INTRODUCTION
Process synthesis or conceptual process design is concerned with the identification
of the best flowsheet structure to perform a given task. Three major approaches
are traditionally available in the literature to address this class of problem:
(1) the heuristics method, notably the hierarchical decomposition of design
decisions procedure; (2) the technique based on thermodynamic targets and physical
insights as exemplified by pinch analysis and (3) the algorithmic approach that
utilizes optimization based on the construction of a superstructure that seeks
to represent all feasible process flowsheets (Seider et
al., 2009).
The intricate complexities associated with process synthesis problem in general
and the refinery design problem in specific necessitates the development and
implementation of a systematic and automated approach that efficiently and rigorously
integrate the elaborate interactions involving the design decision variables.
This study aims to extend the superstructureoptimizationbased approach of
using logical constraints (Raman and Grossmann, 1991,
1992, 1993, 1994)
within a MixedInteger Linear Program (MILP) to incorporate qualitative design
knowledge based on engineering experience and heuristics in modeling the major
process flows in a refinery. These constraints adopt discrete integer decision
variables of the binary 01 type to model the existence of a refinery process
unit and the associated stream piping interconnections (which are effectively
pipelines) in a network structure, in which a value of one for a 01 variable
designates that a unit is present in the optimal structure while the converse
is true for a value of zero. Our work serves to further substantiate that the
use of 01 decision variables offer a more natural and powerful modeling approach
compared to the conventional linear programming technique that employs only
continuous decision variables (Hassan et al., 2011;
Adeosun and Adetunde, 2008; Lan,
2008; Lan et al., 2008). It also affords
the convenience of representing fixedcost charges in the objective function
formulation. A variation in the use of integer variables in optimization model
formulations has been reported elsewhere (Nja and Udofia,
2009).
PROBLEM STATEMENT
We consider the following process synthesis problem of superstructure optimization
for the topology design of a refinery. Given the following data:
(a) fixed production amounts of desired products (b) the available process units
and ranges of their capacities and (c) cost of crude oil and cost structure
for process units, we are to determine the optimal topology or configuration
of the refinery in terms of the selection and sequencing of the streams as well
as the operating levels as represented by the stream flowrates.
PROPOSITIONAL LOGICS AND LOGIC CUTS IN PROCESS SYNTHESIS PROBLEMS
This study is based on the MixedInteger Linear Program (MILP) of Khor
and Elkamel (2010) for determining the optimal topology of a refinery with
environmental considerations. Our emphasis is to conduct an extensive investigation
of employing logical constraints on the design and structural specifications
of a refinery topology design. Logical constraints have been proven to be logic
cuts that serve to reduce the computational expense of solving an MILP by tightening
its linear relaxation and excluding fractional solutions without affecting the
quality of the optimal solution (Hooker et al., 1994).
They are algebraic linear inequalities or equalities formulated by using 01
binary variables to represent discrete decisions for the selection of alternative
tasks corresponding to the process units as well as alternative states corresponding
to the material streams.
SUPERSTRUCTURE OPTIMIZATION FOR SUBSYSTEM OF NAPHTHA PRODUCED FROM THE ATMOSPHERIC DISTILLATION UNIT (ADU)
Figure 1 shows a StateTask Network (STN)based superstructure representation that is sufficiently rich to embed all possible alternative topologies for the subsystem of naphtha produced from the (ADU) of a refinery. A substantial part of the data and information for the associated case study is provided by a refinery in Malaysia through an industrial collaboration that took place in AugustDecember 2008.
Process description for superstructure development: The first processing step in petroleum refining is crude distillation, in which Crude Oil (CR) is distilled into oil fractions with respect to its boiling points. Naphtha constitutes the lighter fractions that are obtained from this process. Depending on the distillation column design as well as the refinery economics, the ADU can produce: (a) light straight run naphtha (LSRN1) and heavy straight run naphtha (HSRN1) or (b) an undifferentiated class of naphtha, typically termed as wild Naphtha (NAP1), for which the 01 structural variables of z_{i} are used to represent these three possible states of naphtha.
In the first case, LSRN1 is mixed with purchased naphtha (PCHN2) and LSRN2
from the hydrotreater HDT1 in a mixer (MIX3).

