INTRODUCTIONCucumis melo L. is a tropical and sub-tropical fruitthat easily decays and rots because of its high-watercontent (70–95%). For the fruit to maintain its qualityand freshness, it has to be handled properly during andafter harvesting. A good quality fruit should be fresh,with a smooth, undamaged, and flawless skin. Comparedto other cucumbers (Cucumis), Cucumis melo has agreener colour, more crunchy texture, higher watercontent, and sweeter taste. In addition, Cucumis melocan be harvested at an earlier stage.Packaging is extremely important in post-harvesthandling. It creates proper condition for the fruit tomaintain its quality during the desired period. Packagingis a container or wrapper that can help to prevent orreduce damage to the packaged/wrapped object. Themain functions of packaging are to keep food productsfrom contamination, to protect them from physicaldamage, and to inhibit their quality degradation.In the Province of Aceh (Indonesia), Cucumis melois usually packaged in traditional manner by usingbanana stem, because banana leaves are cheap, easy tofind, and eco-friendly. The fruit is placed in the middlepart of banana stem, which are then folded into two parts(Fig. 1). Banana stem are able to protect the fruit fromshocks and damage during transportation from producerto consumer. When ripe, the epidermis of Cucumis melocracks, and banana stem help keep its shape and texture.Usually, Cucumis melo is protected with a single layer ofbanana stem.According to Lukman [1], banana stem is part ofbanana pseudo stem. Its structure is very different fromthat of woody plants, because it is an apparent trunkformed by tightly packed, over-leaping stem. The fibreof banana stem are strong and waterproof to both freshand salt water. The packaging of Cucumis melo with avarious amount of banana stem is necessary to preserveits wholeness and texture, because this fruit is easilybroken when ripe. The storage temperature varies fromroom temperature to cold temperature, which is alsoexpected to prolong the shelf life of Cucumis melo.A quick method to find out consumer acceptancetowards the food product is to perform a sensoryassessment by collecting respondents’ opinions on theproduct. This multi-criteria assessment method wascompleted with a weighting assessment approach,Research Article DOI: http://doi.org/10.21603/2308-4057-2019-2-339-347Open Access Available online at http:jfrm.ruA multi-criteria sensory assessment of Cucumis melo (L.) usingfuzzy-Eckenrode and fuzzy-TOPSIS methodsRahmat Fadhil* , Raida AgustinaUniversitas Syiah Kuala, Banda Aceh, Indonesia* e-mail: rahmat.fadhil@unsyiah.ac.idReceived May 26, 2019; Accepted in revised form June 17, 2019; Published October 21, 2019Abstract: The paper introduces a multi-criteria assessment system that can be used for sensory analysis by fuzzy-Eckenrodeand fuzzy-TOPSIS methods. Respondents evaluated the sensory characteristics of Cucumis melo (L.), which included aroma,colour, taste, texture, and overall acceptance, after six days of storage. The product was stored under three different temperatureconditions: 10°C (B1), 14°C (B2), and room temperature (27–30°C) (B3). The product was also stored at three types of packaging:unpackaged stem (A1), packaged fruit with one layer of banana stem (A2), and packaged fruit with two layers of banana stem(A3). The best result was demonstrated by the Cucumis melo that was stored at 14°C and packaged in a two-layered banana stem(A3B2). Both fuzzy-Eckenrode and fuzzy-TOPSIS method provided an easy, fast, and unambiguous calculation of multi-criteriasensory assessment.Keywords: Banana stem, hedonic scale, Cucumis melo (L.), sensory assessment, TOPSIS, EckenrodePlease cite this article in press as: Fadhil R, Agustina R. A multi-criteria sensory assessment of Cucumis melo (L.) usingfuzzy-Eckenrode and fuzzy-TOPSIS methods. Foods and Raw Materials. 