A new approach to strategic objectives ranking based on fuzzy logarithmic least squares method and fuzzy similarity technique

Khanmohammadi, Ehsan; Malmir, Behnam; Safari, Hossein; Zandieh, Mostafa Article A new approach to strategic objectives ranking based on fuzzy logarithmic least squares method and fuzzy similarity technique Operations Research Perspectives Provided in Cooperation with: Elsevier Suggested Citation: Khanmohammadi, Ehsan; Malmir, Behnam; Safari, Hossein; Zandieh, Mostafa (2019) : A new approach to strategic objectives ranking based on fuzzy logarithmic least squares method and fuzzy similarity technique, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 6, pp. 1-14, https://doi.org/10.1016/j.orp.2019.100122 This Version is available at: https://hdl.handle.net/10419/246397 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Contents lists available at ScienceDirect Operations Research Perspectives journal homepage: www.elsevier.com/locate/orp A new approach to strategic objectives ranking based on fuzzy logarithmic least squares method and fuzzy similarity technique Ehsan Khanmohammadi a,⁎ , Behnam Malmir b , Hossein Safari a , Mostafa Zandieh c a Department of Industrial Management, Faculty of Management, University of Tehran, Tehran, Iran b Department of Business Information Technology, Pamplin College of Business, Virginia Tech, Blacksburg, USA c Department of Industrial Management, Faculty of Management and Accounting, Shahid Beheshti University, Tehran, Iran ARTICLE INFO Keywords: Balanced scorecard Strategy map Strategic objectives Fuzzy similarity method Logarithmic least square method ABSTRACT In this paper, we discuss the ratings of strategic objectives in a strategy map. In order to have a more useful strategy map and also improve efficiency in the stage of strategy implementation, the significance level of the strategic objective in a strategic map must be determined. In the current study, objectives have been ranked based on strategic destinations, employing a combination of the fuzzy logarithmic least squares method and fuzzy similarity techniques. First, the former is used to obtain the weight of strategic destinations, and then the objectives are ranked through the latter. For this purpose, a company is selected to implement the proposed model with the collaboration of its senior management. The results of the case study indicate that the most important strategic objectives of this company fall under the aspect of learning and growth, helping the company the most to achieve its strategic destinations in this regard. 1. Introduction The recent daily increases in the complexity of activities and intense business environment changes have made managers realize that traditional planning will no longer be able to solve their problems. Hence, strategic management is introduced as a necessity in organizations, and through its application, managers are seeking appropriate navigation for directing their organizations in today's uncertain environment. Companies possess lots of strategic tools, along with the knowledge of their positive effect on business. Nevertheless, there is still no particular theory or a framework to integrate these tools. Several studies in the last two decades have shown that a 60–80% of companies do not achieve their strategic objectives. According to these studies, the reason for more than 70% of failures lies in the implementation stage [1]. The primary problem of the unimplemented strategies is the gap between the operational and strategic layers. To succeed in strategy implementation, these two layers must be linked, and this is done via balanced scorecards (BSC) in the current study. The BSC method has been regarded beyond a measurement system for more than one decade and has been used as a strategic management system for long-term administration and navigation of organizations. Even though the balanced scorecard method is widely applied in today's organizations, it does embody weaknesses. Kaplan and Norton have defined the four perspectives of financial, customer, internal processes, and learning and growth as the key elements of organization strategy that require measurement. Despite successful achievements of balanced scorecard as a strategic management tool, there have been many criticisms and limitations about this approach (see e.g. [2-6]). Some papers have already proposed ways to resolve problems of this method or deal with its limitations (Table 1). Some of these criticisms refer to the phase of formulating strategic goals. However, McAdam and O'Neil [3] believed that the Balanced Scorecard is more an effective tool for measuring strategies than developing them. Therefore, the SWOT analysis must be considered as an instrument to construct strategic objectives for the balanced scorecard. Another