Multi objective optimization in chemical engineering, developments and applications (2013)

GADE PANDU RANGAIAHDepartment of Chemical and Biomolecular Engineering, National University of Singapore, Singapore Department of Chemical Engineering, Instituto Tecnol´ogico de Aguascal

in Chemical Engineering

GADE PANDU RANGAIAH

Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore

Department of Chemical Engineering,

Instituto Tecnol´ogico de Aguascalientes, Mexico

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Library of Congress Cataloging-in-Publication Data

Multi-objective optimization in chemical engineering : developments and applications / [edited by] Gade Rangaiah, Adri´an Bonilla-Petriciolet.

pages cm

ISBN 978-1-118-34166-7 (hardback)

1 Chemical processes 2 Mathematical optimization 3 Chemical engineering I Rangaiah, Gade Pandu.

II Bonilla-Petriciolet, Adri´an.

Adri´an Bonilla-Petriciolet and Gade Pandu Rangaiah

Shivom Sharma and Gade Pandu Rangaiah

3.4 MOO Applications in the Food Industry, Biotechnology and

4 Performance Comparison of Jumping Gene Adaptations of the Elitist

Shivom Sharma, Seyed Reza Nabavi and Gade Pandu Rangaiah

5 Improved Constraint Handling Technique for Multi-Objective

Shivom Sharma and Gade Pandu Rangaiah

Contents vii

Weiwei Hu, Adeel Butt, Ali Almansoori, Shapour Azarm and Ali Elkamel

Kishalay Mitra

8 Fuzzy Multi-Objective Optimization for Metabolic Reaction Networks

Feng-Sheng Wang and Wu-Hsiung Wu

9 Parameter Estimation in Phase Equilibria Calculations

Sameer Punnapala, Francisco M Vargas and Ali Elkamel

10 Phase Equilibrium Data Reconciliation Using Multi-Objective

Adri´an Bonilla-Petriciolet, Shivom Sharma and Gade Pandu Rangaiah

Contents ix

Mohmmad A Al-Mayyahi, Andrew F.A Hoadley and Gade Pandu Rangaiah

12.3.5 Implementation 341

15 Multi-Objective Optimization of a Hybrid Steam Stripper-Membrane

Krishna Gudena, Gade Pandu Rangaiah and S Lakshminarayanan

16 Process Design for Economic, Environmental and Safety Objectives with

Shivom Sharma, Zi Chao Lim and Gade Pandu Rangaiah

16.5 Optimization using EMOO Program 462

17 New PI Controller Tuning Methods Using Multi-Objective Optimization 479

Allan Vandervoort, Jules Thibault and Yash Gupta

Shapour Azarm, University of Maryland, College Park, USA

Catherine Azzaro-Pantel, Universit´e de Toulouse, Laboratoire de G´enie Chimique, France Adri´an Bonilla-Petriciolet, Department of Chemical Engineering, Instituto Tecnol´ogico

de Aguascalientes, Mexico

Adeel Butt, Department of Chemical Engineering, The Petroleum Institute, Abu Dhabi,

UAE

Ali Elkamel, Department of Chemical Engineering, The Petroleum Institute, Abu Dhabi,

UAE and Department of Chemical Engineering, University of Waterloo, Canada

Krishna Gudena, Department of Chemical and Biomolecular Engineering, National

University of Singapore, Singapore

Yash Gupta, Department of Chemical and Biological Engineering, University of Ottawa,

Canada

Andrew F.A Hoadley, Department of Chemical Engineering, Monash University,

Australia

Weiwei Hu, University of Maryland, College Park, USA

S Lakshminarayanan, Department of Chemical and Biomolecular Engineering, National

University of Singapore, Singapore

Zi Chao Lim, Department of Chemical and Biomolecular Engineering, National University

of Singapore, Singapore

Kishalay Mitra, Department of Chemical Engineering, Indian Institute of Technology,

Hyderabad, India

Seyed Reza Nabavi, Faculty of Chemistry, University of Mazandaran, Iran

Adama Ouattara, Universit´e de Toulouse, Laboratoire de G´enie Chimique, France Karthik Raja Periasamy, Department of Chemical and Biomolecular Engineering,

National University of Singapore, Singapore

Luc Pibouleau, Universit´e de Toulouse, Laboratoire de G´enie Chimique, France

Sameer Punnapala, Department of Chemical Engineering, The Petroleum Institute,

Abu Dhabi, UAE

Gade Pandu Rangaiah, Department of Chemical and Biomolecular Engineering, National

University of Singapore, Singapore

Shivom Sharma, Department of Chemical and Biomolecular Engineering, National

University of Singapore, Singapore

Abhijit Tarafder, Department of Chemistry, University of Tennessee, USA

Jules Thibault, Department of Chemical and Biological Engineering, University of Ottawa,

Haibo Zhang, Department of Chemical and Biomolecular Engineering, National

Univer-sity of Singapore, Singapore

The optimization approach is well established in both academia and in industrial practicewith numerous applications in chemical engineering Several tools are readily availablefor process optimization However, optimization applications often have more than oneobjective, which requires Multi-Objective Optimization (MOO) Since the early 2000s,MOO has grown significantly as an effective and useful approach, especially for processoptimization in chemical engineering In particular, current technologies and requirements

in the petrochemical, chemical, biotechnology, energy and other emerging industries haveimposed new challenges to the field of MOO These challenges are due to the necessity ofsolving complex design-optimization problems that involve several objectives, many deci-sion variables and constraints To date, there have been many theoretical and computationaldevelopments in MOO and its applications for solving these complex problems of modernindustry Yet, in spite of many advances and applications of MOO, there is only one bookspecifically devoted to MOO techniques and their applications in chemical engineering.This earlier book, edited by Rangaiah and published in 2009, describes selected MOOtechniques and a number of application problems

The present book on MOO covers the most recent developments in and novel applications

of MOO, for modeling and solving a variety of challenging case studies in different areas

of chemical engineering In particular, this book covers new MOO methods and ideas thathave not been introduced in earlier MOO books It is a collection of contributions fromthe leading chemical engineering researchers on MOO and its applications Every chapter

in this book has been reviewed anonymously by at least two experts, and then thoroughlyrevised by the respective contributors The review process for chapters co-authored by each

of the editors has been entirely handled by the other editor Through this rigorous review,every attempt has been made to maintain the high-quality and educational value of thecontributions

