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Energy 36 (2011) 4777e4789

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Energy

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Implementing of the multi-objective particle swarm optimizer and fuzzydecision-maker in exergetic, exergoeconomic and environmental optimizationof a benchmark cogeneration system

Hoseyn Sayyaadi*, Meisam Babaie, Mohammad Reza FarmaniFaculty of Mechanical Engineering-Energy Division, K.N. Toosi University of Technology, P.O. Box 19395-1999, No. 15e19, Pardis Str., Mollasadra Ave., Vanak Sq.,Tehran 1999 143344, Iran

a r t i c l e i n f o

Article history:Received 30 September 2010Received in revised form21 April 2011Accepted 11 May 2011

Keywords:CGAM cogeneration systemExergoeconomicsEnviromental impactExergeticsMulti-objective particle swarm optimization(MOPSO)Fuzzy decision-making

* Corresponding author. Tel.: þ98 21 8867 4212; faE-mail addresses: sayyaadi@kntu.ac.ir, hoseynsayya

0360-5442/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.energy.2011.05.012

a b s t r a c t

Multi-objective optimization for design of a benchmark cogeneration system namely as the CGAMcogeneration system is performed. In optimization approach, Exergetic, Exergoeconomic and Environ-mental objectives are considered, simultaneously. In this regard, the set of Pareto optimal solutionsknown as the Pareto frontier is obtained using the MOPSO (multi-objective particle swarm optimizer).The exergetic efficiency as an exergetic objective is maximized while the unit cost of the system productand the cost of the environmental impact respectively as exergoeconomic and environmental objectivesare minimized. Economic model which is utilized in the exergoeconomic analysis is built based on bothsimple model (used in original researches of the CGAM system) and the comprehensive modelingnamely as TTR (total revenue requirement) method (used in sophisticated exergoeconomic analysis).Finally, a final optimal solution from optimal set of the Pareto frontier is selected using a fuzzy decision-making process based on the BellmaneZadeh approach and results are compared with correspondingresults obtained in a traditional decision-making process. Further, results are compared with the cor-responding performance of the base case CGAM system and optimal designs of previous works anddiscussed.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Optimization of thermal systems is one of the most importantsubjects in the energy engineering field. Exergy analysis helps tohighlight the thermodynamic inefficiencies of a system. It is clearthat the thermodynamic improvement of a system withoutconsideration about economical and environmental issues mightbe misleading. Hence, for designing of thermal systems, an inte-grated procedure should be performed to consider all these aspects,simultaneously. Exergoeconomics (thermoeconomics) combinesthe exergy (thermodynamic) analysis with economic principles andincorporates the associated costs of the thermodynamic ineffi-ciencies in the total product cost of an energy system [1]. Exer-goeconomic optimization aims at minimization of the total cost ofan energy system product. In a more comprehensive approachmultiple and even conflicting criteria would be considered simul-taneously. A multi-objective optimization problem requires thesimultaneous satisfaction of a number of different and often

x: þ98 21 8867 4748.adi@gmail.com (H. Sayyaadi).

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conflicting objectives. An example for comparison of differentexergoeconomic methodologies to the design optimization ofa benchmark energy system has been presented for a test casecogeneration system namely as the CGAM1 problem [2e6].

Application of multi-objective optimization method in thermalsystems is not very old. In 2002, Toffolo et al. [7], considered twoobjectives, energetic and economic, in optimization of the CGAMproblem. In their work, the shortcoming of conventional mathe-matical optimization approaches in acquiring of the globaloptimum was relatively maintained using the evolutionary algo-rithms (MOEAs). They presented a Pareto optimum frontier insteadof the single optimum solution of the conventional single-objectiveoptimization. In another work, they improved their work byconsidering the environmental impact as an additional objective totheir previous work and introduced a three objectives optimizationproblem [8]. Unit damage costs were devoted to NOx, CO emissionsand environmental objective was introduced in cost term. However

1 CGAM is adopted from initials of four researchers who defined the benchmarkcogeneration system. These researchers are Cristopher Frangopolous, GeorgeTsatsaronis, Antonio Valero and Michael Von Spakovsky.

Nomenclature

_CP the unit cost of the system product ($ hr�1)_Cenv the cost of the environmental impact ($ hr�1)cF unit cost of the fuel ($ kW�1)CHPI combined heat and powerCI capital investment ($)_E the rate of exergy (kW)e specific exergy (kJ kg�1)max maximum operator in the fuzzy logicsmin minimum operator in the fuzzy logicsMOEAs multi objective evolutionary algorithmsMOPSO multi objective particle swarm optimizer_m flow rate (kg s�1)OMC operating and maintenance costP pressure (kPa)PEC purchase equipment cost ($)r pressure ratioT temperature (�C or K)TRR total revenue requirement_W power (kW)_Zk the total cost rate of kth component including the

capital investment and operating-maintenance cost($ hr�1)

_ZCIk the rate of capital investment of kth component($ hr�1)_ZkOM the rate of operating and maintenance cost of kth

component($ hr�1)

DTP pinch temperature difference in the HRSG (�C or K)DP pressure difference (kPa)

Greek lettersg decision variableh efficiencys annual operating hours (hr)3 exergetic efficiencyhsc compressor isentropic efficiencyhst gas turbine isentropic efficiencym membership function

Subscripts0 index for the first year of the system operation,

restricted dead state1,2,.,12 states 1,2,.,12 at the CGAM systemac air compressoraph air pre-heatercc combustion chamberNDIM non-dimensionalec economizerev evaporatorhrsg heat recovery steam generatori,j element ith, jthk component kthL levelized

H. Sayyaadi et al. / Energy 36 (2011) 4777e47894778

their work still suffers from some shortcoming arose from thesimplification in the selection of decision variables and constraints.Among the five decision variables considered in the original defi-nition of the original CGAMproblem, they chose only three decisionvariables including compressor pressure ratio (rcp), temperature ofthe combustion products entering the gas turbine (T4) and effi-ciency of the air pre-heater (3ap). Further, they utilized a simpleeconomic model instead of comprehensive economic models suchas the total revenue requirement (TRR) method. On the other handthey did not present any decision making process to find a finaloptimal solution from the Pareto optimum solutions.