2019;7(2):339–347. DOI: http://doi.org/10.21603/2308-4057-2019-2-339-347.Copyright © 2019, Fadhil et al. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 InternationalLicense (http://creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix,transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license.Foods and Raw Materials, 2019, vol. 7, no. 2E-ISSN 2310-9599ISSN 2308-4057340Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347which is usually used in decision making. Therefore,this article introduces a multi-criteria assessmentsystem that performs a sensory analysis by using fuzzy-Eckenrode and the fuzzy-TOPSIS (Technique for OrderPerformance by Similarity to Ideal Solution) methods.According to the system, the respondents evaluatedeach product and rated its level of acceptance accordingto a multi-criteria sensory assessment, which includedaroma, colour, taste, texture, and overall acceptance.Fuzzy logic. Fuzzy logic is a development of the settheory, where each member has a degree of membershipthat ranges in value between 0 and 1. It means that fuzzysets can represent interpretation of each value accordingto the opinion, or decision, and its probability. Rating 0represents ‘wrong’, rating 1 represents ‘right’, and there arestill other numbers between the ‘right’ and ‘wrong’ [2, 3].In fuzzy sets, there are two attributes. The first oneis linguistic attribute: it is a naming of a group whichrepresents a certain situation or condition by usinga natural language such as ‘cold’, ‘cool’, ‘normal’, or‘warm’. The second attribute is numeric: it is a value(number) which shows a measure of a variable, such as10, 30, 50, etc. [4]. Membership function is a curve thatdefines how each point in the input room is mapped intothe membership value (degree of membership between 0and 1). If U states universal sets and A is fuzzy functionsets in U, so A can be stated as sorted pair as following [2]:{(x (x)) x U} A Α = ,μ ∈ (1)where (x) A μ is a membership function that gives valueof degree of membership x to fuzzy set A, which is::U →[0,1] A μ (2)In a fuzzy set, there are several membershipfunctions of a new fuzzy set, which result from basicoperation of the fuzzy set, i.e.:Intersection: A Ç B = min (mA[x], mB[y]) (3)Union: A È B = max (mA[x], mB[y]) (4)Complement: ~ A = 1 – mA[x] (5)Membership function is stated as follows:0; x ≤ a or x ≥ cμ (x) = (b – a) / (x – a); a ≤ x ≤ b (6)(b – x) / (c – b); b ≤ x ≤ cIn a fuzzy system, there is a linguistic variable. Thisis a variable that has a value in verbal form in a naturallanguage. Each linguistic variable is related to a certainmembership function. Figure 2 gives an example ofmembership function.Fuzzy-Eckenrode. The Eckenrode method wasinitially known as a weighting multiple criteria method,which was introduced by Robert T. Eckenrode fromDunlop and Association, Inc. in 1965 and has beenwidely used until today [5–8]. The Eckenrode methodis simpler and more efficient in determining theimportance weight in a decision [9–11]. The Eckenrodeweighting analysis method is one of weighing methodsused in determining the degree of importance, or Weight(B), from each Criteria (K) and Sub-criteria (SK), whichhave been set in decision making [12]. This weightdetermination is perceived as very important becauseit affects the final total value of each chosen decision.The concept used in this weighting method is by doinga change of order to value where, for instance, firstorder (1) has the highest rate (value) and the fifth order(5) has the lowest rate.Fuzzy-TOPSIS. TOPSIS belongs to the MultipleAttribute Decision Making (MADM), which was firstlyintroduced by Yoon, Yoon et al. and Hwang et al. [13–15].It has been widely applied in various studies related todecision making, such as Kumar et al., Han et al., Tyagi,Estrella et. al., Roszkowska et al., Selim et al. [16–21].TOPSIS can only be implemented for a criterion whoseweight has