This book is organized into three parts Part I (Chapters 1–3) provides the introduction,one important application of MOO, and an overview of chemical engineering applications ofMOO since the year 2007 New algorithm developments and state-of-the-art techniques usedfor solving MOO problems are presented in Part II (Chapters 4–8) Finally, Chapters 9–17,

in Part III, deal with various MOO application studies from thermodynamics, ical, environmental, biofuels and other chemical engineering areas These illustrate theapplicability and advantages of MOO in process systems engineering within chemicalengineering A number of chapters have exercises at the end, and the material in somechapters is complemented by relevant and useful programs/files available on the book’sweb site (http://booksupport.wiley.com; enter the book’s title, editor names or ISBN toaccess this)

petrochem-Multi-Objective Optimization in Chemical Engineering will be useful for researchers,

practitioners and postgraduate students interested in the area of MOO Chapters can bereadily adopted as part of advanced courses on optimization for senior undergraduate andpostgraduate students They will also allow the readers to adapt and apply available tech-niques to their processes or specific problems In general, readers can choose the chapters

of interest and read them independently

We are grateful to all the contributors and the reviewers of the chapters for their eration in meeting the requirements and schedule to finalize the book In particular, wethank Prof S.K Gupta, Prof J Thibault and Prof A.F.A Hoadley for their timely help

coop-in reviewcoop-ing some chapters authored by the editors Special thanks are due to ShivomSharma and Gudena Krishna, who assisted us in preparing and submitting the final files tothe publisher Finally, we would like to thank Ms Sarah Tilley, Ms Emma Strickland and

Ms Rebecca Stubbs of John Wiley & Sons, Ltd, for their cooperation and promptness inproducing this book

Research in MOO will continue to be an active area in chemical engineering, and wehope that this book will contribute to further developments in this topic

Gade Pandu Rangaiah

National University of Singapore, Singapore

Adri´an Bonilla-Petriciolet

Instituto Tecnol´ogico de Aguascalientes, M´exico

October 2012

Part I

Overview

Multi-Objective Optimization in Chemical Engineering: Developments and Applications, First Edition.

Edited by Gade Pandu Rangaiah and Adri´an Bonilla-Petriciolet.

© 2013 John Wiley & Sons, Ltd Published 2013 by John Wiley & Sons, Ltd.

Introduction

Adri´an Bonilla-Petriciolet 1 and Gade Pandu Rangaiah 2

Aguascalientes, Mexico

National University of Singapore, Singapore

1.1 Optimization and Chemical Engineering

Optimization is important for process modeling, synthesis, design, operation and retrofitting

of chemical, petrochemical, pharmaceutical, energy and related processes Usually, ical engineers need to optimize the design and operating conditions of industrial processsystems to improve their performance, costs, profitability, safety and reliability Processsystem optimization is challenging because chemical engineering application problems areoften complex, nonlinear and large, have both equality and inequality constraints and/orinvolve both continuous and discrete decision variables The mathematical relationshipsamong the objective to be optimized (also known as the performance criterion), constraintsand decision variables establish the difficulty and complexity of the optimization problem,

chem-as well chem-as the optimization method that should be used for its solution In particular, the type

of search space (i.e., continuous or discrete), the properties of the objective function (e.g.,convex or non-convex, differentiable or nondifferentiable), and the presence and nature ofconstraints (e.g., equality or inequality, linear or nonlinear) are the principal characteristics

to classify an optimization problem (Biegler and Grossmann, 2004)

The classes of optimization problems commonly found in engineering applicationsinclude linear programming, quadratic programming, nonlinear programming, combinato-rial optimization, dynamic optimization, mixed integer linear/nonlinear programming, opti-mization under uncertainty, bi-level optimization, global optimization and multi-objective

Multi-Objective Optimization in Chemical Engineering: Developments and Applications, First Edition.

Edited by Gade Pandu Rangaiah and Adri´an Bonilla-Petriciolet.

© 2013 John Wiley & Sons, Ltd Published 2013 by John Wiley & Sons, Ltd.

4 Multi-Objective Optimization in Chemical Engineering

optimization (Floudas, 2000; Diwekar, 2003; Biegler and Grossmann, 2004; Floudas et al.,

2005) These types of optimization problems are found in almost all application areassuch as modeling, synthesis, design, operation and control of chemical and related pro-cesses, and a wide variety of numerical methods have been used to solve them (e.g., Luus,

2000; Edgar et al., 2001; Tawarmalani and Sahinidis, 2002; Diwekar, 2003; Biegler and Grossmann, 2004; Grossmann and Biegler, 2004; Floudas et al., 2005; Ravindran et al.,

2006; Rangaiah, 2009 and 2010)

Application problems may have multiple optima, and it may be essential to find the globaloptimum or the best solution Depending on their convergence properties, optimizationmethods can be classified as local or global They may also be classified as deterministic orstochastic methods depending on whether their search is deterministic (often using gradient

of the objective function and other properties of the problem) or stochastic (employingrandom numbers) Local methods are computationally efficient and suitable for finding alocal optimum These search strategies have been exploited commercially as can be seenfrom their implementation in common software and process simulators such as Solver tool

in Excel, optimization tool-box in Matlab, GAMS, Aspen Plus and Hysys Current progress

in computational capabilities has prompted an increasing and considerable attention on theincorporation of global optimization methods in commercial software For example, anevolutionary search engine is now available in the Solver tool Global methods are morelikely to find the global optimum

To date, research contributions in optimization for chemical engineering have focusedprimarily on theoretical and algorithmic advances including the development of reliableand efficient strategies and their application for solving challenging and important chemicalengineering problems The majority of these contributions deal with optimization problemshaving only one objective function In general, optimization problems in chemical engineer-ing and in other disciplines involve more than one objective function related to performance,economics, safety and reliability, which have to be optimized simultaneously since theseobjective functions may be fully or partially conflicting over the range of interest Exam-ples of conflicting objectives are: capital investment versus operating cost; cost versussafety; quality versus recovery/cost; and environmental impact versus profitability Multi-objective optimization (MOO), also known as multi-criteria optimization, is necessary tofind the optimal solution(s) in the presence of tradeoffs among two or more conflictingobjectives

Multi-objective optimization has therefore been studied and applied to solve a variety

of challenging and important problems in chemical engineering (Bhaskar et al., 2000;

Rangaiah, 2009; Chapter 3 in this book) In a perspective paper on issues and trends in the

teaching of process and product design, Biegler et al (2010) noted that an important goal

in process design is optimization for multiple objectives such as profit, energy tion and environmental impact In another perspective paper on sustainability in chemicalengineering education, identifying a core body of knowledge, Allen and Shonnard (2012)have included process optimization as one of the computer-aided tools for environmentally-conscious design of chemical processes; within process optimization, they have listed multi-objective, mixed integer and nonlinear optimization Both these perspectives from eminentresearchers attest the growing importance and need for MOO in chemical engineering.Even though research in the application of MOO in engineering has grown significantly,there is only one book specifically devoted to MOO techniques and their applications

consump-in chemical engconsump-ineerconsump-ing (Rangaiah, 2009); it describes selected MOO techniques anddiscusses many applications MOO and its applications are growing with new developmentsand interesting applications being reported continually The present book covers the mostrecent developments in MOO methods and novel applications of MOO for modeling, designand operation of chemical, petrochemical, pharmaceutical, energy and related processes Inshort, the present book complements the previous book on MOO in chemical engineering.The remainder of this chapter is organized as follows Section 1.2 provides the basicconcepts and definitions used in MOO Section 1.3 discusses MOO briefly in the context

of chemical engineering Finally, section 1.4 presents an overview of all the chapters

in this book

1.2 Basic Definitions and Concepts of Multi-Objective Optimization

In this section, basic definitions and key concepts in MOO are introduced briefly The reader

is referred to earlier publications (e.g., Deb, 2001; Coello Coello et al., 2002; Rangaiah,