Fig. 1: 
STNbased superstructure representation on case study of
refinery topology for naphtha processing subsystem 
The output from MIX3, i.e., LSRN4, can undergo two processes: (a) it is used
as a feedstock for the Isomerization Unit (ISO) and (b) it is sold as a final
product. Isomerization Yields Isomerate (ISO), one of the blending components
for gasoline (GSLN). Meanwhile, HSRN1 is mixed with naphtha from the cracking
of heavier fractions in MIX1 before being sent to HDT1 to be desulfurized.
HDT1 produces hydrogen sulfide gas (H2S1), liquefied petroleum gas (LPG1),
desulfurized naphtha (LSRN2, HSRN3, NAP4) and fuel gas (FG1). H2S1 is sent
to the Sulfur Recovery Unit (SRU) where Sulfur (S) is extracted and finally
sold. All LPG (LPG123) are sent to MIX6 and subsequently to the LPG recovery
unit (LPG), from which treated LPG (LPG5) is sold. Similar to the ADU outputs,
the desulfurized naphtha from HDT1 can be classified as light (LSRN2) and
heavy (HSRN3) or wild (NAP4). HSRN3 is sent to a mixer (MIX4), possibly
with purchased naphtha (PCHN32) and/or naphtha from the hydrocracker (HCR3).
The output of MIX4 (HSRN5) is the feedstock for the Reformer (REF). FG1 goes
to the Fuel Gas Header (FGH), supplying fuel gas (FG5) to the entire refinery.
In the case that NAP4 is produced from HDT1, it is also mixed with purchased
naphtha (PCHN31) and/or naphtha from the hydrocracker (HCR4) in MIX5 whose
output of NAP5 is sent to the reformer. The products from the reformer are
Hydrogen Gas (H2), Fuel Gas (FG3), LPG (LPG2) and reformate (REFs). H2 is
a feed to the HDT while reformate is used as a gasoline blending component.
FG3 is sent to the FGH.
In the second case involving NAP1 exiting ADU, the processing route is similar to the first case in that NAP1 is mixed with naphtha from cracking processes in MIX2 before being hydrotreated in HDT2. The products from HDT2 are H2S2, LPG3, desulfurized naphtha of LSRN3, HSRN4 and NAP3 and FG2. Each product has the exact same route as the products from HDT1. Other than distillation, naphtha is also produced from the cracking of distillation bottoms in the Visbreaker (VIS), Coker (COK), Catalytic Cracker (FCC), Hydrocracker (HCR). VIS has the lowest severity while COK has the highest.
A few assumptions are taken into consideration in developing the superstructure:
• 
The intermediate products from the Visbreaker (VIS), delayed
Coker (COK), Fluidized Catalytic Cracker (FCC) and Hydrocracker (HCR) are
assumed to be heavy naphtha (that is, heavier fractions of naphtha) 
• 
It is assumed that the API for medium and heavy crude oils is >33°
whereas for light crude oil, the API is >33° 
The processing of medium and heavy crude oils typically require more severe
processes, hence COK, FCC and HCR are enforced as possible external sources
of naphtha in such a case whereas VIS and FCC are the possible external naphtha
sources for the processing of light crude oils which require less severe processing.
General formulation of logical constraints for process synthesis problems: Based on the depicted superstructure of processing alternatives for naphtha exiting the ADU in Fig. 1, we consider the following design specification: MIX3 is selected if and only if LSRN1 or LSRN3 is produced. We contemplate the use of two logical relations and comment on some possible pitfalls.
First, using a combination of the logical or operator and the equivalence logic relation in the following form:
This is equivalent to the following two logic propositions:
_{}
By employing the following three steps involving the De Morgan’s theorem, these yields:
Thus, we obtain the following algebraic constraints:
_{}
However, the pitfall to using this formulation is that it allows the 01 variables to be satisfied for the case of (z_{LSRN 1}, z_{LSRN 3}, y_{MIX 3}) = (1, 1, 1). This violates the physics of the problem stipulating that either LSRN1 or LSRN3 (only) is selected in the optimal configuration.
Consider now the use of the logical relation exclusive or as given by the following:
Translating this logic proposition into its equivalent algebraic constraints
form, the proposition corresponds to:
However, there are three possible pitfalls in the use of this logical relation
which are all attributable to the logical constraint Eq. 8.
First, this constraint compels either LSRN1 stream or LSRN3 stream to be selected
even if there is no crude oil feed. Second, the two linear inequalities Eq.
7 and 8 enforce that y_{MIX3} = 1 which mandates
the MIX3 unit to be selected under all circumstances; in other words, it requires
MIX3 to be a permanent feature of a refinery topology which violates the physical
problem. Third, this logic proposition is not satisfied for the case of (z_{LSRN1},
z_{LSRN3}, y_{MIX3}) = (0, 0, 0) which is the hypothetical
case of no crude oil feed is available.
Thus, the constraints given by Eq. 5 best enforce the design specification that MIX 3 is selected if and only if LSRN1 or LSRN3 is produced.
In our computational experiments, it is perhaps noteworthy to highlight the following frequentlyencountered form of logic proposition in developing logical constraints on design specifications and structural specifications for synthesis problems. The logic form is generally given as:
which is equivalent to:
Transforming these logic propositions into inequalities yields:
The MILP model formulation is summarized as follows:
Logical constraints for processing alternatives of naphtha in refineries:
In summary, the following are the rest of the complete set of logical statements
and their associated logic propositions for the subsystem of naphtha produced
from ADU. For simplicity, note that the abbreviations iff stands for if and
only if and is stands for is/are selected. Parentheses are used to improve
readability:
• 
ADU is iff (HDT1 or HDT2) Is:




• 
(HDT1 or HDT2) is iff SRU is:




• 
(HSRN3 or HSRN4) is iff MIX4 is:




• 
(NAP3 or NAP4) is iff MIX5 is:




• 
(HDT1 or HDT2) is iff (MIX3 and MIX4), or (MIX3 and MIX5), or MIX5
Is:



• 
(HSRN5 or NAP5) is iff REFu is:




• 
HDT1 or HDT2) is iff LPG is:




• 
(FG1 or FG2 or FG3 or FG4) is iff FGH is:




• 
ISO is iff HDT1 is:




Generalized Disjunctive Programming (GDP) formulation: Generalized Disjunctive
Programming (GDP), with Raman and Grossmann (1993, 1994)
as its proponents, is an alternative modeling framework that has been found
to be amenable in translating physical intuition into more formal mathematical
expressions particularly in chemicalengineeringrelated problems, as it is
more recently substantiated by Furman and Androulakis (2008).
Since there will be conditional tasks or equipment that may be selected in the
final refinery topology, the use of GDP is of particular interest, since process
synthesis problems naturally lead to models where the solution space is disjoint
and there is a strong logic on the connectivity among the different tasks (Raman
and Grossmann, 1993, 1994).
To develop a GDP formulation, it is necessary to identify the conditional constraints
from among those that must hold for all synthesis alternatives. The conditional
constraints are represented with disjunctions and assigned a Boolean variable
that represents its existence. Although disjunctions and logic propositions
are useful in modeling design alternatives especially in chemical engineering
applications, they cannot be included in conventional mathematical programming
models (such as MixedInteger Programs (MIP)) without reformulations into logical
constraints in algebraic equality of inequality forms. This is one of the reasons
that calls for the adoption of GDP as it is able to handle disjunctions and
logic propositions directly in which, design alternatives in terms of design
and structural specifications can be formulated in the more intuitive representation
of logic propositions while constraints with discrete variables are represented
through disjunctions. Furthermore, the solution strategy of convex hull reformulation
of GDP into mixedinteger programs avoids the use of bigM logical constraints
which present weak relaxation, thus yielding a tighter linear programming relaxation
(Turkay and Grossmann, 1996). The GDP formulation is
summarized as follows:
COMPUTATIONAL EXPERIENCE
Computational experiments and numerical studies on the proposed modeling approach within the mixedinteger optimization framework of MILP and GDP formulations for the flowsheet superstructure optimization problem considered in this study are coded and implemented using GAMS 22.8 modeling language platform. Two design scenarios as distinguished by the API gravity of the crude oil charge to the ADU are considered in the computational experiments conducted, namely for light crude oil mixture as characterized by API > 33° and heavy crude oil mixture as characterized by API = 33°. Both scenarios are developed based on conventional distillation column design and refinery economics, in which products from ADU are either: (1) separated into light straight run naphtha (LSRN1) and heavy straight run naphtha (HSRN1), or (2) separated in the form of an undifferentiated class of naphtha (NAP1) typically termed as wild naphtha in the industry.
DISCUSSION OF COMPUTATIONAL RESULTS
The optimal refinery topology or configuration generated by the MILP and the
GDP are, as expected, identical for both design scenarios. As depicted in Fig.
2 and 3, the optimal solutions of both models select an
identical processing route comprising the same process units and material streams
and their interconnections but at different processing levels. It is noteworthy
that the optimal topology generated by each formulation agrees reasonably well
with the topology of realworld existing refineries reported in standard references
on the refining industry (Hydrocarbon Processing, 2008;
Gary et al., 2007; Meyers, 2003;
Maples, 2000). A closer inspection of Fig.
2 and 3 reveals that it is economically optimal to build
a reformer and a hydrotreater within a single site based on the particular economics
that is investigated in the numerical example.
The optimal objective function value of total investment cost which accounts
for a summation of the fixed capital costs, variable operating costs and the
raw material costs of purchasing the required crude oil slate, has been validated
by comparing the values tabulated in Table 1 against industrial
data available in the open literature.
Table 1: 
Optimal objective function values 