criticism is that the implementation stage of the balanced scorecard is qualitative, and the acceptance rate of results could be increased through quantitative methods. On the other hand, in strategy implementation phase, over 70% of organizations’ strategic initiatives face failure [7]. When strategic initiatives are not implemented, an organization is not able to achieve its vision. Some of the reasons that organizations fail to conduct their initiatives are limitations in time and budget during times of crisis. In order to reduce the deviation of an organization's codified strategic plan in such times, it is necessary that organizations’ strategic objectives are ranked, and the limited budget is used more efficiently through implementing the most important objectives. The main purpose of this research is to present a comprehensive model in order to translate organizational perspective into strategic https://doi.org/10.1016/j.orp.2019.100122 Received 14 December 2018; Received in revised form 5 September 2019; Accepted 5 September 2019 ⁎ Corresponding author. E-mail address: [email protected] (E. Khanmohammadi). Operations Research Perspectives 6 (2019) 100122 Available online 06 September 2019 2214-7160/ © 2019 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/). T goals and performance indicators, and also implementing it across all organizational layers using BSC tool. The proposed method removes criticisms of the balanced scorecard to some extent. This is done through adding the SWOT analysis to the development stage of the strategy. The strategic objectives prioritizing is considered as a multi–- criteria decision making problem and so, the strategic destinations which is similar to the long-term company objectives or goals of Fred & Forest's model [48] and the strategic themes of company in Kaplan & Norton's model [1], were assumed as the criteria for this problem. The Strategic objectives are ranked based on how much they could help the company, so it can achieve its destinations. Therefore, the strategic objectives were ranked using the proposed fuzzy similarity method and the weights of strategic destinations were calculated using fuzzy logarithmic least squares method. The main contributions of this research include: (a) presenting a comprehensive methodology to formulate organizational strategies using the BSC tool, and (b) proposing a new methodology to rank the organizational goals based on their effectiveness in reaching organizational strategic destinations, which are, in fact, a translation of organizational vision. In the times of crisis, organizations are not able to follow all their objectives or preform initiatives. This is when organizations fail to achieve their destinations and vision. Based on the proposed approach when an organization faces budget deficit, dedicating a large portion of budget to more critical objectives may result in the least deviation from organizational strategic destinations. Given the uncertainty in goal ranking process, logarithmic least squares method and similarity technique were used in fuzzy environment. The ranking method used in this fuzzy development research is Deng's similarity method introduced above. The rest of this paper is organized as follows: The literature review section discusses the use of BSC method. Section 3 presents the proposed methodology along with the fuzzy logarithmic least squares method and the proposed intuitionistic fuzzy similarity-based method. Section 4 thoroughly investigates an application of the proposed approach. Finally, managerial implications and conclusions are presented in Section 5 and 6, respectively. 2. Literature review Norton and Kaplan [49] developed the balanced scorecard in 1992. The most important strength of BSC is to make a balance in integrating financial and non-financial indexes, which translates the organization's strategic objectives into a set of related criteria [1]. Scorecard designers believe that a company's growth requires good financial conditions. The financial perspective of a company originates from customer perspective, which is formed by factors such as customer satisfaction, Table 1 Previous research on the balanced scorecard. Research field Tools and methods Research area Reference article Formulating strategic objectives Balanced scorecard General [8] Balanced scorecard Metalworking firm [9] Balanced scorecard Health Care Organizations [10] Balanced scorecard Ministry of Youth and Sports of Turkey [11] Balanced Scorecard Indian handmade carpet industry [12] Scenario planning for strategy map An Iranian post company [13] Balanced Scorecard The Spanish electricity sector [14] Balanced Scorecard Passenger transport companies [15] Prioritizing strategic objectives/ strategy The technique for order of preference by similarity to ideal