Research Article Multiobjective Optimization of a Benfield ...amine solutions such as DEA, MDEA, and DGA in a gas sweetening process. Rahimpour and Kashkooli [ , ] developed a mathematical

Hindawi Publishing CorporationJournal of Industrial MathematicsVolume 2013 Article ID 260918 11 pageshttpdxdoiorg1011552013260918

Research ArticleMultiobjective Optimization of a Benfield HiPure GasSweetening Unit

Richard Ochieng12 Abdallah S Berrouk1 and Ali Elkamel3

1 Department of Chemical Engineering Petroleum Institute PO Box 2533 Abu Dhabi UAE2Abu Dhabi Gas Industries Ltd PO Box 66 Abu Dhabi UAE3Department of Chemical Engineering University of Waterloo Waterloo ON Canada N2L 3G1

Correspondence should be addressed to Abdallah S Berrouk aberroukpiacae

Received 1 June 2013 Accepted 5 September 2013

Academic Editor Jinwen Ma

Copyright copy 2013 Richard Ochieng et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We show how a multiobjective bare-bones particle swarm optimization can be used for a process parameter tuning andperformance enhancement of a natural gas sweetening unit This has been made through maximization of hydrocarbon recoveryand minimization of the total energy of the process as the two objectives of the optimization A trade-off exists between these twoobjectives as illustrated by the Pareto front This algorithm has been applied to a sweetening unit that uses the Benfield HiPureprocess Detailed models of the natural gas unit are developed in ProMax process simulator and integrated to the multi-objectiveoptimization developed in visual basic environment (VBA) In this study the solvent circulation rates stripper pressure and reboilerduties are considered as the decision variables while hydrogen sulfide and carbon dioxide concentrations in the sweetened gasare considered as process constraints The upper and lower bounds of the decision variables are obtained through a parametricsensitivity analysis of the models The Pareto sets show a significant improvement in hydrocarbon recovery and a decent reductionin the heat consumption of the process

1 Introduction

As global energy demand rises natural gas now plays animportant strategic role in world energy supply It is thecleanest and most hydrogen rich of all the hydrocarbonenergy sources and it has high energy efficiencies for powerenergy Natural gas resources exploited and discovered areplentiful however they contain complex contaminants suchas CO

2 H2S Mercaptans and other sulfur compounds

Excessive amounts of these contaminants in natural gasstreamswill lead to low gas heating value andor cause seriousenvironmental hazards to the consumers In LNG plantslarge amounts of carbon dioxide and sulfur compounds mayaffect the quality of LNGproducts or pose serious operationalproblems in the cryogenic columns [1 2] Therefore oneof the major purposes of the sweetening units is to purifyraw natural gas to meet both sale gas and liquefactionspecifications [3]

Unpredicted changes in reservoir conditions combinedwith the tough market competition have forced the natural

gas sweetening business to adopt sophisticated optimizationtechniques to improve the purity of their products increaseproduction capacity and minimize total energy requirementfor unitsrsquo operations

Gas sweetening processes use complex facilities whosedesign and operation basically depend on many parame-ters including gas composition flow rate circulation ratesabsorber temperatures and stripper pressures Accuratemod-eling of such processes involving multicriteria decisionmaking has inspired process engineers to acquire the bestpractices for process automation in theway to realize compet-itive returns on investment Despite the rapid developmentin process automation no systematic methodology can beemployed to continuously guarantee optimality of processoperation However routine update of performance canalways be achieved

Problems associated with the gas treating processes suchas large energy consumption and limited gas recovery haveled to the increasing focus on mathematical modeling and

2 Journal of Industrial Mathematics

simulation of gas sweetening processes often performedto understand effects of design and operation on processenhancement

In most cases optimization of such complex processes isachieved through parametric sensitivity analysis [4 5] Suchtedious procedures may be eliminated through the use ofsystematic approaches which guarantee global optimality ofthe process

Previously flow sheet optimization was achieved throughcalculation of inaccurate gradient derivatives of the processmodel generated inside modular process simulators [6] Thedevelopment of high speed computers has led to increasingfocus on designing and optimizing chemical process usingaccurate and complex stochastic algorithms

Further the decreased computational time has also con-tributed to routinely solving real-world problems involvinglarge realistic nonlinearmodels often encountered in processengineering Researchers such as Aroonwilas et al [7] andPark and Kang [8] simulated CO

2removal with aqueous

amine solutions such as DEA MDEA and DGA in agas sweetening process Rahimpour and Kashkooli [9 10]developed a mathematical model for the absorption of CO

2

by DEA-promoted potash solution and used it to examinethe performance of a split flow absorber More work on themodeling of the individual potassium carbonate and amineprocesses using computer simulations has been reported inthe specialized literature [11ndash15]

Recently engineering optimization has been heavilyinclined towards nature-inspired and complex computationaltechniques such as evolutionary algorithms to achieve thenecessary design and operation standards [16 17] Thesecomplex and accurate optimization techniques have also beengreatly motivated by the market availability of modern high-performance computers

Optimization problems are often computationally expen-sive especially those involving multicriteria decision mak-ing processes In practice real-world engineering problemsinvolve making decisions over multiple objectives such asmaximizing production capacity minimizing energy max-imizing reliability or any other objectives which could beconflicting For solving such problems accurately researcherscall for multiobjective optimization techniques which arenongradient basedThe application of evolutionary optimiza-tion methods to solve complex engineering multi-objectiveoptimization problems has been published in a number ofarticles [18ndash21]

In this paper we show how a classical multi-objectivebare-bone particle swarm optimization (MOBBPSO) can beeffectively utilized in process parameter tuning and perfor-mance analysis for a natural gas sweetening unit throughhydrocarbon recovery maximization and energy minimiza-tion The two objectives are combined and solved as asingle objective problem through the weighted sum methoddescribed by Marler and Arora [22]

Once optimization basis is determined it is necessaryto determine process variables to be optimized Processvariables are the variables that affect the values of theobjective functions These can be divided into decision anddependent variables Decision variables are determinedwhile

dependent variables are influenced by process constraints Inthis study concentrations of H

2S and CO

2in the purified

gas were considered as the dependent variables A parametricsensitivity study is carried out on the process model todetermine the upper and lower bounds of the decisionvariables The bounds are necessary to enable the optimizerto create a search space which obeys the necessary rules ofthumb

The optimization code developed in Visual Basic forApplication (VBA) environment and then integrated withProMax process simulator to tune the solvent circulationrates reboiler duties and regeneration pressures of the gassweetening units so that energy consumption and hydrocar-bon recovery are optimized

This paper is divided into six sections Section 2 intro-duces multi-objective optimization through the weightedsum approach and also describes a brief theory on bare-bones particle swarm optimization Section 3 describes theBenfield HiPure process and the use of ProMax to develop itsmodelThis section also contains a comparison of simulationresults with plant data obtained from ADGAS Section 4describes the parametric sensitivity analysis which explainshow the constraints in Section 5 are obtained The opti-mization results are presented and discussed in Section 6Conclusions on this study including possible future work arediscussed in Section 7 of this article

2 Multiobjective Optimization

A general multi-objective problem can be expressed as a setof equations below

Minimize 119891 (119909) = [1198911(119909) 119891

2(119909) 119891

119896(119909)] (1)

Subject toConstraints

119892119894(119909) le 0 119894 = 1 2 119898 (2)

ℎ119895(119909) = 0 119895 = 1 2 119902 (3)

Bounds

119886119897le 119909119897le 119887119897 119897 = 1 2 119901 (4)

where 119909 = [1199091 1199092 119909

119901]119879 is the vector of decision variables

119891119894 R119899 rarr R 119894 = 1 2 119896 are the objective functions

119892119894 ℎ119895 R119899 rarr R 119894 = 1 2 119898 119895 = 1 2 119902 are the

constraint functions of the problem and 119886119897 119887119897 R119899 rarr R 119897 =

1 2 119901 are the lower and upper bounds of the decisionvariables respectively

Such multiple objective problems are often appearing inmany process engineering problems [23] Inmost cases theseobjectives are generally conflicting and hence preventingsimultaneous optimization of the individual objectives

Themost common approach tomulti-objective optimiza-tion is the weighted sum method (5) which transforms themultiple objective problem into a single objective engineeringproblem [22 23] Objectives are allocated weights and added

Journal of Industrial Mathematics 3

to form a single objective problem (maximization objectivestake a negative sign) Consider

119891 (119909) =119896

sum119894=1

119908119894119891119894(119909) (5)

