Introduction
I: DC J: Depot
4.3 Model Description
This study attempts to simultaneously determine the berthing and crane allocations under two objectives. The wharf is modeled to be discrete, that is, it represents a collection of partitioned sections.
Different types of cranes with different handling rates are considered.
Handling time and the number of cranes to be assigned to the ship are not known in advance. Handling time depends on the type and the number of cranes allocated to a vessel, which is dynamic throughout
8 8 deniz ozdemir and e Vrim ursaVas
the service time. For instance, a vessel can start to be served by only one crane and end up being served by three cranes. Therefore, the ships do not have to wait until a specified number of cranes are avail- able. This prevents suboptimal solutions resulting from misleading crane unavailability assumption.
We now present the bi-objective optimization model for solv- ing simultaneous berth–vessel–crane allocation problem. The basic assumptions of the model can be summarized as follows.
4.3.1 Assumptions
1. There are discrete berths with specified lengths. A vessel may be assigned to any of the available berths as long as the vessel length fits to the berth length.
2. There are cranes with different technology that give service with varying handling rates.
3. Some of the cranes are mobile, in a sense that cranes can be assigned to any berth and any vessel in any order.
4. Crane allocation is dynamic throughout the handling period of a vessel. The number and the type of cranes assigned are flexible, and vessel handling time is dependent on crane allocations.
5. A vessel cannot be given service before its arrival.
6. Each different crane allocation incurs a cost.
7. There are a maximum allowable number of cranes that can be assigned to a vessel.
The indices, parameters, decision variables, and the integer linear pro- gramming model are defined as follows.
4.3.2 Notation Indices
i = (1, …, I) set of vessels
j = (1, …, J) set of cranes, where first p cranes are static and last J–p cranes are assumed to be portable
k = (1, …, K) set of berths t = (1, …, T) time periods
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bi- ob jeC tiVe berth – Cr ane allo Cation
Input Parameters
li: Vessel length including the safety margin for the vessel Qk: Length of berth k
ai: Arrival time of vessel i
Ni0: Number of containers initially on the vessel
U: Maximum number of cranes that can be assigned to a vessel simultaneously
Rj: Container handling rate of jth crane For modeling purposes, we define two constants:
M: Large constants m: Constant 0 ≤ m ≤ 1 Decision Variables
yijtk: 1 if crane j is allocated to vessel i at time t at berth k and 0 otherwise
BVitk: 1 if vessel i is assigned to berth k at time t Nit: Total number of containers on vessel i at time t Δik: 1 if vessel i is assigned to berth k
YHit: 1 if vessel i is served at time t
PHit: 1 if vessel i has remaining containers at time t CRijt: 1 if crane j will start serving vessel i at time t + 1
TempHit: Auxiliary variable that realizes the logical connection between yijtk and YHit
4.3.3 Model
f time PH
f setup CR
i t a it
i j t a
ijt i
i 1
2
( )
( )
∑∑
∑∑∑
=
=
: min : min
l yi⋅ ijtk ≤Qk ∀i j t k, , , (4.1) yijtk i j t
∑
k ≤1 ∀ , , (4.2)9 0 deniz ozdemir and e Vrim ursaVas
yijtk j t
k
i
∑
∑
≤1 ∀ , (4.3)yijtk j t k
∑
i ≤1 ∀ , , (4.4)yijtk i
k t a
j
∑
i∑
∑
=≥1 ∀ (4.5)
yijtk U i t
k
j
∑
∑
≤ ∀ , (4.6)BVitk t k
∑
i ≤1 ∀ , (4.7)yijtk M BV i t k
j
∑
≤ ⋅ itk ∀ , , (4.8)yijt k y CR i j t
k
ijtk k
ijt
∑
+1 −∑
≤ ∀ , , (4.9)Nit+1 ≤M PH⋅ it ∀i t t T, , ≠ (4.10)
Nit R yj ijtk N i t t T
k j
−
∑ ∑
⋅ = i t,+1 ∀ , , ≠ (4.11)Ni T, ≤0 ∀i (4.12)
YHit ≤PHit ∀i t, (4.13) YHit ≤TempHit ∀i t, (4.14)
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bi- ob jeC tiVe berth – Cr ane allo Cation
yijtk m TempH i t
k j
∑
it∑
≥ ⋅ ∀, (4.15)yijtk M TempH i t
k j
∑
it∑
≤ ⋅ ∀ , (4.16)YHit ≥ ⋅m TempHit ∀i t, (4.17)
yijtk m i k
t a j
ik
=i
∑
∑
≥ ⋅Δ ∀ , (4.18)yijtk M i k
k k t a
ik
j i
ʹ
=
∑
ʹ≠∑
∑
≤ ⋅ −(1 Δ ) ∀, (4.19)yijtk M i k
t a j
ik
=i
∑
∑
≤ ⋅Δ ∀ , (4.20)yijtk m i k
k k t a
ik
j i
ʹ
=
∑
ʹ≠∑
∑
≥ ⋅ −(1 Δ ) ∀ , (4.21)yij tk M y p t k
k k j j
ijtk i i
ʹ ʹ ʹ≤ − ʹ≥ +
∑
∑ ∑
∑
1 1 ≤ ⋅⎛⎝⎜⎜1− ⎞⎠⎟⎟ ∀ ≤j , , (4.22)yijtk,Δik,PH YH TempH CR BVit, it, it, ijt, i t k, , ∈{ , }0 1 ∀i j t, , (4.23)
Nit ∃ ∀i t, (4.24)
The first objective f1 minimizes the total time the vessels spend at the port. When all the containers are handled, the handling time is calculated by summing the total number of assignments in the time horizon. To calculate the total time, waiting time of the vessels on
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the bay is also considered. The second objective f2 minimizes the total number of crane setups.
Constraint set (4.1) ensures that the allocation of a vessel does not exceed the quay length. Constraint set (4.2) implies that a vessel can be assigned to at most one berth. Constraint set (4.3) does not allow any crane to be allocated to more than one vessel at multiple berths at time t. Constraint set (4.4) implies that a single vessel can be served by a certain crane at any given time. Constraint set (4.5) ensures that all arriving vessels are served. Constraint set (4.6) guarantees that the total number of cranes allocated in a time period exceeds the maximum number of cranes that can be allo- cated to a vessel. By constraint set (4.7), the number of vessels allo- cated to a berth at a given time is limited to 1. Constraint set (4.8) ensures that the value of BVitk at the considered berth–vessel pair is set to 1 if a vessel is given service at the dock at a given time.
In constraint set (4.9), crane setup indicators are updated. By con- straint set (4.10), a vessel’s PHit value is set to 1, if the vessel has arrived and there are remaining containers. In constraint set (4.11), the number of containers to be handled in each vessel is decreased by the crane handling rate at each period. Constraint set (4.12) ensures that all the containers on the vessel are handled. The logi- cal connection between PHit and YHit is secured by constraint set (4.13). Constraint sets (4.14) through (4.17) formulate the equa- tions for solving the total handling time of each vessel. If an yijtk
assignment exists for a vessel at a given time, the vessel handling time variable, YHit, is set to 1. Constraint sets (4.18) through (4.21) ensure that a vessel is docked at a single berth. Constraint set (4.22) handles the crane passing constraints for static cranes. If a crane j is serving a vessel at berth k, then no other crane with a larger crane id can serve a vessel at any berth that is positioned to its right. In the next section, our solution approach will be discussed.