In a similar multi-objective optimization work with exergetic,economic and environmental objectives, Sayyaadi [9] used a morecomprehensive economic modeling (TRR method) instead of thesimple economic models that were used by Toffolo et al. inRefs. [7,8]. He considered the environmental objective in the costimpact form and integrated it into the exergoeconomic objectivefunction and introduced a new objective namely as thermoen-vironomic objective function. He utilized the evolutionary algo-rithm in order to optimize the thermoenvironomic and exergeticobjectives in a multi-objective optimization approach. Decisionvariables were selected same as those decision variables consideredin the original definition of the CGAM problem. Further, theyintroduced a simple method for decision-making from the Paretofrontier as an example of decision-making process and a finaloptimal solution was introduced.

In comparisonwith previous studies in this field (Refs. [7e9]), thiswork utilizes the faster andmore confidant algorithm in optimizationprocedure namely as multi-objective particle swarm optimizer(MOPSO). This algorithm can overcome the problem of non-feasiblesolution which has been faced in Refs. [7,8]. No decision variables arechanged or fixed, all variables and constraints are in accordance withthe original definition of the CGAM problem. These improvementslead to results which aremore realistic than the corresponding resultsobtained before. Another difference between the present work and

previous work in this area [9], is in shape of Pareto frontier which isacquired in a three dimensional space instead of the two-dimensionalPareto frontier obtained in Ref. [9]. Since the order of magnitude forthe cost impact of the environmental objective ismuch lower than theorder ofmagnitude for the exergoeconomic objective, it is a danger forcancellation of the effect of the environmental objective by its inte-gration into the exergoeconomic objective as performed in Ref. [9].Therefore, the environmental cost impact objective is considered as anindependent objective in thiswork.Moreover, the gas emissionwhichconsidered in the environmental objective of Refs. [7e9]was only duetoNOx andCOemission. The cost impact of CO2 emission is consideredin this work as another source of environmental damage too. Further,the cost damage of unit CO emission ismodified based on new source.

It should be noted that in the three-objective space, the Paretofrontier is a 3D surface so that the decision-making process is moredifficult and complicated than the two-dimensional Pareto frontierof Ref. [9]. Moreover, in multi-objective optimizations of previousworks, therewas a lack of a systematic approach for decision-makingprocess in which a final optimal solution is selected from the Paretofrontier. These problems have been solved by implication of a deci-sion-making process based on the fuzzy logics. In the presentwork, after acquiring of the Pareto optimal frontier, a fuzzy basedapproach namely as the BellmaneZadeh method [10] is employedfor the decision-making process and its results are compared withthe base case design and optimal design obtained based on tradi-tional decision-making method namely as the LINMAP [11,12]method and also optimal results obtained in previous works.

2. Multi-objective optimization and decision-making

A multi-objective optimization problem requires the simulta-neous satisfaction of a number of different and often conflictingobjectives. These objectives are characterized by distinct measuresof performance that may be (in) dependent and/or incommensu-rable. It is required mentioning that there is no combination of

H. Sayyaadi et al. / Energy 36 (2011) 4777e4789 4779

decision variables values that optimizes all the components of theobjective vector, simultaneously. For instance, the objectives cansimultaneously be to minimize the negative environmental impactof the process, maximize the profit and to maximize the safety ofthe process [11e14].

A common way of defining optimality is the Pareto optimalitywhich is defined as follows, an objective vector z*˛Z is a Paretooptimal in the absence of another objective vector z˛Z such that zj� z*j cj˛f1;2;.; kg, andz*j 3zj, for at least one index j. Multi-Objective Particle Swarm Optimization (MOPSO) is an example ofa heuristic approachwhich is a faster andmore confidant algorithmin optimization procedure than traditional meta-heuristic methodsuch as genetic algorithms. Pareto optimal solutions are mathe-matically equal, and a decision-maker is needed to select thepreferred solution. In the case of the decision-maker being able tostate the preferences before the solution process, a priori methodscan be applied. Examples of a priori methods are the value functionmethod [15], lexicographical-ordering [16], goal-programming [17]and BellmaneZadeh approach [10]. BellmaneZadeh approach isa decision-making process in a fuzzy environment for acquiring ofa final optimal solution in multi-objective optimization problems.

The BellmaneZadeh approach permits one to realize an effec-tive (from the computational standpoint) as well as rigorous (fromthe standpoint of obtaining solutions) method for analyzing ofmulti-objective models [18]. Finally, its usage allows one topreserve a natural measure of uncertainties in the decision-makingand to take into account indices, criteria and constraints of quali-tative character. In this work, the fuzzy decision-making is appliedto multi-objective optimization problems.

2.1. Particle swarm optimization

PSO (particle swarm optimization) is an exciting new method-ology in the evolutionary computation that is somewhat similar to thegenetic algorithm in that the system is initialized with a population ofrandom solutions [19]. Unlike other algorithms, however, eachpotential solution (called a particle) is also assigned a randomizedvelocity and then flown through the problem hyperspace.

The algorithmdefines a population of particles (swarm) as randomguesses in the variables’ search space. Then, an iterative process is setby changing the position of particles within the search space toimprove the fitness function quality. The particles evaluate the fitnessfunction value iteratively and remember the location of their bestsuccess. This location is called PBest for each particle. In each iteration,the particles are also able to connect to the particle that contains thebest fitness function value (GBest) and try to follow it.

As indicated, each n dimensional particle within a swarm isa representative candidate solution, Xi ¼ ðxi1; xi2;.; xinÞ, whichcontains the following information:

1) PBest (personal best position): The best position of a particlewhich is obtained so far and includes the best fitness value ofthe objective function.

2) GBest (global best position): Thepositionof a particle that includesthe best fitness value in the swarm Pg ¼ ðpg1; pg2;.; pgnÞ:

3) Particle’s current velocity: The velocity of each particle that is theamount of change in its position, Vi ¼ ðvi1; vi2;.; vinÞ: Eq. (1) isused to compute particles’ new velocity in the next generation:

vijðtþ1Þ ¼wvijðtÞþc1r1hpijðtÞ�xijðtÞ

iþc2r2

hpgjðtÞ�xijðtÞ

i(1)

Where, i¼ 1;2;.;n is the size of the swarm population; w is theinertia weight factor which is often used as a parameter to controlexploration and exploitation in the search space; c1 and c2 are two

positive constant coefficients to specify particles’ tendencies ofusing their cognitive and social experiences; and r1 and r2 are tworandom numbers within the range (0,1).