solution (TOPSIS) General [16] Analytic hierarchical process (AHP) Biopharmaceutical firm [17] AHP and TOPSPS Iranian mining sector [18] Analytic network process (ANP) Pittsburgh-based commercial building company. [19] ANP and zero-one goal programming General [20] Fuzzy AHP A Spanish financial software factory [21] Grey systems theory General [22] Determining quantitative targets for performance measures in the BSC using response surface methodology An oil company [23] Strategy Map Designing Path analysis and structural equation modeling (SEM), General [24] ANP and DEMATEL Science and technology in Taiwan [25] Fuzzy DEMATEL Trading Company in Iran [26] Fuzzy cognitive maps (FCMs) General [27] System dynamic approach General [28] Decision-making trial and evaluation laboratory (DEMATEL) Banking institutions [29] Path analysis University of Science and Technology in Iran [30] AHP and linear programming General [31] ANP General [32] DEMATEL and linear programming General [33] ANP and DEMATEL A manufacturing company [34] Fuzzy DEMATEL and linear programming A particular enterprise of the banking sector [35] Performance evaluation Fuzzy analytic hierarchy process (FAHP), SAW, TOPSIS and VIKOR Banking performance Evaluation [36] fuzzy analytic network process (FANP) A manufacturing firm [37] Non-additive fuzzy integral High technology firm [38] Fuzzy Delphi method (FDM) and ANP to construct a sustainability balanced scorecard In the semiconductor [39] DEMATEL, ANP and VIKOR Three extension education centers of universities [40] DANP (DEMATEL-based ANP) Hoteling in Taiwan [41] The UTASTAR method Health care organizations [42] Fuzzy linguistic method Health care organizations [43] ANP and COPRAS (COmplex PRoportional ASsessment) technique Oil producing companies [44] Fuzzy ANP The recreation and leisure industry [45] Fuzzy ANP, Fuzzy DEMATEL and MOORA European airlines [46] DANP (DEMATEL-based ANP) and VIKOR International airports [47] E. Khanmohammadi, et al. Operations Research Perspectives 6 (2019) 100122 2 profitability, retention, new customer attraction, and organizational image. Meanwhile, the situation of an organization from the customer's perspective depends on the internal processes of that organization. The cost and quality of such processes determine the customer satisfaction and retention. The appropriate conduction of the process depends on the ability or inability of the organization in terms of growth and learning perspectives (physical capital management, human communication and information management). In order to maintain constant growth and improvement of an organization, there should exist a good connection and balance between the organization's four perspectives. After introducing the basics of the strategies, they must be put into practice. In this regard, the first step is creating a strategy map. Without Fig. 1. Stages of the proposed methodology. Table 2 The Group fuzzy pairwise comparison matrix. B ˜ Criteria 1 Criteria 2 … Criteria n Criteria 1 (1, 1, 1) … ( ) l m u l m u ( , , ) , , e e e 121 121 121 12 12 12 12 12 12 … … ( ) l m u l m u ( , , ) , , n n n ne nne nne n 1 1 1 1 1 1 111111 Criteria 2 … ( ) l m u l m u ( , , ) , , e e e 211 211 211 21 21 21 21 21 21 (1, 1, 1) … … ( ) l m u l m u ( , , ) , , n n n ne nne nne n 2 1 2 1 2 1 222222 ⋮ ⋮ ⋮ ⋮ ⋮ Criteria n … ( ) l m u l m u ( , , ) , , n n n n enn enn en 11 11 11 111111 … ( ) l m u l m u ( , , ) , , n n n n enn enn en 21 21 21 222222 … (1, 1, 1) E. Khanmohammadi, et al. Operations Research Perspectives 6 (2019) 100122 3 a strategy map, the balanced scorecard would only be an executive scorecard which reports the important measures for implementation. Being developed around the causal structure that links the four perspectives, a strategy map forces an organization to clarify the value creation logic and its target [50]. It claims that every strategy is a hypothesis of the causal relationships between destinations, starting from learning and growth and ending in financial perspective. The strategy map could be regulated based on a strategic theme – a set of related strategic objectives in the map. Most strategic themes are vertical combinations of objectives that originate from the process perspective – where the strategy is operated. A process-oriented strategic theme could link to the customer and financial consequences, and enabling objectives in the learning and growth aspect in bottom-up and top-down ways respectively [1]. A part of BSC literature includes studies that have combined the balanced scorecard approach with multi-criteria decision-making techniques. Organizations can use these techniques to select the visions, strategies, or resource allocation schemes for implementing their strategies or achieving their goals. They could also evaluate their performance through the