Multi-objective optimization in process plants has beenrecently reported byWu et al [21] Bernier et al [24] Li et al[25] Montazer-Rahmati and Binaee [26] Sun and Lou [27]and Shadiya et al [28] Genetic algorithm [19] has alwaysbeen the most common global optimization algorithmemployed in optimizing process plants Recently particleswarm optimization has gained more interest in obtainingglobal optimum of nonconvex problems [29] Poli et al [30]discussed an overview on the development and applicationof particle swarm optimization to real-world problems andReyes-Sierra and Coella [31] presented a survey on use ofmulti-objective particle swarm optimizers in solvingmultipleobjectives problems Here one of the improved versionsof particle swarm optimization (PSO) commonly known asbare-bone particle swarm optimization (BBPSO) will be uti-lized to optimize the natural gas process PSO has undergoneseveral changes since its introduction in 1995 [32] One ofthese changeswas the use of theGaussian normal distributionofmean and standard deviation to update particle position InBBPSO the exploration search is facilitated by the standarddeviation which tends to zero as the search space progressesMore information on the development and application ofBBPSO is discussed in thework published byZhang et al [33]

3 Case Study

31 Benfield HiPure Process of Natural Gas Sweetening TheBenfieldHiPure process was first described in 1974 by Bensonand Parish [34] This process consists of the Benfield processand an amine unit downstream to scrub the remaining tracesof acid gasesThe integrated schematic of the BenfieldHiPureprocess used in ADGAS plant at Das Island is shown inFigure 1

The hot potassium carbonate absorption system com-prises a split flow absorber and a regenerator with no sidedraws The carbonate absorber and regenerator are bothvertical packed bed columns The treated gas from thecarbonate absorber is fed directly into the amine absorberThe DEA amine system comprises absorber and strippercolumns which are vertical and made of a stack-structuredpacking The rich solution from the absorber is pumped totheDEA regenerator that has no condenserThe overhead gas(from DEA regenerator) is fed to the middle of the carbonateregenerator which does have a condenser Liquid from thecarbonate regenerator condenser is fed to the top of the DEAregenerator as reflux

Sweet gas exiting the DEA absorber will undergo furtherprocessing before it is sent for LNG processing The strippedacid gases from both the carbonate and DEA regeneratorsproceed to a sulfur recovery unit (SRU) where they areprocessed to produce molten liquid sulfur Table 1 gives thetypical operating data of the ADGAS plant

Table 1 Typical ADGAS operating data

Parameter ValueFeed Gas Flow Rate (MMSCFD) 4769Feed Gas Temperature (∘C) 250Feed Gas Pressure (barg) 521H2S Feed Gas Composition () 47CO2 Feed Gas Composition () 21

Hot Potassium Carbonate Unit

Circulation Rate (m3hr) Main 3435Split 12922

Lean Solvent Temperature (∘C) Main 818Split 1170

Lean Solvent Pressure (barg) 514K2CO3 Concentration (wt) 300Promoter Concentration (DEA) (wt) 30

Amine UnitCirculation Rate (m3hr) 1098Lean Solvent Temperature (∘C) 499Lean Solvent Pressure (barg) 537DEA Concentration (wt) 200

32 Simulation Results The process calculations were com-pleted using ProMax V32 [35] The electrolytic propertypackage was used to predict the H

2S and CO

2absorption in

both potassium carbonate and amine units of the BenfieldHiPure processThe TSWEET kinetics model in ProMax wasused to predict the CO

2-aminecarbonate kinetic reactions

taking place in all absorbers TSWEET kinetics was devel-oped by Bryan Research and Engineering for the purpose ofaccurately calculating the relatively slow absorption of CO

2

by amine solutionsTable 2 gives a comparison of the simulation results and

the operating data As can be seen there is a close matchbetween the simulation results with plant data In the nextsection a sensitivity analysis study will be described in orderto show the effect of various operating conditions on theperformance of the process and to motivate the optimizationstudy that will be discussed in a later section of this paper

4 Sensitivity Analysis

Sensitivity analysis gives the best approach to investigate theeffect of process parameters on performance of a chemicalplant Here a scenario tool was developed in ProMax andthe effect of change in decision variables is monitored withrespect to the required gas specification (process constraint)while compromising the respective rules of thumb

Figure 2 shows the effect of change in the carbonatecirculation rate with composition of H

2S and CO

2in the

purified gas As shown in the figure decreasing the carbonatecirculation rate beyond 1400m3hr will rapidly cause anincrease in H

2S and CO

2composition in the purified gas

hence violating the constraints of not more than 1 ppmv H2S

and 50 ppmv CO2


2-C-311Rich DEA

2-C-310Sweet gasKnockout

drum

2-C-318DEA regenerator

2-G-305 ABLean DEA pumpsLV-26

2-C-309DEA absorber

Air cooler

Sweetened gas

2-C-302Carb ABSKnockout

drum

2-C-308Rich carb

Flash drum 2-C-301Lean carbonate flash drum

2-G-303 A B CLean carbonate pumps

2-C-312Carbonate regenerator

Acid gas tosulphur recovery unit

Feednatural gas

2-C 303Carbonate absorber

LV-24

(2 turbines + 1 motor)

lt50ppm CO2lt5ppm H2S

Temp 107∘CPress 08 barg

Temp80∘CPress50barg

Temp 50∘CPress 50barg flash drum


Figure 1 ADGASrsquo Train number 3 gas treating plant Schematic of the Benfield HiPure process

Table 2 Comparison of hot carbonate section with plant data

Components Sour gas feed Carbonate absorber Overhead product Amine absorber Overhead productPlant data Simulation Plant data Simulation

CO2 (ppmv) 470000 5741 5700 1900 2500H2S (ppmv) 210000 7074 6830 040 040Nitrogen (vv) 21 21 23 230 231Methane (vv) 814 854 862 8590 8711Ethane (vv) 56 69 59 690 600Propane (vv) 27 31 29 310 290i-Butane (vv) 04 06 04 060 039n-Butane (vv) 06 04 06 100 064i-Pentane (vv) 01 02 01 020 020n-Pentane (vv) 03 03 03 030 030

On the other hand a carbonate circulation rate of1700m3hr or more will produce sweet gas with almost thesame gas specification Therefore 1400 and 1700m3hr willbe considered as the lower and upper bounds of the carbonatecirculation rate respectively Figure 3 shows that decreasingthe carbonate section reboiler energy beyond 40Gcalhr willinstantly cause the product gas to go off spec thereforethis will act as the lower bound for this decision variableTo minimize waste of energy the upper bound may beconstrained at 50Gcalhr since at this value all the sweet gasproduced meets the required specification

In a similar way Figure 4 is considered to choosethe upper and lower bounds on the circulation rate ofdiethanolamine (decision variable 119909

3) as 90 and 110m3hr

respectively From Figure 5 the lower and upper bounds onthe amine section reboiler duty may lead 10 and 12Gcalhrrespectively

Figures 6 and 7 show the variation of column strippingpressures with H

2S and CO

2composition in the sweet

gas Figure 6 shows that operating the carbonate stripper atpressures above 08 barg may lead to producing sweet gaswhich violates constraints and to enable amoderate stripping


1300 1400 1500 1600 1700 1800

032

034

036

038

040

042

20

30

40

50

60

70

80

90

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2

Carbonate circulation rate (m3hr)

Figure 2 Effect of K2CO3DEA circulation rate on CO

2and H

2S

composition in sweet gas

35 40 45 50 55 60 65 70038

039

040

041

042

043

044

Carbonate reboiler duty (Gcalhr)

0

5

10

15

20

25

30

35

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2

Figure 3 Effect of the reboiler duty of the carbonate section on CO2

and H2S composition in sweet gas

80 100 120 140 160 180

02

04

06

08

10

12

10

20

30

40

50

60

70

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2

DEA circulation rate (m3hr)

Figure 4 Effect of DEA circulation rate on CO2and H

2S composi-

tion in sweet gas

90 105 120 135 15000

02

04

06

08

10

12

Amine reboiler duty (Gcalhr)

325

330

335

340

345

350

355

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2

Figure 5 Effect of the reboiler duty of the amine section on the CO2


060 065 070 075 080 085 09000

02

04

06

08

10

12

14

Carbonate stripper pressure (barg)

0

20

40

60

80

100

120Sw

eet g

as H

2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2

Figure 6 Effect of carbonate stripper pressure on CO2and H

2S


efficiency 07 barg is considered as the lower limit decisionvariable

Figure 7 shows that the variation of the amine stripperpressure with the composition of hydrogen sulfide shows areduction in H

2S removal and exhibits a maximum increase

at about 1 barg and increasing the stripper pressure beyondthis region rapidly increases the H

2S removal On the other

hand an increase in stripper pressure continuously decreasesthe amount of CO

2in the sweet Therefore 09 barg and

12 barg are considered as the lower and upper limit on thedecision variable respectivelyThe lower limit pressure on theamine stripper is always expected to be greater than the upperlimit for the carbonate stripper pressure since the overheadacid gas from the amine regenerator is directly feeding thecarbonate stripper


08 09 10 11 1209725

09730

09735

09740

09745

09750

09755

09760

09765

Amine stripper pressure (barg)

