Introduction

A proper machine selection among feasible alternatives has been a crucial issue for manufacturing firms as it affects the overall performance of manufacturing system. An improperly chosen machine can reduce the productivity, flexibility and responsiveness capabilities of the manufacturing process. There exists a strong correlation between speed, quality and cost of the production and type of the machines utilized (Arslan et al. 2004). Besides, the capital investment in machinery and equipment affects the profitability of the facility during its initial stages of operation. Due to having many alternatives and conflicting objectives, selecting the appropriate machine is a complicated and time-consuming problem to be tackled. In order to make an effective assessment decision makers (DMs) should be experts or be very familiar with the machine features. However, Gerrard (1988) concludes that the role of engineering staff for the final decision on machine selection is 6 % while the rest 94 % belongs to middle and upper management. This result implies that the selection process should be simplified and practical methods should be used.

Suitable machine selection process includes both quantitative and qualitative aspects such as purchasing cost, floor space, working environment, productivity, operation conditions, compatibility, and operating preference to be taken into consideration (Wang et al. 2000). Hence, appropriate machine selection can be regarded as a multi-criteria decision making (MCDM) problem. MCDM refers to making decisions in the presence of multiple, usually conflicting evaluation criteria (Zavadskas et al. 2014). In many decision making problems the DMs’ judgments are not crisp, and it is difficult for the DM to provide exact numerical values for the evaluation criteria or attributes. Furthermore, it is recognized that human assessment on qualitative attributes is always subjective and thus imprecise. Since DMs’ judgments on evaluating machine alternatives inhold ambiguity and multiplicity of meaning, most of the evaluation parameters cannot be provided precisely. Thus, DMs’ judgements and the weights of the criteria are usually expressed in linguistic terms (Kahraman et al. 2009). Fuzzy set theory (Zadeh 1965) has been widely utilized in order to model this kind of uncertainty in human preferences.

This paper aims to propose an integrated approach consisting of fuzzy simple multiattribute rating technique (SMART) method and weighted fuzzy axiomatic design (WFAD) approach to selecting a proper continuous fluid bed tea dryer for a privately-owned tea plant operating in Turkey. The weights of the evaluation criteria are calculated via fuzzy SMART and then WFAD is exploited to rank competing machine alternatives in terms of their overall performance. The rest of the paper is organized as follows. A comprehensive literature research concerning optimal machine selection problem is provided in section “Literature review on machine selection”. In section “Methodology”, the research methodology containing the mathematical models used in the application is described. The real case application is presented in section “Discussion and conclusion”. The results are discussed in this section, as well. Finally, conclusions are given in the last section.

Literature review on machine selection

Suitable machine selection problem for different manufacturing facilities have been studied by various researchers using mathematical models, heuristic algorithms and different MCDM methods. The existing studies have addressed the problem mostly for specific type of environment, namely, flexible manufacturing systems (FMS). Atmani and Lashkari (1998) introduced a model for machine tool selection and operation allocation in FMS using a linear, 0–1 integer programming model. Arslan et al. (2004) presented a DSS including a multi-criteria weighted average (MCWA) method for machine tool selection process and the method is demonstrated with an example. Sarkis (1997) executed Data Envelopment Analysis (DEA) to handle the evaluation of FMS.

The use of fuzzy logic is another largely adopted approach among researchers. Wang et al. (2000) proposed a fuzzy multiple attribute decision making model to select the appropriate machines for FMS. A fuzzy multiple objective programming approach to facilitate decision making in the selection of a FMS is presented by Karsak and Kuzgunkaya (2002). Linguistic variables and triangular fuzzy numbers (TFNs) were used to quantify the vagueness inherent in decision parameters, e.g., increase in market response, improvement in quality, reduction in setup cost, and so forth. Karsak (2008) employed a DEA model for FMS selection that can take into account crisp, ordinal, and fuzzy data. Liu (2008) developed a fuzzy DEA method able to evaluate the performance of FMS alternatives when the input and output data are represented with crisp and fuzzy data. Sivarao et al. (2009a) proposed Fuzzy based Graphical User Interface (GUI) for modeling of laser machining conditions. The models were then compared for their statistical validation by Root Mean Square Error (RMSE) values. Sivarao et al. (2009b) presented a modeling technique and prediction of surface roughness for 2.5 mm Manganese Molybdenum pressure vessel plate by hybrid intelligence, namely, adaptive neuro-fuzzy inference system (ANFIS). Back propagation optimization method was employed to optimize the epoch number and training of data sets. Based on Zadeh’s extension principle a pair of two-level mathematical programs were formulated to calculate the lower and upper bounds of the fuzzy efficiency score of the alternatives.

