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Technical Report

A methodological concept for phase change material selection based onmultiple criteria decision analysis with and without fuzzy environment

Manish K. Rathod ⇑, Hiren V. KanzariaDepartment of Mechanical Engineering, S.V. National Institute of Technology, Surat, 395 007 Gujarat, India

a r t i c l e i n f o

Article history:Received 27 July 2010Accepted 15 February 2011Available online 18 February 2011

a b s t r a c t

Selection of proper phase change material (PCM) plays an important role towards the development of alatent heat thermal energy storage system. Selection of the phase change material is a difficult andrestrained task due to the immense number of different available materials having different characteris-tics. One has to select such PCM which will give the desired thermal performance at minimum cost. Thisstudy deals with two Multiple Attribute Decision-Making (MADM) methods to solve PCM selection prob-lem. These two methods are technique for order preference by similarity to ideal solution (TOPSIS)method and fuzzy TOPSIS method that uses linguistic variable presentation and fuzzy operation. Boththe methods use an analytic hierarchy process (AHP) method to determine weights of the criteria. TOPSISand fuzzy TOPSIS methods are used to obtain final ranking. A problem to evaluate the best choice of PCMused in solar domestic hot water system is considered here to demonstrate the effectiveness and feasi-bility of the proposed model. Empirical results showed that the proposed methods are viable approachesin solving PCM selection problem. TOPSIS is suitable for the use of precise performance ratings while thefuzzy TOPSIS is a preferred technique when the performance ratings are vague and inaccurate.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The storage of thermal energy as a latent heat of phase changematerial (PCM) has generated considerable interest amongresearchers in recent times. It has a capacity to store large amountof heat in the form of latent heat of fusion [1]. Latent heat is ab-sorbed or released when a material melts or solidifies respectively.This gives material an extra heat storing capacity if their meltingpoint is located within the working temperature. Every latent heatthermal energy storage system requires a suitable PCM for use inparticular kind of thermal energy storage application.

Sharma et al. [2], Zalba et al. [3] and Abhat [4] have given over-views of phase change materials (PCMs) used in low temperatureapplications. PCMs can be broadly classified as paraffins, fatty acidsand salt hydrates. Paraffins are readily available, inexpensive andmelt at different temperatures relating to their carbon-chainlength. Fatty acids come from meat byproducts and vegetables.They are renewable and readily available but 2–3 times costly thanparaffins. The oldest and most studied group of PCMs is salthydrates. They have high thermal conductivity and low cost. Allthe group of PCM has its own characteristics, applications, advan-tages, and limitations. For the development of latent heat thermalenergy storage system, the choice of suitable PCM plays an impor-tant role in addition to heat transfer mechanisms.

However, in choosing the right material, there is not always asingle definite criterion of selection. The designers or engineershave to take into account a large number of material selection cri-teria depending upon the application. The ideal phase changematerial to be used for latent heat storage must have the followingcriteria: high sensitive heat capacity and heat of fusion; high den-sity and heat conductivity; chemically inert; non-toxic, non-flam-mable and non-hazardous; reasonable and inexpensive. As nosingle material can have all the required properties for an idealthermal storage media, one has to select such PCM which will givethe desired thermal performance at minimum cost.

The selection of an optimal material for an engineering applica-tion from among two or more alternative materials on the basis oftwo or more attributes or criteria is a Multiple Attribute Decision-Making (MADM) problem. MADM methods perform an importantrole for the decision process for both the small and large problem.In the past, lots of research had been reported for selection ofmaterial using classical MADM methods. The methods are simpleadditive weighted (SAW) method, weighted product method(WPM), technique for order preference by similarity to ideal solu-tion (TOPSIS), Vise Kriterijumska Optimizacija KompromisnoResenje (VIKOR) method, analytical hierarchy process (AHP), graphtheory and matrix representation approach (GTMA), etc. [5–7]. Jeeand Kang [8] proposed TOPSIS method to rank the candidatematerials for which several requirements are consideredsimultaneously. Shanian and Savadogo [9] applied TOPSIS methodas multiple criteria decision support analysis for material selection

0261-3069/$ - see front matter � 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.matdes.2011.02.040

⇑ Corresponding author. Tel.: +91 261 2201966; fax: +91 261 2228394.E-mail address: [email protected] (M.K. Rathod).