2009) for more details on these topics Formally, MOO refers to simultaneous optimization(i.e., maximization and/or minimization) of two or more objective functions, which areoften in conflict with one another This optimization problem can be stated as follows:

Optimize f1(x), f2(x), , f n (x) (1.1)subject to

x l < x < x u

(1.2)

where n is the number of objective functions to be simultaneously optimized, x is the vector

The feasible space, F is the set of vectors x that satisfy all the constraints and bounds in

Equation 1.2

In MOO, we are interested in determining the set of values of x that yields the best

compromise solutions for all the specified objective functions A single solution that taneously optimizes conflicting objectives is not feasible Instead, a set of solutions is foundwith the following characteristic: improvement of any one of the objectives is not possiblewithout worsening one or more of other objectives in the optimization problem These opti-mal solutions are referred to as the Pareto-optimal solutions (named after Italian economist,Vilfredo Pareto) They provide quantitative tradeoffs among the objectives involved

improve some objective function without causing a simultaneous deterioration in at leastone other objective function The Pareto-optimal solutions are also called non-dominatedsolutions In this context, the concept of domination implies that, given two solutions S1and S3, S1 dominates S3 if S1 is at least as good as S3 in all objectives and better in atleast one (see Figure 1.1(a)) If neither of the solutions dominates the other, then both arenon-dominated to each other (e.g., S1 and S2 in Figure 1.1(a)) The determination of the

6 Multi-Objective Optimization in Chemical Engineering

con-Pareto-optimal front (i.e., the set of non-dominated solutions) is the main goal in MOO

A process engineer can establish and understand tradeoffs and process performance usingthe MOO results The selection of a solution from the Pareto-optimal front depends onthe decision maker’s preferences, knowledge about the studied problem and also optimalvalues of decision variables Therefore, the decision maker, based on his/her expertise andintuition, needs to choose the most appropriate solution for implementation or particularregions of the tradeoff surface for further exploration

In general, a good Pareto-optimal front should show two desirable characteristics: thenon-dominated solutions are distributed evenly, and they cover a wide range of values ofobjectives under study However, finding such a Pareto-optimal front can be very difficultespecially for large problems with non-continuous and non-convex search spaces In MOO,the concept of a local minimum is replaced by a local Pareto-optimal front, whose presencemay cause problems in the convergence of MOO methods to the global Pareto-optimalfront

The Pareto-optimal fronts can be concave, convex or may consist of both concave andconvex sections including discontinuities Figure 1.1 illustrates these for the case of abi-objective optimization problem Better non-dominated solutions are obtained by MOOmethods for problems having convex Pareto fronts than for those having concave Paretofronts The Pareto-optimal fronts with discontinuities are common in engineering problems,

and are more complex to analyze The problem dimension can affect the size and shape ofthe Pareto-optimal set, and consequently determines the performance of MOO methods.Further, the complexity of and difficulty of solving multi-objective problems as well as thedifficulty of analyzing their solutions are likely to increase with number of objectives.There are several types of algorithms used for solving MOO problems to find the Pareto-optimal solutions (Miettinen, 1999; Rangaiah, 2009) These include different types ofweighted methods (e.g., global criterion, weighted sum, weighted min-max, weightedproduct, exponential weighted), goal programming methods, the bounded objective function

2002; Marler and Arora, 2009) Methods to solve MOO problems can be classified indifferent ways, for example, depending on the decision-maker’s preference (i.e., methodswith a priori, posteriori and without articulation of preferences) or whether one or manynon-dominated solutions are obtained in one run

the decision maker, and find one non-dominated solution in one run By changing thepreference, one can find more non-dominated solutions but this requires more than one run.Many of these were proposed before 1990, and so can be considered as classical methods.They generally transform a MOO problem into a single-objective optimization problem,which can then be solved by a suitable deterministic or stochastic method Methods withposteriori or without articulation of preferences can find many non-dominated solutions

in one run These have been developed after 1990 and can be termed “modern methods.”Many of them use stochastic global optimization methods such as genetic algorithms,differential evolution and particle swarm optimization There are also interactive methods,which incorporate the decision-maker’s preference during the search for non-dominatedsolutions A comprehensive review of MOO methods can be found in Miettinen (1999),

Coello Coello et al (2002) and Marler and Arora (2009).

The available MOO methods have their own strengths and weaknesses for solvingapplication problems, and it is important to identify and understand them for two reasons:one is to choose and use the appropriate method for the application on hand and another

is for developing new and more robust MOO techniques In particular, the study anddevelopment of stochastic methods has been an active research area in MOO since theearly 1990s because these strategies can find multiple non-dominated solutions in a singlerun These methods do not require any assumptions on the objective functions and theirmathematical characteristics Stochastic MOO methods include adaptations of simulatedannealing, genetic algorithms, evolutionary approaches, tabu search, differential evolutionand particle swarm optimization for multiple objectives One stochastic MOO solver,namely, elitist nondominant sorting genetic algorithm (NSGA-II) has been used for solvingmany chemical engineering application problems (see Chapter 3) because of its readyavailability and effectiveness The convergence performance of classical MOO methodsdepends on the shape and continuity of the Pareto-optimal front Stochastic MOO methodsare less sensitive to the characteristics of the optimization problem (e.g., type of objectivefunctions, decision variables and constraints) and the Pareto-optimal front

The performance of MOO methods can be quantified using different metrics based oncomputational requirement (such as CPU time and number of function evaluations), thecloseness of the obtained non-dominated solutions to the true/exact Pareto-optimal front(known only for benchmark problems) and the spread of the non-dominated solutions found

8 Multi-Objective Optimization in Chemical Engineering

Table 1.1 Summary of relevant journal articles on MOO of chemical engineering

applications.

Period

Number ofjournalpapers Major application areas of MOO ReferenceBefore the

year 2000

≈ 30 Process design and control, chemical

reaction engineering, biochemicalengineering, waste treatment andpollution control, electrochemicalprocess

Bhaskar et al.