There are also hybrid methods in the literature integrating different MCDM models that are effective and practical in guiding machine selection problem. Tabucanon et al. (1994) proposed a two-stage decision support system (DSS) for the selection process of alternative CNC turning centers. The presented approach combines the Analytic Hierarchy Process (AHP) method with the rule-based technique for creating expert systems. Önüt et al. (2008) proposed a hybrid fuzzy approach by integrating fuzzy AHP and fuzzy TOPSIS in order to select the optimal vertical CNC machining centers for a manufacturing company in Istanbul, Turkey. The fuzzy AHP was used to obtain the criteria weights and subsequently fuzzy TOPSIS was performed for ranking the alternatives. Taha and Rostam (2011) adopted a DSS for machine tool selection in flexible manufacturing cell using fuzzy AHP and artificial neural network (ANN). The ANN was used to verify the results of fuzzy AHP and to predict the alternatives’ ranking. Hybrid use of FAHP and Grey Relational Analysis (GRA) approaches was employed by Samvedi et al. (2012) for the selection of a machine tool from a given set of alternatives. Fuzzy AHP was used to calculate the priority weights of the criteria and GRA was next employed to rank the alternatives.

Initiated by Suh (1990), Axiomatic Design (AD) theory provides a systematic search process via the design space to minimize the random search process and attain the best design solution among many alternatives (Kulak and Kahraman 2005a). The AD was developed as an effective DSS for FMS designers in determining the appropriate FMS configuration at the design stage in the study of Babic (1999). In the fuzzy case, we have incomplete information about the system and design ranges. The Fuzzy Information Axiom (FIA) has been developed by Kulak and Kahraman (2005a, b) to solve MCDM problems having linguistic information. In Kulak and Kahraman (2005a), the evaluation of the alternatives and the definition of functional requirements (FRs) were defined by TFNs. The proposed approach was applied to multiattribute comparison of advanced manufacturing systems (AMS). Kulak et al. (2005) used the unweighted and weighted multi-attribute AD approaches to the selection among punching machines while investing in a manufacturing system. In the study, the weighted information axiom approach for the MCDM problems was first time proposed in the literature. Kulak (2005) proposed a DSS considering both technical and economic criteria in material handling equipment selection problem. In the developed DSS called FUMAHES, the final decision for the best equipment selection was provided via using the information axiom of the design principles.

The existing studies addressed the machine selection problem mostly for specific type of environment, namely, FMS. However, new machine selection should be applicable to many manufacturing environments. In order to fill this gap in the literature a novel approach integrating fuzzy SMART and WFAD methods is proposed in this study so as to select a suitable continuous fluid bed tea dryer for a privately-owned tea plant. The presented methodology aims to provide a robust decision support tool to the DMs of manufacturing industries, specifically to those of the tea industry. To the best of our knowledge, fuzzy SMART and WFAD methods have never been combined to solve any MCDM problem to date. In addition, we have not come across any paper in the literature dealing with machine selection problem in the tea industry world-wide.

Methodology

In this section, crisp AD, fuzzy AD and WFAD methods will be defined briefly.

Principles of axiomatic design

That there are fundamental axioms which drive the design process is the basic postulate of AD approach. Two axioms were identified by analyzing common elements always present in good designs, be they product, process, or systems design. The first design axiom is the Independence Axiom and the second one is the Information Axiom (Suh 1995).

Axiom 1

The Independence Axiom

“Maintain the independence of the FRs”.

FRs are defined as the minimum set of independent requirements must be satisfied in the design process.

Axiom 2

The Information Axiom

“Minimize the information content”.