Materials and Design 32 (2011) 3578–3585

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of metallic bipolar plates for polymer electrolyte fuel cell. Chan andTong [10] used gray relation analysis approach (GRA) for multi cri-teria selection method. Rao [11] presented a logical procedure formaterial selection for a given engineering application. The proce-dure was based on an improved compromise ranking method.Rao and Davim [12] presented a logical procedure based on a com-bined TOPSIS and AHP method for material selection applicable toengineering design. Triantaphyllou and Mann [13] examined someof the practical and computational issues involved when the AHPmethod is used in engineering applications. Dagdeviren et al.[14] develops an evaluation model based on the AHP and TOPSISmethod for the optimal selection of weapon in fuzzy environment.Chaterjee et al. [15] proposed VIKOR, a compromise ranking meth-od and ELECTRE, an outranking method to solve material selectionproblem. Fayazbakhsh et al. [16] proposed Z-transformation meth-od for normalization of material properties to overcome the short-coming of modified digital logic (MDL) method used for materialselection in mehcanical design. Khabbaz et al. [17] proposed a fuz-zy logic approach for material selection. However, the procedureneeds many fuzzy IF-THEN rules. Jahan et al. [18] reviewed thevariety of quantitative selection procedures developed for screen-ing and choosing materials. They also proposed a linear assignmenttechnique which is relatively simpler than other MADM methodsfor material selection problem [19]. However, the main weaknessof the suggested method is that it may not be as precise as otherMADM methods when the material selection is based on quantita-tive material properties. Rao and Patel [20] proposed a novelMADM method for material selection for a considered design prob-lem. They used fuzzy logic to convert the qualitative attributes intoquantitative attributes.

The present study is aimed to propose a systematic evaluationmodel for the selection of a best suitable phase change materialused in latent heat thermal energy storage unit. The PCM selectionproblem is a MADM problem where many criteria should be con-sidered in decision making among a set of available alternatives.This problem also contains subjectivity, uncertainty and ambiguityin assessment process. TOPSIS is more efficient in dealing with thecorporeal attributes and the number of alternatives to be assessed.It has rational and understandable logic. TOPSIS gives a solutionthat is not only closest to the hypothetically best, but which is alsofarthest from the hypothetically worst. However, TOPSIS methodrequires an efficient tool to evaluate the relative importance of dif-ferent criteria with respect to the objective and AHP provides sucha tool. AHP is a powerful and flexible decision making process toget priorities and make the best decision when both tangible andnon tangible aspects of a decision need to be considered [12].Therefore, the present study employs AHP method to determinethe importance weights of evaluation criteria, and TOPSIS/fuzzyTOPSIS to obtain the performance ranking of the feasiblealternatives.

2. Methodology

2.1. Evaluation of weights: AHP method

The AHP is developed by Saaty [21] to solve complex problemsinvolving multiple criteria. It is useful to determine the relativeimportance of a set of criteria in a MADM problem. The AHP meth-od requires three steps: (i) developing structure of the model witha goal or objective, (ii) assessing the decision-maker evaluations bypair-wise comparison, (iii) using the eigenvector method to yieldweights for criteria. An advantage of the AHP over other MADMmethods is that AHP is designed to incorporate tangible as wellas non tangible factors. These factors are an important part ofthe decision process [22].

In the first step, a complex decision problem is structured as ahierarchy with an overall objective at the top level, the multiplecriteria that define the alternatives at the second level and thedecision alternatives at the third level [23]. Once the problemhas been decomposed and the hierarchy is constructed, the secondstep starts in order to determine the relative importance of the cri-teria within each level. To do so, one has to construct a pair-wisecomparison matrix using a scale of relative importance. The indexof importance is defined as shown in Table 1, according to Saaty’s[21] 1–9 scale. Third step is to ensure that the evaluation of thepair-wise comparison matrix is reasonable and acceptable per-forming consistency check.

The complete procedure of AHP method is as follows [22].

(1) Construct a pair-wise comparison matrix using a scale of rel-ative importance. Let C = {Cj|j = 1, 2, . . . , n} be the set of cri-teria. The result of the pair-wise comparison on n criteriacan be summarized in an (n � n) evaluation matrix A. Theevery element aij (i, j = 1, 2, . . . , n) denotes the comparativeimportance of criteria i with respect to criteria j. A criteriacompared with itself is always assigned the value 1 so themain diagonal entries of the pair-wise comparison matrixare all 1.

A ¼

1 a12 . . . a1n

a21 1 . . . a2n

. . . . . . . . . . . .

an1 an2 . . . 1

26664

37775 aji ¼ 1=aij; aij – 0 ð1Þ

(2) Find the relative normalized weight (Wi) of each criteria bycalculating the geometric mean of ith row and normalizingthe geometric means of rows in the comparison matrix.

GMi ¼ fai1 � ai2 � ai3 � . . .� aijg1=n ð2Þ

Wi ¼GMiPj¼nj¼1GMi

ð3Þ

(3) Obtain matrix X which denote an n-dimensional columnvector describing the sum of the weighted values for theimportance degrees of alternatives, then X = A �W, where

W ¼ ½W1;W2;W3; . . . ;Wn�T ð4Þ

X ¼ A �W ¼

1 a12 . . . a1n

a21 1 . . . a2n

. . . . . . . . . . . .

an1 an2 . . . 1

26664

37775

W1

W2

. . .

Wn

26664

37775 ¼

c1

c2

. . .

cn

26664

37775 ð5Þ

(4) Calculate the consistency values (CV) for the cluster of alter-natives represented by the vector

CVi ¼ci

Wi

(5) Find out the maximum eigenvalue kmax that is the average ofthe consistency values.

Table 1Relative importance of factors [7].