Each particle’s new position in the search space is calculated by:

xijðt þ 1Þ ¼ xijðtÞ þ�vijðt þ 1Þ � 1

�(2)

In PSO, the size of swarm population is fixed and the members ofthe swarm can only be adjusted by their PBest and GBest to producethe off-spring. Each off-spring replaces its parent if its fitness isbetter than its parent’s fitness at each iteration. In the Multi-objective Particle Swarm Optimization (MOPSO), however, a setof solutions called non-dominated solutions are found during theprocess of the optimization using the concept of the Paretooptimality.

The recent MOPSO works applied the concept of Pareto domi-nance to select Pareto solutions in a swarm as leaders to convergethe solutions to the Pareto frontier. Some proposed methods usedextra conditions such as diversity estimation for reaching betterdiversity of solutions along the Pareto frontier. In this work,a MOPSO algorithm with an external archive to store the non-dominated solutions which are found during the process of opti-mization is applied. The leaders of swarms that guide the particlesto the Pareto frontier are selected from the top portion of thisarchive at each iteration. Moreover, the concept of MAXIMIN[19,20] is applied here to improve the convergence and diversity ofthe Pareto optimal solutions. The MAXIMIN fitness for a decisionvector X can be evaluated by following steps. First the minimalvalue from set ffiðXÞ � fiðYÞjci˛f1;.;mgg is obtained by callingthe min function:

mini¼1;.;mffiðXÞ � fiðYÞg (3)

When m is the number of objective functions. Two loops ofcomparison containing the min first stepping through all theobjectives from 1 to m, and then the max looping through allcandidate solutions in the population from 1 to N, except X, takeplace using Eq. (3). Another loop will be required to obtain allPareto solutions. In this loop each solution in the populationwill bechecked from 1 to N by Eq. (4):

fmaximin ¼ maxj¼1;.;N;XsY�mini¼1;.;mffiðXÞ � f ðYÞg� (4)

The comparison among the particles and their PBests is based onfully connected approach to increase the selection pressure towardthe true Pareto frontier. In order to reduce the cost of computationduring the process, a dynamic fitness inheritance technique whichis proposed in Ref. [21] is utilized. The following formulas calculatethe new position of a particle in the objective space using thefitness inheritance technique:

VFiðtÞ ¼ c1r1hFPBestðiÞðtÞ � FiðtÞ

iþ c2r2

hFGBestðiÞðtÞ � FiðtÞ

i(5)

FiðtÞ ¼ Fiðt � 1Þ þ VFiðtÞ (6)

Where Fi(t), FPBest(i)(t) and FGBest(i)(t) are ith objective function valuefor the current particle, and its PBest and GBest values respectively.

The parameter pt, called inheritance, indicates the proportion ofparticles that their objective function values must be inheritedinstead of evaluation at each iteration where t is the iteration(generation) number of the optimization process. To determine theamount of pt, following nonlinear function is used when Itr is thetotal number of iterations:

pt ¼ pðsÞ ¼ s2; s ¼ tItr

(7)

H. Sayyaadi et al. / Energy 36 (2011) 4777e47894780

It is important to note that as the Pareto optimal solutions at the end oftheoptimizationprocessmustbe truevaluesof theobjective functions,the inherited objective functions values cannot enter into the finalexternal repository [19]. Fig. 1 shows the flow diagram of the MOPSO.

In this study, the MOPSO algorithm uses 100 particles and 200iterations. The constant coefficients of c1 ¼ 2 and c2 ¼ 2 are appliedto consider both social and cognitive characteristics of each particleequally. In this work, at first, we applied the inertia weight factor,wk, proposed in Ref. [22] in which is adjusted dynamicallythroughout the optimization process [22].

wk ¼ wmax ��wmax �wmin

kmax

�k (8)

Where k and kmax are current and maximum iteration numbers,respectively andwmax andwmin are set to 0.9 and 0.1.We also applieda constant value of inertia factor, w ¼ 1, to our MOPSO algorithm.Comparing the obtained results using dynamic and constant inertiafactor value, we have noticed that it does not have serious effect onthe quality of our solutions. Hence, the inertia factor value,w ¼ 1, isused to control the exploration and exploitation of the swarm in thesearch space. To control the magnitude of velocity, maximumamount of velocities are adjusted to the subtraction of lowerboundaries from upper boundaries multiplied by 0.1. If a particlegoes beyond the variables’ boundaries, then its value is reintegratedinto the lower or upper boundary and its velocity ismultiplied by�1having the effect of searching the opposite direction.

Fig. 1. The flow diagram

2.2. Fuzzy decision-making

2.2.1. Different aspects of decision-makingThe subject of decision-making is, as the name suggests, the

study of how decisions are actually made and how they can bemade better or more successfully. That is, the field is concerned, ingeneral, with both descriptive theories and normative theories.Applications of fuzzy sets within the field of decision making have,for the most part, consisted of fuzzifications of the classical theoriesof decision making. While decision making under conditions of riskhave been modeled by probabilistic decision theories and gametheories, fuzzy decision theories attempt to deal with the vague-ness and non-specificity inherent in human formulation of pref-erences, constraints, and goals. Several classes of decision-makingproblems are usually recognized [23]. In multi-objective decisionproblems, relevant alternatives are evaluated according toa number of criteria. Each criterion induces a particular ordering ofthe alternatives, and we need a procedure by which to constructone overall preference ordering. The lack of clarity in the concept of“optimal solution” is the basic methodological complexity insolving multi-objective problems. When applying the Bell-maneZadeh approach [10], this concept is defined with reasonablevalidity: the maximum degree of implementing goals serves asa criterion of optimality. This conforms to the principle of guaran-teed result and provides constructive lines in obtaining harmonioussolutions [18].

of the MOPSO [22].

H. Sayyaadi et al. / Energy 36 (2011) 4777e4789 4781

2.2.2. BellmaneZadeh approachWhen using the BellmaneZadeh approach, each Fj(X) is replaced

by a fuzzy objective function or a fuzzy set

Aj ¼nX;mAj

ðXÞo

X˛L; j ¼ 1;2;.; k (9)

Where mAjðXÞ is a membership function of Aj [24].

