Table II. Data generation for the SCBSP...

Table II.

Data generation for the SCBSP mathematical model

Parameter	Selected Value
Number of arriving vessels: m (vessels)	Values are changed based on the problem instance
Number of available berthing positions: n (berthing positions)	Values are changed based on the problem instance
Number of periods: p (periods)	Values are changed based on the problem instance
Arrival time of vessel v at the MCT: $θ_{v}^{a}, v \in V$ (period)	$θ_{v + 1}^{a} = r o u n d^{+} (θ_{v}^{a} + Δ θ) \forall v \in V$ ^a
Average inter-arrival time of vessels: Δθ (periods)	Δθ = exp[2]^b
Period when berthing position b is available for the first time in the planning horizon: $θ_{b}^{b e r}, b \in B$ (period)	$θ_{b}^{b e r} = 0 \forall b \in B$
Periods required for handling vessel v at its preferred berthing position: $\tilde{θ_{v}^{h t}}, v \in V$ (periods)	$\tilde{θ_{v}^{h t}} = r o u n d (U [10; 24]) \forall v \in V$ ^c,d
Periods required for handling vessel v at berthing position b: $θ_{v b}^{h t}, v \in V, b \in B$ (periods)	$θ_{v b}^{h t} = r o u n d^{+} (\tilde{θ_{v}^{h t}} \cdot (1 + 0.03 \cdot a b s (B_{v}^{p r} - B_{v}^{a s}))) \forall v \in V, b \in B$
Negotiated departure period for vessel v: $θ_{v}^{d}, v \in V$ (period)	$θ_{v}^{d} = r o u n d (θ_{v}^{a} + \tilde{θ_{v}^{h t}} \cdot U [1.20; 1.50]) \forall v \in V$
Draft of vessel v: $D_{v}^{v e s}, v \in V$ (feet)	[41.0; 41.0; 42.7; 47.6; 49.9; 50.9]
Depth of berthing position b: $D_{b}^{b e r}, b \in B$ (feet)	$D_{b}^{b e r} = 1.20 \cdot \min_{v} (D_{v}^{v e s}) + U [0; 1.20 \cdot \max_{v} (D_{v}^{v e s}) - \min_{v} (D_{v}^{v e s})] \forall b \in B$
Length of vessel v: $L_{v}^{v e s}, v \in V$ (feet)	[820.2; 951.4; 935.0; 984.3; 1200.8; 1312.3]
Length of berthing position b: $L_{b}^{b e r}, b \in B$ (feet)	$L_{b}^{b e r} = 1.20 \cdot \min_{v} (L_{v}^{v e s}) + U [0; 1.20 \cdot \max_{v} (L_{v}^{v e s}) - \min_{v} (L_{v}^{v e s})] \forall b \in B$
Minimum vertical clearance requirement for vessel v (feet): $P_{v}^{v e r}, v \in V$ (feet)	$P_{v}^{v e r} = U [4; 8] \forall v \in V$
Minimum horizontal clearance requirement for vessel v: $P_{v}^{h o r}, v \in V$ (feet)	$P_{v}^{h o r} = U [50; 100] \forall v \in V$
Unit waiting cost for vessel v: $φ_{v}^{w t}, v \in V$ (US$/period)	$φ_{v}^{w t} = U [1000; 5000] \forall v \in V$
Unit handling cost for vessel v: $φ_{v}^{h t}, v \in V$ (US$/period)	$φ_{v}^{h t} = U [50000; 70000] \forall v \in V$
Unit late departure cost for vessel v: $φ_{v}^{l t}, v \in V$ (US$/period)	$φ_{v}^{l t} = U [5000; 10000] \forall v \in V$
Large positive number: Υ	10000

Parameter	Selected Value
Number of arriving vessels: m (vessels)	Values are changed based on the problem instance
Number of available berthing positions: n (berthing positions)	Values are changed based on the problem instance
Number of periods: p (periods)	Values are changed based on the problem instance
Arrival time of vessel v at the MCT: $θ_{v}^{a}, v \in V$ (period)	$θ_{v + 1}^{a} = r o u n d^{+} (θ_{v}^{a} + Δ θ) \forall v \in V$ ^a
Average inter-arrival time of vessels: Δθ (periods)	Δθ = exp[2]^b
Period when berthing position b is available for the first time in the planning horizon: $θ_{b}^{b e r}, b \in B$ (period)	$θ_{b}^{b e r} = 0 \forall b \in B$
Periods required for handling vessel v at its preferred berthing position: $\tilde{θ_{v}^{h t}}, v \in V$ (periods)	$\tilde{θ_{v}^{h t}} = r o u n d (U [10; 24]) \forall v \in V$ ^c,d
Periods required for handling vessel v at berthing position b: $θ_{v b}^{h t}, v \in V, b \in B$ (periods)	$θ_{v b}^{h t} = r o u n d^{+} (\tilde{θ_{v}^{h t}} \cdot (1 + 0.03 \cdot a b s (B_{v}^{p r} - B_{v}^{a s}))) \forall v \in V, b \in B$
Negotiated departure period for vessel v: $θ_{v}^{d}, v \in V$ (period)	$θ_{v}^{d} = r o u n d (θ_{v}^{a} + \tilde{θ_{v}^{h t}} \cdot U [1.20; 1.50]) \forall v \in V$
Draft of vessel v: $D_{v}^{v e s}, v \in V$ (feet)	[41.0; 41.0; 42.7; 47.6; 49.9; 50.9]
Depth of berthing position b: $D_{b}^{b e r}, b \in B$ (feet)	$D_{b}^{b e r} = 1.20 \cdot \min_{v} (D_{v}^{v e s}) + U [0; 1.20 \cdot \max_{v} (D_{v}^{v e s}) - \min_{v} (D_{v}^{v e s})] \forall b \in B$
Length of vessel v: $L_{v}^{v e s}, v \in V$ (feet)	[820.2; 951.4; 935.0; 984.3; 1200.8; 1312.3]
Length of berthing position b: $L_{b}^{b e r}, b \in B$ (feet)	$L_{b}^{b e r} = 1.20 \cdot \min_{v} (L_{v}^{v e s}) + U [0; 1.20 \cdot \max_{v} (L_{v}^{v e s}) - \min_{v} (L_{v}^{v e s})] \forall b \in B$
Minimum vertical clearance requirement for vessel v (feet): $P_{v}^{v e r}, v \in V$ (feet)	$P_{v}^{v e r} = U [4; 8] \forall v \in V$
Minimum horizontal clearance requirement for vessel v: $P_{v}^{h o r}, v \in V$ (feet)	$P_{v}^{h o r} = U [50; 100] \forall v \in V$
Unit waiting cost for vessel v: $φ_{v}^{w t}, v \in V$ (US$/period)	$φ_{v}^{w t} = U [1000; 5000] \forall v \in V$
Unit handling cost for vessel v: $φ_{v}^{h t}, v \in V$ (US$/period)	$φ_{v}^{h t} = U [50000; 70000] \forall v \in V$
Unit late departure cost for vessel v: $φ_{v}^{l t}, v \in V$ (US$/period)	$φ_{v}^{l t} = U [5000; 10000] \forall v \in V$
Large positive number: Υ	10000

Notes:

^aNotation round⁺(Val₁) is for the nearest integer larger than Val₁; ^bnotation exp[Val₂] is for the exponentially distributed pseudorandom numbers with an average of Val₂; ^c notation round(Val₃) is for the nearest integer to Val₃; ^dnotation U[Val₄; Val₅] is for the uniformly distributed pseudorandom numbers that range from Val₄ to Val₅

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