Relative importance (aij) Description (i over j)

1 Equal importance3 Moderate importance5 Strong importance7 Very strong importance9 Absolute importance2, 4, 6, 8 Intermediate values

M.K. Rathod, H.V. Kanzaria / Materials and Design 32 (2011) 3578–3585 3579


(6) Calculate the consistency index (CI) = (kmax � n)/(n � 1). Itshould be noted that the quality of the output of the AHPis strictly related to the consistency of the pair-wise compar-ison judgments.

(7) Obtain the random index (RI) for the number of criteria usedin decision making from Table 2 [21].

(8) Calculate the consistency ratio CR = CI/RI. The number 0.1is the accepted upper limit for CR. If the final consistencyratio exceeds this value, the evaluation procedure has tobe repeated to improve consistency. The measurement ofconsistency can be used to evaluate the consistency ofdecision makers as well as the consistency of overall hier-archy [24].

2.2. Evaluation of alternative

2.2.1. The TOPSIS methodThe TOPSIS (Technique for Order Performance by Similarity to

Ideal Solution) was first developed by Hwang and Yoon [25].According to this technique, the best alternative would be theone that is nearest to the positive-ideal (hypothetically best) solu-tion and farthest from the negative ideal (hypothetically worst)solution [12,26]. The positive-ideal solution is a solution that max-imizes the benefit criteria and minimizes the cost criteria. Thesame is reverse for negative ideal solution. There have been lotsof studies in the literature using TOPSIS for the solution of MADMproblems [14,27–30].

The formal TOPSIS procedure consists of following steps.

(1) Establish a decision matrix for the ranking in which columnsindicate criteria or attributes ðC1; C2;C3; . . . CnÞ while rowslist the competing alternatives ðA1;A2;A3; . . . AmÞ.

ð6Þ

An element xij of the matrix indicates the performance ratingof the ith alternative, Ai, with respect to the jth criteria Cj, asshown in Eq. (6).

(2) Calculate normalized rating for each element in the decisionmatrix. The normalized value rij of xij is calculated as definedin Eq. (7).

rij ¼xijffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPi¼m

i¼1 x2ij

q ; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ; n: ð7Þ

(3) Calculate weighted normalized value vij by multiplying thenormalized decision matrix by its associated weights whichis obtained by AHP method.

v ij ¼Wjrij ð8Þ

(4) Determine positive ideal (V+) and negative ideal solutions(V�). The ideal (best) and negative ideal (worst) solutionscan be expressed as:

Vþ ¼Xmax

i

v ij

,j 2 J

!;Xmin

i

v ij

,j 2 J0

!,i ¼ 1;2; . . . ;m

( )

¼ vþ1 ;vþ2 ;v

þ3 ; . . . ;vþn

� �ð9Þ

V� ¼Xmin

i

v ij

,j 2 J

!;Xmax

i

v ij

,j 2 J0

!,i ¼ 1;2; . . . ;m

( )

¼ v�1 ;v�2 ;v�3 ; . . . ;v�n� �

ð10Þ

where J = (j = 1, 2, . . . , n)/j is set of beneficial criteria (larger-the-better type) and J0 = (j = 1, 2, . . . , n) /j is set of non-bene-ficial criteria (small-the-better type).

(5) Obtain separation measures. The separation (distance)between alternatives can be measured by the n-dimensionalEuclidean distance. The separation of each alternative fromthe positive-ideal solution is given as

Sþi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

J¼1v ij � vþ2

j

ri ¼ 1;2 . . . ;m ð11Þ

Similarly, the separation from the negative ideal solution is asfollows:

S�i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

J¼1v ij � v�2

j

ri ¼ 1;2; . . . ;m ð12Þ

(6) Calculate the relative closeness to ideal solution. The relativecloseness of the alternative Aij can be expressed as:

Ri ¼S�i

Sþi þ S�ið13Þ

(7) Choose an alternative with maximum Ri or rank alternativesaccording to Ri in descending order.

2.2.2. The fuzzy TOPSIS methodThe TOPSIS is widely used to tackle ranking problems in real

situations. In the case of qualitative criteria, personal judgmentsare represented with crisp values. It is often difficult for a deci-sion maker to assign a precise performance rating to an alterna-tive which has the qualitative criteria. The merit of using fuzzyapproach is to assign the relative importance of criteria usingfuzzy numbers instead of precise numbers. As a result, fuzzyTOPSIS and its extensions are developed to solve ranking andjustification problems. Before the development of fuzzy TOPSIS,Some basic definitions regarding fuzzy theory are given asfollows.

Definition 1. A fuzzy set ~a in a universe of discourse X ischaracterized by a membership function lã which associates witheach element x in X, a real number in the interval [0, 1]. Thefunction value lã(x) is termed the grade of membership of x in ã[31].

The present study uses triangular fuzzy numbers. A triangularfuzzy number ã can be defined by a triplet (a1, a2, a3). The member-ship function is defined as

Table 2Random index (RI) values [21].