combination of the four perspectives of balanced scorecard and its indicators with these techniques. To investigate the literature of BSC, four types of studies can be taken into consideration: (a) performance evaluation, (b) strategy map designing, (c) formulating strategic objectives, and (d) prioritizing strategic objectives/strategy. The first category includes methods which evaluate an organization based on BSC structure. In this approach, different units or sections of an organization are ranked based on BSC, using organizations multicriteria decision-making methods. Some of the most significant papers in this regard are listed in Table 1. The second research category includes strategy map designing papers. These papers use different methods to determine the causal relationships between strategy map objectives. One of the most important approaches in this regard is DEMATEL. According to this approach, the experts determine direct relations between the objectives. After performing the method procedures and by taking the direct and indirect relationships into account, the general relations between strategic objectives are eventually determined. In order to draw the strategy map, this method is presented in different studies as a definite method [29] or as a fuzzy method (see e.g. [26,51]) or as combined with ANP method (see e.g. [25,34]) or as a combination of DMATEL method and linear programming (see e.g. [33,35]). In these papers, after general determination of relationships between the objectives using DMATEL method, a linear programming model is proposed to select the best relationships. Other methods used for determining relationships are path analysis method [24,30] and Fuzzy cognitive maps method [27]. Formulating strategic objectives constitutes the third category of BSC literature, which investigates the determination of strategic objectives. For instance, Jafari et al. [13] designed different scenarios for the strategic map and investigated several probable conditions based on Net present value (NPV) level of performing each scenario. Their proposed method would help producing the most effective strategic map. In another research, the strategy map was presented based on Financial, Customers, Internal Processes, Learning and Growth, Environmental and Legal perspectives to show the strategies and common objectives of the Spanish electricity supply companies [14]. In their study, Tubis and Werbińska-Wojciechowska formulated Balanced Scorecard model for Polish passenger transport sector [15]. The last category consists of articles discussing the prioritizing strategic objectives/strategy. Huang et al. [17] ranked the strategic map objectives using AHP method based on BSC perspectives. In their model, the perspectives were selected as the first level criteria and objectives as the sub-criteria. This research was conducted in the field of biopharmaceutical. In another study, goals and KPIs were ranked using AHP method, but the experiment was performed in a fuzzy environment [21]. In Fouladgar et al.’s [18] study, the organizational strategies were first formulated using SWOT tool and then ranked based on BSC indicators. For this purpose, they combined AHP and TOPSIS [52] methods. Tjader et al. [19] ranked BSC indicators by the use of ANP method. Initially, they determined the relationships between indicators and then identified the most inportant indicators. In Wudhikarn's [20] proposed methodology, first formulated the strategies, then each strategic objective and KPI of each strategy are defined. Next, KPIs are ranked by the use of ANP method and finally budgeting is done using zero-one goal programming method. Jahantigh et al. [22] presented a combined method based on focus group interviews and grey system theory as qualitative and quantitative methods, respectively. Farokhi and Roghanian [23] conducted another study in the field of strategic map to set strategic objective targets using response surface methodology. Table 1 presents the most important studies based on the four aforementioned fields. The current study focuses on the two fields of formulating strategic objectives and prioritizing strategic objectives, simultaneously. For this purpose, a comprehensive method is presented to formulate the strategic objectives. Next the objectives are ranked in line with the organizational strategic destinations and vision. After building the map and ranking the Fig. 2. Degree of conflict between A i and A ± . Fig. 3. The internal-external matrix of Alpha Company. E. Khanmohammadi, et al. Operations Research Perspectives 6 (2019) 100122 4 objectives, the process of transforming the vision to action plane will be fully proceeded by drawing the map and presenting the initiatives. One of the important criticisms about the BSC is that it does not provide a