e Scientic World Journal 3
2.3.2. Time Variables
𝑖V
≥0,
𝑖V
≥0, ∈,V ∈.
(2)
𝑖V
is the start time of operation V if it is assigned to priority
slot ;
𝑖V
=0otherwise.
𝑖V
is the duration of operation V if it is assigned to
priority slot ;
𝑖V
=0otherwise.
2.3.3. Operation Variables
𝑡
𝑖V
≥0,
𝑖V𝑐
≥0, ∈,V ∈,∈.
(3)
𝑡
𝑖V
is the total volume of crude transferred during operation
V if it is assigned to priority slot ;
𝑡
𝑖V
=0otherwise.
𝑖V𝑐
isthevolumeofcrudetransferred during operation
V if it is assigned to priority slot ;
𝑖V𝑐
=0otherwise.
2.3.4. Resource Variables
𝑡
𝑖𝑟
,
𝑖𝑟𝑐
, ∈,∈,∈.
(4)
𝑡
𝑖𝑟
is the total accumulated level of crude in tank ∈
𝑆
∪
𝐶
before the operation was assigned to priority-slot .
𝑖𝑟𝑐
is the accumulated level of crude in tank ∈
𝑆
∪
𝐶
before the operation was assigned to priority-slot .
2.4. Objective Function. e objective is to maximize the
gross margins of the distilled crude blends. Let
𝑐
be the
individual gross margin of crude ,
max
𝑖∈𝑇
𝑟∈𝑅
𝐷
V∈𝐼
𝑟
𝑐∈𝐶
𝑐
⋅
𝑖V𝑐
.
(5)
2.5. General Constraints. It should be noted that the crude
composition of blends in tanks is tracked instead of their
properties. e distillation specications are later enforced by
calculating a posteriori the properties of the blend in terms
of its composition. For instance, in the problem, a blend
composedof50%ofcrudeAand50%ofcrudeBhasasulfur
concentration of 0.035 which does not meet the specication
forcrudemixXnorforcrudemixY.
2.5.1. Assignment Constraints. In the SOS model, exactly one
operation has to be assigned to each priority slot,
V∈𝑊
𝑖V
=1, ∈.
(6)
2.5.2. Variable Constraints. Variable constraints are given by
their denitions. Start time, duration, and global volume
variables are dened with big-constraints,
𝑖V
+
𝑖V
≤⋅
𝑖V
,∈,V ∈,
𝑡
𝑖V
≤
𝑡
V
⋅
𝑖V
,∈,V ∈,
𝑡
𝑖V
≥
𝑡
V
⋅
𝑖V
,∈,V ∈.
(7)
Crude volume variables are positive variables whose sum
equals the corresponding total volume variable,
𝑐∈𝐶
𝑖V𝑐
=
𝑡
𝑖V
.
(8)
Total and crude level variables are dened by adding to
the initial level in the tank all inlet and outlet transfer volumes
of operations of higher priority than the considered priority
slot,
𝑡
𝑖𝑟
=
𝑡
0𝑟
+
𝑗∈𝑇,𝑗<𝑖
V∈𝐼
𝑟
𝑡
𝑖V
−
𝑗∈𝑇,𝑗<𝑖
V∈𝑂
𝑟
𝑡
𝑖V
,
∈,∈,
(9)
𝑖𝑟𝑐
=
𝑜𝑟𝑐
+
𝑗∈𝑇,𝑗<𝑖
V∈𝐼
𝑟
𝑖V𝑐
−
𝑗∈𝑇,𝑗<𝑖
V∈𝑂
𝑟
𝑖V𝑐
,
∈,∈,∈.
(10)
2.5.3. Sequencing Constraints. Sequencing constraints
restrict the set of possible sequences of operations.
Cardinality and unloading sequence constraints are specic
cases of sequencing constraints. More complex sequencing
constraints will also be discussed later.
2.5.4. Cardinality Constraint. Each crude oil marine vessel
has to unload its content exactly once. ∑
𝑖∈𝑇
∑
V∈𝑂
𝑟
𝑖V
=1, ∈
𝑉
. e total number of distillation operations is bounded by
𝐷
and
𝐷
in order to reduce the cost of CDU switches,
𝐷
≤
𝑖∈𝑇
V∈𝑊
𝐷
𝑖V
≤
𝐷
.
(11)
2.5.5. Unloading Sequence Constraint. Marine vessels have to
unload in order of arrival to the renery. Considering two
vessels
1
,
2
∈
𝑉
,
1
<
2
signies that
1
unloads before
2
,
𝑗∈𝑇,𝑗<𝑖
V∈𝑂
𝑟
2
𝑗V
+
𝑗∈𝑇,𝑗≥𝑖
V∈𝑂
𝑟
1
𝑗V
≤1.
(12)
2.5.6. Scheduling Constraints. Scheduling constraints restrict
the values taken by time variables according to logistics rules.
2.5.7. Nonoverlapping Constraint. A nonoverlapping con-
straint between two sets of operations
1
⊂and
2
⊂
states that any pair of operations (V
1
,V
2
)⊂
1
×
2
must not
be executed simultaneously.
Unloading operations must not overlap,
V∈𝑊
𝑈
𝑖V
+
𝑖V
≤
V∈𝑊
𝑈
𝑗V
+⋅1−
V∈𝑊
𝑈
𝑗V
,
,∈, <.
(13)