been known or calculated before, becausethere is a step in TOPSIS which involves the processof multiplication of criterion weight and the alternativevalue of the criterion.In many situations, the data available is insufficientfor a real life problem, because human assessment,which is considered as preference, is unclear, and thepreference cannot be estimated with exact numericvalue. The verbal expression, e.g. ‘low’, ‘medium’,‘high’, etc., is considered as a representation of thedecision maker. Thus, fuzzy logic is necessary inFigure 1 Cucumis melo packaged in banana stem Figure 2 Membership function341Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347making a structured decision of the preference maker.The Fuzzy theory helps to measure the uncertaintyassociated with human judgement, which is subjective.Therefore, evaluation is necessary to be done in anenvironment. According to Ningrum et al. and Fadhil etal., fuzzy logic can help improve failure, which happenswhen only Eckenrode or TOPSIS method is used [4, 22].STUDY OBJECTS AND METHODSThis study used Cucumis melo (L.) which washarvested in two months after planting. The harvestedCucumis melo was cleaned by washing and then storedunder three different conditions: without bananastem packaging (A1), with one layer of banana stempackaging (A2), and with two layers of banana stempackaging (A3). Cucumis melo was then stored for sixdays under three temperature regimes: 10°C (B1), 14°C(B2), and at room temperature (27–30°C) (B3).Procedure of assessment. The multi-criteria sensoryassessment of Cucumis melo included aroma, colour,taste, texture, and overall acceptance (Table 1). Theattribute weight of respondents’ assessment toward themulti-criteria was determined according to the hedonicscale. The hedonic scale is a preference of respondent’sopinion based on likes or dislikes that are converted intonumber (Table 2).The framework of this study included four steps: (1)selection of respondents and criteria, (2) determinationof criterion weight of the assessment by using thefuzzy-Eckenrode method, (3) determination of the bestalternative of all treatments by using fuzzy-TOPSIS,and (4) recommendation of the best acceptance from allrespondents. Figure 3 shows the complete framework.Combinations of storage conditions were as follows:A1B1: without banana stem-packaging at 10°C;A1B2: without banana stem-packaging at 14°C;A1B3: without banana stem-packaging at 27–30°C;A2B1: with one layer of banana stem-packaging at 10°C;A2B2: with one layer of banana stem-packaging at 14°C;A2B3: with one layer of banana stem-packaging at27–30°C;A3B1: with two layers of banana stem-packaging at10°C;A3B2: with two layers of banana stem-packaging at14°C;A3B3: with two layers of banana stem-packaging at27–30°C.Fuzzy-Eckenrode method. According to theEckenrode weight calculation method, the respondentswere asked to make a rating (e.g. from R1 until Rn,where n ranking, j = 1, 2, 3,…, n, ranking j = Rj) foreach criterion (criterion i is notated with Ki, which ispresented in a number of n criteria, i = 1, 2, 3,…, n) [11].Table 3 shows the obtained data. Next, Ni was calculatedbased on Pij and Rn-j.Rj = ranking order at j, j = 1, 2, 3,…, nKi = criterion type i, i = 1, 2, 3,…, nPij = number of respondents who chose ranking j forcriterion iRn-j = multiplier factor j, which was obtained from thereduction of number of criteria or number of ranking(which is n) with the rank order on the column. Forinstance, if there are five criteria, so the multiplier factorfor column of 3rd rank (if j = 3) is n–j = 5–3 = 2Bi = weight of criterion i.Ni = Gj=1 Prij x Rn-j, j = 1, 2, 3,…, n. (7)Total Score = Gi=1 Ni, i = 1, 2, 3,…, n. (8)Table 1 Attributes of multi-criteria sensory assessmentof Cucumis meloAttribute Assessment considerationAroma (C1) Typical, no sour smellColour (C2) Yellowish-greenTaste (C3) Sweet and not sourTexture (C4) Solid, not