Table I. Nomenclature adopted for the...

Table I.

Nomenclature adopted for the mathematical model

Model component
Type	Nomenclature	Description
Sets	V = {1, …, m}	the set of arriving vessels at the MCT (vessels)
	B = (1, …, n}	the set of available berthing positions at the MCT (berthing positions)
	T = {1, …, p)	the set of discrete periods (periods)
Decision Variables	x_vbt, v ∈ V, b ∈ B, t ∈ T	=1 if vessel v is assigned for service at berthing position b at period t (=0 otherwise)
Auxiliary Variables	$θ_{v b t}^{i t}, v \in V,$ b∈B, t∈T	the number of idling periods for berthing position b between service of vessel v and its immediate predecessor served during period (t – 1) (periods)
	q_vbt, v ∈ V, b ∈ B, t ∈ T	=1 if vessel v leaves berthing position b at period t (=0 otherwise)
	$θ_{v}^{s t}, v \in V$	the start service period for vessel v (period)
	$θ_{v}^{f t}, v \in V$	the finish service period for vessel v (period)
	$θ_{v}^{w t}, v \in V$	the number of periods that vessel v should wait for service (periods)
	$θ_{v}^{l t}, v \in V$	the number of periods that vessel v leaves later than the negotiated departure period (periods)
Parameters	m	the total number of arriving vessels at the MCT to be served (vessels)
	n	the total number of available berthing positions at the MCT (berthing positions)
	p	the total number of periods in the planning horizon (periods)
	$θ_{v}^{a}, v \in V$	the period when vessel v arrives at the MCT (period)
	$θ_{b}^{b e r}, b \in B$	the period when berthing position b is available for the first time in the considered planning horizon (period)
	$\tilde{θ_{v}^{h t}}, v \in V$	the periods required for handling vessel v at its preferred berthing position (periods)
	$θ_{v b}^{h t}, v \in V,$ b∈B	the periods required for handling vessel v at berthing position b (periods)
	$θ_{v}^{d}, v \in V$	the negotiated departure period for vessel v (period)
	$D_{v}^{v e s}, v \in V$	the draft of vessel v (feet)
	$D_{b}^{b e r}, b \in B$	the depth of berthing position b (feet)
	$L_{v}^{v e s}, v \in V$	the length of vessel v (feet)
	$L_{b}^{b e r}, b \in B$	the length of berthing position b (feet)
	$P_{v}^{v e r}, v \in V$	the minimum vertical clearance for vessel v (feet)
	$P_{v}^{h o r}, v \in V$	the minimum horizontal clearance for vessel v (feet)
	$φ_{v}^{w t}, v \in V$	the unit waiting cost component for vessel v (US$/period)
	$φ_{v}^{h t}, v \in V$	the unit handling cost component for vessel v (US$/period)
	$φ_{v}^{l t}, v \in V$	the unit late departure cost component for vessel v (US$/period)
	Υ	a large positive number

Model component
Type	Nomenclature	Description
Sets	V = {1, …, m}	the set of arriving vessels at the MCT (vessels)
	B = (1, …, n}	the set of available berthing positions at the MCT (berthing positions)
	T = {1, …, p)	the set of discrete periods (periods)
Decision Variables	x_vbt, v ∈ V,b ∈ B, t ∈ T	=1 if vessel v is assigned for service at berthing position b at period t (=0 otherwise)
Auxiliary Variables	$θ_{v b t}^{i t}, v \in V,$ b∈B, t∈T	the number of idling periods for berthing position b between service of vessel v and its immediate predecessor served during period (t – 1) (periods)
	q_vbt, v ∈ V,b ∈ B, t ∈ T	=1 if vessel v leaves berthing position b at period t (=0 otherwise)
	$θ_{v}^{s t}, v \in V$	the start service period for vessel v (period)
	$θ_{v}^{f t}, v \in V$	the finish service period for vessel v (period)
	$θ_{v}^{w t}, v \in V$	the number of periods that vessel v should wait for service (periods)
	$θ_{v}^{l t}, v \in V$	the number of periods that vessel v leaves later than the negotiated departure period (periods)
Parameters	m	the total number of arriving vessels at the MCT to be served (vessels)
	n	the total number of available berthing positions at the MCT (berthing positions)
	p	the total number of periods in the planning horizon (periods)
	$θ_{v}^{a}, v \in V$	the period when vessel v arrives at the MCT (period)
	$θ_{b}^{b e r}, b \in B$	the period when berthing position b is available for the first time in the considered planning horizon (period)
	$\tilde{θ_{v}^{h t}}, v \in V$	the periods required for handling vessel v at its preferred berthing position (periods)
	$θ_{v b}^{h t}, v \in V,$ b∈B	the periods required for handling vessel v at berthing position b (periods)
	$θ_{v}^{d}, v \in V$	the negotiated departure period for vessel v (period)
	$D_{v}^{v e s}, v \in V$	the draft of vessel v (feet)
	$D_{b}^{b e r}, b \in B$	the depth of berthing position b (feet)
	$L_{v}^{v e s}, v \in V$	the length of vessel v (feet)
	$L_{b}^{b e r}, b \in B$	the length of berthing position b (feet)
	$P_{v}^{v e r}, v \in V$	the minimum vertical clearance for vessel v (feet)
	$P_{v}^{h o r}, v \in V$	the minimum horizontal clearance for vessel v (feet)
	$φ_{v}^{w t}, v \in V$	the unit waiting cost component for vessel v (US$/period)
	$φ_{v}^{h t}, v \in V$	the unit handling cost component for vessel v (US$/period)
	$φ_{v}^{l t}, v \in V$	the unit late departure cost component for vessel v (US$/period)
	Υ	a large positive number

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