(2000)

From 2000 to

mid-2007

≈ 100 Process design and operation, petroleum

refining and petrochemicals,biotechnology and food technology,pharmaceuticals, polymerization

Masuduzzamanand Rangaiah(2009)From 2007 to

mid-2012

≈ 230 Process design and operation, petroleum

refining, petrochemicals,polymerization, power generation,pollution control, renewable energy,hydrogen production, fuel cells

Chapter 3 of thisbook

Analysis of MOO results has been mainly focused on the values of objective functions (i.e.,

in the objective function space shown in Figure 1.1) It is equally important to review andunderstand the trends of values of decision variables corresponding to the non-dominatedsolutions as one of these has to be selected and implemented to achieve the desired tradeoffsolution for the application under study

1.3 Multi-Objective Optimization in Chemical Engineering

In chemical engineering, the presence of several conflicting objectives to be optimizedsimultaneously is a common situation and, consequently, MOO applications have grownconsiderably since the late 1990s In fact, the importance of this optimization approach isreflected by a significant increase in the number of papers published in different journals—see Table 1.1 Recent chemical engineering applications of MOO are summarized inChapter 3 of this book This rapidly growing interest in the chemical engineering com-munity has prompted the development of new MOO methods, concepts and novel processapplications

Reported MOO of chemical engineering applications include scheduling, productionplanning and management of chemical processes, process design and simulation of unitoperations (e.g., crystallization and distillation), chemical reaction engineering, pollutionprevention and control, industrial waste management, water recycling and wastewater min-imization, supply chain with environmental considerations, biorefinery process design and

integration (Bhaskar et al., 2000; Masuduzzaman and Rangaih, 2009) In particular, novel

chemical engineering applications combine economic objectives with process performancemetrics (such as conversion and energy consumption) and also environmental objectives

obtained, for example, from life-cycle analysis These applications include new emergingareas such as the design of renewable energy systems and the distributed energy resources

planning (see Chapter 3) As stated by Garcia et al (2012), the inclusion of environmental

concerns as optimization targets for process design in chemical engineering and other fieldshas increased the application and uses of MOO tools

In summary, MOO is playing an important role in chemical engineering, and a variety

of MOO techniques can be used for chemical engineering applications There is no doubtthat the number and type of MOO of chemical engineering applications will increase inthe coming years In fact, many chemical engineering problems that consider only oneobjective can be reformulated as MOO problems to develop a more realistic approach totheir solution Thus, MOO can be used to quantify and understand the tradeoffs among theconflicting objectives in the optimization of a chemical process

1.4 Scope and Organization of the Book

This book is organized in three parts Part I consists of Chapters 1 to 3 and provide anoverview to MOO and its chemical engineering applications Chapters 4 to 8, in Part II, coverdevelopments in MOO; although these are contributed by chemical engineering researchers,they are applicable to and useful in other disciplines too The focus of Chapters 9 to 17,

in Part III, are on MOO applications in chemical engineering Chapters 2 to 17 are brieflysummarized in the following paragraphs

Chapter 2 addresses the optimization of pooling problems for two objectives using the

ε-constraint method, contributed by Zhang and Rangaiah It describes pooling problems,

two objectives Pooling problems are optimization problems of importance in petroleumrefineries They are likely to have multiple minima, and so a global optimization method isrequired to find the optimal solution The solution of pooling problems for single objectivehas been studied using many deterministic global optimization algorithms However, therehas been no attempt to solve the pooling problems for multiple objectives Hence, in this

along with a recent stochastic global optimization algorithm, namely, integrated differentialevolution (IDE) Further, a new formulation that does not involve equality constraints isdescribed and used Many pooling problems from the literature are optimized for twoobjectives, and the results demonstrate the potential of MOO for finding tradeoff solutionsfor pooling problems In short, this chapter illustrates the application of a popular classical

Multi-objective optimization has found numerous applications in chemical engineering,

particularly since the late 1990s Earlier, Bhaskar et al (2000) have reviewed applications

of MOO in chemical engineering Masuduzzaman and Rangaiah (2009) have reviewedreported applications of MOO in chemical engineering from the year 2000 until middle

of 2007 In Chapter 3, Sharma and Rangaiah summarize about 230 articles on MOO inchemical engineering and related areas, published from the year 2007 until June 2012, undersix groups: (1) process design and operation, (2) petroleum refining, petrochemicals andpolymerization, (3) food industry, biotechnology and pharmaceuticals, (4) power generationand carbon dioxide emissions, (5) renewable energy, and (6) hydrogen production and fuel

10 Multi-Objective Optimization in Chemical Engineering

cells The first group and the last three groups have seen significant increase in the number

of papers published since 2007

Part II on MOO developments begins with Chapter 4, where Sharma, Nabavi and iah analyze the performance of jumping gene adaptations of elitist non-dominated sortinggenetic algorithm (NSGA-II), which has been used to optimize many process design andoperation problems for two or more objectives In order to improve the performance

Ranga-of this algorithm, jumping gene concept from natural genetics has been incorporated inNSGA-II Several jumping-gene adaptations have been proposed and used to solve math-ematical and application problems in different studies In Chapter 4, four jumping-geneadaptations are selected and comprehensively evaluated on a number of bi-objective uncon-strained and constrained test functions Three quality metrics, namely, generational distance,spread and inverse generational distance are employed to evaluate the distribution and con-vergence of the obtained Pareto-optimal solutions at selected intermediate generations andthe final generation Additionally, a search termination criterion based on the improvement

in the Pareto-optimal front, has been described and used to check convergence of NGSA-IIwith the selected jumping-gene adaptations

In Chapter 5, Sharma and Rangaiah discuss an improved constraint handling techniquefor MOO and its application to two fermentation processes Constraints besides bounds areoften present in MOO problems in chemical engineering; these arise from mass and energybalances, equipment limitations, and operation requirements Penalty function and feasi-bility approaches are the popular constraint handling techniques for solving constrainedMOO problems by stochastic global optimization (SGO) techniques, such as genetic algo-rithms and differential evolution This chapter briefly reviews selected applications of theseconstraint-handling approaches in chemical engineering In the penalty-function approach,solutions are penalized based on constraint violations; its performance depends on thepenalty factor, which necessitates selection of a suitable value for the penalty factor fordifferent problems Generally, the feasibility approach is good for solving problems withinequality constraints due to their large feasible regions It gives higher priority to a feasiblesolution over an infeasible solution, but this limits the diversity of the search Feasible searchspace is extremely small for equality-constrained problems and so the feasibility approachmay not be effective for handling equality constraints The approach of adaptive relaxation

of constraints in conjunction with feasibility approach, addresses this issue by relaxingfeasible search space dynamically This approach has been found to be better and effectivefor solving SOO problems with equality and inequality constraints by SGO techniques InChapter 5, a modified adaptive relaxation with feasibility approach is explored for solvingconstrained MOO problems by stochastic optimizers, and its performance is compared withthat of feasibility approach alone For this, the modified adaptive relaxation with feasibilityapproach is incorporated in the multi-objective differential evolution (MODE) algorithmand tested on two benchmark functions with equality constraints Finally, MODE with theproposed constraint handling approach is applied to optimize two fermentation processesfor multiple objectives