method for formulating the objectives of the strategy map. In the proposed methodology, the organization's perspective is translated into strategic goals. The objectives of the map are based on the Swot matrix and formulated to achieve strategic goals. On the other hand, during the crisis, organizations are not able to focus on all their goals. For this reason, in the proposed methodology, strategic destinations were ranked using the FLLSM method of Fuzzy AHP methods and map objectives were ranked using the fuzzy similarity method. Organizations are able to focus on more important destinations and objectives. In this study, Deng's similarity method has been developed to rank the objectives. Deng [53] elucidated why his proposed model is more effective than that of TOPSIS method. Intuitionistic Fuzzy Numbers (IFN) was employed in this proposed method on account of uncertainty in the strategic management realm. 3. Methodology For a successful strategy implementation, the strategic layer must be linked to the executive layer. This starts with designing the balanced scorecard on the foundation of strategy mapping. In this study, a comprehensive approach is provided to develop and prioritize the strategic objectives of a strategy map, which is summarized in Fig. 1. In this study, first the strengths, weaknesses, opportunities and threats affecting the company are identified. Then, considering the obtained Fig. 4. Generic strategy, strategic destinations, and strategic objectives. Table 3 Linguistic variable for completing the pairwise comparisons. Defuzzy Positive triangular fuzzy number reciprocal triangular fuzzy number 9 (Extremely Strong) ES (9,9,9) RES (1/9,1/9,1/9) 8 (Intermediate) IVS (7,8,9) RIVS (1/9,1/8,1/7) 7 (Very Strong) VS (6,7,8) RVS (1/8,1/7,1/6) 6 (Intermediate) IS (5,6,7) RIS (1/7,1/6,1/5) 5 (Strong) S (4,5,6) RS (1/6,1/,1/4) 4 (Intermediate) IMS (3,4,5) RIMS (1/5,1/4,1/3) 3 (Moderately Strong) MS (2,3,4) RMS (1/4,1/3,1/2) 2 (Intermediate) IES (1,2,3) RIES (1/3,1/2,1) 1 (Equally Strong) EQS (1,1,1) EQS (1,1,1) Table 4 Decision makers’ pairwise comparison results between strategic destinations. SD.1 SD.2 SD.3 SD.4 SD.5 SD.6 SD.1 (EQS, EQS, EQS, EQS, EQS) (IES, EQS, EQS,RIES,RIES) (IMS,IES,IES, EQS,IES) (S,IMS,MS,IES,MS) (MS,IES,MS,MSMIMS) (MS,MS,MS,MS,IMS) SD.2 (RIES, EQS, EQS,IES,IES) (EQS, EQS, EQS, EQS, EQS) (IES,MS,IES,IES,MS) (IMS,S,MS,MS,IMS) (MS,MS,MS,MS,IMS,S) (IES,IMS,MS,IMS,S) SD.3 (RIES,RIES,RIES,E,RIES) (RIES,RMS,RIES,RIES,RMS) (EQS, EQS, EQS, EQS, EQS) (IMS,MS,IES,IES,IES) (IES,E,IES,MS,MS) (MS,IES,IES,MS,IMS) SD.4 (RS,RIMS,RMS,RIES,RMS) (RIMS,RS,RMS,RMS,RIMS) (RIMS,RMS,RIES,RIES,RIES) (EQS, EQS, EQS, EQS, EQS) (RMS,RIMS, EQS,IES,MS) (RIES,RMS, EQS,IES,MS) SD.5 (RMS,RIES,RMS,RMS,RIMS) (RMS,RMS,RMS,RIMS,RS) (RIES, EQS,RIES,RMS,RMS) (MS,IMS, EQS,RIES,RMS) (EQS, EQS, EQS, EQS, EQS) (IES,IES, EQS, EQS, EQS) SD.6 (RMS,RMS,RMS,RMS,RIMS) (RIES,RIMS,RMS,RIMS,RS) (RMS,RIES,RIES,RMS,RMS) (IES,MS, EQS,RIES,RMS) (RIES,RIES, EQS, EQS, EQS) (EQS, EQS, EQS, EQS, EQS) E. Khanmohammadi, et al. Operations Research Perspectives 6 (2019) 100122 5 matrix (SWOT), the strategic position and generic strategy of the company are determined, and strategic objectives are specified at the four perspectives of the balanced scorecard model. The next step is to prioritize the objectives based on the organization's strategic destinations as follows. A group fuzzy pairwise comparison matrix is formed, and the criteria weights are obtained through the fuzzy logarithmic least squares method (FLLSM). The strategic objectives priority problem is solved through considering the destinations as the criteria, using the fuzzy similarity technique (FSimilarity). Deng introduced this method in 2007 [53] and Safari et al. [54] reformed it [54]. This study presents the proposed fuzzy similarity method. In the last step, the cause and effect relationships between the strategic objectives are obtained and the strategy map is drawn. 3.1. The fuzzy logarithmic least squares method There are two categories of the techniques used to extract the weights of the fuzzy pairwise comparisons. The first one includes techniques in which the obtained weights are in the form of real numbers, like the fuzzy preference programming [55] and logarithmic fuzzy preference programming [56] methods, and the weights in the second are extracted as fuzzy numbers, such as the LAMBDA-Max approach [57], the linear goal programming method [58,59], credibility theory [60], and FLLSM [61]. In this study, the FLLSM method is applied; a method able to extract weights from the group pairwise comparison matrix as fuzzy numbers by solving a non-linear programming problem as shown below: = + + + + = + = = … = = = = = = = Min Z w w a w w a w w a Subject to w w w w w w w w w w i n ln ln ln ln ln ln ln ln ln 1 1 1 ( ) 2 0, 