watery, no wrinklesOverall acceptance (C5) Yellowish-green in colour, solid,and sweetTable 2 Assessment of preference according to hedonic scaleScore Preference5 Like very much4 Like3 Neither like nor dislike2 Dislike1 Dislike very muchFigure 3 Research frameworkTabel 3 Calculation of criterion weight accordingto the Eckenrode methodCriteria Rank Score WeightR1 R2 ...... Rj ...... RnK1 P11 P12 ...... P1n N1 B1K2 P21 P22 ...... P2n N2 B2...... ...... ...... ...... ...... ...... ......Ki Pij......Kn Pn1 Pn2 ...... Pmn Nn BnMultiplierfactorRn-1 Rn-2 ...... Rn-j Rn-n TotalScore1.00342Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347Then, criterion weight Bi (which are B1, B2,B3,…, Bn) was calculated, where i = 1, 2, 3,…, 3, byusing the following formula:Bi = (Ni/Total Score) (9)To find the level of importance of each sub-criterionwithin a criterion, the respondents were also askedto rank each sub-criterion within a criterion. Then,by using the same procedure, the weight of each subcriterionwas calculated (B1i, the weight of sub-criterion1 in criterion i). Thus, the weighted weight (BT) fromsub-criterion 1 in criterion i was obtained, which wasBT1 = B1i·Bi. Then, to find the score of each criterion, therespondents were asked to rate each sub-criterion withineach criterion [23].The assessment of each sub-criterion was calculatedby using a geometric mean formula according to theassessment result from all respondents, which wasmultiplied with the weighted weight of each subcriterion.Each criterion (K1 to K5) was calculated bysumming up the total score of all sub-criteria in eachcriterion. To assess the weighting by the respondents, thefuzzy-Eckenrode method was applied with the value ofpreference, as shown in Table 4.Fuzzy-TOPSIS method. The analysis with thefuzzy-TOPSIS method included the following tasks [24]:To rank the fuzzy from each decision made, Dk;(k = 1, 2, 3,…, k) can be represented as triangularfuzzy number ˜Rk; (k = 1, 2, 3,…, K) with membershipfunction μ˜R (x).To produce an appropriate alternative, to determinethe criteria of evaluation, and to organise the groupof decision-maker. It was assumed that there were malternatives, n criteria of evaluation, and decision k.To choose the linguistic variable according to theweight of criterion importance =𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)andalternative linguistic rankings on criterion (𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖) inTriangular Fuzzy Number (TFN).To do a weight aggregation of each criterion toobtain fuzzy weight aggregate (𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)) in criterion Cj and todetermine the fuzzy aggregate value from alternative Aion each criterion Cj.𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑥𝑥̃… 𝑥𝑥̃], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛](10)i = 1, 2,.., m; and j = 1, 2,..., n𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣𝑗𝑗+= max {𝑣𝑣𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)(11)j = 1, 2,..., nTo build a fuzzy decision matrix.𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑗𝑗𝑢𝑢𝑖𝑖𝑖𝑖)𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖),𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)(12)To do normalisation of the decision matrix, where:𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃ 𝑤𝑤𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)i = 1, 2,..., m; and j = 1, 2,..., n (13)Calculating𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)can be done with:𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)(14)where𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴12𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛̃21 𝑥𝑥̃22 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)= max𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖).To determine the weight normalisation of thefuzzy decision matrix. Based on