The information axiom states that among those designs that satisfy the independence axiom, the design that has the smallest information content is the best design. Axiom 2 is the criterion for the selection of the optimum design solutions from among those that satisfy Axiom 1. Information is defined in terms of the information content, \(I_{i}\), which is related in its simplest form to the probability of satisfying a given FR. \(I_{i}\) determines the design with the highest probability of success as the best design (Kulak et al. 2005).

Information content, \(I_{i}\), for a given \(FR_{i}\) is expressed as follows:

$$\begin{aligned} I_i =\log _2 \left( {\frac{1}{p_i }} \right) \end{aligned}$$
(1)

where p is the probability of achieving the \(FR_{i}\). In any design situation, the probability of success is specified by what the designer wants to attain in terms of tolerance (i.e., design range) and what the system is capable of delivering (i.e., system range). As shown in Fig. 1 (Kim et al. 1991) the overlap between the designer-specified “design range” and the system capability range “system range” is the region where the acceptable solution exists. Thus, in the case of uniform probability distribution function, \(p_{i}\) can be written as (Kim et al. 1991):

$$\begin{aligned} \hbox {p}_\mathrm{i} =\frac{\hbox {System range}}{\hbox {Common range}} \end{aligned}$$
(2)

Hence, the information content is equivalent to Eq. (3).

$$\begin{aligned} \hbox {I}_\mathrm{i} =\log _2 \left( {\frac{\hbox {System range}}{\hbox {Common range}}} \right) \end{aligned}$$
(3)
Fig. 1
figure 1

Probability distribution of a system parameter

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In case \({{\textit{FR}}}_{i}\) is a continuous random variable, the probability of achieving \({\textit{FR}}_{i}\) in the design range can be denoted as,

$$\begin{aligned} p_i =\int _{dr^{l}}^{dr^{u}} {p_s } (\hbox {FR}_{i}) \hbox {dFR}_{\mathrm{i}} \end{aligned}$$
(4)

where \(p_{s}({FR}_{i})\) is the system pdf (probability density function) for \({FR}_{i}\). The Eq. (4) provides the probability of success by integrating the system pdf over the whole design. The area of the common range \((A_{cr})\) depicted in Fig. 2 (Kulak and Kahraman 2005a, b) is equal to the probability of success \(\hbox {p}_{\mathrm{i}}\) (Suh 1990). Therefore, the information content is equal to

$$\begin{aligned} I_i =\log _2 \left( {\frac{1}{A_{cr} }} \right) \end{aligned}$$
(5)
Fig. 2
figure 2

Design range, system range, common range and probability density function of a FR

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In the literature, there are many applications of AD methodology to design products (Suh 1995), systems, (Suh 1997) and software (Kim et al. 1991).

Fuzzy axiomatic design approach

Kulak and Kahraman (2005a, b) introduced the FIA as a novel approach which has been used for MCDM problems under fuzzy environment. The crisp information axiom approach explained above can be used as a solution of decision-making problems where available information is suitable to be modeled by probability theory. However, this approach cannot be used where available information is qualitative and linguistic. Zadeh’s fuzzy set theory (Zadeh 1965) offers a mathematical system form and helps to deal this kind of vagueness by using linguistic variables or fuzzy numbers. TFN is the most utilized fuzzy number type in MCDM literature due to its simplicity and ease of implementation. A TFN, \(\tilde{\hbox {A}}\) can be defined as a triplet \(\tilde{\hbox {A}}=(\upalpha _1, \upalpha _2 ,\upalpha _3)\) of crisp numbers with \(\upalpha _1 <\upalpha _2 <\upalpha _3\) and can be denoted with membership function \(\mu _\mathrm{A}\)(x) as shown in (Fig. 3) (Chen and Wang 2009: 235).

$$\begin{aligned} \upmu _\mathrm{A} (\mathrm{x})=\left\{ {\begin{array}{ll} (\hbox {x}-\upalpha _1 )/(\upalpha _2 -\upalpha _1 ),&{} \upalpha _1 \le \hbox {x}\le \upalpha _2 \\ (\upalpha _{3} -\hbox {x})/(\upalpha _3 -\upalpha _2 ),&{} \upalpha _2 \le \hbox {x}\le \upalpha _3 \\ 0, &{}\hbox {otherwise} \\ \end{array}} \right. \end{aligned}$$
(6)
Fig. 3
figure 3