1, , i n j i j n k eij iLj Uijk LiMj Mijk M i Uj Lijk U iLj i j n j U i Uj i j n j L i n iM i n iLi U j Uj Mj L 1 1, 1 2 2 2 1, 1, 1 1 (9) Table 5 Weights of strategic destinations. Strategic destinations SD1.Annual 2% market share increase SD2.increasing Profit SD3.excellence in the award level SD4. Development of two new super-beneficial products annually SD5. Annual 2% increase in exports SD6.Setting up abroad production Weight (0.22,0.27,0.31) (0.25,0.31,0.35) (0.12,0.16,0.21) (0.06,0.07,0.09) (0.08,0.09,0.01) (0.06,0.07,0.09) Table 6 The importance and weights of decision makers. Decision makers Linguistic variable Weight 1 L9 0.183673 2 L7 0.15873 3 L5 0.113379 4 L4 0.090703 5 L4 0.090703 6 L4 0.090703 7 L4 0.090703 8 L4 0.090703 9 L4 0.090703 Table 7 Linguistic terms used to derive the alternative ratings and the importance of the decision makers. Linguistic Variable IFNs Very Very-Low L1 [0.1,0.9] Very-Low L2 [0.1,0.75] Low L3 [0.25,0.65] Middle-Low L4 [0.4,0.5] Medium L5 [0.5,0.4] Middle-High L6 [0.6,0.3] High L7 [0.7,0.2] Very-High L8 [0.8,0.1] Very Very-High L9 [0.9,0.1] Extremely-High L10 [1,0.0] E. Khanmohammadi, et al. Operations Research Perspectives 6 (2019) 100122 6 When the group fuzzy paired comparison B ˜ Table 2) is formed, the fuzzy weights of criteria can be extracted without aggregating comments by solving Model ((9).Where =a l m u ˜( , , ) ijk ijk ijk ijk are triangular fuzzy judgments with =a a u m l ˜ ˜ (1/ , 1/ , 1/ ) ijk jik jik jik jik 1 for = =i j n i j k e, 1, ..., , , 1, ..., ij ,nindicates the number of criteria, and e ij indicates the number of decision makers. For Matrix B ˜ , there should exist a normalized triangular fuzzy weight vector = =W w w w w w w w w(˜, ..., ˜) (( ˜,˜,˜), ...,( ˜,˜,˜)) n TL M U n L n M n UT 11 1 1 where =a l m u w w ˜( , , ) ˜/˜ ijk ijk ijk ijk i j . In times of inconsistency a w w ˜ ˜ /˜ ijk i j which in fact the error equals: =a w w error ˜ ˜ /˜ ijk i j .the above equation can be shown as: =a w w errorln ˜ln( ˜/˜) ijk i j . The objective function of model (9) aims to minimize the error, and limitations of normalizing model provide output weights of the model. The weights derived from solving the model =W w w w ˜( , , ) i L i M i U are defuzzified using the following equation: = + + =W w w w i n 1 4( 2 ), 1, ..., ii L i M i U (10) 3.2. Intuitionistic fuzzy sets Intuitionistic fuzzy sets (IFS) have been applied in a variety of areas, including decision-making problems [62-69] and pattern recognition [70,71]. Atanassov [72] first introduced them by extending classical fuzzy set theory (FST) in 1986, and they have turned into a popular method to handle vagueness. Definition 1. An intuitionistic fuzzy set Ain a finite set Xis defined as [72,73]: =A x µ x x x X{ , ( ), ( ) | } AA (1) In whichμ A (x), ν A (x): X→ [0, 1] are membership and non-membership functions and must meet the following constraint [72,73]: +µ x x0 ( ) ( ) 1 AA (2) IFS has another parameter calledπ A (x), which is referred to as the intuitionistic fuzzy index. It demonstrates the degree of hesitation whether xis a member of Aor not [72,73]: =x µ x x( ) 1 ( ) ( ) AAA (3) Therefore, for every x∈X: x0 ( ) 1 A (4) A smallπ A (x) denotes higher certainty aboutx, and vice versa. We can easily deduce that, if all the elements of the universe have =µ x x( ) 1 ( ) AA , the IFS is reduced to an ordinary fuzzy set [74]. Definition 2. Atanassov [72] has defined the following relationships for intuitionistic fuzzy numbers =A x µ x x x X{ , ( ), ( ) | } AA , =B x µ x x x X{ , ( ), ( ) | } BB , =C x µ x x x X{ , ( ), ( ) | } CC and the real number n [72,73]: + = +B C x µ x µ x µ x µ x x x x X{ , ( ) ( ) ( ). ( ), ( ). ( ) | } B C B C B C (5) = +B C x µ x µ x x x x x x X. { , ( ). ( ), ( ) ( ) ( ). ( ) | } B C B C B C (6) =nA x µ x x x X{ , 1 (1 ( )) , ( ( )) | } A nAn (7) Definition 3. Assume that s: F(X) × F(X) → [0, 1] is a mapping that fits in the following four relations for any intuitionistic fuzzy sets A ˜1 , A ˜2 and A ˜3 of F(X)[75]: (1) s A A0 ( ˜,˜) 1, 1 2 (2) = =s A A A A(˜,˜) 1 if and only if ˜ ˜ , 1 2 1 2 (3) =s A A s A A and(˜,˜) ( ˜,˜) 1 2 2 1 (4) A A A s A A s A A and s A A s A Aif ˜ ˜ ˜ , then ( ˜,˜) ( ˜,˜) ( ˜,˜) ( ˜,˜). 