different importanceon each criterion, the fuzzy decision of the weightednormalisation matrix can be arranged as:𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)with, i = 1, 2,..., m; and j = 1, 2,..., n (15)where:𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, ̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)with, i = 1, 2,..., m; and j = 1, 2,..., n (16)To determine fuzzy positive ideal solution (FPIS) S+and fuzzy negative ideal solution (FNIS) S-:𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = ̃ 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)(17)𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)(18)where:𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , ̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)= max𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, ̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖, 𝑢𝑢𝑗𝑗∗ )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, ̃2… . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, ̃2… . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖𝑢𝑢)and𝑥𝑥̃𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖]𝑚𝑚𝑟𝑟̃𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖]𝑚𝑚𝑣𝑣̃𝑖𝑖 = 𝑟𝑟̃𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, ̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖)= min𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚 𝑚𝑚𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , ̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)with𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = 𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)areTFN normalisation weight.To calculate the interval between each alternativevalue and the value of FPIS (Fuzzy Positive IdealSolution) and FNIS (fuzzy negative ideal solution).𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖= 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)(19)𝑖𝑖𝑖𝑖 1𝑘𝑘 𝑖𝑖𝑖𝑖1 𝑖𝑖𝑖𝑖2 𝑖𝑖𝑖𝑖𝑘𝑘 𝑗𝑗𝑗𝑗1 𝑗𝑗2 𝑗𝑗𝑘𝑘1 2 𝑛𝑛𝐷𝐷 ̃=12𝑚𝑚𝑛𝑛𝑛𝑛𝑚𝑚1 𝑚𝑚2 𝑚𝑚𝑚𝑚𝑊𝑊 ̃𝑛𝑛𝑅𝑅 ̃𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 ̃𝑖𝑖𝑖𝑖 𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ 𝑗𝑗max 𝑖𝑖𝑖𝑖 𝑉𝑉 ̃𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚𝑖𝑖𝑖𝑖 ̃𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖+ 𝑛𝑛− 𝑛𝑛𝑗𝑗max 𝑖𝑖𝑖𝑖and 𝑗𝑗min 𝑖𝑖𝑖𝑖with 𝑗𝑗= 13 1 2 1 2 21+ 𝑖𝑖𝑖𝑖𝑗𝑗𝑛𝑛 𝑗𝑗=11− 𝑖𝑖𝑖𝑖𝑗𝑗𝑛𝑛 𝑗𝑗=1𝑖𝑖 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− (𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)(20)𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)(21)To calculate the closeness coefficient (CCi) and theranking according to the coefficient value obtained usingthe following equation:𝑥𝑥̃𝑖𝑖 =1𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖2 + . + 𝑥𝑥̃𝑖𝑖𝑖𝑖𝑘𝑘 ]𝑤𝑤̃𝑗𝑗=1𝑘𝑘[𝑤𝑤̃𝑗𝑗1 + 𝑤𝑤̃𝑗𝑗2 + … . + 𝑤𝑤𝑗𝑗𝑘𝑘]𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛𝐷𝐷 ̃=𝐴𝐴1𝐴𝐴2𝐴𝐴𝑚𝑚[𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚], 𝑊𝑊 ̃= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖 𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑚𝑚𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , 𝑢𝑢𝑖𝑖𝑖𝑖𝑈𝑈𝑗𝑗∗ , )𝑈𝑈𝑗𝑗∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1+, 𝑣𝑣̃2+, … . , 𝑣𝑣̃𝑛𝑛+)𝑆𝑆− = (𝑣𝑣̃1−, 𝑣𝑣̃2−, … . , 𝑣𝑣̃𝑛𝑛−)𝑣𝑣̃𝑗𝑗+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √13 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)𝑑𝑑1+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝑑𝑑1− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖−𝑑𝑑𝑖𝑖++𝑑𝑑𝑖𝑖− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)(22)Table 4 Scale of weighting comparison among criteriaof fuzzy-Eckenrode methodScale Annotation TFN membership function~1 Very unimportant 1, 1, 2~2 Less important 1, 2, 3~3 Neutral 2, 3, 4~4 Important 3, 4, 5~5 Very important 4, 5, 5343Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347To rate each alternative by the respondents, we usedthe fuzzy-TOPSIS method with preference value, as inTable 5.Figure 4 illustrates the procedure