Criteria RI Criteria RI

3 0.52 7 1.354 0.89 8 1.45 1.11 9 1.456 1.25 10 1.49

3580 M.K. Rathod, H.V. Kanzaria / Materials and Design 32 (2011) 3578–3585


l~aðxÞ ¼

0; x � a1x�a1

a2�a1; a1 < x � a2

a3�xa3�a2

; a2 < x � a3

0; x > a3

8>>>><>>>>:

ð14Þ

Definition 2. Let ã = (a1, a2, a3) and ~b = (b1, b2, b3) are twotriangular fuzzy numbers, then the vertex method is defined tocalculate the distance between them, as Eq. (15):

dð~a; ~bÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13

a1 � b21 þ a2 � b2

2 þ a3 � b23

h irð15Þ

The basic operations on fuzzy triangular numbers are as follows:For multiplication,

~a� ~b ¼ ða1 � b1; a2 � b2; a3 � b3Þ ð16Þ

For addition,

~aþ ~b ¼ ða1 þ b1; a2 þ b2; a3 þ b3Þ ð17Þ

On the basis of brief fuzzy theory mentioned above, TOPSIS stepscan be followed as [31,32]:

(1) Choose the linguistic values (~xij, i = 1, 2, . . . , m, j = 1, 2, . . . , n)for alternatives with respect to criteria and the appropriatelinguistic variables ( ~wi) for the weight of the criteria. Thefuzzy linguistic rating (~xij) preserves the property that theranges of normalized triangular fuzzy numbers belong to[0, 1]; thus, there is no need for normalization.

(2) Calculate the weighted normalized fuzzy decision matrix.(3) Determine positive ideal (V+) and negative ideal solutions

(V�). The fuzzy ideal (best) and fuzzy negative ideal (worst)solutions can be expressed as:

Vþ ¼Xmax

i

~v ij

,j 2 J

!;Xmin

i

~v ij

,j 2 J0

!,i ¼ 1;2; . . . ;m

( )

¼ f~vþ1 ; ~vþ2 ; ~vþ3 ; . . . ; ~vþn gð18Þ

V� ¼Xmin

i

~v ij

,j 2 J

!;Xmax

i

~v ij

,j 2 J0

!,i ¼ 1;2; . . . ;m

( )

¼ f~v�1 ; ~v�2 ; ~v�3 ; . . . ; ~v�n gð19Þ

(4) Calculate separation measures. The separation (distance) ofeach alternative from V+ and V� can be currently calculatedas

Sþi ¼Xn

j¼1

d ~v ij; ~vþi i ¼ 1;2 . . . ;m ð20Þ

Similarly, the separation from the negative ideal solution is asfollows:

S�i ¼Xn

j¼1

d ~v ij; ~v�j i ¼ 1;2; . . . ;m ð21Þ

(5) Calculate the relative closeness to ideal solution as describedbefore.

Ri ¼S�i

Sþi þ S�ið22Þ

(6) Rank preference order. Choose an alternative with maxi-mum Ri or rank alternatives according to Ri in descendingorder.

Now a case study of PCM selection problem is used for the dem-onstration of the above mentioned method.

3. Selection of phase change material for solar water system

The most important groups of PCM with a solid–liquid phasechange are salt hydrates and paraffins. However scientists disagreeon which group is most promising. For instance, Sharma et al. [2]state that salt hydrates are the most important PCM group whileKim and Dzal [33] has shown that paraffins are most promising.Hence selection of the phase change material (PCM) plays animportant role in addition to heat transfer mechanisms. The focushere is on the selection of PCM used in solar domestic hot watersystem. A suitable PCM is chosen so that large amount of solar en-ergy can be stored during daytime and then this energy can beused for water heating at a later time. In the present case study,nine PCMs are considered which can be used in solar domestichot water system as latent heat storage materials. Now to demon-strate the proposed procedure of PCM selection, composed of AHPand TOPSIS/fuzzy TOPSIS, three basic stages are to be followed: (1)identify the criteria to be used in the model, (2) AHP computations,(3) evaluation of alternatives with TOPSIS/fuzzy TOPSIS and deter-mination of the final rank.

3.1. Identification of the criteria for PCM selection

The objective is to evaluate the best choice of PCM used in solardomestic hot water system to store large amount of heat. The PCMswhich can be used in the above system should have melting point30–60 �C [34]. The performance ratings for the nine alternativeswith respect to the six criteria are summarized in Table 3. Thealternatives and the criteria are presented here for demonstrationpurpose only. The criteria used here are latent heat (LH), density(D), specific heat for solid (Cp(s)), specific heat for liquid (Cp(l)),thermal conductivity (K) and cost (C) which influences the selec-tion of PCM for a given application. In the present case, the objec-tive value of cost is qualitative type (i.e., quantitative value is notavailable). To convert these into quantitative value, a ranked valuejudgment on a fuzzy conversion scale is adopted. An eleven-pointscale is considered which represents the material selection criteriaon a qualitative scale using fuzzy logic. The same is presented inTable 4. The latent heat, density, specific heat for solid, specificheat for liquid, and thermal conductivity are beneficial criteria,i.e., higher values are desired. The remaining is non-beneficial cri-teria, i.e. cost. Table 5 which represents decision matrix shows theobjective data of the criteria with fuzzy score.