A TFN

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In the fuzzy case, there is incomplete information regarding the system and design ranges. The system and design range for a certain criterion will be expressed by using “over a number”, “around a number” or “between two numbers”. Triangular or trapezoidal fuzzy numbers can represent these kinds of expressions (Çelik et al. 2009c). We now have a function of triangular or trapezoidal fuzzy number while we have a probability density function in the crisp case. So, the common area is the intersection area of triangular or trapezoidal fuzzy numbers. The common area between design range and system range is depicted in Fig. 4. (Kulak and Kahraman 2005b)

Fig. 4
figure 4

The common area of system and design ranges

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Thus, information content is formulated as below.

$$\begin{aligned} \hbox {I}_\mathrm{i} =\log _2 \left( {\frac{\hbox {TFN of system design}}{\hbox {Common area}}} \right) \end{aligned}$$
(7)

Weighted fuzzy AD approach

For cases where the DM wishes to assign a different weight \((w_{j})\) for each criterion, Kulak and Kahraman (2005b) proposed the following equation to be used.

$$\begin{aligned} I_{ij} =\left\{ {\begin{array}{ll} \left[ {\log _2 \left( {\frac{1}{p_{ij} }} \right) } \right] ^{1/w_j },&{} 0\le I_{ij} \le 1 \\ \left[ {\log _2 \left( {\frac{1}{p_{ij} }} \right) } \right] ^{1/w_j },&{} 0\le I_{ij} \le 1 \\ w_j, &{} I_{ij} =1 \\ \end{array}} \right\} \end{aligned}$$
(8)

The Eq. (8) is utilized for both the weighted crisp and the WFAD approaches. In the literature, there are a few applications of FIA. The research problems considered in these studies cover: determining most usable seat design solution (Eraslan et al. 2006), teaching assistant selection problem (Kahraman and Cebi 2009), selection of the best alternative among shipyards (Çelik et al. 2009c), final shipping investment decision (Çelik et al. 2009a), ensuring the competitiveness requirements for major Turkish container ports (Çelik et al. 2009b) and evaluation of ergonomic compatibilities of AMS (Maldonado et al. 2013).

The fundamentals of SMART

SMART is a compensatory method of MCDM introduced by Edwards in 1971. This method has been designed to provide a simple way to implement the beginnings of MAUT. According to Edwards and Barron (1994), the SMART is “by far the most common method actually used in real, decision-guiding multiattribute utility measurements”. SMART is well-accepted in MCDM literature as it involves both quantitative and qualitative decision criteria. The SMART begins with the identification of alternatives and the determination of the criteria to be used for evaluating those alternatives. Then, the procedure covers ordering the criteria according to importance, obtaining criteria weights based upon that ordering, scoring the alternatives according to the criteria and synthesizing the scores for each of the alternatives (Seydel 2006).

SMART utilizes the simple additive weighting (SAW) method so as to attain total values for individual alternatives helping to rank them according to order of preference (Edwards 1977; Edwards and Barron 1994). In this method, a score is obtained by adding the contribution from each criterion. The total score for each alternative can be calculated by multiplying the normalized value of each criterion for the alternatives with the importance weight of the criterion and then summing these products over all the criteria as formulated below (Akgün et al. 2010).

$$\begin{aligned} \hbox {S}_{\mathrm{i}} =\sum _{\hbox {j}=1}^{\mathrm{n}} {\hbox {w}_{\mathrm{j}} \hbox {r}_{\mathrm{ij}} \hbox { i}=1,\ldots , \hbox {m}} \end{aligned}$$
(9)

where \(\hbox {S}_{\mathrm{i}}\) is the total score of alternative i, \(\hbox {w}_{\mathrm{j}}\) is the importance weight of criterion j, \(\hbox {r}_{\mathrm{ij}}\) is the normalized rating of the alternative i for the criterion j, m is the number of alternatives and n is the number of criteria. Finally, the alternative with the highest score is selected as the preferred one.