A robust multi-objective genetic algorithm (RMOGA) with online approximation underinterval uncertainty is the subject of Chapter 6 by Hu, Butt, Almansoori, Azarm andElkamel Optimization of chemical processes is usually multi-objective, constrained andhas uncertainty in the process inputs, variables and/or parameters This uncertaintycan produce undesirable variations in the objective and/or constraints The traditional

multi-objective genetic algorithm (MOGA) assumes that all inputs are deterministic ever, optimal solutions obtained by it can be sensitive to input uncertainty and degrade thesolutions The goal in RMOGA is to obtain solutions that are optimum while also beingrelatively insensitive to uncertainty For this, one nested approach and another sequentialapproach are presented in Chapter 6 In both of them, a measure of robustness is consid-ered using a worst-case analysis, which assumes that the uncertainty in inputs is expressed

How-by an interval with known lower and upper bounds In the nested approach, an upperlevel problem identifies and improves candidate solutions, while a lower level subproblemevaluates their robustness In the sequential approach, the MOO problem is first solved toobtain optimal solutions, and then the robustness of each optimal solution is evaluated Bothnested and sequential RMOGA can be computationally costly To ease the computationalcost, an online approximation-assisted method is used in both approaches The purpose ofthe approximation is to replace the computationally intensive evaluation of objectives andconstraints with a surrogate model (which is computationally much less intensive) whileadaptively improving the accuracy of the approximation as the search progresses Onenumerical example and a petroleum refinery example are used to demonstrate and comparethe applicability of the two RMOGA approaches

Another technique to handle uncertainty in nonlinear process models is presented inChapter 7 by Mitra Among the various preventive uncertainty handling techniques, thechance-constrained programming (CCP) has gained considerable interest in recent timesdue to certain advantages of its usage over its competitors The CCP is different fromdeterministic optimization since the former has a stochastic component attached to it Thecomplexity involved in propagating the uncertainties in stochastic parameters to the corre-sponding constraints and objective functions of the deterministic equivalent optimizationformulation is one of the key challenges in CCP In Chapter 7, various facets of CCP hasbeen presented and explained through examples of different types Problem formulationusing CCP under different scenarios has been discussed and demonstrated with examplesfrom the literature and the real world It has been also shown how stochastic componentpresent in the CCP formulation leads to solution reliability which has an inverse relationshipwith solution quality

Chapter 8, the last in Part II, is on fuzzy MOO for metabolic reaction networks bymixed-integer hybrid differential evolution (MIHDE) by Wang and Wu In the optimiza-tion of metabolic reaction networks, designers have to manage the nature of uncertaintyresulting from qualitative characters of metabolic reactions, for example the possibil-ity of enzyme effects A deterministic approach does not give adequate representation

of metabolic reaction networks with uncertain characters Fuzzy optimization lations can be applied to cope with this problem Chapter 8 introduces a generalizedfuzzy MOO problem (GFMOOP) for finding the optimal engineering interventions onmetabolic network systems considering the resilience phenomenon and cell viability con-straints This approach first formulates a constrained MOO problem that considers theresilience effects and minimum set of manipulated enzymes simultaneously by combin-ing the concepts of minimization of metabolic adjustment (MOMA) and regulatory on/offminimization (ROOM) In addition, the nonlinear kinetic equations were included in theoptimization formulation, and so it was formulated as a constrained mixed-integer nonlinearprogramming (MINLP) problem Mixed-integer hybrid differential evolution (MIHDE) wasextended to solve constrained MINLP problems through the implementation of constraint

formu-12 Multi-Objective Optimization in Chemical Engineering

handling techniques The fuzzy goal attainment approach implemented in MIHDE wasused to solve GFMOOPs for the identification of optimal genetic manipulation strategies

on metabolic reaction networks, and its effectiveness is discussed in Chapter 8

Chemical engineering applications of MOO, in Part III, begin with Chapter 9 by pala, Vargas and Elkamel, on parameter estimation in phase-equilibrium modeling Phase-equilibrium calculations play a vital role in the design, development, operation, optimizationand control of chemical processes Equations of state or activity coefficient models are nor-mally tuned to match certain properties in order to give an accurate description of thephase behavior This chapter introduces the application of MOO for parameter estimationwherein a model is simultaneously fit to two or more conflicting properties As an example,the parameters of NRTL activity coefficient model are estimated by fitting the parame-ters to vapor-liquid equilibrium data and heat of mixing (excess enthalpy) Particle swarmoptimization is used for this MOO

Punna-Chapter 10 by Bonilla-Petriciolet, Sharma and Rangaiah considers another application ofMOO to phase equilibrium data modeling In this chapter, MOO is applied for simultaneousparameter estimation and data reconciliation of vapor-liquid equilibrium using the error-in-variable formulation and activity coefficient models Multi-objective differential evolutionwith a tabu list is used for obtaining the Pareto-optimal front of data reconciliation problemswith three and four objectives The application of some criteria of interest in thermodynamicmodeling is illustrated to characterize the solutions obtained from the Pareto-optimal fronts

of reconciled phase equilibrium data The results show that MOO is an alternative andreliable approach for performing data reconciliation in phase equilibrium modeling.Al-Mayyahi, Hoadley and Rangaiah describe multi-objective process synthesis withembedded energy integration in Chapter 11 Energy integration decreases energy costs ofindustrial processes by increasing heat recovery and reducing utilities consumption Severalpotential opportunities for improving the energy efficiency and, consequently, reducing

implementing heat integration within a single process unit or among different refining

objectives has not been widely covered In Chapter 11, MOO has been implemented for

an integrated model of a crude distillation (CDU) and fluidized catalytic cracking (FCC)

and economic objectives The CDU includes the atmospheric distillation unit (ADU), thevacuum distillation unit (VDU) and the crude preheat train, whilst the FCC model includesthe reactor/regenerator section, the feed preheat train, the main fractionator and flue gasheat and power recovery sections Pinch analysis is used to maximize the heat recoverywithin the integrated model and evaluate the distribution of utilities related to emissions.The Pareto-optimal results including optimal operating conditions are presented and theirsignificant features are discussed

In Chapter 12, Azzaro-Pantel and Pibouleau describe ecodesign of chemical processeswith MOGAs Process synthesis is a complex activity involving many decision makersand multiple levels of decision steps From these many alternatives, the designers want

to select the one that best suits both economic and environmental criteria This chaptershows that MOO and multiple choice decision making (MCDM) techniques can be usefulfor the ecodesign of a process Two examples illustrate the determination of eco-friendlyand cost-effective designs: the so-called Williams and Otto process and the well-known

benchmark process for hydrodealkylation (HDA) of toluene to produce benzene Thischapter deals with the definition of various objectives for designing eco-efficient processes,

by considering simultaneously ecological and economic features An improved variant ofNSGA-II is implemented for solving the resulting MOO problems The environmentalburdens are evaluated by means of a decision support tool dedicated to the management

of plant utilities and to the emission control of pollutants After finding the Pareto-optimalsolutions, a MCDM technique is used to discover the most interesting tradeoff designalternatives

