IEEE Robotics & Automation Magazine - June 2015 - 73

Problem P1: Maximize J =

/ Qj =0 / i ! DS fij .

(2)

Subject to
1) Fi ( MIN ) # fij # Fi ( MAX ), 6i ! DS and j ! BUK.
2)

detailed schedule can be found. Starting at the initial state, a
charging tank, say CTK1, is released for the first time, one can assign it to feed a distiller, say DS i . It is known from the given refining schedule that the oil type and amount for FPi1 are COTi1

/ i ! DS fij # Fp (MAX), 6j ! BUK.

3) During any bucket, the oil available for processing by
each distiller is enough.
Notice that, by Constraint 2, the schedulability condition 2 is
guaranteed. By solving Problem P1, the amount of crude oil required for processing by each distiller during the scheduling horizon is determined. With crude oil that is available during each
bucket in the charging tanks, pipeline, storage tanks, and tankers
known, the crude oil can be assigned to a distiller using a transportation problem model at Stage 2. To do so, the oil processing
effectiveness of type j by distiller i can be described by a cost coefficient C ij . If it is the most effective, C ij is set small; otherwise it
is large. If distiller i is unable to process crude-oil type j, a big M
is set for it. For a system with three distillers, two buckets, H oil
types, and one tanker, the problem for Stage 2 can be briefly described by a transportation problem model as shown in Table 1,
where D ij and Vij denote a demand and supply, respectively. The
readers can refer to [27] and [28] for further details.
Since a pipeline should be full of oil all the time, this requirement must be met when a schedule is completed at the
end of the scheduling horizon. Thus, crude oil should be assigned to the pipeline too. The oil assigned to the dummy represents the remaining oil in the pipeline at the end of horizon.
Using the results obtained from the problem at Stage 2, the
oil is divided into a number of parcels for each distiller. Notice
that in modeling the problem at Stage 2 the crude oil comes
from different sources and a source is treated as a supplier.
However, for this problem, different sources do not represent
different oil types. Hence, some parcels of crude oil from different sources that are assigned to a same distiller may be the
same type. Therefore, it is profitable to merge parcels with the
same oil type if possible so that the cost resulting from oil-type
switches in oil transportation and charging tank switches in
distiller feeding is reduced. This can be done by checking the
available time of each parcel. Because of the limited number of
crude-oil types, one can do the parcel merging easily. Notice
that by doing so we minimize the number of operational
switches, though we cannot ensure an optimal solution.
Detailed Scheduling
Since the schedulability conditions are modeled as constraints
in the refining scheduling problem, the realizability of an
obtained refining schedule is ensured. Then, given a refining
schedule, one needs to find a detailed schedule to realize it. In
a detailed schedule an FPij is realized by a number of ODs. In
other words, an FPij in a refining schedule should be divided
into m ODFs, i.e., ODFij 1, ODFij 2, f, and ODFijm , which
should be included in a detailed schedule. By using this
detailed schedule, the schedulability conditions are satisfied.
Since the set of safe states for a system can be identified by
using the schedulability conditions given a refining schedule, a

Schedulability
Condition

A Charging
Tank CTK
Is Released

Assign
CTK to
Distiller DSi

Create an
ODF for
DSi

Create
an ODT

Safe State
Sk

Safe State
Sk+1

Figure 8. The creation of ODs in finding a detailed schedule.

Table 2. The costs for processing oil types by different distillers.
Distiller 1

Distiller 2

Distiller 3

Crude-oil number 1

Crude-oil number 2

Crude-oil number 3

Crude-oil number 4

Crude-oil number 5

Crude-oil number 6

Table 3. Oil in charging tanks at the initial state.

Tank

Type of
Capacity Oil in
(Ton)
Tank

Amount
(Ton)

Distiller in
Feeding

Tank number 34,000
129

Crude-oil
number 3

27,000

DS1

Tank number 34,000
128

Crude-oil
number 2

30,000

DS2

Tank number 34,000
116

Crude-oil
number 4

27,000

DS3

Tank number 34,000
117

Crude-oil
number 5

30,000

Tank number 34,000
115

Crude-oil
number 5

25,000

Tank number 34,000
127
Tank number 20,000
182
Tank number 20,000
180
Tank number 20,000
181

JUNE 2015

IEEE ROBOTICS & AUTOMATION MAGAZINE

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - June 2015