3.2. Calculate the weights of criteria

After forming the decision hierarchy for the problem, theweights of the criteria to be used in evaluation process are calcu-lated by using AHP method. In this stage, an individual pair-wisecomparison matrix is developed by using the scale given in Table1. Let the decision maker makes the following assignments:

Latent heat (LH) is considered strongly more important thandensity (D) in PCM selection. So a relative importance value of 5is assigned to LH over D (i.e., a12 = 5) and a relative importance va-lue of 1/5 is assigned to D over LH (i.e., a21 = 1/5). Likewise, LH is



considered as more important than cost (C). So a relative impor-tance value of 9 is assigned to LH over C (i.e., a16 = 5) and a relativeimportance value of 1/9 is assigned to C over LH (i.e., a61 = 1/9).Similarly, the relative importance among other criteria can be ex-plained. However, it may be added that, in actual practice, thesevalues of relative importance can be judiciously decided by theuser/experts depending on the requirements. The assigned valuesin this paper are for demonstration purpose only.

The normalized weights of each criterion are calculated follow-ing the procedure presented in Section 2.1. The results obtainedfrom the computations based on the pair-wise comparison matrix

are presented in Table 6. The consistency ratio of the pair-wisecomparison matrix is calculated as 0.0725 which is much less thanallowed CR value of 0.1. Hence the weights are shown to be consis-tent and they are used in the selection process.

3.3. Evaluation of alternatives

At this stage of decision procedure, the decision matrix is to beestablished by comparing alternatives under each of the criteriaseparately.

3.3.1. The TOPSIS methodThe decision matrix from Table 5 is used for the TOPSIS analysis.

Based on the TOPSIS procedure mentioned in Section 2.2.1, eachelement is normalized by Eq. (7). The resulting normalized decisionmatrix for the TOPSIS analysis is shown as Table 7.

The second step is to form the weighted decision matrix usingweights of the criteria calculated by AHP method. To obtain therank of alternatives the steps 3–5 are followed as explained in Sec-tion 2.2.1. The results are summarized in Table 8.

Finally the rank is given to alternatives according to relativecloseness to ideal solution (Ri). The alternative materials can be ar-ranged in descending order as A1–A9–A7–A8–A4–A5–A6–A2–A3. It isconcluded that the material designated as A1 i.e. Calcium chloridehexahydrate, the first best choice for the given design applicationunder the given conditions considering six criteria.

3.3.2. The fuzzy TOPSIS methodThe decision matrix from Table 3 is again used for the fuzzy

TOPSIS analysis. In order to transform the performance ratings tofuzzy linguistic variables as discussed in the previous section, theperformance ratings in Table 5 are normalized into range of[0, 1]. The normalized rating for each element in the decision ma-trix can be calculated as follows.

(i) For the beneficial criteria:

rij ¼ ½xij �minfxijg�=½maxfxijg �minfxijg� ð23Þ

Table 3Summary of performance rating.

Phase changematerial (PCM)

Material selection criteria

Latent Heat,J/kg (LH)

Density,kg/m3 (D)

Specific heat,kJ/kg K (Cp(s))

Specific heat,kJ/kg K (Cp(l))

Thermal conductivityW/m K (K)

Cost (C)

1 Calcium chloride hexahydrate 169.98 1560 1.46 2.13 1.09 Very low2 Stearic acid 186.5 903 2.83 2.38 0.18 Very high3 p116 190 830 2.1 2.1 0.21 Low4 RT 60 214.4 850 0.9 0.9 0.2 Very low5 Paraffin wax RT 30 206 789 1.8 2.4 0.18 Low6 n-Docosane 194.6 785 1.93 2.38 0.22 Low7 n-Octadecane 245 773.22 0.3767 2.267 0.14 Low8 n-Nonadecane 222 775.8 1.7189 1.921 0.142 High9 n-Eicosane 247 776.33 0.7467 2.377 0.13836 Low

Table 4Conversion of linguistic terms into fuzzy scores(11-point scale) [7].

Linguistic term Crisp score

Exceptionally low 0.045Extremely low 0.135Very low 0.255Low 0.335Below average 0.410Average 0.500Above average 0.590High 0.665Very high 0.745Extremely high 0.865Exceptionally high 0.955

Table 5Objective data of the criteria with fuzzy score (decision matrix).

LH D Cp(s) Cp(l) K C

A1 169.98 1560 1.46 2.13 1.09 0.255A2 186.5 903 2.83 2.38 0.18 0.745A3 190 830 2.1 2.1 0.21 0.335A4 214.4 850 0.9 0.9 0.2 0.255A5 206 789 1.8 2.4 0.18 0.335A6 194.6 785 1.93 2.38 0.22 0.335A7 245 773.22 0.3767 2.267 0.14 0.335A8 222 775.8 1.7189 1.921 0.142 0.665A9 247 776.33 0.7467 2.377 0.138 0.335

Table 6Weights of the criteria obtained from AHP method.