38

39

40

41

42

43

44

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2

Figure 7 Effect of amine stripper pressure on CO2and H

2S


5 Process Optimization

The optimization of the gas sweetening unit involves twoobjectives maximization of hydrocarbon recovery and min-imization of the heat required to run the plant perunit timeThis problem considers no variation in sour gas conditions(flow rate temperature pressure and composition)

Themathematical expression of the objective functions ofthe optimization problem can be cast as

Maximize 1198911(119909) =Hydrocarbon Recovery

Hydrocarbon Recovery = (119872119878minus119872119886

119872119891

) lowast 100 (6)

Minimize 1198912(119909) = Total Heat

Total Heat

= sum(119876Reboilers + 119876Pumps minus 119876Coolers minus 119876Condensers) (7)

The first objective function represented by (6) is the per-centage amount of sweet gas produced from the gas facilityexcluding CO

2and H

2S This is also used as the measure

of hydrocarbon content (C1to normal-C

5) since the feed

gas contained no traces of mercaptans or another sulfurcompounds 119872

119878and 119872

119886are the mass flow rate of the

sweetened gas and acid gas (CO2and H

2S) respectively119872

119891

is the mass flow rate of the sour gas fed to the gas facilityEquation (7) represents the second objective function whichis the total heat required to produce 119872

119878kg of sweet gas in a

given unit of timeGas sweetening plants require high amounts of energy

supply to produce ultrasweet gas needed in LNG plantsfor safe process operation and high efficient hydrocarbonseparation in the downstream operations

To maximize the sweet gas purity (high hydrocarboncontent) higher amounts of heat need to be applied toabsorb a substantial amount of the acid gases However

Table 3 The upper and lower bound of the operating variables

Process parameter LB119894

UB119894

1199091

1400 17001199092

40 501199093

90 1101199094

10 121199095

07 081199096

09 12

due to the high costs of energy to run the plant facilityenergy minimization is often recommended to reduce theplant operating costs Therefore due to the two conflictingobjectives process operators often strive for the search ofa trade-off between energy minimization and hydrocarbon(pure sweet gas) maximization

Here these two objectives are modeled and combinedthrough the weighted summethod to form a single objectiveThe weights are selected interactively to produce preferenceinformation needed during the progressive optimizationprocess

The constraints of the optimization problem are

119910H2S lt 1 (8)

119910CO2

lt 50 (9)

119871119861119894le 119909119894le 119880119861119894 (10)

Equations (8) and (9) provide constraints on the maximumcomposition of H

2S and CO

2allowable in the sweetened gas

respectively Ultrasweet gas of utmost 1 and 50 ppmv of H2S

and CO2 respectively is used as feed to the liquefied natural

gas plant 119871119861119894and119880119861

119894in (10) are the upper and lower bounds

on the various conditions of the processThese bounds are imposed on the K

2CO3DEA Cir-

culation rate (1199091) carbonate section reboiler duty (119909

2)

DEA circulation rate (1199093) amine section reboiler duty (119909

4)

carbonate stripper pressure (1199095) and amine stripper pressure

(1199096) From the simulation results and the sensitivity analysis

of the previous section appropriate bounds that define thesearch space are summarized in Table 3 These bounds arechosen based on a physical understanding of the problem soas not to compromise desired operating conditions

51 Simulation Optimization Framework Simulation opti-mization entails finding optimal settings of the input vari-able(s) that is values of 119909

1 1199092 119909

6 which optimize

the output variable(s) of the simulation model Optimizersdesigned for simulation embody the principle of executingseparately from themodel In such a context the optimizationproblem is defined outside the complex system The outputof the simulation model is used by an optimizer to providenew input values of the decision variables as the search forthe optimal solution proceeds On the basis of both currentand past evaluations the method decides upon a new set ofinput values

Provided that a feasible solution exists the optimiza-tion procedure ideally carries out a special search until an


Simulationmodel

Output Optimizationengine

New input values

Figure 8 Integrated simulation optimization framework

8862 8864 8866 8868 8870525

530

535

540

545

550

Min

imiz

atio

n of

f1(x)

Maximization of f2(x)

Figure 9 Pareto optimal front obtained from the simultaneousoptimization of 119891

1(119909) and 119891

2(119909)

appropriate termination criterion is satisfied A sequential-like approach can be employed by coupling the solution of theoptimization problem to an optimization routine (Figure 8)Every time the optimization routine needs a function evalua-tion by this approach a call is made to the simulation modelIn this work optimization was performed through the bare-bone particle swarm optimization as discussed next

52 Particle Swarm Optimization PSO is inspired by thesocial behavior of some biological organisms especially thegrouprsquos ability of some animal species to locate a desirableposition at the given area It was proposed first by Kennedyand Eberhart [32] In the PSO a swarm consists of a set ofparticles and each particle represents a potential solutionof an optimization problem Particles are placed in thesearch space of the function and each evaluates the objectivefunction at its current location

Each individual in the basic particle swarm is composedof 119889 the dimensions of the optimization problem (numberof decision variables) the current position 119909

119894 the previous

best position 119901119894 and the velocity update V

119894 Kennedy [36]

conducted some experiments using PSO variants which dropthe velocity termandupdate a particle usingGaussian normaldistribution of mean and standard deviation The use ofGaussian normal distribution in the basic PSO algorithmto update the particle is known as bare-bones particleswarm optimization (BBPSO) Unlike the PSO the BBPSOis parameter-free and is suitable for those real applicationproblems where the information on parameters such as

inertia weights and acceleration coefficients of particles islacking or hard to obtain

The bare-bones particle swarm optimization (BBPSO)proposed by Kennedy [36] is a simple version of PSO TheBBPSO algorithm does not use the particle velocity but usesa Gaussian sampling based on 119875119892best119896 and 119875best119896

119894 and the

position equation is replaced by

119909119889119896+1119894

= 119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

(11)

Equation (11) shows that the position of each particle israndomly selected from the Gaussian distribution with theaverage of the personal best position and the global bestposition

In BBPSO the exploration search is facilitated by 120590119889119896119894

term which tends to zero as the search progresses andmore emphasis will be put on the exploitation of the searchKennedy [36] further proposed an alternative version ofBBPSO also referred to as BBExp in the present paper wherethe 119889th dimension of the 119894th particle is updated as follows

119909119889119896+1119894

=

119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

if 119880 [0 1] lt 05

119875best119889119896119894

Otherwise(12)

Equation (12) justifies that since there is 50 chance that the119889th dimension of the particle changes to the correspondingpbest position the BBExp inclines to search for pbest posi-tionsThus BBExp is said to be biased towards exploiting thepbest positions as shown in Algorithm 1

6 Results and Discussion

Figure 9 shows a Pareto set of optimal solutions obtainedfrom the formulated problem ((6)ndash(12)) The figure showsthat increasing heat of the process (119891

2)will reduce the amount

of H2S and CO

2in the sweet gas (119891

1) meaning that the gas

will contain more of the hydrocarbons than the acid gasesEach point on the Pareto set is associated with a set of

decision variables A unique solution can be obtained usinghigh-level decision making by managementThe closeness ofthe Pareto set shows that a unique solution can also be easilygenerated using this method

The percentage recovery of hydrocarbons may be inter-preted as a small change along the hydrocarbon recoveryaxis However if translated into an equivalent amount ofgas per unit time the change is significant That is to saya 01 increase in hydrocarbon recovery is equivalent toapproximately 94 tonnes of hydrocarbon per day

Figures 10 and 11 show a comparison of the Paretooptimal solutions with the nonoptimal operating regionThe


1652 1653 1654 1655 1656 1657860

865

870

875

880

885f1(x)

x1

(a)

1652 1653 1654 1655 1656 165750

52

54

56

58

60

62

64

66

f2(x)

x1

(b)

465 470 475 480 485 490860

865

870

875

880

885

890

f1(x)

x2

(c)

465 470 475 480 485 49052

54

56

58

60

62

64

66

f2(x)

x2

(d)

1000 1001 1002 1003 1004 1005 1006 1007860

865

870

875

880

885

890

f1(x)

x3

(e)

1000 1001 1002 1003 1004 1005 1006 100750525456586062646668

f2(x)

x3

(f)

OptimalNon-optimal

f1(x)

x4

102 104 106 108 110 112 114 116860

865

870

875

880

885

890

(g)

OptimalNon-optimal

f2(x)

x4

104 106 108 11050

52

54

56

58

60

62

64

66

(h)

Figure 10 Pareto optimal set with regard to decision variables 1199091 1199092 1199093 and 119909

4with objectives 119891

1(119909) and 119891

2(119909)


070 075 080 085 090860

865

870

875

880

885

890f1(x)

x5

(a)

070 075 080 085 09050

52

54

56

58

60

62

64

66

f2(x)

x5

(b)

070 075 080 085 090 095 100 105 110860

865

870

875

880

885

890

f1(x)

x6

OptimalNon-optimal

(c)