of the analysis.Selection of respondents. A total of 10 respondentswere chosen to do a multi-criteria sensory assessment ofCucumis melo. The respondents were selected accordingto several criteria. The potential respondents had to:1. like Cucumis melo, raw or processed;2. be experienced in sensory assessment;3. be healthy, as flu, cough, mouth ulcers, etc. canbother the sensory assessment process;4. be able to distinguish colours.RESULTS AND DISCUSSIONDetermination of assessment criteria weight.A hedonic scale was used to evaluate the results ofdetermination of respondents’ assessment of criteriaweight towards multi-criteria which were considered inthe sensory assessment. After that, they were translatedinto fuzzy logic functions (Table 6).As for the data of respondents’ assessment towardscriteria of importance weight determination fromeach sensory attribute, the values of lower bound(low), middle (medium), and upper bound (upper) werearranged as summarised on Table 7. The next step wasto calculate the score and the weight of each criterion.Figure 5 represents a radar diagram.According to the respondents’ assessment of thecriteria with the help of the fuzzy-Eckenrode method,the order of criteria weight was obtained from thehighest to the lowest: (1) overall acceptance, 0.216; (2)colour, 0.211; (3) aroma, 0.203; (4) taste, 0.191; and (5)texture 0.176.Determination of the best alternative. Thepriority of the best alternative from the multi-criteriasensory assessment of Cucumis melo was determinedby summarising all respondents’ preferences. Thepreferences were chosen based on the mode number,i.e. the value that appears most often from each choiceof material treatment. The mode number was chosenby the respondents. The next step was to arrange thematrix of the respondents’ assessment on all alternatives(Table 8). The data of respondents’ assessment was thentransformed into TFN linguistic data, as presented inTable 9.After that, we formulated the normalised weightmatrix on each alternative. The value normalisation canbe done by using Eqs. (13) and (14). Table 10 shows theresults of the TFN value normalisation.Table 5 Comparison scale of determinationof the fuzzy-TOPSIS method alternativeScale TFN LinguisticsDislike very much (STS) 1, 1, 2Dislike (TS) 1, 2, 3Neither like nor dislike (N) 2, 3, 4Like (S) 3, 4, 5Like very much (SS) 4, 5, 5Figure 4 Steps of the fuzzy-TOPSIS method analysisTable 6 Respondents’ weighting score of criteria based on thefuzzy-Eckenrode methodNo Criteria Order1 2 3 4 51 C1 ~4 ~3 ~1 ~1 ~12 C2 ~3 ~4 ~1 ~1 ~13 C3 ~3 ~1 ~4 ~1 ~14 C4 ~2 ~2 ~4 ~1 ~15 C5 ~5 ~2 ~1 ~1 ~1Nilai(Gcriteria-order)4 3 2 1 0Table 7 TFN value of experts’ weighting on criteria of the fuzzy-Eckenrode methodNo Criteria 1 2 3 4 5 Score Weightl m u l m u l m u l m u l m u1 C1 4 5 5 1 2 3 1 1 2 1 1 2 1 1 2 82 0.2062 C2 3 4 5 1 2 3 1 2 3 1 1 2 1 1 2 84 0.2113 C3 1 2 3 1 1 2 3 4 5 1 2 3 1 1 2 76 0.1914 C4 1 1 2 1 2 3 3 4 5 1 2 3 1 1 2 70 0.1765 C5 3 4 5 1 2 3 1 1 2 1 2 3 1 1 2 86 0.216Nilai (Gcriteria-order) 4 3 2 1 0 398 1,000l = lower, m = middle, u = upper344Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347Table 8 Matrix of experts’ assessment on alternativesAlternatives CriteriaC1 C2 C3 C4 C5A1B1 3 2 1 1 3A1B2 3 2 2 1 3A1B3 2 2 2 1 1A2B1 4 3 3 2 4A2B2 4 4 3 2 4A2B3 2 2 2 1 1A3B1 5 4 5 4 5A3B2 5 5 5 5 5A3B3 1 1 1 1 1Then, we arranged the matrix of multiplicationbetween criteria weights and normalisation value of eachalternative. This process can be done by using Eqs. (15)and (16). Table 11 summarises the results of the matrixmultiplication.The next step was to determine the positive idealsolution value (FPIS) S+ and the negative ideal solutionvalue (FNIS) S–. When determining both values, thecharacteristic of data available should be taken intoconsideration. To obtain both groups of values, one canuse Eqs. (17) and (18). Table 12 demonstrates FPIS