Fig. 2: 
Optimal refinery topology of the naphtha processing subsystem
for light crude oil charge (API > 33°) 

Fig. 3: 
Optimal refinery topology of the naphtha processing subsystem
for heavy crude oil charge (API = 33°) 
For instance, the annualized total investment cost of the Bharat Petroleum
Corporation Limited Refinery in Mahul, Bombay, India was estimated to be approximately
RM2700 million which by inspection, is of relatively reasonable accuracy with
our results (Hydrocarbon Processing, 2006). It is noteworthy
that in this respect, the GDP model offers a more costeffective solution by
registering a lower total investment cost.
Table 2: 
Model sizes and computational statistics 

The associated model sizes and computational statistics are reported in Table
2. As observed from the table, a GDP formulation typically offers a smaller
model size relative to its MILP counterpart and hence, is likely to be more
amenable for implementing extensions to this study that encompasses design features
required for a more complex refinery.
In the final analysis, the strength of this study includes improved computational performance through incorporating engineering insights in process synthesis problems by using logical constraints on certain design and structural specifications. This is accomplished within a conventional MILP framework that presents the advantage of considering the effects of all relevant constraints simultaneously, thus affording a global perspective to the model. However, since the typical algorithm for MILP requires solving an LP subproblem at each node of the search tree, all the constraints must be linear equalities or inequalities. This imposes a restriction on the expressiveness of MILP as a modeling language as some problems may require a very large number of variables and constraints. Hence, this gives rise to our attempt to adopt the alternative modeling framework of GDP.
CONCLUSION
This study attempts to extend the existing optimization modeling strategies
of integrating qualitativebased information in synthesis problems by using
logical constraints. The novelty of the proposed approach lies in the application
of logical constraints that enforce certain design specifications and structural
specifications on the interconnectivity of the process units and materials streams
in determining an optimal refinery topology. These logical constraints have
been proven to be logic cuts that are algebraic linear inequalities or linear
equalities that serve to reduce the computational time of an MILP or GDP by
providing information that increases the efficiency of the enumeration procedure
employed in the algorithms of the associated model solvers. In addition, this
study also provides some insights on a generalized form of the logical constraints
for synthesis and design problems and on how to identify possibly inconsistent
integer constraints derived from logic propositions. On the overall, the proposed
modeling approach offers a potential for assessing an optimal oil refinery topology
via a discrete optimization approach.
ACKNOWLEDGMENT
The main author gratefully thanks Ali Elkamel for conceiving the initial ideas that have led to an extended study as in the present form and Aida Azwana Sabidi, the undergraduate research assistant in this study.