Fuzzy SMART

Although SMART has been successfully applied in MCDM problems (Barla 2003; Seydel 2006) this approach is ineffective when dealing with the inherent imprecision of linguistic valuation in the decision-making. Kwong et al. (2002) integrated fuzzy set theory into SMART in order to assess the performance of suppliers. The supplier assessment forms were used first to establish the scores of individual assessment items and then the scores were input to a fuzzy expert system for the determination of supplier recommendation index. Chou and Chang (2008) proposed a novel approach integrating fuzzy logic with SMART to solve MCDM problems in group decision making.

The proposed approach comprised of the following steps (Chou and Chang 2008).

Step 1. Form a decision committee and specify evaluation criteria

Group size also influences the effectiveness of group decision making. Yetton and Botter (1983) suggested groups of five as the most effective closely followed by groups of seven.

Step 2. Assessment and aggregation of the fuzzy weights

DMs settle the relative importance of decision criteria using linguistic weighting variables and then the aggregate fuzzy weight of each criterion are calculated. Let \(\tilde{{W}}_{{il}} =({a}_{{il}} ,{b}_{{il}} ,{c}_{{il}} )\), (\(\hbox {i} = 1,\ldots , \hbox {m}; l = 1,{\ldots },\hbox {n}\)) be the TFN corresponding to the linguistic variable given to criterion l by \(\hbox {DM}_{\mathrm{i}}\). The aggregate fuzzy criterion weight, \(\tilde{W}_l =(a_l ,b_l ,c_l)= (l = 1,\ldots ,m)\) of criteria l designated by the committee of m DMs is computed as follows:

$$\begin{aligned} \tilde{W}_l =\frac{1}{m}\otimes \sum _{i=1}^m {\tilde{W}_{il} } \end{aligned}$$
(10)

where \(a_l =\frac{1}{m}\otimes \sum \nolimits _{i=1}^m {a_{il}} , b_l =\frac{1}{m}\otimes \sum \nolimits _{i=1}^m {b_{il} }\) and \(c_l =\frac{1}{m}\otimes \sum \nolimits _{i=1}^m {c_{il} }\)

Step 3. Defuzzification of the fuzzy weights of criteria

As the aggregate fuzzy criteria weights obtained through Eq. (2) are fuzzy numbers, a defuzzification method is entailed to converting the fuzzy numbers to crisp values. There exists four popular defuzzification approach in the literature each of which has its advantages and disadvantages. These are the centroid method (or center of area-COA), mean of maximal, \(\upalpha \)-cut method, and signed distance method (Yao and Chiang 2003). Since the fuzzy SMART requires cardinal weights that are normalized to sum to 1, the crisp value of weight for criterion l, denoted as \(\hbox {W}_{l}\) is defined as:

$$\begin{aligned} W_l =\frac{d(\tilde{W}_l )}{\sum \nolimits _{l=1}^n {d(\tilde{W}_l )} },\hbox {l}=1,\ldots ,n \end{aligned}$$
(11)

where \(\sum \nolimits _{l=1}^n {\tilde{W}_l} =1\). In here, \(d(\tilde{W}_l)\) denotes the defuzzification of \(\tilde{W}_l\).

Step 4. Computation of the aggregated fuzzy ratings of criteria

DMs exploit linguistic rating variables to assess fuzzy ratings of alternatives with respect to decision criteria and then compute aggregated fuzzy ratings and form the fuzzy rating matrix. Let \(\tilde{x}_{ijkl} =(a_{ijkl} ,b_{ijkl} ,c_{ijkl} )\), (\(\hbox {i}=1,\ldots ,\hbox {m}; \hbox {j}=1,{\ldots },\hbox {p}; \hbox {k}=1,{\ldots },\hbox {r}; \hbox {l} = 1,\ldots , \hbox {n}\)) be the linguistic rating assigned to the alternative \(\hbox {A}_{j}\) for qualitative/subjective criterion \(\hbox {C}_{l}\) by \(\hbox {DM}_{\mathrm{i}}\). Similarly, let \(\hbox {v}_{\mathrm{ijkl} }= (\hbox {d}_{\mathrm{ijkl}}, \hbox {e}_{ijkl}, \hbox {f}_{\mathrm{ijkl}}) (\hbox {i}=1,\ldots ,\hbox {m}; \hbox {j}=1,{\ldots },\hbox {p; k}=1,{\ldots },\hbox {r}; \hbox {l} = 1,\ldots , \hbox {n}\)) be the TFN (or crisp) cost or benefit value assessed to the alternative \(\hbox {A}_{j}\) for quantitative/objective criterion \(\hbox {C}_{l}\) by \(\hbox {DM}_{\mathrm{i}}\). The following formulization is applied to normalizing the quantitative value.

