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```sout, j, p
)
amount of state produced from unit j at time point p
mu
(
sin, j, p
)
amount of state used in or enters unit j at time point p
y
(
s∗in, j, p
)
binary variable associated with usage of state s at time point p
d (sout, p) amount of state delivered to customers at time point p
Parameters
a
(
s∗in, j
)
duration parameter associated with variable batch size
b
(
s∗in, j
)
duration parameter associated with variable batch size
18 2 Short-Term Scheduling
tL
(
s∗in, j
)
minimum processing time in unit j corresponding to a particular task
tU
(
s∗in, j
)
maximum processing time in unit j corresponding to a particular task
VUj maximum design capacity of a particular unit j
VLj minimum design capacity of a particular unit j
H time horizon of interest
τ
(
s∗in, j
)
mean processing time for a state
Q0s (s) initial amount of state s stored
QUs (s) maximum amount of state s stored within the time horizon of interest
CP (s) Selling price of product s, s = product
Capacity Constraints
VLj y
(
s∗in, j, p
)
≤
∑
sin, j
mu
(
sin, j, p
) ≤ VUj y
(
s∗in, j, p
)
,∀j ∈ J, p ∈ P,∀sin, j ∈ Sin, j
(2.1)
This constraint implies that the total amount of all the states consumed at time
point p is limited by the capacity of the unit which consumes the states. The U
and L superscripts denote the upper and the lower bound on capacity. According to
constraint (2.1), states will be consumed in a particular unit jif the corresponding
effective state is used at time point p.
Material Balances
∑
sin, j
mu
(
sin, j, p− 1
) = ∑
sout, j
mp
(
sout, j, p
)
,∀p ∈ P, p > p1, j ∈ J,
∀sin, j ∈ Sin, j, sout, j ∈ Sout, j
(2.2)
qs (s, p1) = Q0s (s)− mu (s, p1) , s
= product,∀s ∈ S (2.3)
qs (s, p) = qs (s, p− 1)− mu (s, p) , s = feed,∀s ∈ S,∀p ∈ P, p > p1 (2.4)
qs (s, p) = qs (s, p− 1)+ mp (s, p)− mu (s, p) , s
= product, feed,∀s ∈ S
∀p ∈ P, p > p1 (2.5)
qs (s, p1) = Q0s (s)− d (s, p1) , s = product,∀s ∈ S (2.6)
qs (s, p) = q (s, p− 1)+ mp (s, p)− d (s, p) , s = product, byproduct,∀s ∈ S
∀p ∈ P, p > p1
(2.7)
2.3 Mathematical Model 19
Constraint (2.2) is the material balance around a particular unit j. It implies that
the sum of the masses for all the input states used at time point p –1 should be
equal to the sum of the masses for all the output states produced at time point p.
Constraint (2.3) states that the amount of state s stored at the first time point, is
the difference between the amount stored before the beginning of the process and
that being utilised at the first time point. Constraint (2.4) only applies to the feed,
since it is the state that is only used in the process. Constraint (2.5) only applies
to intermediates, since they are both produced and used in the process. Constraints
(2.6) and (2.7) only apply to products and byproducts, since they are the only states
that have to be taken out of the process as shown by the terms d(s, p).
Duration Constraints (Batch Time as a Function of Variable
Batch Size)
In this section the duration constraints are modelled as a function of batch size. The
following constraints show how this effect is modelled in the proposed approach
using the SSN representation.
tp
(
sout, j, p
)= tu
(
s∗in, j, p−1
)
+a
(
s∗in, j
)
y
(
s∗in, j, p−1
)
+b
(
s∗in, j
) ∑
sin, j
mu (s, p−1),
∀j ∈ J,∀p ∈ P, p > p1,∀sin, j ∈ Sin, j, sout, j ∈ Sout, j
(2.8)
b
(
s∗in, j
)
=
tU
(
s∗in, j
)
− tL
(
s∗in, j
)
VUj − VLj
,∀j ∈ J,∀s∗in, j ∈ Sin, j (2.9)
a
(
s∗in, j
)
= τ
(
s∗in, j
) (
1− υ
(
s∗in, j
))
= tL
(
s∗in, j
)
,∀j ∈ J,∀s∗in, j ∈ Sin, j (2.10)
tU
(
s∗in, j
)
= τ
(
s∗in, j
) (
1+ υ
(
s∗in, j
))
,∀j ∈ J,∀s∗in, j ∈ Sin, j (2.11)
The parameter for variable batch time is defined by constraint (2.9). This gives
the amount of time required to process a unit amount of a batch corresponding
to a particular effective state in a corresponding unit operation. Constraint (2.10)
denotes the minimum processing time for the effective state in the corresponding
unit operation. This is, in essence, the minimum residence time of a batch within a
unit operation. In constraints (2.10) and (2.11), υ
(
s∗in, j
)
is the percentage variation
in processing time based on operational experience.
