Table AIII Indices, parameters, decision...

Table AIII

Indices, parameters, decision variables and constrains for CSP₃

Symbol	Description
Decision variables
XH_f,i,j	If the means of transport f travels from point i to point j, then X_f,i,j=1, otherwise X_f,i,j=0 (f∈F, i∈O∪C, j∈O∪C)
YH_f,i,j,h	If the means of transport f travels from point i to point j carrying shipment/pallet h, then Y_f,i,j,h=1, otherwise Y_f,i,j,h=0 (f∈F, i∈O∪C, j∈O∪C, h∈H)
FH_h,f	If shipment/pallet h is delivered by means of transport f than FH_h,f=1, otherwise FH_h,f=0 (h∈H, f∈F)
Constraints
Arrival and departure means of transport from the delivery point
$\sum_{j \in O \cup C} X H_{f, i, j} = \sum_{j \in O \cup C} X H_{f, j, i} \forall f \in F, i \in O \cup C$	(C15)
If no shipments h are to be carried on the route i,j, a means of transport does not travel that route – linking the decision variable XH_f,i,j with decision variable YH_f,i,j,h
$\begin{array}{l} X H_{f, i, j} ⩽ \sum_{h \in H} Y H_{f, i, j, h} \forall f \in F, i \in O \cup C, j \in O \\ Y H_{f, i, j, h} ⩽ X H_{f, i, j} \forall f \in F, i \in O \cup C, j \in O \cup C, h \in H \end{array}$	(C16)
The given means of transport carries no more shipments than allowable as per its capacity volume
$\sum_{h \in H} U_{h} \cdot Y H_{f, i, j, h} ⩽ V F_{f} \forall f \in F, i \in O \cup C, j \in O \cup C$	(C17)
Shipment being delivered to customer
$\sum_{f \in F} \sum_{i \in O \cup C} {YH}_{f, i, j, h} = {DOS}_{h, j} \forall h \in H, j \in O$	(C18)
Means of transport f carries at most one course/route in the planning period
$\sum_{j} X H_{f, c, j} ⩽ 1 \forall f \in F, c \in C$	(C19)
Means of transport f carries the course/route from this distribution center to which it was assigned
$\sum_{j} X H_{f, c, j} ⩽ {CF}_{c, f} \forall f \in F, c \in C$	(C20)
Items picked up/delivered to a delivery point
$\sum_{j \in O} \sum_{f \in F} Y H_{f, j, i, h} - \sum_{j \in O} \sum_{f \in F} Y H_{f, i, j, h} = D O S_{h, i} \forall i \in O, h \in H$	(C21)
Shipment/pallet can be taken up only from the distribution center, where it is prepared
$\sum_{f \in F} \sum_{i \in O} \sum_{c \in C} D O C_{h, c} \cdot Y H_{f, c, i, h} = 1 \forall h \in H$	(C22)
What means of transport the shipment is delivered to the customer
$Y H_{f, i, j, h} = F H_{h, f} \forall f \in F, i \in O, j \in O, h \in H$	(C23)
The given shipment/pallet is delivered by only one means of transport
$\sum_{f \in F} F H_{f, h} = 1 \forall h \in H$	(C24)
A means of transport carrying only such shipment/pallet, which can carry because of the type of products in the shipment/pallet
$F H_{h, f} ⩽ D O F_{h, f} \forall h \in H, f \in F$	(C25)
Binarity
$\begin{array}{l} Y H_{f, i, j, h} = {0, 1} \forall f \in F, i \in O \cup C, j \in O \cup C, h \in H \\ X H_{f, i, j} = {0, 1} \forall f \in F, i \in O \cup C, j \in O \cup C \end{array}$	(C26)

Symbol	Description
Decision variables
XH_f,i,j	If the means of transport f travels from point i to point j, then X_f,i,j=1, otherwise X_f,i,j=0 (f∈F, i∈O∪C, j∈O∪C)
YH_f,i,j,h	If the means of transport f travels from point i to point j carrying shipment/pallet h, then Y_f,i,j,h=1, otherwise Y_f,i,j,h=0 (f∈F, i∈O∪C, j∈O∪C, h∈H)
FH_h,f	If shipment/pallet h is delivered by means of transport f than FH_h,f=1, otherwise FH_h,f=0 (h∈H, f∈F)
Constraints
Arrival and departure means of transport from the delivery point
$\sum_{j \in O \cup C} X H_{f, i, j} = \sum_{j \in O \cup C} X H_{f, j, i} \forall f \in F, i \in O \cup C$	(C15)
If no shipments h are to be carried on the route i,j, a means of transport does not travel that route – linking the decision variable XH_f,i,j with decision variable YH_f,i,j,h
$\begin{array}{l} X H_{f, i, j} ⩽ \sum_{h \in H} Y H_{f, i, j, h} \forall f \in F, i \in O \cup C, j \in O \\ Y H_{f, i, j, h} ⩽ X H_{f, i, j} \forall f \in F, i \in O \cup C, j \in O \cup C, h \in H \end{array}$	(C16)
The given means of transport carries no more shipments than allowable as per its capacity volume
$\sum_{h \in H} U_{h} \cdot Y H_{f, i, j, h} ⩽ V F_{f} \forall f \in F, i \in O \cup C, j \in O \cup C$	(C17)
Shipment being delivered to customer
$\sum_{f \in F} \sum_{i \in O \cup C} {YH}_{f, i, j, h} = {DOS}_{h, j} \forall h \in H, j \in O$	(C18)
Means of transport f carries at most one course/route in the planning period
$\sum_{j} X H_{f, c, j} ⩽ 1 \forall f \in F, c \in C$	(C19)
Means of transport f carries the course/route from this distribution center to which it was assigned
$\sum_{j} X H_{f, c, j} ⩽ {CF}_{c, f} \forall f \in F, c \in C$	(C20)
Items picked up/delivered to a delivery point
$\sum_{j \in O} \sum_{f \in F} Y H_{f, j, i, h} - \sum_{j \in O} \sum_{f \in F} Y H_{f, i, j, h} = D O S_{h, i} \forall i \in O, h \in H$	(C21)
Shipment/pallet can be taken up only from the distribution center, where it is prepared
$\sum_{f \in F} \sum_{i \in O} \sum_{c \in C} D O C_{h, c} \cdot Y H_{f, c, i, h} = 1 \forall h \in H$	(C22)
What means of transport the shipment is delivered to the customer
$Y H_{f, i, j, h} = F H_{h, f} \forall f \in F, i \in O, j \in O, h \in H$	(C23)
The given shipment/pallet is delivered by only one means of transport
$\sum_{f \in F} F H_{f, h} = 1 \forall h \in H$	(C24)
A means of transport carrying only such shipment/pallet, which can carry because of the type of products in the shipment/pallet
$F H_{h, f} ⩽ D O F_{h, f} \forall h \in H, f \in F$	(C25)
Binarity
$\begin{array}{l} Y H_{f, i, j, h} = {0, 1} \forall f \in F, i \in O \cup C, j \in O \cup C, h \in H \\ X H_{f, i, j} = {0, 1} \forall f \in F, i \in O \cup C, j \in O \cup C \end{array}$	(C26)

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