andFNIS values.After that, the interval between each alternativevalue and FPIS and FNIS was calculated by usingEqs. (19), (20), and (21). The results of the intervalcalculation between alternative value toward FPIS andFNIS can be observed from Table 13 and Table 14.We evaluated the criteria distance value to the fuzzypositive ideal solution (FPIS) and the fuzzy negative idealsolution (FNIS) according to comparison of d+ and d–.It showed preference of product acceptance on a radardiagram (Fig. 6). For instance, the treatment of CucumisFigure 5 Radar diagram of criteria weightmelo without packaging at temperature of 10°C (A1B1)had such d+ and d– values that showed the biggestdistance from the positive ideal and the negative ideal.The final step was to calculate the closenesscoefficient (CCi) of each alternative by usingEq. (22). From the calculation result, we obtainedranking from the highest to the lowest (Table 15). Thebiggest coefficient value was the main alternative, whichwas suggested to be chosen or prioritised, comparedto other alternatives based on respondents’ preference(product acceptance).According to the closeness coefficient (CCi), analternative ranking can be arranged from the biggest tothe lowest as follows: two-layer banana stem-packagingat 14°C (A3B2), two-layer banana stem-packaging at10°C (A3B1), one-layer banana stem-packaging at 14°C(A2B2), one-layer banana stem-packaging at 10°C(A2B1), without banana stem packaging at 14°C (A1B2),one-layer banana stem-packaging at room temperature(A2B3), without banana stem packaging at 10°C (A1B1),without banana stem packaging at room temperatureTable 9 Matrix of respondents’ assessment on alternative in TFN scaleAlternatives CriteriaAroma (0.191,0.206, 0.211)Colour (0.206,0.211, 0.216)Taste (0.176,0.191, 0.206)Texture (0.176,0.176, 0.191)Overall acceptance(0.204, 0.216, 0.216)A1B1 (2, 3, 4) (1, 2, 3) (1, 1, 2) (1, 1, 2) (2, 3, 4)A1B2 (2, 3, 4) (1, 2, 3) (1, 2, 3) (1, 1, 2) (2, 3, 4)A1B3 (1, 2, 3) (1, 2, 3) (1, 2, 3) (1, 1, 2) (1, 1, 2)A2B1 (3, 4, 5) (2, 3, 4) (2, 3, 4) (1, 2, 3) (3, 4, 5)A2B2 (3, 4, 5) (3, 4, 5) (2, 3, 4) (1, 2, 3) (3, 4, 5)A2B3 (1, 2, 3) (1, 2, 3) (1, 2, 3) (1, 1, 2) (1, 1, 2)A3B1 (4, 5, 5) (3, 4, 5) (4, 5, 5) (3, 4, 5) (4, 5, 5)A3B2 (4, 5, 5) (4, 5, 5) (4, 5, 5) (4, 5, 5) (4, 5, 5)A3B3 (1, 1, 2) (1, 1, 2) (1, 1, 2) (1, 1, 2) (1, 1, 2)A1B1: without banana stem-packaging at 10°CA1B2: without banana stem-packaging at 14°CA1B3: without banana stem-packaging at 27–30°CA2B1: with one layer of banana stem-packaging at 10°CA2B2: with one layer of banana stem-packaging at 14°CA2B3: with one layer of banana stem-packaging at 27–30°CA3B1: with two layers of banana stem-packaging at 10°CA3B2: with two layers of banana stem-packaging at 14°CA3B3: with two layers of banana stem-packaging at 27–30°C345Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347Table 10 Matrix of TFN scale normalisationAlternative CriteriaAroma(0.191, 0.206, 0.211)Colour(0.206, 0.211, 0.216)Taste(0.176, 0.191, 0.206)Texture(0.176, 0.176, 0.191)Overall acceptance(0.204, 0.216, 0.216)A1B1 (0.40, 0.60, 0.80) (0.20, 0.40, 0.60) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40) (0.40, 0.60, 0.80)A1B2 (0.40, 0.60, 0.80) (0.20, 0.40, 0.60) (0.20, 0.40, 0.60) (0.20, 0.20, 0.40) (0.40, 0.60, 0.80)A1B3 (0.20, 0.40, 0.60) (0.20, 0.40, 0.60) (0.20, 0.40, 0.60) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40)A2B1 (0.60, 0.80, 1.00) (0.40, 0.60, 0.80) (0.40, 0.60, 0.80) (0.20, 0.40, 0.60) (0.60, 0.80, 1.00)A2B2 (0.60, 0.80, 1.00) (0.60, 0.80, 1.00) (0.40, 0.60, 0.80) (0.20, 0.40, 0.60) (0.60, 0.80, 1.00)A2B3 (0.20, 0.40, 0.60) (0.20, 0.40, 0.60) (0.20, 0.40, 0.60) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40)A3B1 (0.80, 1.00, 1.00) (0.60, 0.80, 1.00) (0.80, 1.00, 1.00) (0.60, 0.80, 1.00) (0.80, 1.00, 1.00)A3B2 (0.80, 1.00, 1.00) (0.80, 1.00, 1.00) (0.80, 1.00, 1.00) (0.80, 1.00, 1.00) (0.80, 1.00, 