Duration Constraints (Batch Time Independent of Batch Size)
In a situation where duration is constant regardless of the batch size, the duration
constraint assumes the following form.
20 2 Short-Term Scheduling
tp
(
sout, j, p
) = tu
(
s∗in, j, p− 1
)
+ τ
(
s∗in, j
)
y
(
s∗in, j, p− 1
)
, ∀p ∈ P, p > p1,
∀ sout, j ∈ Sout, j, s∗in, j ∈ Sin, j
(2.12)
Sequence Constraints
tu
(
sin, j, p
) ≥ ∑
sin, j
∑
sout, j
∑
p′≤p
(
tp
(
sout, j, p′
)− tu
(
sin, j, p′ − 1
))
,
∀j ∈ J, p′ ∈ P, p > p1,∀sout, j ∈ Sout, j, sin, j ∈ Sin, j
(2.13)
tu
(
sin, j, p
) ≥ tp
(
sout, j, p′
)− H
(
2− y
(
s∗in, j, p
)
− y
(
s∗in, j, p′ − 1
))
,
∀j ∈ J, p, p′ ∈ P, p′ > p1, p′ ≤ p,∀sout, j ∈ Sout, j, sin, j ∈ Sin, j, s∗in, j → sout, j
(2.14)
tu
(
sin, j, p
) ≥ tp
(
sout, j′ , p
′
)
− H
(
2− y
(
s∗in, j, p
)
− y
(
s∗
in, j′ , p
′ − 1
))
,
∀j ∈ J, p, p′ ∈ P, p′ > p1, p′ ≤ p,∀sin, j ∈ Sin, j, sin, j = sout, j′ , s∗in, j′ → sout, j′
(2.15)
Constraints (2.13) and (2.14) imply that state s can only be used in a particular
unit, at any time point, after all the previous states have been processed. Constraint
(2.13) is only relevant in situations where more than one task can be conducted in
one unit, otherwise it is redundant in the presence of constraints (2.14) and (2.15).
Constraint (2.15) stipulates that a state can only be processed at a particular time
point p in a particular unit j after it has been produced from another unit j′. In case
of a recycle, j is the same as j′. It is worthy of note that constraints (2.14) and
(2.15) are only applicable to intermediates, since they are the only states that are
both produced and used. For clarity of notation which is adopted throughout this
textbook, s → s′ implies that state s′ is formed from a task that uses state s as its
input.
Assignment Constraint
The assignment constraint is aimed at ensuring that only one task is conducted in
a unit at any time point. It is, therefore, apparent that the assignment constraint is
only necessary if more than one task can be performed in a given unit. Otherwise, it
is also redundant.
∑
s∗in, j
y
(
s∗in, j, p
)
≤ 1,∀p ∈ P, j ∈ J, s∗in, j ∈ Sin, j (2.16)
2.4 Literature Examples 21
Time Horizon Constraints
tu
(
sin, j, p
) ≤ H,∀sin, j ∈ Sin, j, p ∈ P, j ∈ J (2.17)
tu
(
sout, j, p
) ≤ H,∀sout, j ∈ Sout, j, p ∈ P, j ∈ J (2.18)
Constraints (2.17) and (2.18) respectively stipulate that the usage or production
of state should be within the time horizon of interest.
Storage Constraints
qs (s, p) ≤ QU (s) ,∀s ∈ S, p ∈ P (2.19)
Constraint (2.19) states that the amount of state s stored at each time point cannot
exceed the maximum allowed.
Objective Function
The objective function for this formulation is the maximisation of product through-
put or revenue.
Maximize
∑
s
∑
p
CP (s) d (s, p), s = product, s ⊂ S, p ∈ P (2.20)
2.4 Literature Examples
2.4.1 First Literature Example
In this section, the above mathematical model is applied to a literature example
shown in Fig. 2.2 (Ierapetritou and Floudas, 1998). The SSN representation is given
in Fig. 2.3b. Table 2.1 gives data for this example. 5 time points and a 12-h time
horizon were used. Using less time points leads to a suboptimal solution with an
objective value of 50, and using more time points than 5 did not improve the
solution. It is worthy of note that, in this particular example, constraint (2.13) is
redundant as mentioned earlier, since each unit is only performing one task.
Capacity Constraints
State s1
mu (s1,p) ≤ 100y (s1, p) ,∀p ∈ P
State s2
mu (s2, p) ≤ 75y (s2, p) ,∀p ∈ P
State s3
mu (s3, p) ≤ 50y (s3, p) ,∀p ∈ P
22 2 Short-Term Scheduling
Table 2.1 Data for the literature example (Ierapetritou and Floudas, 1998)
Unit Capacity Suitability Mean processing time (τ)
Unit 1 100```
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