A final decision is defined by the Bellman and Zadeh model asthe intersection of all fuzzy criteria and constraints and is repre-sented by its membership function. A fuzzy solution D with settingup the fuzzy sets (10) is turned out as a result of the intersectionD ¼ Xk

j¼1Aj with a membership function

mDðXÞ ¼ Xkj¼1mAj

ðXÞ ¼ minj¼1;.;k

mAjðxÞ X˛L (10)

Using Eq. (11), it is possible to obtain the solution proving themaximum degree as follows,

max mDðXÞ ¼ max minX˛L j¼1;.;k

mAjðxÞ (11)

X0 ¼ arg maxX˛L

minj¼1;.:;k

mAjðxÞ (12)

To obtain Eq. (11), it is necessary to build membershipfunctionsmAjðXÞ, j ¼ 1, ., k reflecting a degree of achieving “own”optimas by the corresponding FjðXÞ;X˛L, j¼ 1,., k. This is satisfiedby the use of the membership functions [25]. The membershipfunction of objectives and constraints, linear or nonlinear, can bechosen depending on the context of problem. One of possible fuzzyconvolution schemes is presented below [26].

� Initial approximation for X-vector is chosen. Maximum(minimum) values for each criterion Fj(X) are established viascalar maximization (minimization). Results are denoted as‘‘ideal’’ points fX0

j ; j ¼ 1;.;mg:� The matrix table {T}, where the diagonal elements are ‘‘ideal’’points, is defined as follows:

266F1�X01

F2�X01

. Fn

�X01

� � �

377

fTg ¼666664F1 X0

2 F2 X02 . Fn X0

2:::F1�X0n

F2�X0n

. Fn

�X0n

777775

(13)

� Maximum and minimum bounds for the criteria are defined:

Fmini ¼ min

jFj�X0j

; i ¼ 1;.;n (14a)

Fmaxi ¼ max

jFj�X0j

; i ¼ 1;.;n (14b)

� The membership functions are assumed for all fuzzy goals asfollows. For minimized objective functions

mFiðXÞ ¼

8>>><>>>:

0 if FiðxÞ>Fmaxi ;

Fmaxi � Fi

Fmaxi � Fmin

i

if Fmini < Fi � Fmax

i ;

1 if FiðxÞ � Fmini

(15a)

For maximized objective functions

mFiðXÞ ¼

8>>><>>>:

1 if FiðxÞ>Fmaxi ;

Fi � Fmaxi

Fmaxi � Fmin

i

if Fmini < Fi � Fmax

i ;

0 if FiðxÞ � Fmini

(15b)

� Fuzzy constraints are formulated:

GjðXÞ � Gmaxj þ dj; j ¼ 1;2;.; k (16)

Where dj is a subjective parameter that denotes a distance ofadmissible displacement for the bound Gmax

j of the jth constraint.Corresponding membership functions are defined in followingmanner:

mGiðXÞ ¼

8>>><>>>:

0 if GiðxÞ>Gmaxi

1�GjðxÞ�Gmax

j

djif GiðxÞ �Gmax

i

1 if Gmaxi < GiðXÞ � Gmax

i þdj

(17)

� A final decision is determined as the intersection of all fuzzycriteria and constraints represented by its membership func-tions. This problem is reduced to the standard nonlinearprogramming problems: to find the such values of X and k thatmaximize k subject to

l � mFi ; i ¼ 1;2;.;nl � m ; j ¼ 1;2;.; k (18)

Gj

The solution of the multi-criteria problem discloses the meaning ofthe optimality operator and depends on the decision-maker’sexperience and problem understanding.

To visualize the decision-making process, a hypothetical casewith two objective functions is described here. Fig. 2 shows thefuzzy membership functions of two hypothetical objectives. In thisfigure, magnitudes of the membership function corresponding toeach optimal solution of the Pareto frontier are illustrated. Thehorizontal axis is dedicated to an index for each set of decisionvariables (each optimal solution of the Pareto frontier) and thevertical axis shows the degree of membership function of eachobjective corresponding to the proposed optimal solution of thePareto frontier. For example for a case with index number#15(optimal solution number #15 on the Pareto frontier), themembership functions for objectives #1 and #2 are 0.5 and 1,respectively. According to Eq. (8) the intersection of these twoobjectives, should be considered. Employing of the minimumoperator, leads to 0.5 for membership function of at index number#15. This procedure is applied for all set of decision variables and allintersected values will be collected in a vector. Then application ofthe maximum operator (Eq. (9)), the biggest value for intersectedpoints is selected as a desired value for membership function of thefinal optimal solution. The argument related to this membershipfunctionwill be selected as the final optimal solution. In the case ofhypothetical objectives illustrated in Fig. 2, the maximummembership function is located on intersection of membershipgraphs of two objectives. This is located at index number #30 withmembership function of 0.73.

3. Application of the algorithm to a case study

3.1. The CGAM problem

In 1990, a group of concerned specialists in the field of Exer-goeconomics (C. Frangopoulos, G. Tsatsaronis, A. Valero, andM. VonSpakovsky) decided to compare their methodologies by solving

Fig. 2. Fuzzy membership functions of two hypothetical objectives.

H. Sayyaadi et al. / Energy 36 (2011) 4777e47894782

a predefined and simple problem of optimization: the CGAMproblem, which was named from the first initials of the partici-pating investigators [2]. The CGAM problem originally is aneconomic optimization of a simple cogeneration system whichinvolves physical, thermodynamic, and economic models. Itassumes ideal gas behavior and constant heat capacities. The CGAMProblem designs a cogeneration plant which delivers 30 MW ofelectricity and 14 kg s�1 of saturated steam at 20 bar. The instal-lation consists of an AC (air compressor), APH (air pre-heater), CC(combustion chamber), GT (gas turbine), and HRSG (heat recoverysteam generator). The air pre-heater uses thermal energy from thecombustion gas leaving the turbine to heat the air entering thecombustion chamber. The structure of the cogeneration plant isillustrated in Fig. 3. The HRSG is composed of an EC (economizer)section where the feed water is heated and an EV (evaporator)section where the heated water is vaporized into steam. Thetemperature profiles of the air pre-heater and the HRSG are shownin Fig. 3b and c, respectively. Other specifications and operatingcondition of the CGAM problem for the base case design are [1]:

Fig. 3. (a) Schematic flow diagram of the CGAM cogeneration plant [7]; (b) Tem

T1 ¼ 298.15 K; P1 ¼ 1.01325 bar; T8 ¼ 298.15 K, P8 ¼ 20 bar;T10 ¼ 298.15 K; P10 ¼ 12 bar; T3 ¼ 850 K; T4 ¼ 1520 K; P2/P1 ¼ 10;hsc ¼ 0.86; hst ¼ 0.86.