1.00)A3B3 (0.20, 0.20, 0.40) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40)Table 11 Matrix of multiplication of criteria weights and alternative normalisation valuesAlternatives CriteriaAroma(0.203, 0.204, 0.209)Colour(0.196, 0.203, 0.204)Taste(0.188, 0.196, 0.203)Texture(0.188, 0.188, 0.196)Overall acceptance(0.204, 0.209, 0.209)A1B1 (0.08, 0.12, 0.17) (0.04, 0.08, 0.13) (0.04, 0.04, 0.08) (0.04, 0.04, 0.08) (0.08, 0.13, 0.17)A1B2 (0.08, 0.12, 0.17) (0.04, 0.08, 0.13) (0.04, 0.08, 0.12) (0.04, 0.04, 0.08) (0.08, 0.13, 0.17)A1B3 (0.04, 0.08, 0.13) (0.04, 0.08, 0.13) (0.04, 0.08, 0.12) (0.04, 0.04, 0.08) (0.04, 0.09, 0.13)A2B1 (0.11, 0.16, 0.21) (0.08, 0.13, 0.17) (0.07, 0.11, 0.16) (0.04, 0.07, 0.11) (0.12, 0.17, 0.22)A2B2 (0.11, 0.16, 0.21) (0.12, 0.17, 0.22) (0.07, 0.11, 0.16) (0.04, 0.07, 0.11) (0.12, 0.17, 0.22)A2B3 (0.04, 0.08, 0.13) (0.04, 0.08, 0.13) (0.04, 0.08, 0.12) (0.04, 0.04, 0.08) (0.04, 0.04, 0.09)A3B1 (0.15, 0.21, 0.21) (0.12, 0.17, 0.22) (0.14, 0.19, 0.21) (0.11, 0.14, 0.19) (0.16, 0.22, 0.22)A3B2 (0.15, 0.21, 0.21) (0.16, 0.21, 0.22) (0.14, 0.19, 0.21) (0.14, 0.18, 0.19) (0.16, 0.22, 0.22)A3B3 (0.04, 0.04, 0.08) (0.04, 0.04, 0.09) (0.04, 0.04, 0.08) (0.04, 0.04, 0.08) (0.04, 0.04, 0.09)Table 12 Positive ideal solution and negative ideal solution valuesCriteria Aroma Colour Taste Texture Overall acceptanceS (+) (0.21, 0.21, 0.21) (0.22, 0.22, 0.22) (0.21, 0.21, 0.21) (0.19, 0.19, 0.19) (0.22, 0.22, 0.22)S (–) (0.04, 0.04, 0.04) (0.04, 0.04, 0.04) (0.04, 0.04, 0.04) (0.19, 0.19, 0.19) (0.22, 0.21, 0.22)Table 13 Intervals between criteria value and FPISFPIS(d+)Criteria d+Aroma Colour Taste Texture OverallacceptanceA1B1 0.096 0.136 0.156 0.143 0.095 0.626A1B2 0.096 0.136 0.133 0.143 0.095 0.603A1B3 0.134 0.136 0.133 0.143 0.135 0.681A2B1 0.062 0.096 0.097 0.122 0.059 0.436A2B2 0.062 0.060 0.097 0.122 0.059 0.400A2B3 0.134 0.136 0.133 0.143 0.160 0.706A3B1 0.034 0.060 0.039 0.057 0.030 0.219A3B2 0.034 0.030 0.039 0.030 0.030 0.162A3B3 0.158 0.161 0.156 0.143 0.160 0.778Table 14 Interval between criteria value and FNISFPIS(d-)Criteria d-Aroma Colour Taste Texture OverallacceptanceA1B1 0.093 0.057 0.027 0.143 0.095 0.416A1B2 0.093 0.057 0.056 0.143 0.095 0.445A1B3 0.057 0.057 0.056 0.143 0.135 0.449A2B1 0.131 0.094 0.090 0.122 0.059 0.496A2B2 0.131 0.134 0.090 0.122 0.059 0.536A2B3 0.057 0.057 0.056 0.143 0.160 0.474A3B1 0.154 0.134 0.147 0.057 0.030 0.521A3B2 0.154 0.158 0.147 0.030 0.030 0.518A3B3 0.027 0.026 0.027 0.143 0.160 0.384(A1B3), and two-layer banana stem-packaging at roomtemperature (A3B3) (Fig. 7).The analysis with fuzzy-TOPSIS approach showedthat the respondents preferred Cucumis melo storedin a two-layer banana stem packaging at 14°C (A3B2).Since the scores were fairly close between Cucumis melostored in a two-layer banana stem packaging at 14°C(A3B2) and Cucumis melo stored in a two-layer bananastem packaging at 10°C (A3B1), both products werefavored by consumers (respondents’ preferences).CONCLUSIONAccording to the consumer assessment of all typesof the six-day storage of Cucumis melo, the optimal346Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347storage conditions involved packaging with two layersof banana stem at the temperature of 14°C (A3B2). Thefuzzy-Eckenrode and fuzzy-TOPSIS methods were veryhelpful in calculating the results of the multi-criteriasensory assessment through weighing. They made theFigure 6 Evaluation of d+ and d–Figure 7 Alternative ranking of Cucumis melo productacceptanceprocess of determining consumers’ acceptance easier,faster, and more certain.CONFLICT OF INTERESTThe authors declare no conflict of interest.ACKNOWLEDGEMENTSWe thank our laboratory assistants at the Post-Harvest Engineering Laboratory, Faculty of Agriculture,Syiah Kuala University, especially Riza Rahmah, fortheir support and assistance in organising respondentsand material preparation for this research.