070 075 080 085 090 095 100 105 11050

52

54

56

58

60

62

64

66

f2(x)

x6

OptimalNon-optimal

(d)Figure 11 Pareto optimal set with regard to decision variables 119909

5and 119909


1(119909) and 119891

2(119909)

nonoptimal operating conditions give 8635 hydrocarbonrecovery and 6436Gcalhr of energy which shows a nonproper region of operation Applying this optimizationmodelwill approximately increase hydrocarbon recovery by 25(sim235 tonnes of hydrocarbonday) and minimize heat energyup to 16 of the non-optimal region

To obtain optimal operating conditions for a desiredpercentage recovery and the corresponding minimal energyfromFigure 9 the decision variables of the respective optimalset can be read off from Figures 10 and 11 Another setof Pareto front is generated when each objective is plottedagainst all the decision variables at the optimal solutionsThisstudy provides an opportunity to process operators to choosethe operating conditions needed to optimize both objectives

Figures 10-11 show also a distinction between the optimaland non-optimal operating process variables of ADGASplant and its consistency is illustrated on all process vari-ables In these figures the minimization and maximizationobjectives give a Pareto set below and above the non-optimalsets respectivelyMaximization of hydrocarbon recovery and

minimization of energy are justified in the plots 1198911(119909) and

1198912(119909) versus 119909

1 119909

6with the Pareto sets above and below

the non-optimal set respectivelyThedifference in the optimaland non-optimal Pareto gives the amount of savings madeon implementing this optimization approach The Pareto seton the carbonate circulation rate (119909

1) takes on values close

to the upper limit of its bounds The carbonate reboiler duty(1199092) also takes on optimal values near the upper bound of

the constraint Unlike decision variable 1199091 the optimal values

of 1199092vary significantly between its midpoint and the upper

limit The optimal value of decision variable 1199093takes values

existing midway of the upper and lower bounds Decisionvariables 119909

4 1199095 and 119909

6span the entire feasible region For

all cases involving either 1198911(119909) or 119891

2(119909) all variables give rise

to the same trend

7 Conclusion

This work presented a study to optimize a natural gassweetening process with the objectives of maximization


Start up(i) Choose priori weights 119908

119894

(ii) Combine objective functions using weighted sum method(iii) Randomly generate an initial population NP(iv) 119896 = 0Iteration

For each particle 119894 = 1 to NPFor 119889 = 1 to119863

120583119889119896119894

=(119875best119889119896

119894+ 119875119892best119889119896

119894)

2

120590119889119896119894

=10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best119889119896

119894

10038161003816100381610038161003816If 119880[0 1] lt 05Then119909119889119896+1119894

= 119873(120583119889119896119894 120590119889119896119894)

Else119909119889119896+1119894

= 119875best119889119896119894

End ifEnd for

For each dimension 119889 = 1 to119863IF 119909119889119896+1119894

lt 119909119889min OR 119909119889119896+1119894

lt 119909119889max Then119909119889119896+1119894

= 119909119889min + 119880[0 1] lowast (119909119889

max minus 119909119889

min)End ifEnd forif119891(119909119889119896

119894) lt 119891(119875best119889119896

119894)Then

119875119892best119896119894= 119909119896+1119894

End if119870 = 119896 + 1

Until termination criteria is satisfiedEnd for

Print Pgbest 119891119894(Pgbest)

Choose new weights 119908119894

Print a Pareto-front

Algorithm 1 The pseudocode of BBExp-PSO algorithm

hydrocarbon recovery and minimization of heat energyrequired to run the process ProMax simulation tool wasused tomodel the process and the simulation results matchedwell the plant data The multiobjective optimization usingmultiobjective bares-bone particle swarm indicates that atrade-off exists between the two objectives and plant oper-ation can be optimized up to a 25 (sim235 tonnesday)increase in hydrocarbon recovery and 16 decrease in heatrequired for the processoperation The results also showedthat the integration of an evolutionary algorithm in a processsimulator can serve as an optimization tool box and cansignificantly improve the operation of the process Futurework will consider metamodeling of natural gas plants toovercome the large computational time involved in runningintegrated models of process simulators with evolutionaryalgorithms

Acknowledgments

The authors would like to acknowledge GASCO ADGASandAlHosnGas operating companies of AbuDhabiNationalOil Company (ADNOC) for their technical and financialsupport

References

[1] S A Ebenezer and J S Gudmundsson Removal of carbondio-xide from natural gas for LNG production [PhD thesis]Norwegian University of Science and Technology TrondheimNorway 2005

[2] A Kohl and R Nielsen Gas Purification Gulf Houston TexUSA 1997

[3] R Hubbard ldquoThe role of gas processing in the natural-gas valuechainrdquo Journal of Petroleum Technology vol 61 no 8 Article ID118535 pp 65ndash71 2009

[4] B Hassan and T Shamim ldquoParametric and exergetic analysisof a power plant with CO

2and capture using chemical looping

combustionrdquo in Proceedings of the International Conference onClean and Green Energy Singapore 2012

[5] R Mohammad L Schneiders and J Neiderer ldquoCarbondioxidecapture from power plants part I a parametric study of thetechnical performance based on monoethanolaminerdquo Interna-tional Journal of Greenhouse Gas Control vol 1 no 1 pp 37ndash462007

[6] D Wolbert X Joulia B Koehret and L T Biegler ldquoFlowsheetoptimization and Ootimal sensitivity analysis using exactderivativesrdquo Internal Report PA 15213 Engineering DesignResearch Center Carnegie Mellon University Pittsburgh PaUSA 1993


[7] A Aroonwilas A Chakma P Tontiwachwuthikul and AVeawab ldquoMathematical modelling of mass-transfer and hydro-dynamics in CO

2absorbers packed with structured packingsrdquo

Chemical Engineering Science vol 58 no 17 pp 4037ndash40532003

[8] K Y Park and T-W Kang ldquoComputer simulation of H2S and

CO2absorption processesabsorption processesrdquoKorean Journal

of Chemical Engineering vol 12 no 1 pp 29ndash35 1995[9] M R Rahimpour and A Z Kashkooli ldquoEnhanced carbon

dioxide removal by promoted hot potassium carbonate in asplit-flow absorberrdquo Chemical Engineering and Processing vol43 no 7 pp 857ndash865 2004

[10] M R Rahimpour and A Z Kashkooli ldquoModeling and simul-ation of industrial carbon dioxide absorber using amine-promoted potash solutionrdquo Iranian Journal of Science andTechnology B vol 28 no 6 pp 653ndash666 2004

[11] K Y Park and T F Edgar ldquoSimulation of the hot carbonate pro-cess for removal of CO

2andH

2S frommedium Btu gasrdquo Energy

progress vol 4 no 3 pp 174ndash181 1984[12] P Patil Z Malik and M Jobson ldquoRetrofit design for gas sweet-

ening processesrdquo in Proceedings of the IChemE Symposium 152pp 460ndash467 2006

[13] J C Polasek and J A Bullin ldquoDesign and optimization of integ-rated amine sweetening claus sulfur and tail gas cleanup unitsby computer simulationrdquo Internal Report Bryan Research ampEngineering Bryan Tex USA 1990

[14] J C Polasek J A Bullin and S T Donnely ldquoAlternative flowschemes to reduce capital and operating costs of amine sweet-ening unitsrdquo Internal Report Bryan Research amp EngineeringBryan Tex USA 2006

[15] R H Weiland M Rawal and R G Rice ldquoStripping of carbon-dioxide from monoethanolamine solutions in a packed col-umnrdquo AIChE Journal vol 28 no 6 pp 963ndash973 1982

[16] H Pierreval and L Tautou ldquoUsing evolutionary algorithms andsimulation for the optimization of manufacturing systemsrdquo IIETransactions vol 29 no 3 pp 181ndash189 1997

[17] E Zitzler and LThiele ldquoMultiobjective optimization using evo-lutionary algorithms a comparatice case studyrdquo in Proceedingsof the 5th International Conference on Parallel Problem Solvingfrom Nature (PPSN rsquo98) Amsterdam The Netherlands 1998

[18] S Dehuri and S-B Cho ldquoMulti-criterion Pareto based particleswarm optimized polynomial neural network for classificationa review and state-of-the-artrdquo Computer Science Review vol 3no 1 pp 19ndash40 2009

[19] A KonakD CoitW Smith and E Alice ldquoMulti-objective opti-mization using genetic algorithm a tutorialrdquo Reliability Engi-neering amp System Safety vol 91 pp 992ndash100 2006

[20] L Mian S Azarm and V Aute ldquoAmulti-objective genetic algo-rithm for robust design optimizationrdquo in Proceedings of theGenetic and Evolutionary Computation Conference (GECCOrsquo05) pp 771ndash778 New York NY USA June 2005

[21] W Wu Y Liou and Y Zhou ldquoMultiobjective optimization ofa hydrogen production system with low CO

2emissionsrdquo Indus-

trial and Engineering Chemistry Research vol 51 no 6 pp2644ndash2651 2012

[22] R T Marler and J S Arora ldquoThe weighted sum method formulti-objective optimization new insightsrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 853ndash862 2010