3.2. The thermodynamic model

The utilized thermodynamic model is developed based on thefollowing basic assumptions [2,7,8]:

1. All processes are steady state.2. The principle of the ideal-gas mixture is applied for the air and

combustion products.3. The fuel is natural gas and it is assumed to be 100% methane.

The methane is an ideal gas.4. Heat loss from the combustion chamber is considered to be 2%

of the fuel lower heating value. All other components areconsidered adiabatic.5.Constant pressure loss ratios are considered in thecomponents.

6. The restricted dead state is P0 ¼ 1.013 bar and T0 ¼ 25 �C. 7.3%and 5% pressure losses are assumed for the air and gases in theair pre-heater, respectively. 8.5% pressure losses are assumedfor the gases in the HRSG and combustion chamber. Detail forthe comprehensive thermodynamic modeling of the CGAMproblem can be found in Refs. [1,27].

3.3. The exergoeconomic model

The governing equation of exergoeconomic model for the costbalancing of a component of an energy system is as follows [1],

Xnj¼1

cj _Ej

�k;inþ _Z

CIk þ _Z

OMk ¼

Xmj¼1

cj _Ej

�k;out (19)

Where cj is the unit cost of exergy for the jth stream to/from thecomponent, _Ej is exergy flow for the jth stream to/from thecomponent and _Z

CIk and _Z

OMk are the related cost of the capital

perature profile of the air pre-heater; (c) Temperature profile the HRSG [7].

H. Sayyaadi et al. / Energy 36 (2011) 4777e4789 4783

investment and operating and maintenance for the kth componentobtained in the economic model. In Eq. (17), n and m are the totalnumber of inlet and outlet exergy to/from the kth component,respectively. In this paper for exergoeconomic modeling of systemwe have used two different economic models. First the simple andeasy economicmodel as used in Refs. [2e8] is implemented and ourresults are compared with the optimal design of Ref. [3] as a basecase. Then a comprehensive economic modeling according to theTRR method [1] as used in Ref. [9] was implemented in order tocompare the final results with the corresponding results obtainedin Ref. [9]. Detail for simple economic modeling can be found in[2,7,8]. The comprehensive economic model developed based onthe TRR (total revenue requirement) method is explained in detailin Refs. [1,9].

3.4. The combustion pollutants

The original CGAM problem does not perform calculations onthe formation of pollutants within the combustion chamber. Asimple model, based on semi-analytical correlations [28], is addedhere to determine pollutant emissions, which are essential for thesetup of the environmental objective function. The adiabatic flametemperature, Tpz, in the primary zone of the combustion chamber isderived from the expression by Gulder [29]. The pollutant emis-sions in grams per kilogram of fuel are obtained based on thefollowing expression [9]:

_mNOx¼ 0:15� 1016s0:5e

�71100Tpz

�p0:053 ðDp3=p3Þ0:5

(20)

_mCO ¼ 0:179� 109e

7800Tpz

�p23sðDp3=p3Þ0:5

(21)

Where s is the residence time in the combustion zone (s is assumedconstant and is equal to 0.002 s); Tpz is the primary zonecombustion temperature; p3 is the combustor inlet pressure; Dp3/p3 is the non-dimensional pressure drop in the combustor (Dp3/p3 ¼ 0.05 as in the CGAM problem [2]). Note that the primary zonetemperature is used in the NOx correlation instead of the stoi-chiometric temperature, since the maximum attainable tempera-ture in premixed flames is Tpz, as pointed out by Lefebvre [30].

3.5. Objective functions, decision variables and constraints

Three objective functions of the multi-criteria optimizationproblem are the “total exergetic efficiency” (to be maximized), the“total cost rate of products” (to be minimized) and the “environ-mental impact” (to be minimized). The third objective functionexpresses the environmental impact as the total pollution damagecost ($ s�1) due to CO2, NOx and CO emissions is calculated bymultiplying their associated flow rates into their correspondingunit damage cost (cCO2

and cNOxare equal to 6.853 $ kg�1CO2 and

0.02086 $ kg�1NOx, respectively [8]). Further, the unit damage costof the CO emission is added to environmental objective function.This cost is estimated to be about 31% of NOx emission cost [31]. Themathematical formulations of objective functions are as follows,

Exergetic objective:

3tot ¼_WNET þ _msteamðe9 � e8Þ

_mfuelefuel(22)

Exergoeconomic objective:

_CP;tot ¼ _CF;tot þ _ZCItot þ _Z

OMtot (23)

Environmental cost impact objective:

_Cenv ¼ cCO2_mCO2

þ cNOx_mNOx

þ cCO _mCO (24)

The flow rate for carbon dioxide used in Eq. (24) is obtainedfrom solution of combustion process in the combustion chamber asdetailed in Refs. [1,27,31]. In this paper, for keeping the consistencywith the original CGAM problem [2], the same decision variableshave been selected as follows,

i. The compressor pressure ratio, p2/p1.ii. Isentropic efficiency of the compressor, hsc.iii. Isentropic efficiency of the turbine, hst.iv. Temperature of the air entering the combustion chamber, T3.v. Temperature of the combustion products entering the gas

turbine, T4.

With employing of the MOPSO algorithm, there is no need toreduce the number of decision variables for overcoming theoccurrence of non-feasible solutions.

Although the decision variables may be varied in optimizationprocedure, each decision variable is normally required to be withina given range as follows,

6 � p2=p1 � 16 (25)

0:6 � hsc � 0:9 (26)

0:6 � hst � 0:92 (27)

700 � T3 � 1000K (28)

1200 � T4 � 1550K (29)

The heat exchange between hot and cold streams in the air pre-heater and in the HRSG should satisfy the following feasibilityconstraints,

Air pre-heater:

T5>T3 (30)

T6>T2 (31)

HRSG:

DTP ¼ T7P � T9>0 (32)

T6 ¼ T9 þ DTP (33)

T7 ¼ T8 þ DTP (34)

T7P>T8P (35)

T7 � 378:15K (36)

The last constraint is an additional constraint with respect to theoriginal CGAMproblem imposed on the exhaust gases temperature,which must not fall below 378.15 K (105 �C). This limitation isconsidered to prevent the condensation of the water vapor exist inthe combustion products at the outlet section of economizer. Thecondensation of water vapor in the presence of carbon dioxide mayleads to the formation of carbonic acid which is corrosive materialand can damage the economizer surface.

Fig. 4. Pareto optimal frontier obtained by the NSGA-II.