Tarafder presents modeling and MOO of a chromatographic system in Chapter 13.Chromatography is a separation technique, which plays a crucial role in the downstream

of several pharmaceutical and fine chemical industries The chromatographic units in theseindustries handle gram to kilogram scale of very high-value products per day, and they may

be required to be redesigned after a period of time, depending on the changes in the productlines or other requirements The current industrial practice mostly relies on empiricalmethods to develop the operating conditions; but, given the high cost of products, there is ahuge incentive of applying model-based MOO studies in improving the performance of suchprocesses Chapter 13 describes the development of a model-based optimization program,and then demonstrates the ways of conducting optimization studies with this model Theexample chosen for this chapter is the separation of enantiomers in batch processes Inthe industries, there is a strong requirement for faster separation of enantiomers, but thathas to sacrifice the product recovery as the sample mixture may not get enough time toseparate entirely But, as recovery is a critically important parameter for process economics,the situation leads to an optimization problem having conflicting objectives Chapter 13provides the basic understanding of developing a mathematical model to simulate such

a system, formulate the objective functions, identify the constraints and the most usefuldecision variables, and finally, with the help of a genetic algorithm, determine the Pareto-optimal solutions

Estimation of crystal size distribution by image thresholding based on MOO is thesubject of Chapter 14 by Periasamy and Lakshminarayanan Crystallization process can

be effectively controlled by monitoring the crystal size distribution (CSD), which can

be estimated using particle vision and measurement (PVM) images Image segmentationbased on thresholding is critical in this regard Generally, the threshold is selected byoptimizing a single objective Based on the type of thresholding used, segmentation can

be improved Hence, in this work, optimum threshold is calculated by solving a MOOproblem The two objectives used are within-class variance and overall probability of error.This MOO problem is solved based on the plain aggregating approach and simulatedannealing by assigning appropriate weights to each objective function The MOO-basedthresholding overcomes the limitations and outperforms the thresholding performed byeither of the single objectives The segmented images are further processed by means offeature extraction to estimate the CSD The algorithm was tested on a set of artificiallygenerated crystallization images, and its accuracy was calculated by comparing the CSDestimated to the data used to generate the artificial images This accuracy was found to bearound 90% for images in which about 20–25 particles exist

In Chapter 15 by Gudena, Rangaiah and Lakshminarayanan, a hybrid steam-strippermembrane process for continuous bioethanol purification is optimized for multipleobjectives Several ethanol-water separation technologies for continuous recovery and

14 Multi-Objective Optimization in Chemical Engineering

purification of bioethanol from fermentation broth are discussed in the literature Recently,

a hybrid steam-stripping membrane-separation process is proposed and shown to reduceenergy consumption for separation by nearly half when compared to the conventional dis-tillation process This chapter discusses detailed modeling of the hybrid stripper-membrane

Important objectives, namely operating cost per unit of ethanol produced, ethanol purity(as there is no consensus on the limit for water in bioethanol as a fuel in different countries)and ethanol loss in the waste stream, are considered Pareto-optimal solutions obtained forthese conflicting objectives are presented and discussed

The design of the cumene process for economic, environmental and safety (EES) tives is described in Chapter 16 by Sharma, Lim and Rangaiah Safety is very important

objec-in the process objec-industry, but it has received much less attention than economic objectives

in process optimization Although it is difficult to quantify process safety at the nary design stage, several safety indices have been proposed to assess inherent safety ofchemical processes In this chapter, these safety indices are reviewed, and one of them ischosen for MOO of the cumene process design Integrated inherent safety index (I2SI),material loss from the cumene process and total capital cost, respectively, are used assafety, environmental and economic objectives for MOO Three bi-objective and one tri-objective optimization problems for the cumene process are solved using NSGA-II Thenon-dominated solutions obtained are presented and discussed These are useful for betterunderstanding of tradeoffs among the EES objectives and for selecting a suitable design ofcumene process

prelimi-In the last chapter of Part III and also the book (Chapter 17), Vandervoort, Thibault andGupta develop new proportional-integral (PI) controller tuning methods for processes rep-resented by a first-order plus dead time transfer function The developed methods involveapproximating the Pareto-optimal domain associated with the minimization of three perfor-mance criteria: the integral of the time-weighted absolute error, the integral of the squares

of the differences in the manipulated variable, and the settling time Two tuning ods were developed, achieving optimal controller performance by specifying either one

meth-of the controller input parameters or the desired values meth-of the performance criteria Thedeveloped controller tuning methods were compared to several previously developed con-troller correlations Finally, the developed tuning methods were applied to a fourth-orderprocess subjected to a set point change and a disturbance, and shown to provide excellentperformance

In summary, chemical engineers working in industry will find the introductory chapters

in Part I and the application chapters in Part III beneficial when using MOO in applicationsrelated to their jobs Methodological developments in MOO covered in part II will be ofparticular interest to researchers from diverse fields who are interested in MOO Chemi-cal engineering students, particularly those learning or pursuing research in optimizationincluding MOO, will find all chapters in this book useful in their studies Many chapters inthis book have exercises at the end, and some chapters provide useful programs / files onthe book web site Depending on their background and interest, readers can choose to readthe entire book, one or more parts, or particular chapters

Increasing importance and significance of MOO in chemical engineering studies and

practice can be seen from the article by Garica et al (2012) on teaching mathematical modeling software for MOO in chemical engineering courses, and the article by Lee et al.

(2008) on optimizing process plants for more than one objective Availability of

Excel-based MOO programs (e.g., Sharma et al., 2012) will further facilitate MOO of more

applications in chemical engineering We hope the book in your hand will help to increasethe use of MOO in both academia and industrial practice We also anticipate the availability

of selected modern methods of MOO for generating Pareto-optimal fronts for chemicalengineering applications, in commercial process simulators such as Aspen Plus and AspenHysys in the coming years

References

Alarcon-Rodriguez, A., Ault, G and Galloway, S., Multi-objective planning of distributedenergy resources: a review of the state-of-the-art, Renewable and Sustainable EnergyReviews, 14 (2010) 1353–1366

Allen, D.T and Shonnard, D.R., Sustainability in chemical engineering education: fying a core body of knowledge, AIChE Journal, 58 (2012) 2296–2302

identi-Bhaskar, V., Gupta, S.K and Ray, A.K., Applications of multiobjective optimization inchemical engineering, Reviews in Chemical Engineering, 16 (2000) 1–54

Biegler, L.T and Grossmann, I.E., Retrospective on optimization, Computer and ChemicalEngineering, 28 (2004) 1169–1192

Biegler, L.T., Grossmann, I.E and Westerberg, A.W., Issues and trends in the teaching ofprocess and product design, AIChE Journal, 56 (2010) 1120–1125

Coello Coello, C.A., Veldhuizen, D.A.V and Lamont, G.B., Evolutionary Algorithms forSolving Multi-objective Problems, Kluwer Academic, 2002

Deb, K., Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons,Ltd, 2001

Diwekar, U.M., Introduction to Applied Optimization, Kluwer Academic, 2003

Edgar, T.F., Himmelblau, D.M and Lasdon, L.S., Optimization of Chemical Processes,second edition, McGraw-Hill, 2001

