient MC_p,c
$M C_{p, c} = \sum_{g \in G} A_{p, g} \cdot L_{c, g} \forall p \in P, c \in C$	(C8)
Determination of the size of the shipment/pallet for product type g delivered to the customer o from the distribution center c
$\sum_{k \in K} \sum_{p \in P} \sum_{s \in S} V P_{p} \cdot A_{p, g} \cdot D_{u, g} \cdot H S_{s, u} X_{k, p, c, o, s} = U O_{c, o, g} \forall c \in C, o \in O, g \in G$	(C9)
Aggregation and unification attributes the shipment/pallet and its renumbering
$h = [c, o, g] \forall c \in C, o \in O, g \in G : U O_{c, o, g} > 0 U_{h} = U O_{c, o, g} \forall c \in C, o \in O, g \in G, u \in U : h = [c, o, g]$	(C10)
Determination of the coefficient DOS_h,o
$D O S_{h, o} = 1 \forall c \in C, o \in O, g \in G, h \in H : h = [c, o, g]$	(C11)
Determination of the coefficient DOC_h,c
$D O C_{h, c} = 1 \forall c \in C, o \in O, g \in G, h \in H : h = [c, o, g]$	(C12)
Determination of the coefficient DOG_h,g
$D O G_{h, g} = 1 \forall c \in C, o \in O, g \in G, h \in H, p \in P : h = [c, o, g], A_{p, g} = 1$	(C13)
Determination of the coefficient DOF_h,f
$D O F_{h, f} = 1 \forall h \in H, f \in F, g \in G, u \in U : D O G_{h, g} = 1, D_{u, g} = 1, H F_{f, u} = 1$	(C14)

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