[23] R T Marler and J S Arora ldquoSurvey of multi-objective optimi-zation methods for engineeringrdquo Structural and Multidisci-plinary Optimization vol 26 no 6 pp 369ndash395 2004

[24] E Bernier F Marechal and R Samson ldquoMulti-objective designoptimization of a natural gas-combined cycle with carbondioxide capture in a life cycle perspectiverdquo Energy vol 35 no 2pp 1121ndash1128 2010

[25] H Li FMarechal M Burer andD Favrat ldquoMulti-objective op-timization of an advanced combined cycle power plant includ-ing CO

2separation optionsrdquo Energy vol 31 no 15 pp 3117ndash

3134 2006[26] M M Montazer-Rahmati and R Binaee ldquoMulti-objective opti-

mization of an industrial hydrogen plant consisting of a CO2

absorber using DGA and a methanatorrdquo Computers and Chem-ical Engineering vol 34 no 11 pp 1813ndash1821 2010

[27] L Sun and H H Lou ldquoA strategy for multi-objective optimiza-tion under uncertainty in chemical process designrdquo ChineseJournal of Chemical Engineering vol 16 no 1 pp 39ndash42 2008

[28] O O Shadiya V Satish and K A High ldquoProcess enhancementthrough waste minimization and multiobjective optimizationrdquoJournal of Cleaner Production vol 31 pp 137ndash149 2012

[29] K E Parsopoulos and M N Vrahatis ldquoRecent approaches toglobal optimization problems through particle swarm opti-mizationrdquo Natural Computing vol 1 pp 230ndash235 2002

[30] R Poli J Kennedy and T Blackwell ldquoParticle swarm optimiza-tionrdquo Swarm Intelligence vol 1 pp 33ndash35 2007

[31] M Reyes-Sierra and C A Coello ldquoMulti-objective particelswarm optimizers a survey of the state-of-the-artrdquo Interna-tional Jounal of Computing Intelligenece Research vol 2 pp 287ndash300 2006

[32] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conefercence on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[33] H Zhang D Kennedy G Rangaiah and A Bonilla-PetricioletldquoNovel bare-bones particle swarm optimization and its perfor-mance formodeling vapor-liquid equilibriumdatardquo Fluid PhaseEquilibria vol 301 no 1 pp 33ndash45 2011

[34] H E Benson and R W Parrish ldquoHiPure process removesCO2H2Srdquo Hydrocarbon Processing vol 53 no 4 pp 81ndash82

1974[35] Bryan Research and Engineering ldquoProMax V32rdquo 2010 http

wwwbrecomhomeaspx[36] J Kennedy ldquoBare bones particle swarmsrdquo in Proceedings of the

IEEE Swarm Intelligence Symposium pp 80ndash87 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


simulation of gas sweetening processes often performedto understand effects of design and operation on processenhancement

In most cases optimization of such complex processes isachieved through parametric sensitivity analysis [4 5] Suchtedious procedures may be eliminated through the use ofsystematic approaches which guarantee global optimality ofthe process

Previously flow sheet optimization was achieved throughcalculation of inaccurate gradient derivatives of the processmodel generated inside modular process simulators [6] Thedevelopment of high speed computers has led to increasingfocus on designing and optimizing chemical process usingaccurate and complex stochastic algorithms

Further the decreased computational time has also con-tributed to routinely solving real-world problems involvinglarge realistic nonlinearmodels often encountered in processengineering Researchers such as Aroonwilas et al [7] andPark and Kang [8] simulated CO

2removal with aqueous

amine solutions such as DEA MDEA and DGA in agas sweetening process Rahimpour and Kashkooli [9 10]developed a mathematical model for the absorption of CO

2

by DEA-promoted potash solution and used it to examinethe performance of a split flow absorber More work on themodeling of the individual potassium carbonate and amineprocesses using computer simulations has been reported inthe specialized literature [11ndash15]

Recently engineering optimization has been heavilyinclined towards nature-inspired and complex computationaltechniques such as evolutionary algorithms to achieve thenecessary design and operation standards [16 17] Thesecomplex and accurate optimization techniques have also beengreatly motivated by the market availability of modern high-performance computers

Optimization problems are often computationally expen-sive especially those involving multicriteria decision mak-ing processes In practice real-world engineering problemsinvolve making decisions over multiple objectives such asmaximizing production capacity minimizing energy max-imizing reliability or any other objectives which could beconflicting For solving such problems accurately researcherscall for multiobjective optimization techniques which arenongradient basedThe application of evolutionary optimiza-tion methods to solve complex engineering multi-objectiveoptimization problems has been published in a number ofarticles [18ndash21]

In this paper we show how a classical multi-objectivebare-bone particle swarm optimization (MOBBPSO) can beeffectively utilized in process parameter tuning and perfor-mance analysis for a natural gas sweetening unit throughhydrocarbon recovery maximization and energy minimiza-tion The two objectives are combined and solved as asingle objective problem through the weighted sum methoddescribed by Marler and Arora [22]

Once optimization basis is determined it is necessaryto determine process variables to be optimized Processvariables are the variables that affect the values of theobjective functions These can be divided into decision anddependent variables Decision variables are determinedwhile

dependent variables are influenced by process constraints Inthis study concentrations of H

2S and CO

2in the purified

gas were considered as the dependent variables A parametricsensitivity study is carried out on the process model todetermine the upper and lower bounds of the decisionvariables The bounds are necessary to enable the optimizerto create a search space which obeys the necessary rules ofthumb

The optimization code developed in Visual Basic forApplication (VBA) environment and then integrated withProMax process simulator to tune the solvent circulationrates reboiler duties and regeneration pressures of the gassweetening units so that energy consumption and hydrocar-bon recovery are optimized

This paper is divided into six sections Section 2 intro-duces multi-objective optimization through the weightedsum approach and also describes a brief theory on bare-bones particle swarm optimization Section 3 describes theBenfield HiPure process and the use of ProMax to develop itsmodelThis section also contains a comparison of simulationresults with plant data obtained from ADGAS Section 4describes the parametric sensitivity analysis which explainshow the constraints in Section 5 are obtained The opti-mization results are presented and discussed in Section 6Conclusions on this study including possible future work arediscussed in Section 7 of this article

2 Multiobjective Optimization

A general multi-objective problem can be expressed as a setof equations below

Minimize 119891 (119909) = [1198911(119909) 119891

2(119909) 119891

119896(119909)] (1)

Subject toConstraints

119892119894(119909) le 0 119894 = 1 2 119898 (2)

ℎ119895(119909) = 0 119895 = 1 2 119902 (3)

Bounds

119886119897le 119909119897le 119887119897 119897 = 1 2 119901 (4)

where 119909 = [1199091 1199092 119909

119901]119879 is the vector of decision variables

119891119894 R119899 rarr R 119894 = 1 2 119896 are the objective functions

119892119894 ℎ119895 R119899 rarr R 119894 = 1 2 119898 119895 = 1 2 119902 are the

constraint functions of the problem and 119886119897 119887119897 R119899 rarr R 119897 =

1 2 119901 are the lower and upper bounds of the decisionvariables respectively

Such multiple objective problems are often appearing inmany process engineering problems [23] Inmost cases theseobjectives are generally conflicting and hence preventingsimultaneous optimization of the individual objectives

Themost common approach tomulti-objective optimiza-tion is the weighted sum method (5) which transforms themultiple objective problem into a single objective engineeringproblem [22 23] Objectives are allocated weights and added



119891 (119909) =119896

sum119894=1

119908119894119891119894(119909) (5)


3 Case Study












2S and CO

2absorption in




2






2S and CO

2in the


2S and CO



and 50 ppmv CO2


2-C-311Rich DEA


drum



2-C-309DEA absorber

Air cooler

Sweetened gas


drum

2-C-308Rich carb





Feednatural gas


LV-24
















2S and CO




1300 1400 1500 1600 1700 1800

032

034

036

038

040

042

20

30

40

50

60

70

80

90

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



2and H

2S


35 40 45 50 55 60 65 70038

039

040

041

042

043

044


0

5

10

15

20

25

30

35

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



80 100 120 140 160 180

02

04

06

08

10

12

10

20

30

40

50

60

70

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



2S composi-

tion in sweet gas

90 105 120 135 15000

02

04

06

08

10

12


325

330

335

340

345

350

355

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



060 065 070 075 080 085 09000

02

04

06

08

10

12

14


0

20

40

60

80

100

120Sw

eet g

as H

2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2


2S











08 09 10 11 1209725

09730

09735

09740

09745

09750

09755

09760

09765


38

39

40

41

42

43

44

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2


2S







119872119891

) lowast 100 (6)


Total Heat



2and H



5) since the feed


119878and 119872




119891








UB119894

1199091

1400 17001199092

40 501199093

90 1101199094

10 121199095

07 081199096

09 12




119910H2S lt 1 (8)

119910CO2

lt 50 (9)