Criteria GM Weights (W) X = A �W CV

LH 4.718 0.4901 3.3550 6.8456D 1.611 0.1674 1.0811 6.4587Cp(s) 0.508 0.0528 0.3295 6.2462Cp(l) 0.508 0.0528 0.3295 6.2462K 2.030 0.2109 1.3540 6.4200C 0.251 0.0261 0.1698 6.5027

kmax ¼ Avg ðCVÞ 6.4532

Table 7Normalized decision matrix for TOPSIS analysis.


A1 0.2700 0.5621 0.2864 0.3313 0.9070 0.1962A2 0.2962 0.3254 0.5552 0.3702 0.1498 0.5733A3 0.3018 0.2991 0.4120 0.3267 0.1747 0.2578A4 0.3405 0.3063 0.1766 0.1400 0.1664 0.1962A5 0.3272 0.2843 0.3531 0.3733 0.1498 0.2578A6 0.3091 0.2829 0.3787 0.3702 0.1831 0.2578A7 0.3892 0.2786 0.0739 0.3526 0.1165 0.2578A8 0.3526 0.2796 0.3372 0.2988 0.1182 0.5118A9 0.3923 0.2797 0.1465 0.3698 0.1151 0.2578

Weight 0.4901 0.1674 0.0528 0.0528 0.2109 0.0261



(ii) For the non-beneficial criteria:

rij ¼ ½maxfxijg � xij�=½maxfxijg �minfxijg� ð24Þ

Now, Table 5 can be transformed into Table 9. The next step isto establish the decision matrix using fuzzy linguistic variables.The membership functions of these linguistic variables are shownin Fig. 1.

The triangular numbers related with these variables are shownin Table 10. Hence Table 9 is converted into Table 11 as explainedby the following example. If the numeric rating is 0.80, then its fuz-zy linguistic variable is ‘‘H’’. The fuzzy linguistic variable is thentransformed into the triangular fuzzy numbers which are equiva-lent of linguistic variables as shown in Table 12. The fuzzy criteria

weight is also presented in Table 12. This is the first step of the fuz-zy TOPSIS analysis.

After the fuzzy evaluation matrix is determined, The secondstep is to obtain a fuzzy weighted decision table. Using the criteriaweights calculated by AHP, the weighted evaluation matrix isestablished applying fuzzy multiplication Eq. (16). The resultingfuzzy weighted decision matrix is shown as Table 13.

According to Table 13, it is noted that the elements ~v ij, are nor-malized positive triangular fuzzy numbers and their ranges belongto closed interval [0, 1]. Thus the fuzzy positive-ideal solutions(FPIS, V+) and the fuzzy negative ideal solution (FNIS, V�) can be de-fined as ~vþj = (1, 1, 1) and ~v�j = (0, 0, 0) for beneficial criteria and~vþj = (0, 0, 0) and ~v�j = (1, 1, 1) for non-beneficial criteriarespectively.

For the fourth step, the separation (distance) of each alternativefrom V+ and V� can be currently calculated using Eqs. (20) and (21).The next step is to calculate the relative closeness to ideal solutionas described by Eq. (22).

In order to illustrate steps 5 and 6 calculation, a sample calcu-lation for first alternative is presented here.

Sþ1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13½ð1� 0Þ2 þ ð1� 0:07Þ2 þ ð1� 0:21Þ2�

r

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13½ð1� 0:11Þ2 þ ð1� 0:27Þ2 þ ð1� 0:45Þ2�

r

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13½ð1� 0Þ2 þ ð1� 0:05Þ2 þ ð1� 0:16Þ2�

r


r

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13½ð1� 0:26Þ2 þ ð1� 0:45Þ2 þ ð1� 0:65Þ2�

r


r¼ 4:1881

Table 8TOPSIS analysis results.

LH D Cp(s) Cp(l) K C Sþi S�i Ri

A1 0.1323 0.0941 0.0151 0.0175 0.1913 0.0051 0.0617 0.1745 0.7390A2 0.1452 0.0545 0.0293 0.0195 0.0316 0.0150 0.1714 0.0328 0.1605A3 0.1479 0.0501 0.0218 0.0172 0.0369 0.0067 0.1668 0.0299 0.1522A4 0.1669 0.0513 0.0093 0.0074 0.0351 0.0051 0.1656 0.0382 0.1875A5 0.1604 0.0476 0.0186 0.0197 0.0316 0.0067 0.1697 0.0357 0.1740A6 0.1515 0.0474 0.0200 0.0195 0.0386 0.0067 0.1651 0.0324 0.1639A7 0.1907 0.0466 0.0039 0.0186 0.0246 0.0067 0.1752 0.0600 0.2552A8 0.1728 0.0468 0.0178 0.0158 0.0249 0.0134 0.1747 0.0437 0.2000A9 0.1923 0.0468 0.0077 0.0195 0.0243 0.0067 0.1749 0.0618 0.2612

Vþj 0.1923 0.0941 0.0293 0.0197 0.1913 0.0051

V�j 0.1323 0.0466 0.0039 0.0074 0.0243 0.0150

Table 9Normalized decision matrix for fuzzy TOPSIS analysis.