$$\begin{aligned} \tilde{x}_{ijkl} =\frac{\tilde{v}_{ijkl} - \min \nolimits _{jk} (d_{ijkl} )}{\max \nolimits _{jk} (f_{ijkl} )-\min \nolimits _{jk} (d_{ijkl} )}\otimes 10 \end{aligned}$$
(12)

where \(\tilde{x}_{ijkl} \) denotes the normalized fuzzy rating of fuzzy benefit \(\tilde{v}_{ijkl} \).

$$\begin{aligned} \tilde{x}_{ijkl} =\frac{\mathop {\max }\limits _{jk} (f_{ijkl} )-\tilde{v}_{ijkl} }{\mathop {\max }\limits _{jk} (f_{ijkl} )-\mathop {\min }\limits _{jk} (d_{ijkl} )}\otimes 10 \end{aligned}$$
(13)

where \(\tilde{x}_{ijkl} \) denotes the normalized fuzzy rating of fuzzy cost \(\tilde{v}_{ijkl} \).

The aggregated fuzzy rating, denoted as \(\tilde{x}_{jkl} =(a_{jkl} ,b_{jkl} ,c_{jkl})(\hbox {j}=1,{\ldots },\hbox {p}; \hbox {k}=1,{\ldots },\hbox {r}; \hbox {l} = 1,\ldots , \hbox {n}\)) is computed by:

$$\begin{aligned} \tilde{x}_{jkl} =\frac{1}{m}\otimes \sum _{i=1}^m {\tilde{x}_{ijkl} } \end{aligned}$$
(14)

Then, the aggregated fuzzy rating of alternative \(\hbox {A}_{j}\) for criterion \(\hbox {C}_{j}\) can be obtained by,

$$\begin{aligned} \tilde{x}_{jk} =\sum _{l=1}^n {W_l } \otimes \sum _{i=1}^m {\tilde{x}_{ikl} } \end{aligned}$$
(15)

which can subsequently be represented as \(\tilde{x}_{jk} =(a_{jk} ,b_{jk} ,c_{jk} )\) (\(\hbox {j}=1,{\ldots },\hbox {p}; \hbox {k}=1,{\ldots },\hbox {r}\)).

Thus, the fuzzy rating matrix \(\tilde{D}\) can be expressed in a matrix format as,

$$\begin{aligned} \tilde{D}=\left[ {x_{ij} } \right] _{mxn} \end{aligned}$$
(16)

Step 5. Computation of the total fuzzy values of alternatives

The total fuzzy value vector \(\tilde{T}\) can be then achieved by multiplying the fuzzy rating matrix \(\tilde{D}\) by the corresponding weight vector W respectively, i.e.

$$\begin{aligned} \tilde{T}=\tilde{D}\otimes W^{T}=\left[ {\tilde{f}_j } \right] _{mx1} \end{aligned}$$
(17)

where \(\tilde{f}_j =(a_j ,b_j ,c_j )\) (\(\hbox {j}=1,{\ldots },\hbox {p}\)).

Step 6. Defuzzification of the total fuzzy values

The total crisp values of alternatives \(d(\tilde{f}_j )\) can be calculated using the following managerial defuzzification equation,

$$\begin{aligned} d(\tilde{f}_j )=(a_j^{*} a_j +(1-a_j )^{*} c_j ,0 \le a_j \le 1 \end{aligned}$$
(18)

where \(\tilde{f}_j =(a_j ,b_j ,c_j )(\hbox {j}=1,{\ldots },\hbox {p}) a_j\) is the risk coefficient of alternative \(A_{j}\) and \(d(\tilde{f}_j)\) is the defuzzified value of the total fuzzy value of alternative \(A_{j}\). Then, the alternative with maximum \(d(\tilde{f}_j)\) is selected as the optimal one.