1 2 3 1 3 1 2 1 3 2 3 Table 8 The ratings of the alternatives. C01 C02 C 03 C 04 C 05 C 06 A1 (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L8, L8, L8, L8, L8, L8, L8, L8, L8) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A2 (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L7, L7, L7, L8, L7, L8, L8, L8, L8) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A3 (L7, L7, L8, L7, L8, L7, L7, L7, L7) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L6, L6, L6, L7, L6, L7, L7, L7, L7) (L6, L6, L6, L6, L7, L6, L6, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A4 (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L8, L8, L8, L8, L8, L8, L8, L8, L8) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A5 (L8, L8, L8, L8, L8, L8, L8, L8, L8) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A6 (L6, L6, L6, L6, L7, L6, L6, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L6, L6, L6, L6, L7, L6, L6, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A7 (L6, L6, L7, L6, L7, L6, L6, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L6, L6, L6, L6, L7, L6, L7, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A8 (L7, L7, L7, L7, L8, L7, L7, L7, L7) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L7, L7, L7, L7, L7, L7, L8, L7, L7) (L7, L7, L8, L7, L8, L7, L8, L7, L7) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A9 (L6, L6, L5, L6, L6, L6, L6, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L5, L5, L5, L5, L6, L5, L5, L5, L5) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A10 (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L7, L7, L8, L7, L8, L7, L8, L7, L7) (L6, L6, L6, L6, L6, L6, L7, L6, L6) (L8, L8, L8, L8, L8, L8, L8, L8, L8) A11 (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L8, L8, L8, L8, L8, L8, L8, L8, L8) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L6, L6, L6, L6, L6, L6, L7, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A12 (L5, L5, L5, L5, L5, L5, L5, L5, L7) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L7, L7, L7, L7, L7, L7, L7, L7, L7) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L7, L8, L8, L8, L8, L8, L7, L8, L8) A13 (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L4, L5, L5, L6, L5, L5, L4, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A14 (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A15 (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L7, L7, L7, L7, L7, L7, L7, L7, L7) (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L1, L1, L1, L1, L1, L1, L1, L1, L1) A16 (L5, L5, L5, L5, L5, L6, L5, L5, L5) (L1, L1, L1, L1, L1, L1, L1, L1, L1) (L7, L6, L7, L7, L7, L8, L7, L7, L7) (L6, L6, L6, L6, L6, L7, L6, L6, L6) (L6, L6, L6, L6, L6, L7, L6, L6, L6) (L7, L7, L7, L7, L7, L8, L7, L7, L7) A17 (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L7, L8, L7, L8, L7, L8, L7, L8, L8) (L6, L7, L6, L7, L6, L8, L6, L7, L7) (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L6, L6, L6, L7, L6, L7, L6, L7, L7) A18 (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L7, L7, L6, L7, L6, L8, L7, L7, L7) (L5, L6, L5, L6, L5, L6, L5, L6, L6) (L5, L5, L5, L5, L5, L5, L5, L5, L5) (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L5, L6, L5, L7, L5, L6, L5, L7, L7) A19 (L6, L6, L6, L6, L6, L6, L7, L6, L6) (L7, L8, L7, L8, L7, L8, L8, L8, L8) (L6, L7, L6, L7, L6, L7, L7, L7, L7) (L7, L7, L7, L7, L7, L7, L8, L7, L7) (L6, L6, L6, L6, L6, L6, L7, L6, L6) (L6, L6, L6, L7, L6, L7, L7, L7, L7) A20 (L4, L4, L4, L4, L4, L4, L4, L4, L4) (L6, L7, L6, L7, L6, L8, L6, L7, L7) (L6, L6, L6, L6, L6, L8, L6, L6, L6) (L5, L5, L5, L5, L5, L5, L5, L5, L5) (L6, L6, L6, L6, L6, L6, L6, L6, L6) (L6, L6, L6, L7, L6, L7, L6, L7, L7) A21 (L4, L4, L4, L4, L4, L4, L4, L4, L4) (L8, L8, L8, L8, L8, L8, L8, L8, L8) (L4, L5, L4, L6, L4, L6, L4, L6, L6) (L4, L4, L4, L4, L4, L4, L4, L4, L4) (L7, L7, L7, L7, L7, L7, L7, L7, L7) (L7, L7, L7, L8, L7, L8, L7, L8, L8) E. Khanmohammadi, et al. Operations Research Perspectives 6 (2019) 100122 7 Satisfying these constraints, sand s A A(˜,˜) 1 2 will be called the similarity function of intuitionistic fuzzy sets and the similarity degree between the intuitionistic fuzzy sets A ˜1 and A ˜2 , respectively. The cosine similarity measure which is based on Bhattacharya's distance [76,77] between any two intuitionistic fuzzy sets A ˜1 and A ˜2 on the finite universal set =X X X X{ , , ..., } n1 2 can be established as follows [78]: = + + + + + + = Sim A A µ x µ x x x x x µ x x x µ x x x (˜,˜) (( ( ) ( ) ( ) ( ) ( ) ( ))/ / ( ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) ni n AiAi A i A i A i A i Ai A i A i Ai A i A i 1 2 1 1 2 2 2 2 2 2 1 2 1 2 1 2 11 1 1 222 2 1 2 (8) 3.3. Proposed Intuitionistic fuzzy similarity-based method Deng [53] offered a method to rank alternatives in a multi-criteria problem through the combination of gradient and variable values. In Deng's approach, each alternative is considered as a vector in an ndimensional space. Positive and negative ideal vectors are defined, and the alternatives are ranked based on the similarity between alternativeA i and the ideal vectors ± Ai . The highly preferable option should have the highest degree of similarity in the positive ideal solution and the lowest degree of similarity in the negative ideal solution (Fig. 2). In this article, a fuzzy similarity method is presented based on Deng's model, consisting of the following steps: Having =A A A A{ , , ..., } m1 2 as a set of alternatives