119871119861119894le 119909119894le 119880119861119894 (10)


2S and CO




gas plant 119871119861119894and119880119861



2CO3DEA Cir-


2)


4)





1 1199092 119909

6 which optimize




Simulationmodel


New input values


8862 8864 8866 8868 8870525

530

535

540

545

550

Min

imiz

atio

n of

f1(x)



1(119909) and 119891

2(119909)




119894 the previous


119894 Kennedy [36]




119894 and the


119909119889119896+1119894

= 119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

(11)




119909119889119896+1119894

=

119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

if 119880 [0 1] lt 05

119875best119889119896119894

Otherwise(12)





of H2S and CO








1652 1653 1654 1655 1656 1657860

865

870

875

880

885f1(x)

x1

(a)

1652 1653 1654 1655 1656 165750

52

54

56

58

60

62

64

66

f2(x)

x1

(b)

465 470 475 480 485 490860

865

870

875

880

885

890

f1(x)

x2

(c)

465 470 475 480 485 49052

54

56

58

60

62

64

66

f2(x)

x2

(d)

1000 1001 1002 1003 1004 1005 1006 1007860

865

870

875

880

885

890

f1(x)

x3

(e)

1000 1001 1002 1003 1004 1005 1006 100750525456586062646668

f2(x)

x3

(f)

OptimalNon-optimal

f1(x)

x4

102 104 106 108 110 112 114 116860

865

870

875

880

885

890

(g)

OptimalNon-optimal

f2(x)

x4

104 106 108 11050

52

54

56

58

60

62

64

66

(h)



1(119909) and 119891

2(119909)


070 075 080 085 090860

865

870

875

880

885

890f1(x)

x5

(a)

070 075 080 085 09050

52

54

56

58

60

62

64

66

f2(x)

x5

(b)

070 075 080 085 090 095 100 105 110860

865

870

875

880

885

890

f1(x)

x6

OptimalNon-optimal

(c)

070 075 080 085 090 095 100 105 11050

52

54

56

58

60

62

64

66

f2(x)

x6

OptimalNon-optimal


5and 119909


1(119909) and 119891

2(119909)





1198912(119909) versus 119909

1 119909









4 1199095 and 119909




to the same trend

7 Conclusion




119894



120583119889119896119894

=(119875best119889119896

119894+ 119875119892best119889119896

119894)

2

120590119889119896119894

=10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best119889119896

119894

10038161003816100381610038161003816If 119880[0 1] lt 05Then119909119889119896+1119894

= 119873(120583119889119896119894 120590119889119896119894)

Else119909119889119896+1119894

= 119875best119889119896119894

End ifEnd for


lt 119909119889min OR 119909119889119896+1119894

lt 119909119889max Then119909119889119896+1119894

= 119909119889min + 119880[0 1] lowast (119909119889

max minus 119909119889


119894) lt 119891(119875best119889119896

119894)Then

119875119892best119896119894= 119909119896+1119894

End if119870 = 119896 + 1







Acknowledgments


References



















2andH









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891 (119909) =119896

sum119894=1

119908119894119891119894(119909) (5)


3 Case Study












2S and CO

2absorption in




2






2S and CO

2in the


2S and CO



and 50 ppmv CO2


2-C-311Rich DEA


drum



2-C-309DEA absorber

Air cooler

Sweetened gas


drum

2-C-308Rich carb





Feednatural gas


LV-24
















2S and CO




1300 1400 1500 1600 1700 1800

032

034

036

038

040

042

20

30

40

50

60

70

80

90

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



2and H

2S


35 40 45 50 55 60 65 70038

039

040

041

042

043

044


0

5

10

15

20

25

30

35

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



80 100 120 140 160 180

02

04

06

08

10

12

10

20

30

40

50

60

70

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



2S composi-

tion in sweet gas

90 105 120 135 15000

02

04

06

08

10

12


325

330

335

340

345

350

355

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



060 065 070 075 080 085 09000

02

04

06

08

10

12

14


0

20

40

60

80

100

120Sw

eet g

as H

2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2


2S











08 09 10 11 1209725

09730

09735

09740

09745

09750

09755

09760

09765


38

39

40

41

42

43

44

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2


2S







119872119891

) lowast 100 (6)


Total Heat



2and H



5) since the feed


119878and 119872




119891








UB119894

1199091

1400 17001199092

40 501199093

90 1101199094

10 121199095

07 081199096

09 12




119910H2S lt 1 (8)

119910CO2

lt 50 (9)

119871119861119894le 119909119894le 119880119861119894 (10)


2S and CO




gas plant 119871119861119894and119880119861



2CO3DEA Cir-


2)


4)





1 1199092 119909

6 which optimize




Simulationmodel


New input values


8862 8864 8866 8868 8870525

530

535

540

545

550

Min

imiz

atio

n of

f1(x)



1(119909) and 119891

2(119909)




119894 the previous


119894 Kennedy [36]




119894 and the


119909119889119896+1119894

= 119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

(11)




119909119889119896+1119894

=

119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

if 119880 [0 1] lt 05

119875best119889119896119894

Otherwise(12)





of H2S and CO








1652 1653 1654 1655 1656 1657860

865

870

875

880

885f1(x)

x1

(a)

1652 1653 1654 1655 1656 165750

52

54

56

58

60

62

64

66

f2(x)

x1

(b)

465 470 475 480 485 490860

865

870

875

880

885

890

f1(x)

x2

(c)

465 470 475 480 485 49052

54

56

58

60

62

64

66

f2(x)

x2

(d)

1000 1001 1002 1003 1004 1005 1006 1007860

865

870

875

880

885

890

f1(x)

x3

(e)

1000 1001 1002 1003 1004 1005 1006 100750525456586062646668

f2(x)

x3

(f)

OptimalNon-optimal

f1(x)

x4

102 104 106 108 110 112 114 116860

865

870

875

880

885

890

(g)

OptimalNon-optimal

f2(x)

x4

104 106 108 11050

52

54

56

58

60

62

64

66

(h)



1(119909) and 119891

2(119909)


070 075 080 085 090860

865

870

875

880

885

890f1(x)

x5

(a)

070 075 080 085 09050

52

54

56

58

60

62

64

66

f2(x)

x5

(b)

070 075 080 085 090 095 100 105 110860

865

870

875

880

885

890

f1(x)

x6

OptimalNon-optimal

(c)

070 075 080 085 090 095 100 105 11050

52

54

56

58

60

62

64

66

f2(x)

x6

OptimalNon-optimal


5and 119909


1(119909) and 119891

2(119909)





1198912(119909) versus 119909

1 119909









4 1199095 and 119909




to the same trend

7 Conclusion




119894



120583119889119896119894

=(119875best119889119896

119894+ 119875119892best119889119896

119894)

2

120590119889119896119894

=10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best119889119896

119894

10038161003816100381610038161003816If 119880[0 1] lt 05Then119909119889119896+1119894

= 119873(120583119889119896119894 120590119889119896119894)

Else119909119889119896+1119894

= 119875best119889119896119894

End ifEnd for


lt 119909119889min OR 119909119889119896+1119894

lt 119909119889max Then119909119889119896+1119894

= 119909119889min + 119880[0 1] lowast (119909119889

max minus 119909119889


119894) lt 119891(119875best119889119896

119894)Then

119875119892best119896119894= 119909119896+1119894

End if119870 = 119896 + 1







Acknowledgments


References



















2andH









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2-C-311Rich DEA


drum



2-C-309DEA absorber

Air cooler

Sweetened gas


drum

2-C-308Rich carb





Feednatural gas


LV-24
















2S and CO




1300 1400 1500 1600 1700 1800

032

034

036

038

040

042

20

30

40

50

60

70

80

90

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



2and H

2S


35 40 45 50 55 60 65 70038

039

040

041

042

043

044


0

5

10

15

20

25

30

35

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



80 100 120 140 160 180

02

04

06

08

10

12

10

20

30

40

50

60

70

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



2S composi-

tion in sweet gas

90 105 120 135 15000

02

04

06

08

10

12


325

330

335

340

345

350

355

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



060 065 070 075 080 085 09000

02

04

06

08

10

12

14


0

20

40

60

80

100

120Sw

eet g

as H

2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2


2S











08 09 10 11 1209725

09730

09735

09740

09745

09750

09755

09760

09765


38

39

40

41

42

43

44

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2


2S







119872119891

) lowast 100 (6)


Total Heat



2and H



5) since the feed


119878and 119872




119891








UB119894

1199091

1400 17001199092

40 501199093

90 1101199094

10 121199095

07 081199096

09 12




119910H2S lt 1 (8)

119910CO2

lt 50 (9)

119871119861119894le 119909119894le 119880119861119894 (10)


2S and CO




gas plant 119871119861119894and119880119861



2CO3DEA Cir-


2)


4)





1 1199092 119909

6 which optimize




Simulationmodel


New input values


8862 8864 8866 8868 8870525

530

535

540

545

550

Min

imiz

atio

n of

f1(x)