Case study

Attention has often been directed to aspects of cost reduction, fuel savings, etc. in designing tea factories, but often and quite wrongly, at the expense of quality. The vital roles of temperature and time of processing are frequently overlooked by even experienced factories (Hampton 1992). Having being plucked from the tea bush the tea leaves undergo various processing stages, such as withering, cutting, tearing and curling (CTC), fermentation, drying and finally packing. The main objectives of drying are to arrest enzymic reaction as well as oxidation, removing moisture from the leaf particles and to produce a stable product with good keeping quality. The drying operation in the tea industry does not merely remove the moisture content since there are many quality factors that can be adversely affected by incorrect selection of drying conditions and drying equipments (Panchariya et al. 2002).

This paper mainly concentrates on multiple criteria evaluation of proper continuous fluid bed tea dryer alternatives for a privately-owned tea company operating in Turkish tea industry for 20 years. The firm produces 15.000 tons of black tea in a year in its three manufacturing facilities. Because the drying process plays a crucial role on the quality of tea, a comprehensive evaluation framework that is illustrated step-by-step in Fig. 5 has been established.

Fig. 5
figure 5

The flowchart of the proposed methodology

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In the initial phase of the application, a decision committee comprised of two engineers and three cross-functional managers working for the company was constructed. Subsequently, a detailed interview was carried out with the committee to settle evaluation criteria and dryer alternatives. In this stage, a questionnaire concerning the assessment of the qualitative and quantitative criteria for the fluid dryer was prepared and conducted. Consequently, the number of the alternatives was reduced to five as \(\hbox {A}_{1}, \hbox {A}_{2},\hbox {A}_{3},\hbox {A}_{4}\) and \(\hbox {A}_{5}\) and 9 evaluation criteria were determined. These criteria can be grouped into benefit criteria and cost criteria as shown in Table 1.

Table 1 List of decision criteria for machine selection
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Fuzzy SMART phase

After identifying the evaluation criteria the DMs assessed the importance weight of these criteria using the linguistic variables demonstrated in Table 2.

Table 2 Linguistic variables for the relative importance weights and corresponding TFNs
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Subsequently, these assigned fuzzy values are aggregated using Eq. (10) and the fuzzy weights of criteria can then be obtained as shown in Table 3. In order to achieve crisp weight values for the criteria centroid method is adopted due to its simplicity and widespread use. A TFN, \(\tilde{\hbox {A}}=(\upalpha _1 ,\upalpha _2 ,\upalpha _3 )\) is defuzzified by using the following centroid method equation:

$$\begin{aligned} \hbox {d}(\tilde{\hbox {A}})=\frac{1}{3} (\alpha _{1} +\alpha _{2} +\alpha _{3}) \end{aligned}$$
(19)

In addition, normalized weight values are also calculated by using Eq. (11) and included in Table 3.

Table 3 Linguistic and aggregated importance weights of the criteria
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As can be seen in Table 3, \(\hbox {C}_{1}\) (capacity) turns out to be the most important criterion for assessing dryer alternatives (\(W_{l}= 0.162\)) while space occupied is the least important one (\(W_{l}= 0.032\)).

Weighted fuzzy axiomatic design phase

The first step of the WFAD phase is the determination of the FRs for each criterion. The decision committee rated the FRs using linguistic variables. The following numerical approximation systems were adopted to systematically transform linguistic variables to their corresponding fuzzy numbers. The system includes five conversion scales as depicted in Figures 6 and 7 (Kulak and Kahraman 2005a). Besides, the membership functions of FRs are shown for both intangible and tangible criteria in Figs. 8 and 9, respectively.

The FRs defined by DMs is exhibited in Table 4.