LH D Cps Cpl K C

A1 0.00 1.00 0.44 0.82 1.00 1.00A2 0.21 0.16 1.00 0.99 0.04 0.00A3 0.26 0.07 0.70 0.80 0.08 0.84A4 0.58 0.10 0.21 0.00 0.06 1.00A5 0.47 0.02 0.58 1.00 0.04 0.84A6 0.32 0.01 0.63 0.99 0.09 0.84A7 0.97 0.00 0.00 0.91 0.00 0.84A8 0.68 0.00 0.55 0.68 0.00 0.16A9 1.00 0.00 0.15 0.98 0.00 0.84

Wj 0.4901 0.1674 0.0528 0.0528 0.2109 0.0261

Table 10Transformation for fuzzy membership functions [31].

Linguistic variables Fuzzy numbers

Very low (VL) (0.00, 0.10, 0.25)Low (L) (0.15, 0.30, 0.45)Medium (M) (0.35, 0.50, 0.65)High (H) (0.55, 0.70, 0.85)Very high (VH) (0.75, 0.90, 1.00)

Table 11Decision matrix with fuzzy linguistic variables.


A1 VL VH M H VH VHA2 VL VL VH VH VL VLA3 L VL VH H VL HA4 M VL VL VL VL VHA5 M VL M VH VL HLA6 L VL M VH VL HA7 VH VL VL VH VL HA8 H VL M H VL VLA9 VH VL VL VH VL H

Wj H M VL VL M VL

0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9 10.30

0

0.5

1 VL L M H VH

Fig. 1. Fuzzy triangular membership functions [31].



S�1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13½ð0� 0Þ2 þ ð0� 0:07Þ2 þ ð0� 0:21Þ2�

r

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13½ð0� 0:11Þ2 þ ð0� 0:27Þ2 þ ð0� 0:45Þ2�

r


r


r

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13½ð0� 0:26Þ2 þ ð0� 0:45Þ2 þ ð0� 0:65Þ2�

r


r¼ 2:0298

R1 ¼S�1

Sþ1 þ S�1¼ 2:0298

4:1881þ 2:0298¼ 0:3264

The resulting fuzzy TOPSIS analyses are summarized in Table 14.Based on Ri values, the alternatives are arranged in descending or-der. The ranking of the alternatives in descending order are A1–

A9–A7–A8–A5–A3–A6–A2–A4. The fuzzy TOPSIS results indicate thatA1 is the best alternative with R value of 0.3264.

4. Conclusions

Proper selection of the phase change material leads to efficientutilization of latent heat thermal energy storage system. Most ofthe researchers use PCM in the particular application dependingon their experience or availability of the material. However, severalalternatives must be considered and evaluated in terms of manydifferent conflicting criteria in a PCM selection problem. Therefore,an effective evaluation approach is essential to improve decisionquality.

The present study proposes two MADM methods-TOPSIS andfuzzy TOPSIS, in solving the PCM selection problem. Both the meth-ods use AHP to assign weights to the criteria which are used inPCM selection. When precise performance ratings are available,the TOPSIS method is considered to be a viable approach in solvingmaterial selection problem. When performance ratings are vagueand inaccurate, then the fuzzy TOPSIS is a preferred technique.The present study illustrates the feasibility of the fuzzy based TOP-SIS method for the instance of fuzzy inputs.

Table 12Fuzzy decision matrix and fuzzy attribute weights.


A1 (0.00, 0.10, 0.25) (0.75, 0.90, 1.00) (0.35, 0.50, 0.65) (0.55, 0.70, 0.85) (0.75, 0.90, 1.00) (0.75, 0.90, 1.00)A2 (0.00, 0.10, 0.25) (0.00, 0.10, 0.25) (0.75, 0.90, 1.00) (0.75, 0.90, 1.00) (0.00, 0.10, 0.25) (0.00, 0.10, 0.25)A3 (0.15, 0.30, 0.45) (0.00, 0.10, 0.25) (0.55, 0.70, 0.85) (0.55, 0.70, 0.85) (0.00, 0.10, 0.25) (0.55, 0.70, 0.85)A4 (0.35, 0.50, 0.65) (0.00, 0.10, 0.25) (0.00, 0.10, 0.25) (0.00, 0.10, 0.25) (0.00, 0.10, 0.25) (0.75, 0.90, 1.00)A5 (0.35, 0.50, 0.65) (0.00, 0.10, 0.25) (0.35, 0.50, 0.65) (0.75, 0.90, 1.00) (0.00, 0.10, 0.25) (0.55, 0.70, 0.85)A6 (0.15, 0.30, 0.45) (0.00, 0.10, 0.25) (0.35, 0.50, 0.65) (0.75, 0.90, 1.00) (0.00, 0.10, 0.25) (0.55, 0.70, 0.85)A7 (0.75, 0.90, 1.00) (0.00, 0.10, 0.25) (0.00, 0.10, 0.25) (0.75, 0.90, 1.00) (0.00, 0.10, 0.25) (0.55, 0.70, 0.85)A8 (0.55, 0.70, 0.85) (0.00, 0.10, 0.25) (0.35, 0.50, 0.65) (0.55, 0.70, 0.85) (0.00, 0.10, 0.25) (0.00, 0.10, 0.25)A9 (0.75, 0.90, 1.00) (0.00, 0.10, 0.25) (0.00, 0.10, 0.25) (0.75, 0.90, 1.00) (0.00, 0.10, 0.25) (0.55, 0.70, 0.85)

Wj (0.55, 0.70, 0.85) (0.35, 0.50, 0.65) (0.00, 0.10, 0.25) (0.00, 0.10, 0.25) (0.35, 0.50, 0.65) (0.00, 0.10, 0.25)

Table 13Fuzzy-weighted decision matrix.