Floudas, C.A., Deterministic Global Optimization: Theory, Methods and Applications,Kluwer Academic, 2000

Floudas, C.A., Akrotirianakis, I.G., Caratzoulas, S., Meyer, C.A and Kallrath, J., Global

Engineering, 29 (2005) 1185–1202

Garcia, N., Ruiz-Femenia, R and Caballero, J.A., Teaching mathematical modeling ware for multi-objective optimization in chemical engineering courses Education forChemical Engineers, 7 (2012) e56–e67

soft-Grossmann, I.E and Biegler, L.T., Part II Future perspective on optimization, Computersand Chemical Engineering, 28 (2004) 1193–1218

Lee, E.S.Q., Ang, A.Y.W and Rangaiah, G.P., Optimize your process plant for more thanone objective, Chemical Engineering, September (2008), 58–64

L´opez-Jaimes, A and Coello Coello, C.A., Multi-objective evolutionary algorithms: areview of the state-of-the-art and some of their applications in chemical engineering,

in G.P Rangaiah, Multi-objective Optimization Techniques and Applications in ChemicalEngineering, World Scientific, Singapore, 2009

Luus, R., Iterative Dynamic Programming, Chapman & Hall, 2000

16 Multi-Objective Optimization in Chemical Engineering

Marler, R.T and Arora, J.S., Multi-objective Optimization: Concepts and Methods forEngineering, VDM Verlag, 2009

Masuduzzaman and Rangaiah, G.P., Multi-objective optimization applications in chemicalengineering, in Multi-objective Optimization Techniques and Applications in ChemicalEngineering, World Scientific, Singapore, 2009

Miettinen, K.M., Non-linear multi-objective optimization, Kluwer Academic Publishers,Boston, 1999

Rangaiah, G.P (Ed.), Multi-objective Optimization: Techniques and Applications in ical Engineering, World Scientific, 2009

Chem-Rangaiah, G.P (Ed.), Stochastic Global Optimization: Techniques and Applications inChemical Engineering, World Scientific, 2010

Ravindran, A., Ragsdell, K.M and Reklaitis, G.V., Engineering Optimization: Methodsand Applications, Second Edition, John Wiley & Sons, Ltd, 2006

Sharma, S., Rangaiah, G.P and Cheah, K.S., Multi-objective optimization using MS Excelwith an application to design of a falling-film evaporator system, Food and BioproductsProcessing, 90 (2012) 123–134

Tawarmalani, M and Sahinidis, N.V., Convexification and Global Optimization in tinuous and Mixed-integer Nonlinear Programming: Theory, Algorithms, Software andApplications, Kluwer Academic, Dordrecht, 2002

Optimization of Pooling Problems

for Two Objectives Using

Haibo Zhang and Gade Pandu Rangaiah

Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore

2.1 Introduction

Pooling and blending problems are important optimization problems in petroleum

refiner-ies with huge cost savings potential (Baker and Lasdon, 1985; Rigby et al., 1995) In

petroleum refineries, final products are often made by mixing flows from pools or feedstreams with different sulfur content, octane number and/or density Pooling occurs whensource/feed streams are mixed in storage tanks (pools) before producing final products Thesource streams can be intermediate products from different distillation units, reformers andcatalytic crackers, and/or additives like ethanol, and therefore have different compositions

and properties (DeWitt et al., 1989) The resulting streams from pools are then dispatched

to different final products to meet the specified requirements The pools enhance the ational flexibility of the process by providing intermediate storage On the other hand,blending is direct mixing of source/feed streams into final products, and hence blendingproblems do not involve pools (i.e., storage tanks) Pooling problems are also encountered

oper-in waste-water treatment (Bagajewicz, 2000), supply-chaoper-in operations and communications(Misener and Floudas, 2009)

Pooling problems have been studied for a single objective, namely, cost minimization(i.e., profit maximization) by optimal allocation of source/feed streams to pools and then

Multi-Objective Optimization in Chemical Engineering: Developments and Applications, First Edition.

Edited by Gade Pandu Rangaiah and Adri´an Bonilla-Petriciolet.

© 2013 John Wiley & Sons, Ltd Published 2013 by John Wiley & Sons, Ltd.

18 Multi-Objective Optimization in Chemical Engineering

blending streams from pools to final products, subject to constraints arising from massbalances, quality balances, feed availability and product requirements Because of the non-linearities and nonconvexities in the optimization problem, a global optimization algorithm

is required for solving pooling problems

Many studies have focused on the solution of pooling problems, starting from Haverly

(1978), who used recursive linear programming Next, Lasdon et al (1979) applied

suc-cessive linear programming for solving pooling problem Later, Baker and Lasdon (1985)applied successive linear programming to improve blending schemes at Exxon Company.The solution of pooling problems led to annual savings of more than 30 million dollars in

the 1980s (DeWitt et al., 1989) Reviews of global optimization methods and their

appli-cations to pooling problems can be found in Floudas and Gounaris (2009) and Misenerand Floudas (2009) All these attempts to solve pooling problems have used deterministicalgorithms for single objectives only

In general, many real-world problems involve several conflicting objectives It is oftendifficult to formulate these problems into a single-objective optimization (SOO) problem.The conflicting objectives in multi-objective optimization (MOO) problems lead to a set ofoptimal solutions called Pareto-optimal solutions, which are equally good for the specifiedobjectives These solutions provide better understanding of the tradeoff among objectivesand many choices to the decision-maker for choosing one of them for implementation Overthe last two decades, MOO field has grown significantly and many chemical engineeringapplications of it have been reported (Cheah and Rangaiah, 2009; Jaimes and CoelloCoello, 2009; Masuduzzaman and Rangaiah, 2009; Chapter 3 in this book) One approach

to solving a MOO problem is to transform it into a single-objective problem that can be

problem is converted into a SOO problem for one primary objective and the other objectivesare treated as constraints

Khosla et al (2007) have studied fuel oil blending for two and three objectives such as

profit, quality give-away, production rate, use of light products and calorific value, using theelitist nondominant sorting genetic algorithm (NSGA-II) with and without jumping-geneadaptations However, until now, pooling problems have not been optimized for multipleobjectives Hence, in this chapter, pooling problems are optimized for two objectives usingthe ε-constraint method Another purpose of this chapter is to illustrate the ε-constraint

method The resulting SOO problem is solved using a recent stochastic global optimizationalgorithm, namely, integrated differential evolution (IDE) (Zhang and Rangaiah, 2012).Many benchmark pooling problems from the literature are optimized for two objectives:cost (or profit) and product quality, which conflict The results demonstrate the usefulness

Comparison of several deterministic and stochastic global optimization algorithms hasbeen discussed in the literature (Nocedal and Wright, 2006; Srinivas and Rangaiah, 2006;

Weise 2008; Mashinchi et al., 2011; Exler et al., 2008) In general, stochastic methods

are more robust, require little or no assumption on the characteristics of the optimizationproblem, and yet provide a high probabilistic convergence to the global optimum Further,they are usually simple in principle and easy to implement and use Hence, in this chapter,IDE is used for finding the global optimum of the SOO problem

In pooling problems, objective functions and/or constraints are nonlinear with respect todecision variables; they are also non-convex and have multiple optima These characteristics

and the presence of many equality and inequality constraints add to the difficulty of findingthe global optimum Mathematical formulation of pooling problems can mitigate thesedifficulties by changing nonlinear terms, size of the search space and/or number of equal-ity/inequality constraints In this chapter, we describe and use an improved formulation

named, r-formulation, for pooling problems.