1(119909) and 119891

2(119909)




119894 the previous


119894 Kennedy [36]




119894 and the


119909119889119896+1119894

= 119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

(11)




119909119889119896+1119894

=

119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

if 119880 [0 1] lt 05

119875best119889119896119894

Otherwise(12)





of H2S and CO








1652 1653 1654 1655 1656 1657860

865

870

875

880

885f1(x)

x1

(a)

1652 1653 1654 1655 1656 165750

52

54

56

58

60

62

64

66

f2(x)

x1

(b)

465 470 475 480 485 490860

865

870

875

880

885

890

f1(x)

x2

(c)

465 470 475 480 485 49052

54

56

58

60

62

64

66

f2(x)

x2

(d)

1000 1001 1002 1003 1004 1005 1006 1007860

865

870

875

880

885

890

f1(x)

x3

(e)

1000 1001 1002 1003 1004 1005 1006 100750525456586062646668

f2(x)

x3

(f)

OptimalNon-optimal

f1(x)

x4

102 104 106 108 110 112 114 116860

865

870

875

880

885

890

(g)

OptimalNon-optimal

f2(x)

x4

104 106 108 11050

52

54

56

58

60

62

64

66

(h)



1(119909) and 119891

2(119909)


070 075 080 085 090860

865

870

875

880

885

890f1(x)

x5

(a)

070 075 080 085 09050

52

54

56

58

60

62

64

66

f2(x)

x5

(b)

070 075 080 085 090 095 100 105 110860

865

870

875

880

885

890

f1(x)

x6

OptimalNon-optimal

(c)

070 075 080 085 090 095 100 105 11050

52

54

56

58

60

62

64

66

f2(x)

x6

OptimalNon-optimal


5and 119909


1(119909) and 119891

2(119909)





1198912(119909) versus 119909

1 119909









4 1199095 and 119909




to the same trend

7 Conclusion




119894



120583119889119896119894

=(119875best119889119896

119894+ 119875119892best119889119896

119894)

2

120590119889119896119894

=10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best119889119896

119894

10038161003816100381610038161003816If 119880[0 1] lt 05Then119909119889119896+1119894

= 119873(120583119889119896119894 120590119889119896119894)

Else119909119889119896+1119894

= 119875best119889119896119894

End ifEnd for


lt 119909119889min OR 119909119889119896+1119894

lt 119909119889max Then119909119889119896+1119894

= 119909119889min + 119880[0 1] lowast (119909119889

max minus 119909119889


119894) lt 119891(119875best119889119896

119894)Then

119875119892best119896119894= 119909119896+1119894

End if119870 = 119896 + 1







Acknowledgments


References



















2andH









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1300 1400 1500 1600 1700 1800

032

034

036

038

040

042

20

30

40

50

60

70

80

90

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



2and H

2S


35 40 45 50 55 60 65 70038

039

040

041

042

043

044


0

5

10

15

20

25

30

35

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



80 100 120 140 160 180

02

04

06

08

10

12

10

20

30

40

50

60

70

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



2S composi-

tion in sweet gas

90 105 120 135 15000

02

04

06

08

10

12


325

330

335

340

345

350

355

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2



060 065 070 075 080 085 09000

02

04

06

08

10

12

14


0

20

40

60

80

100

120Sw

eet g

as H

2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2


2S











08 09 10 11 1209725

09730

09735

09740

09745

09750

09755

09760

09765


38

39

40

41

42

43

44

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2


2S







119872119891

) lowast 100 (6)


Total Heat



2and H



5) since the feed


119878and 119872




119891








UB119894

1199091

1400 17001199092

40 501199093

90 1101199094

10 121199095

07 081199096

09 12




119910H2S lt 1 (8)

119910CO2

lt 50 (9)

119871119861119894le 119909119894le 119880119861119894 (10)


2S and CO




gas plant 119871119861119894and119880119861



2CO3DEA Cir-


2)


4)





1 1199092 119909

6 which optimize




Simulationmodel


New input values


8862 8864 8866 8868 8870525

530

535

540

545

550

Min

imiz

atio

n of

f1(x)



1(119909) and 119891

2(119909)




119894 the previous


119894 Kennedy [36]




119894 and the


119909119889119896+1119894

= 119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

(11)




119909119889119896+1119894

=

119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

if 119880 [0 1] lt 05

119875best119889119896119894

Otherwise(12)





of H2S and CO








1652 1653 1654 1655 1656 1657860

865

870

875

880

885f1(x)

x1

(a)

1652 1653 1654 1655 1656 165750

52

54

56

58

60

62

64

66

f2(x)

x1

(b)

465 470 475 480 485 490860

865

870

875

880

885

890

f1(x)

x2

(c)

465 470 475 480 485 49052

54

56

58

60

62

64

66

f2(x)

x2

(d)

1000 1001 1002 1003 1004 1005 1006 1007860

865

870

875

880

885

890

f1(x)

x3

(e)

1000 1001 1002 1003 1004 1005 1006 100750525456586062646668

f2(x)

x3

(f)

OptimalNon-optimal

f1(x)

x4

102 104 106 108 110 112 114 116860

865

870

875

880

885

890

(g)

OptimalNon-optimal

f2(x)

x4

104 106 108 11050

52

54

56

58

60

62

64

66

(h)



1(119909) and 119891

2(119909)


070 075 080 085 090860

865

870

875

880

885

890f1(x)

x5

(a)

070 075 080 085 09050

52

54

56

58

60

62

64

66

f2(x)

x5

(b)

070 075 080 085 090 095 100 105 110860

865

870

875

880

885

890

f1(x)

x6

OptimalNon-optimal

(c)

070 075 080 085 090 095 100 105 11050

52

54

56

58

60

62

64

66

f2(x)

x6

OptimalNon-optimal


5and 119909


1(119909) and 119891

2(119909)





1198912(119909) versus 119909

1 119909









4 1199095 and 119909




to the same trend

7 Conclusion




119894



120583119889119896119894

=(119875best119889119896

119894+ 119875119892best119889119896

119894)

2

120590119889119896119894

=10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best119889119896

119894

10038161003816100381610038161003816If 119880[0 1] lt 05Then119909119889119896+1119894

= 119873(120583119889119896119894 120590119889119896119894)

Else119909119889119896+1119894

= 119875best119889119896119894

End ifEnd for


lt 119909119889min OR 119909119889119896+1119894

lt 119909119889max Then119909119889119896+1119894

= 119909119889min + 119880[0 1] lowast (119909119889

max minus 119909119889


119894) lt 119891(119875best119889119896

119894)Then

119875119892best119896119894= 119909119896+1119894

End if119870 = 119896 + 1







Acknowledgments


References



















2andH









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










08 09 10 11 1209725

09730

09735

09740

09745

09750

09755

09760

09765


38

39

40

41

42

43

44

Swee

t gas

H2S

cont

ent (

ppm

v)

Swee

t gas

CO

2co

nten

t (pp

mv)

H2SCO2


2S







119872119891

) lowast 100 (6)


Total Heat



2and H



5) since the feed


119878and 119872




119891








UB119894

1199091

1400 17001199092

40 501199093

90 1101199094

10 121199095

07 081199096

09 12




119910H2S lt 1 (8)

119910CO2

lt 50 (9)

119871119861119894le 119909119894le 119880119861119894 (10)


2S and CO




gas plant 119871119861119894and119880119861



2CO3DEA Cir-


2)


4)





1 1199092 119909

6 which optimize




Simulationmodel


New input values


8862 8864 8866 8868 8870525

530

535

540

545

550

Min

imiz

atio

n of

f1(x)



1(119909) and 119891

2(119909)




119894 the previous


119894 Kennedy [36]




119894 and the


119909119889119896+1119894

= 119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

(11)




119909119889119896+1119894

=

119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

if 119880 [0 1] lt 05

119875best119889119896119894

Otherwise(12)





of H2S and CO








1652 1653 1654 1655 1656 1657860

865

870

875

880

885f1(x)

x1

(a)

1652 1653 1654 1655 1656 165750

52

54

56

58

60

62

64

66

f2(x)

x1

(b)

465 470 475 480 485 490860

865

870

875

880

885

890

f1(x)

x2

(c)

465 470 475 480 485 49052

54

56

58

60

62

64

66

f2(x)

x2

(d)

1000 1001 1002 1003 1004 1005 1006 1007860

865

870

875

880

885

890

f1(x)

x3

(e)

1000 1001 1002 1003 1004 1005 1006 100750525456586062646668

f2(x)

x3

(f)

OptimalNon-optimal

f1(x)

x4

102 104 106 108 110 112 114 116860

865

870

875

880

885

890

(g)

OptimalNon-optimal

f2(x)

x4

104 106 108 11050

52

54

56

58

60

62

64

66

(h)



1(119909) and 119891

2(119909)


070 075 080 085 090860

865

870

875

880

885

890f1(x)

x5

(a)