A1 (0.00, 0.07, 0.21) (0.11, 0.27, 0.45) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21) (0.26, 0.45, 0.65) (0.00, 0.09, 0.25)A2 (0.00, 0.07, 0.21) (0.00, 0.03, 0.11) (0.00, 0.09, 0.25) (0.00, 0.09, 0.25) (0.00, 0.05, 0.16) (0.00, 0.01, 0.06)A3 (0.08, 0.21, 0.38) (0.00, 0.03, 0.11) (0.00, 0.07, 0.21) (0.00, 0.07, 0.21) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21)A4 (0.19, 0.35, 0.55) (0.00, 0.03, 0.11) (0.00, 0.01, 0.06) (0.00, 0.01, 0.06) (0.00, 0.05, 0.16) (0.00, 0.09, 0.25)A5 (0.19, 0.35, 0.55) (0.00, 0.03, 0.11) (0.00, 0.05, 0.16) (0.00, 0.09, 0.25) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21)A6 (0.08, 0.21, 0.38) (0.00, 0.03, 0.11) (0.00, 0.05, 0.16) (0.00, 0.09, 0.25) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21)A7 (0.41, 0.63, 0.85) (0.00, 0.03, 0.11) (0.00, 0.01, 0.06) (0.00, 0.09, 0.25) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21)A8 (0.30, 0.49, 0.72) (0.00, 0.03, 0.11) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21) (0.00, 0.05, 0.16) (0.00, 0.01, 0.06)A9 (0.41, 0.63, 0.85) (0.00, 0.03, 0.11) (0.00, 0.01, 0.06) (0.00, 0.09, 0.25) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21)

Table 14Fuzzy TOPSIS analysis.

LH D Cp(s) Cp(l) K C Sþi S�i Ri

A1 (0.00, 0.07, 0.21) (0.11, 0.27, 0.45) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21) (0.26, 0.45, 0.65) (0.00, 0.09, 0.25) 4.1881 2.0298 0.3264A2 (0.00, 0.07, 0.21) (0.00, 0.03, 0.11) (0.00, 0.09, 0.25) (0.00, 0.09, 0.25) (0.00, 0.05, 0.16) (0.00, 0.01, 0.06) 4.5942 1.5697 0.2547A3 (0.08, 0.21, 0.38) (0.00, 0.03, 0.11) (0.00, 0.07, 0.21) (0.00, 0.07, 0.21) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21) 4.5963 1.5823 0.2561A4 (0.19, 0.35, 0.55) (0.00, 0.03, 0.11) (0.00, 0.01, 0.06) (0.00, 0.01, 0.06) (0.00, 0.05, 0.16) (0.00, 0.09, 0.25) 4.6198 1.5171 0.2472A5 (0.19, 0.35, 0.55) (0.00, 0.03, 0.11) (0.00, 0.05, 0.16) (0.00, 0.09, 0.25) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21) 4.4682 1.7121 0.2770A6 (0.08, 0.21, 0.38) (0.00, 0.03, 0.11) (0.00, 0.05, 0.16) (0.00, 0.09, 0.25) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21) 4.6003 1.5755 0.2551A7 (0.41, 0.63, 0.85) (0.00, 0.03, 0.11) (0.00, 0.01, 0.06) (0.00, 0.09, 0.25) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21) 4.2720 1.9115 0.3091A8 (0.30, 0.49, 0.72) (0.00, 0.03, 0.11) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21) (0.00, 0.05, 0.16) (0.00, 0.01, 0.06) 4.2661 1.8928 0.3073A9 (0.41, 0.63, 0.85) (0.00, 0.03, 0.11) (0.00, 0.01, 0.06) (0.00, 0.09, 0.25) (0.00, 0.05, 0.16) (0.00, 0.07, 0.21) 4.2720 1.9115 0.3091

Vþ1 (1, 1, 1) (1, 1, 1) (1, 1, 1) (1, 1, 1) (1, 1, 1) (0, 0, 0)V�1 (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (1, 1, 1)



TOPSIS and fuzzy TOPSIS methods lead to the choice of A1 i.e.calcium chloride hexahydrate as the preferred alternative. Thematerials A9, A7, and A8 are the second, third and fourth best alter-natives materials used in the solar water heater system. Other thanthese four alternatives, the preferences vary between the twomethods. The systematic evaluation of the MADM problem can re-duce the risk of a poor selection of the material. In short, the pro-posed methodology provides a systematic approach to narrowdown the number of alternatives.

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