and =C C C C{ , , ..., } n1 2 as a set of criteria, the intuitionistic fuzzy similaritybased method is performed through the following procedure: Step 1: Deciding the weights of the decision group, which consist of ldecision makers. The intuitionistic fuzzy numbers reflect the importance of the decision makers, which is considered as a linguistic term. If the rate the kth decision maker is illustrated in the intuitionistic fuzzy number =E µ[ , , ] kkk k , its weight is obtained as: = + + + =+ ( ) ( ) µ µ k kk µ µ k l kk µ µ 1 k kk k kk (11) and = = 1 k l k 1 . Step 2. Construction of the aggregated intuitionistic fuzzy decision matrix according to the decision makers' opinions and their weights obtained from the last step. With =× D d( ) kij km n ( ) ( ) as the intuitionistic fuzzy decision matrix of each decision maker and ={ , , ..., } n1 2 as the weight of each decision maker, the next step of the process is to construct the aggregated intuitionistic fuzzy decision matrix. In this regard, all the individual decision opinions must be integrated into one single matrix, for which the IFWA operator proposed by [79] is utilized. We have =× D d( ) ij m n , where: = = = = = = = d IFWA d d d d d d d µ µ ( , , ..., ) ... 1 1 , ( ) , 1 ( ) ij ij ij ij l ij ij ij lij l k l ij k k l ij k k l ij k k l ij k (1) (2) ( ) 1(1) 2(2) 3(3) ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) k k k k (12) Also, = = =d µ x x x i m j n( ( ), ( ), ( ))( 1, 2, ..., ; 1, 2, ..., ) ij Aj A j A j ii i . The aggregated intuitionistic fuzzy decision matrix is thus calculated: = … D d d d d m n nm 11 1 1 (13) Step 3. Finding the weights of the criteria. The differences between the importance of criteria are represented by the different weights (Wj), which are calculated through the FLSSM method and multiplied by the decision matrix as a scalar. Step 4. Construction of the aggregated weighted intuitionistic fuzzy decision matrix. Based on the definitions provided in (14) and (15) (Krassimir T [72]), the aggregated weighted intuitionistic fuzzy decision matrix is created according to (16): = =D W D x µ x x x X. { , 1 (1 ( )) , ( ( )) | } AwAw ii (14) And = +x µ x x( ) 1 (1 (1 ( )) ( ( )) ) A w AwAw . iii (15) = … D d d d d m n nm 11 1 1 (16) Where = = ( ) d µ µ x x x( , , ) ( ), ( ), ( ) ij ij ij ij A W j A W j A W j ii i is an element. Step 5. Calculation of the two intuitionistic fuzzy positive-ideal and negative-ideal solutions. Table 9 The aggregated intuitionistic fuzzy decision matrix. C01 C02 C03 C04 C05 C06 A1 (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.8, 0.07, 0.13) (0.1, 0.9, 0) (0.1, 0.9, 0) A2 (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.75, 0.1, 0.15) (0.1, 0.9, 0) (0.1, 0.9, 0) A3 (0.72, 0.12, 0.15) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.65, 0.16, 0.19) (0.61, 0.19, 0.2) (0.1, 0.9, 0) A4 (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.8, 0.07, 0.13) (0.1, 0.9, 0) A5 (0.8, 0.07, 0.13) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) A6 (0.61, 0.19, 0.2) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.61, 0.19, 0.2) (0.1, 0.9, 0) A7 (0.62, 0.19, 0.19) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.6, 0.19, 0.21) (0.62, 0.19, 0.19) (0.1, 0.9, 0) A8 (0.71, 0.13, 0.16) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.71, 0.13, 0.16) (0.73, 0.12, 0.15) (0.1, 0.9, 0) A9 (0.59, 0.2, 0.21) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.51, 0.25, 0.24) (0.1, 0.9, 0) A10 (0.6, 0.19, 0.21) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.73, 0.12, 0.15) (0.61, 0.19, 0.2) (0.8, 0.07, 0.13) A11 (0.1, 0.9, 0) (0.1, 0.9, 0) (0.8, 0.07, 0.13) (0.1, 0.9, 0) (0.61, 0.19, 0.2) (0.1, 0.9, 0) A12 (0.52, 0.24, 0.24) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.7, 0.13, 0.17) (0.1, 0.9, 0) (0.78, 0.09, 0.14) A13 (0.1, 0.9, 0) (0.1, 0.9, 0) (0.51, 0.25, 0.25) (0.1, 0.9, 0) (0.6, 0.19, 0.21) (0.1, 0.9, 0) A14 (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.1, 0.9, 0) A15 (0.6, 0.19, 0.21) (0.1, 0.9, 0) (0.1, 0.9, 0) (0.7, 0.13, 0.17) (0.6, 0.19, 0.21) (0.1, 0.9, 0) A16 (0.51, 0.25, 0.24) (0.1, 0.9, 0) (0.7, 0.13, 0.17) (0.61, 0.19, 0.2) (0.61, 0.19, 0.2) (0.71, 0.13, 0.16) A17 (0.6, 0.19, 0.21) (0.76, 0.09, 0.15) (0.67, 0.15, 0.18) (0.6, 0.19, 0.21) (0.6, 0.19, 0.21) (0.64, 0.17, 0.19) A18 (0.6, 0.19, 0.21) (0.69, 0.13, 0.17) (0.55, 0.22, 0.23) (0.5, 0.25, 0.25) (0.6, 0.19, 0.21) (0.59, 0.19, 0.22) A19 (0.61, 0.19, 0.2) (0.77, 0.09, 0.15) (0.66, 0.15, 0.18) (0.71, 0.13, 0.16) (0.61, 0.19, 0.2) (0.65, 0.16, 0.19) A20 (0.4, 0.31, 0.29) (0.67, 0.15, 0.18) (0.62, 0.18, 0.2) (0.5, 0.25, 0.25) (0.6, 0.19, 0.21) (0.64, 0.17, 0.19) A21 (0.4, 0.31, 0.29) (0.8, 0.07, 0.13) (0.5, 0.25, 0.26) (0.4, 0.31, 0.29) (0.7, 0.13, 0.17) (0.74, 0.1, 0.15) E. Khanmohammadi, et al. Operations Research Perspectives 6 (2019) 100122 8