The remainder of this chapter is organized as follows Description and formulation ofpooling problems is presented in section 2.2 by considering typical problems Section 2.3

of this method to pooling problems is described in section 2.4 The results are presentedand discussed in section 2.5 Finally, section 2.6 concludes this chapter with a summary ofthe findings

2.2 Pooling Problem Description and Formulations

2.2.1 p-Formulation

The bilinear programming formulation, known as the p-formulation, was formulated by

Haverly (1978), who also studied the solution of pooling problems using recursive linear

programming Nomenclature for the p-formulation of a pooling problem network is

pre-sented in Figure 2.1(a); for illustration, one of the benchmark pooling problems, Ben-Tal 4

(Ben-Tal et al., 1994) is shown in Figure 2.1(b).

are mixed in one or more pools (Figure 2.1) There can be an upper limit on the availability

again in the product tanks The final products from the pooling network should satisfy the

function is the difference between the cost of source/feed streams and the revenue fromselling the final products The constraints are mainly from mass balances about the pools,quality balances about the pools, raw material availability, product demands and quality

requirements Details of the p-formulation of pooling problems can be found in Misener

and Floudas (2009)

For example, BT-4 problem shown in Figure 2.1(b) involves four source streams, twopools and two products Three of the four source streams are mixed in one pool The secondpool has only one source stream, which is referred as bypass stream—it may go through

a pool or mixed directly to the products All streams going from this pool to the productswill have the same qualities as its single source stream, and so the presence or absence

of a pool does not affect the problem formulation and results except for introducing an

additional X variable Considering a pool for each bypass stream is helpful for the general

problem formulation Hence, a pool is considered for each bypass stream in this chapter (seeFigures 2.1b and 2.3) Data shown in Figure 2.1(b) follow the symbols in Figure 2.1(a) Forexample, (6, 3) on top of source stream 1 are the price and quality of the source stream 1,(9, 2.5) on top of product stream 1 are the price and quality requirement of the productstream, and (100) below the product stream 1 is the upper bound on the demand for theproduct stream 1 Note that (50) below the source stream 2 is the availability of this stream

20 Multi-Objective Optimization in Chemical Engineering

Source

Product k

λi,w = w th quality of ith source

A i U = Upper bound on i th source availability

X i,j = Flow rate of ith source to jth pool

(9, 2.5)

(X1,1)

(200) (15, 1.5)

(6, 3)

(15, 1)

(50) (16, 1)

Decision variables and their values at the global optimum for the Ben-Tal 4 problem are:

2.2.2 r-Formulation

The new formulation, named, the r-formulation, reduces the number of decision variables,

constraints and search space in the optimization problem In this formulation, new decision

In the above, K and P are, respectively, the number of products and the number of pools

reduces the search space, the number of decision variables and constraints

Here, N is the total number of source/feed streams Equation 2.5 takes into account the

decision variables and constraints

22 Multi-Objective Optimization in Chemical Engineering

The optimization problem using r-formulation is simplified to as follows Objective

2.2–2.3 respectively Decision variables and their bounds in the optimization problem are:

Thus, the optimization problem using r-formulation has only two sets of inequality

constraints (Equations 2.9 and 2.10), and the decision variables are normalized between 0and 1 (Equations 2.11 and 2.12) Depending on the interconnections in the pooling network,

further reduce the number of decision variables

The SOO problem for this pooling problem using r-formulation is as follows Decision

(9, 2.5)

(r1,1)

(200) (15, 1.5)

(6, 3)

(15, 1)

(50) (16, 1)

(10, 2)

Figure 2.2 Pooling network with variables for the r-formulation: Ben-Tal4 problem.

variables in this problem are r’s and R’s; and q, X and Y are given by the explicit equations

(2.2)–(2.7) So, the latter are dependent variables calculated using the values of decision

variables, r’s and R’s Explicit equations for calculating the dependent variables using

Values of decision variables at the global optimum of the Ben-Tal 4 problem are:

network (i.e., dependent variables) can be calculated easily using Equations 2.13a–2.13c

Another more complicated pooling problem, namely Ben-Tal 5 (Ben-Tal et al., 1994) shown in Figure 2.3, is considered for illustrating r-formulation for pooling problems This

problem involves five source streams, four pools, five product streams and two qualities;one source stream is a bypass stream shown with pool 4 The data (6, 3, 1) on top of sourcestream 1 are the price, quality 1 and quality 2 of the source stream 1, respectively, (18,2.5, 2) above the product stream 1 are price, quality 1 and quality 2 of product stream 1,respectively, and (100) below the product stream 1 is the upper limit on the demand forproduct stream 1 For clarity, only the decision variables at pools 1 and 4 are shown inFigure 2.3

As before, decision variables are r’s and R’s in r-formulation (see Table 2.1) Explicit equations for calculating the dependent variables: q, X and Y are given by generic

24 Multi-Objective Optimization in Chemical Engineering

Table 2.1 Decision variables, dependent variables and their optimal values for Ben-Tal 5 problem.

(with optimal values in brackets) (with optimal values in brackets)

(6, 3, 1) (r1,1 )

(r2,1) (r3,1)

(R1,1) (R1,2) (R1,3) (R1,4) (R1,5)

(R4,1)(R4,2)(R4,3) (R4,4) (R4,5)

(16, 1, 3)

(15, 1, 2.5)

(12, 1.5, 2.5)

(10, 2, 2.5) (50)

Equations 2.2–2.7, and are not shown here for conciseness but they can be found inthe Excel file on the accompanying web site for the book at http://booksupport.wiley.com.The objective function and constraints are:

of products

(2.14b)(1q11+ 3q21+ 2.5q31+ 2.5q41)Y11+ (1q11+ 3q21+ 2.5q31+ 2.5q41)Y21

of products

(2.14c)

X31+ X32+ X33≤ 50 Availability of the source stream 3 (2.14d)The first set of constraints (Equation 2.14b) is for quality 1 requirements on products,and the second set of constraints (Equation 2.14c) is for quality 2 requirements on products.The third constraint (Equation 2.14d) is the availability of the source stream 3 only The

variables and dependent variables at one global minimum are given in Table 2.1

Previous studies on pooling problems have all focused on SOO In this chapter, ing problems are solved for two objectives to find Pareto-optimal solutions that provideinsights into tradeoffs between objectives and many optimal choices for the decision-maker