070 075 080 085 09050

52

54

56

58

60

62

64

66

f2(x)

x5

(b)

070 075 080 085 090 095 100 105 110860

865

870

875

880

885

890

f1(x)

x6

OptimalNon-optimal

(c)

070 075 080 085 090 095 100 105 11050

52

54

56

58

60

62

64

66

f2(x)

x6

OptimalNon-optimal


5and 119909


1(119909) and 119891

2(119909)





1198912(119909) versus 119909

1 119909









4 1199095 and 119909




to the same trend

7 Conclusion




119894



120583119889119896119894

=(119875best119889119896

119894+ 119875119892best119889119896

119894)

2

120590119889119896119894

=10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best119889119896

119894

10038161003816100381610038161003816If 119880[0 1] lt 05Then119909119889119896+1119894

= 119873(120583119889119896119894 120590119889119896119894)

Else119909119889119896+1119894

= 119875best119889119896119894

End ifEnd for


lt 119909119889min OR 119909119889119896+1119894

lt 119909119889max Then119909119889119896+1119894

= 119909119889min + 119880[0 1] lowast (119909119889

max minus 119909119889


119894) lt 119891(119875best119889119896

119894)Then

119875119892best119896119894= 119909119896+1119894

End if119870 = 119896 + 1







Acknowledgments


References



















2andH









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Simulationmodel


New input values


8862 8864 8866 8868 8870525

530

535

540

545

550

Min

imiz

atio

n of

f1(x)



1(119909) and 119891

2(119909)




119894 the previous


119894 Kennedy [36]




119894 and the


119909119889119896+1119894

= 119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

(11)




119909119889119896+1119894

=

119873((119875best119889119896

119894+ 119875119892best119889119896)2

10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best11988911989610038161003816100381610038161003816)

if 119880 [0 1] lt 05

119875best119889119896119894

Otherwise(12)





of H2S and CO








1652 1653 1654 1655 1656 1657860

865

870

875

880

885f1(x)

x1

(a)

1652 1653 1654 1655 1656 165750

52

54

56

58

60

62

64

66

f2(x)

x1

(b)

465 470 475 480 485 490860

865

870

875

880

885

890

f1(x)

x2

(c)

465 470 475 480 485 49052

54

56

58

60

62

64

66

f2(x)

x2

(d)

1000 1001 1002 1003 1004 1005 1006 1007860

865

870

875

880

885

890

f1(x)

x3

(e)

1000 1001 1002 1003 1004 1005 1006 100750525456586062646668

f2(x)

x3

(f)

OptimalNon-optimal

f1(x)

x4

102 104 106 108 110 112 114 116860

865

870

875

880

885

890

(g)

OptimalNon-optimal

f2(x)

x4

104 106 108 11050

52

54

56

58

60

62

64

66

(h)



1(119909) and 119891

2(119909)


070 075 080 085 090860

865

870

875

880

885

890f1(x)

x5

(a)

070 075 080 085 09050

52

54

56

58

60

62

64

66

f2(x)

x5

(b)

070 075 080 085 090 095 100 105 110860

865

870

875

880

885

890

f1(x)

x6

OptimalNon-optimal

(c)

070 075 080 085 090 095 100 105 11050

52

54

56

58

60

62

64

66

f2(x)

x6

OptimalNon-optimal


5and 119909


1(119909) and 119891

2(119909)





1198912(119909) versus 119909

1 119909









4 1199095 and 119909




to the same trend

7 Conclusion




119894



120583119889119896119894

=(119875best119889119896

119894+ 119875119892best119889119896

119894)

2

120590119889119896119894

=10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best119889119896

119894

10038161003816100381610038161003816If 119880[0 1] lt 05Then119909119889119896+1119894

= 119873(120583119889119896119894 120590119889119896119894)

Else119909119889119896+1119894

= 119875best119889119896119894

End ifEnd for


lt 119909119889min OR 119909119889119896+1119894

lt 119909119889max Then119909119889119896+1119894

= 119909119889min + 119880[0 1] lowast (119909119889

max minus 119909119889


119894) lt 119891(119875best119889119896

119894)Then

119875119892best119896119894= 119909119896+1119894

End if119870 = 119896 + 1







Acknowledgments


References



















2andH









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1652 1653 1654 1655 1656 1657860

865

870

875

880

885f1(x)

x1

(a)

1652 1653 1654 1655 1656 165750

52

54

56

58

60

62

64

66

f2(x)

x1

(b)

465 470 475 480 485 490860

865

870

875

880

885

890

f1(x)

x2

(c)

465 470 475 480 485 49052

54

56

58

60

62

64

66

f2(x)

x2

(d)

1000 1001 1002 1003 1004 1005 1006 1007860

865

870

875

880

885

890

f1(x)

x3

(e)

1000 1001 1002 1003 1004 1005 1006 100750525456586062646668

f2(x)

x3

(f)

OptimalNon-optimal

f1(x)

x4

102 104 106 108 110 112 114 116860

865

870

875

880

885

890

(g)

OptimalNon-optimal

f2(x)

x4

104 106 108 11050

52

54

56

58

60

62

64

66

(h)



1(119909) and 119891

2(119909)


070 075 080 085 090860

865

870

875

880

885

890f1(x)

x5

(a)

070 075 080 085 09050

52

54

56

58

60

62

64

66

f2(x)

x5

(b)

070 075 080 085 090 095 100 105 110860

865

870

875

880

885

890

f1(x)

x6

OptimalNon-optimal

(c)

070 075 080 085 090 095 100 105 11050

52

54

56

58

60

62

64

66

f2(x)

x6

OptimalNon-optimal


5and 119909


1(119909) and 119891

2(119909)





1198912(119909) versus 119909

1 119909









4 1199095 and 119909




to the same trend

7 Conclusion




119894



120583119889119896119894

=(119875best119889119896

119894+ 119875119892best119889119896

119894)

2

120590119889119896119894

=10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best119889119896

119894

10038161003816100381610038161003816If 119880[0 1] lt 05Then119909119889119896+1119894

= 119873(120583119889119896119894 120590119889119896119894)

Else119909119889119896+1119894

= 119875best119889119896119894

End ifEnd for


lt 119909119889min OR 119909119889119896+1119894

lt 119909119889max Then119909119889119896+1119894

= 119909119889min + 119880[0 1] lowast (119909119889

max minus 119909119889


119894) lt 119891(119875best119889119896

119894)Then

119875119892best119896119894= 119909119896+1119894

End if119870 = 119896 + 1







Acknowledgments


References



















2andH









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










070 075 080 085 090860

865

870

875

880

885

890f1(x)

x5

(a)

070 075 080 085 09050

52

54

56

58

60

62

64

66

f2(x)

x5

(b)

070 075 080 085 090 095 100 105 110860

865

870

875

880

885

890

f1(x)

x6

OptimalNon-optimal

(c)

070 075 080 085 090 095 100 105 11050

52

54

56

58

60

62

64

66

f2(x)

x6

OptimalNon-optimal


5and 119909


1(119909) and 119891

2(119909)





1198912(119909) versus 119909

1 119909









4 1199095 and 119909




to the same trend

7 Conclusion




119894



120583119889119896119894

=(119875best119889119896

119894+ 119875119892best119889119896

119894)

2

120590119889119896119894

=10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best119889119896

119894

10038161003816100381610038161003816If 119880[0 1] lt 05Then119909119889119896+1119894

= 119873(120583119889119896119894 120590119889119896119894)

Else119909119889119896+1119894

= 119875best119889119896119894

End ifEnd for


lt 119909119889min OR 119909119889119896+1119894

lt 119909119889max Then119909119889119896+1119894

= 119909119889min + 119880[0 1] lowast (119909119889

max minus 119909119889


119894) lt 119891(119875best119889119896

119894)Then

119875119892best119896119894= 119909119896+1119894

End if119870 = 119896 + 1







Acknowledgments


References



















2andH









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894



120583119889119896119894

=(119875best119889119896

119894+ 119875119892best119889119896

119894)

2

120590119889119896119894

=10038161003816100381610038161003816119875best

119889119896

119894minus 119875119892best119889119896

119894

10038161003816100381610038161003816If 119880[0 1] lt 05Then119909119889119896+1119894

= 119873(120583119889119896119894 120590119889119896119894)

Else119909119889119896+1119894

= 119875best119889119896119894

End ifEnd for


lt 119909119889min OR 119909119889119896+1119894

lt 119909119889max Then119909119889119896+1119894

= 119909119889min + 119880[0 1] lowast (119909119889

max minus 119909119889


119894) lt 119891(119875best119889119896

119894)Then

119875119892best119896119894= 119909119896+1119894

End if119870 = 119896 + 1







Acknowledgments


References



















2andH









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















2andH









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Multiobjective Optimization of a Benfield ...amine solutions such as DEA, MDEA, and DGA in a gas sweetening process. Rahimpour and Kashkooli [ , ] developed a mathematical