Fig. 5. Pareto optimal frontier obtained by the MOPSO.

H. Sayyaadi et al. / Energy 36 (2011) 4777e47894784

4. Result and discussion

Multi-objective optimization of the CGAM system with objec-tives expressed by Eqs. (22)e(24) with aforementioned five deci-sion variables described in Section 3.5 (same decision variables asindicated in the original definition of the CGAM problem [2]) hasbeen performed. The optimization constraints as expressed byEqs. (25)e(36) are considered.

As indicated before, the exergoeconomic modeling has beenperformed based on both the simple economic model and totalrevenue requirement method. The former modeling has been usedin order to compare obtained results based on the multi-objectivefuzzy approach with corresponding results of the basic CGAMproblem (indicated in Ref. [2]). The TRR economic model is usedbecause it is a comprehensive economic model which considerersall facts evolved in the economic behaviors of a real thermal system.In this stage the improvement on results are discussed in compar-ison of recent research performed by one of authors in Ref. [9].

In order to provide a comparison of the obtained results by theMOPSO with other similar algorithms, we applied a heuristicmethod called Non-dominated Sorting Genetic Algorithm (NSGA-II) [32]. NSGA-II shows high performance and has been widelyutilized in various multi-objective design problems [33e35]. InFigs. 4 and 5, all non-dominated solutions for 200 iterations and100 populations’ size are plotted. As can be seen, MOPSO results areclearly better than NSGA-II in terms of diversity (the distribution ofthe points on the Pareto frontier is more uniformly distributed andthe MOPSO could find much more non-dominated solutions thanNSGA-II).

Fig. 6 is 3D view of the Pareto frontier in the three objectivesspace. For better insight of the 3D Pareto frontier, two differentviews of these optimal solutions are shown in Fig. 6a and b. Inobtaining the Pareto frontier as indicated in Fig. 6, the thermody-namic and economic modelings of are obtained based on simplemodels presented in early research [2e8] on the CGAM system.

The exergetic efficiency is dimensionless but other objectiveshave the dimension of $ hr�1. In order to show all objectives onsame coordinates as performed in Fig. 6, exergoeconomic andenvironmental objectives should be changed to non-dimensionalforms. Consequently following equations are used to obtain thedimensionless exergoeconomic objective ( _CP;NDIM) and dimen-sionless environment objective function ( _Cenv;NDIM):

_CP;NDIM ¼ _CP �MIN

_CP���

MAX_CP��MIN

_CP��

(37)

_Cenv;NDIM ¼ _Cenv�MIN

_Cenv

���MAX

_Cenv

��MIN_Cenv

��(38)

Maximum and minimum values of the exergoeconomic and envi-ronment objective functions which are obtained from the Paretofrontier are used in Eqs. (37) and (38). These values based on simplethermodynamic and economic models are:

"MIN

_CP�¼ 1309:7$$hr�1 MAX

_CP�¼ 3892:5$$hr�1

MIN_Cenv

�¼ 404:1$$hr�1 MAX_Cenv

�¼ 677:3$$hr�1

#(39)

Points on the surfaces of the Pareto frontiers (Fig. 6a and b)represent the best possible tradeoffs among the three objectives.There is not any absolute optimum inwhich all objectives have hadthe desirable value. Some points have the higher efficiency and aremore close to thermodynamic optimum, others have lower cost andthey are more cost effective. As it can be seen, the problem of non-feasible solutions is overcome and the shape of the Pareto optimalsolution with MOPSO algorithm is more appropriate in contrastwith previous studies [7e9]. Moreover there is no necessity for

changing decision variables or the range of constraints as it wasdone in Refs. [7,8]. It indicates that implementing of this algorithmin real optimization problems with numerous streams andcomponents might leads to reasonable results. Accuracy and speedof the algorithm in convergence process is another importantadvantage which should not be forgotten.

Now the final optimal solution must be selected from theobtained Pareto frontier (Fig. 6). As explained, the fuzzy decision-making process is implemented to select the final optimum. Thefuzzification of objective functions is performed using theminimum and maximum values of objectives as indicated in Eq.(15). Matrix T indicates the values which are necessary for fuzzifi-cation in Eq. (15) as follows,

fTg ¼

264FminCost ¼ 1309:7 $$hr�1 Fmax

Cost ¼ 3895:2 $$hr�1

FminEff ¼ 0:5110 Fmax

Eff ¼ 0:5380

FminEnv ¼ 404:1 $$hr�1 Fmax

Env ¼ 677:3 $$hr�1

375 (40)

With applying the BellmaneZadeh approach, the fuzzy decision-maker selects a final optimum point from the obtained Paretofrontier. The selected final optimal solution is marked on the Paretofrontier as indicated in Fig. 6a and b.

Table 1 compares the results obtained by the combination of themulti-objective optimization and fuzzy decision-maker with thebase case system. Thebase case system is optimumsolutionobtainedby the single-objective exergoeconomic optimization of earlyresearches [3]. The results of Table 1 are obtained while thermody-namic and economicmodeling is performed based on simplemodelsof Ref. [3]. As indicated in this table the value of the exergoeconomic

Fig. 6. The set of Pareto optimal solutions obtained based on the simple thermodynamic and economic models (a) view #1; (b) view #2.

H. Sayyaadi et al. / Energy 36 (2011) 4777e4789 4785

objective of the fuzzy selected optimal solution is increased 20%compared to the value obtained in the exergoeconomic singleobjective solution of Ref. [3]. Instead, magnitudes of exergetic andenvironmental objectives in our case show þ14.7% and þ18.8%improvements in comparison to the base case system, respectively. Itshould be noted that the original CGAM system as presented inRefs. [2e6] had no exergetic and environmental objectives. There-fore in the base case columnof Table 1,magnitudes of these objectivefunctions are calculated by us for the solution that Tsatsaronis andPisa [3] obtained in a single-objective exergoeconomic optimization.

The reason for better exergoeconomic performance of the basecase system (optimal design of Ref. [3]) is due to the fact that in

Table 1Comparison of the optimal solution selected by the fuzzy decision-maker and the correspoas per [3]).

Objective function anddecision variables

Base case design Optimal dof Ref. [3]

Exergetic objective :(%) 46.7 50.7Exergoeconomic objective: ($ hr�1) 1409.6 1303.2Environomic objective: ($ hr�1) 616.7 653.7Decision variables T3 (K) 850.00 914.3

T4 (K) 1520.00 1492.6hsc 0.8600 0.8468hst 0.8600 0.8786p2/p1 10.00 8.52

exergoeconomic single-objective optimization only aims at reduc-tion of the system product without any consideration on otherissues e.g. the limiting source of energy, the environmental damageand so on. In this paper, more realistic approach which considersconflicting objectives is applied. Hence, improvements on otherissues (limiting source of the energy, the environmental impact)has been obtained. It can be claimed that the final optimal design asobtained in this paper better suit various criteria than the base casesystem, even though its exergoeconomic objective is worse thanthe base case system. For validation of our code, in one programrunning, single-objective optimization of the CGAM system wasperformed while simple thermodynamic and economic models

nding results of the base case system (simple thermodynamic and economic models

esign Selected point by thefuzzy decision-maker

Improvement from theoptimal solution of Ref. [3] (%)

53.6 þ14.71693.9 �20.0465.8 þ18.8828.531535.900.88310.909512.48

Fig. 7. The set of Pareto optimal solutions for the case that thermodynamic and economic modeling are performed based the comprehensive thermodynamic ([27]) and the TRReconomic models ([1,9]), respectively.

H. Sayyaadi et al. / Energy 36 (2011) 4777e47894786

were utilized. In other word, we provided same condition asconsidered in Ref. [3]. In this case the optimization leaded to1306.8 $ hr�1 for the exergoeconomic objective. This value iscomparable with the corresponding result obtained in Ref. [3](1303.2 $ hr�1) with 0.3% difference. It means that particle swarmoptimizer works very well.

Simple thermodynamic and economic models used in obtainedresults as indicated in Fig. 6 and Table 1 is used here only in order tocompare our approaches with early works [2e6] of the CGAMproblem. Once more we perform the multi-objective optimizationand fuzzy decision-making while more comprehensive thermo-dynamic [1,27] and economic [9] models are implemented. Thesemodels are used because it is believed that the simple economicmodel cannot consider the complicated economic factors involvedin a real energy system [9]. Further the comprehensive thermo-dynamic model as performed in Refs. [1,27] more accuratelysimulates thermal behaviors of the CGAM system. Fig. 7 is 3D Paretofrontier obtained for the CGAM problem when the system ismodeled based on the comprehensive thermodynamic andeconomic (TRR) models. The final optimal solution selected by thefuzzy decision-maker is marked on Fig. 7.

Table 2 compares the features of the selected optimal design ofthis work with base case system and optimal design presented byone of author in Ref. [9]. The optimal design of the single-objective

Table 2Comparison of the optimal solution selected by the fuzzy decision-maker with correspohensive thermodynamic [1,27] and economic models [9]).

Objective function anddecision variables

Base case design Multi-objective optimaldesign of Ref. [9]

Exergetic objective: 3tot (%) 50.3 51.1Exergoeconomic objective: _CP

($ hr�1)3599.9 3329.4

Environomic objective: _Cenv

($ hr�1)573.6 29.1*(637.8)

Decision variables T3 (K) 850.00 874.00T4 (K) 1520.00 1476.76hsc 0.86 0.8364hst 0.86 0.8702p2/p1 10.00 8.59

exergoeconomic optimization as obtained by Tsatsaronis and Pisa[3] is considered as the base system in the first column of Table 2.However, this single-objective optimum design is modeled basedon our comprehensive thermodynamic and economic models.

Table 2 indicates that in our optimal design 5.2% and 19.0%improvements in the exergetic efficiency and environmental costdamage are obtained, respectively. However, the cost of the systemproduct is increased 4.6%. In comparison to the optimal design ofRef. [9], the exergetic efficiency and environmental cost damage ofour optimal system achieve 3.7% and 27.0% improvement, respec-tively. However, the cost of the system product in our optimalsystem is 12.5% higher than the optimal design of Ref. [9].

In explanation of results indicated in Table 2, it can be said that inour new approach, the environmental objective is relaxed as anindependent objective in the multi-objective process, however inRef. [9], the environmental objective was integrated into the costobjective and it did not treated as an independent objective function.Therefore, in our new approach more weight is considered for theenvironmental cost impact in comparison to other objectives. Thisspecial attention on environmental issues in our approach leads toa relatively higher improvement on the environmental cost impact atthe cost of a higher system product. The difference in the exergeticefficiency of our system and the optimal system of Ref. [9] is due todifferent approaches in the decision-making from the Pareto frontier.

nding results of the base case system and the optimal solution of Ref. [9] (compre-

Selected point by thefuzzy decision-maker

Improvement from thebase case (%)

Improvement from theoptimal design of [9] (%)

52.9 þ5.2 þ3.73746.2 �4.6 �12.5

465.8 þ19.0 þ27.0

843.551505.790.84060.884012.94

Fig. 8. Comparison of the selected final optimal solution using the fuzzy decision-maker with the ideal point and optimal solution selected based on the traditional decision-making(nearest point approach namely as LINMAP decision-making) (comprehensive thermodynamic and economic models).

H. Sayyaadi et al. / Energy 36 (2011) 4777e4789 4787

Decision-making of the final optimal solution of Ref. [9] was based ona traditional engineering judgment approach (see Ref. [9]); however,in this paper a systematic fuzzy approach based on BellmaneZadehmethod is employed. The new fuzzy approach considers a higherweight for the exergetic objective function in comparison to thedecision-making approach of Ref. [9].

In this stage, it is instructive to compare an optimal solutionobtained by the fuzzy decision-maker and an optimal pointselected by a traditional decision-making method namely as theLINMAP decision-making [11,12].

The comparison between various decision-making processes isillustrated in Fig. 8. The traditional process of decision-makingnamely as the LINMAP method used in Refs. [11e13] is performedwith introducing of a hypothetical point on the Pareto diagramnamely as the ideal point. This point is an ideal point in which allobjectives have their corresponding optimum values independentto other objectives. It is clear that it is impossible to have allobjectives at their optimum point, simultaneously. Therefore, theideal point is not located on the Pareto frontier as shown in Fig. 8. Inthe LINMAP method a point that has minimum spacial distancefrom the ideal point (the nearest point to the ideal point) is selecteda desirable final solution. The ideal point, fuzzy selected optimalsolution and the solution selected based on the nearest approach(LINMAP) are marked in Fig. 8.

Table 3 compares specifications of the fuzzy selected optimalsolution, selected optimal solution based on the traditionalapproach (the LINMAP decision-making) and the ideal point. As is

Table 3Comparison of the optimal solution selected by the fuzzy decision-maker with correspsolution approach and ideal point (comprehensive thermodynamic [1,27] and economic

Objective function anddecision variables

Ideal point (IP) Nearest point to the IP(LINMAP decision-maker)

Selected poby the fuzzydecision-ma

Exergetic objective: 3tot(%)

54.3 51.8 52.9

Exergoeconomicobjective: _CP ($ hr�1)

3273.8 3618.0 3746.2

Environomic objective:_Cenv ($ hr�1)

432.4 457.9 465.8

Decision variables T3 (K) 851.58 843.55T4 (K) 1479.22 1505.79hsc 0.8287 0.8406hst 0.8765 0.8840p2/p1 14.19 12.94

indicated in this table, the fuzzy selected solution has less deviationfrom exergetic performance of the ideal point compared to thecorresponding exergetic performance of the traditional optimalsolution. However, in other objectives (the exergoeconomic andenvironmental objectives), the optimal solution selected by theLINMAP has less deviation from the ideal point. The last column ofTable 3 compares improvements obtained by the fuzzy decision-maker compared to the optimal solution selected using the LIN-MAP method. This indicates that the new method has onlyimprovement on the exergetic efficiency objective. On the otherhand, the condition of the optimal solution introduced by theLINMAP is better in other objectives (the exergoeconomic andenvironmental objectives). It means that in this case, the fuzzydecision-maker more weigh the exergetic objective compared tothe weight of the exergetic objective introduced by the LINMAP. Itcan be found that at least in this case, the traditional LINMAPdecision-making approach employed leads to a more appropriatefinal optimal solution although it seems that it is not a systematicdecision-making approach in comparison to the systematic fuzzybased decision-making process.

5. Error analysis

Two methods of error analysis are performed here. In the firstmethod, error analysis has been performed based on the meanabsolute percent error namely as MAPE method. In this regard 21runs for multi-objective optimization have been performed and

onding results of the optimal design selected according to the traditional nearestmodels [9]).

int

ker

Deviation of thefuzzy selectedoptimal from theIP (%)

Deviation of theLINMAP solutionfrom the IP (%)

Improvements of the fuzzy selectedoptimal solution compared to theoptimal solution selected by theLINMAP (%)

�2.5 �4.6 þ2.2

�14.4 �10.5 �3.5

�7.7 �5.9 �1.7

Table 4Magnitude of objective functions for 21 runs of computer code and error analysisbased on MAPE method.

Run number Cost($ hr�1)

Efficiency(%)

Environment($ hr�1)

1 3746.1 52.9 465.92 3698.4 52.5 462.23 3846.9 52.8 450.44 3816.3 53.6 455.75 3618.1 52.1 455.26 3897.4 51.7 454.17 3794.7 53.5 462.08 3773.6 53.9 484.49 3775.9 50.7 468.110 3812.0 52.3 451.611 3765.1 53.7 476.112 3788.4 52.9 451.313 3839.3 54.0 454.914 3708.8 54.4 458.015 3905.3 55.0 449.916 3788.3 54.5 480.317 3851.4 54.7 447.818 3692.3 52.5 480.819 3668.8 51.6 448.720 3653.2 53.5 465.321 3773.2 51.3 456.5Error of each objective 4.47% 3.65% 3.07%Average error of three objectives 3.73%

H. Sayyaadi et al. / Energy 36 (2011) 4777e47894788

a final solution for each run is obtained using the fuzzy decision-making. Then magnitudes of each objective (thermoeconomic,environmental and exergetic objectives) are compared for eachobjective with its corresponding best value (presented as a finalsolution in this paper). Table 4 represents the values of eachobjective obtained in each run. The last row of this table indicateserror of each objective obtained using the MAPE method.

The average error of all objectives as indicated in Table 4 is3.07%.

In the second method of error analysis, whole population of thePareto frontier in each run compared with the correspondingpopulation obtained in the previous run based on correlationcoefficients method which works based on dispersion of eachpopulation of solution compared to the population obtained in theprevious run. In this regard in each run a vector of solutions for eachobjective obtained where the number of vector element is equalwith the total number of solutions of the Pareto frontier for theproposed objective. The disparity of each run for each objective isobtained from the following equation,

Dispersion error of the vector ¼ 1:0� a!$ b!

j a!jj b!j(41)

Where a! and b!

are present and previous vectors of each objective,respectively. The value of 0 for dispersion error means a completecoherence of two vectors and 1.0 indicated a complete incoherencebetween them. In this paper maximum dispersion error of thethermoeconomic, exergetic and environmental objectives areobtained respectively as 1.89%, 1.92% and 2.4% for 21 runs of thecomputer code. Therefore, the average error of three objectives for21 runs is 2.08% based on the correlation coefficients method.

6. Conclusion

Multi-objective optimization of a benchmark cogenerationsystemwas performed with three objective functions including theexergoeconomic objective, exergetic objective and environmentalcost impact objective. The problem of non-feasible solution inapplication of the evolutionary algorithm in previous researchs was

resolved by application of the multi-objective particle swarmoptimization (MOPSO), therefore decision variables were chosen inaccordance with the original definition of the CGAM problem.Special attention was paid on environmental issues of a typicalenergy system through definition of the independent environ-mental objective in comparison to previous works in which envi-ronmental objective was integrated into the exergoeconomic(environomic objective in previous works). Further, environmentalcost impact objective was modified based on the cost impact of allpollutants including NOx, CO, and CO2. A fuzzy approach based onBellmaneZadeh methodology was employed in decision-makingprocess of the multi-objective optimization. The obtained resultswere compared with original works of the CGAM system whilesimple thermodynamic and economic models (as used in originalworks of the CGAM system) were utilized. It was shown thata better exergetic and environmental operation in our optimizedsystem is achieved at the cost of a higher product cost. Moreover,sophisticated thermodynamic and economic modeling was per-formed as were used in recent works. A design with better envi-ronmental sound was obtained compared to previous works thatused comprehensive thermodynamic and economic models.Finally, the final optimal solution which is selected by the fuzzydecision-maker is compared with a solution obtained by thetraditional nearest solution approach namely as the LINMAPdecision-makingmethod. It was found that in this case, the LINMAPdecision-making approach leads to a more appropriate finaloptimal solution although it seems that it is not a systematicdecision-making approach in comparison to the systematic fuzzybased decision-making process.

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