In this study, both FRs and ratings of alternatives were evaluated by five DMs. Therefore, individual fuzzy opinions need to be aggregated to forming group consensus as the basis of group decision. Several approaches have been suggested in the literature for deriving group consensus from individual fuzzy opinions. Wang and Parkan (2006) introduced a novel approach referred to as least squares distance method (LSDM) based on Euclidean distance between fuzzy numbers. The procedure of the proposed method is briefly outlined below (Wang and Parkan 2006).

Fig. 6
figure 6

The numerical approximation system for tangible criteria

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Fig. 7
figure 7

The numerical approximation system for intangible criteria

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Fig. 8
figure 8

Membership functions of FRs of tangible criteria

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Fig. 9
figure 9

Membership functions of FRs of intangible criteria

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Table 4 DMs’ judgements on FRs
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Let \(\tilde{R}_i =(r_{i1} ,\ldots ,r_{im} )\) and \(\tilde{R}_j =(r_{j1} ,\ldots ,r_{jm})\) be two triangular or trapezoidal fuzzy numbers with \(m = 3\) for TFNs and \(m = 4\) for trapezoidal fuzzy numbers and \(w_{i}\) and \(w_{j}\) be their relative weights. The Euclidian distance between \(\tilde{R}_i\) and \(\tilde{R}_j\) is defined as

$$\begin{aligned} d_{ij} =f(w_i \tilde{R}_i ,w_j \tilde{R}_j )=\sqrt{\sum \nolimits _{k=1}^m {(w_i r_{ik} -w_j r_{jk} )^{2}} } \end{aligned}$$
(20)

The fuzzy opinions \(\tilde{R}_1 ,\ldots ,\tilde{R}_n \) are often widely scattered. In order to obtain the maximum agreement the weighted fuzzy opinions \(w_1 \tilde{R}_1 ,\ldots ,w_n \tilde{R}_n \) should move towards one another. This is the premise upon which a collective fuzzy opinion is produced. In line with this condition, the following optimization model is constituted, which minimizes the sum of the squared distances between all pairs of weighted fuzzy opinions:

$$\begin{aligned} Min\,J= & {} \sum _{i=1}^n \sum _{j=1{j}\ne {i}}^n \nonumber \\ {d}_{{ij}}^{2}= & {} \sum _{i=1}^n {\sum _{j=1 {j}\ne {i}}^n {\left[ {\sum _{{k}={1}}^{m} {(w_i r_{ik} -w_j r_{jk} )^{2}} } \right] } } \nonumber \\&\sum _{i=1}^n {w_i =1} w_i \ge 0 \quad { i}=1,\ldots ,{n} \end{aligned}$$
(21)

Let \(W^{*}=(w_1^*,\ldots ,w_n^*)^{T}\) be the optimum solution of Eq. (21). Then,

$$\begin{aligned} W^{*}=\frac{G^{-1}e}{e^{T}G^{-1}e}\ge 0 \end{aligned}$$
(22)

where \(e = (1,{\ldots },1)\) is the transpose of \(e^{T}\) and \(G^{-1}\) is the inverse of G, which is defined as

$$\begin{aligned} G= & {} (g_{ij} )_{mxn} \nonumber \\= & {} \left[ {\begin{array}{cccc} (n-1)\sum \nolimits _{k=1}^m r_{1k}^2 &{} -\sum \nolimits _{{k}={1}}^{m} {r}_{{1k}} r_{2k} &{} \cdots &{} -\sum \nolimits _{{k}={1}}^{m} {{r}_{{1k}} r_{nk}} \\ -\sum \nolimits _{k=1}^m r_{2k} {r}_{{1k}} &{}{ (n-1)}\sum \nolimits _{{k}={1}}^{m} r_{2k}^2 &{} \cdots &{}-\sum \limits _{{k}={1}}^{m} {{r}_{{2k}} r_{nk} } \\ \vdots &{}\vdots &{}\vdots &{} \vdots \\ -\sum \nolimits _{k=1}^m r_{nk} {r}_{{1k}} &{}-\sum \nolimits _{{k}={1}}^{m} r_{nk} {r}_{{2k}} &{}\cdots &{}{(n-1)}\sum \nolimits _{{k}={1}}^{m} {{r}_{{nk}}^{2} } \\ \end{array}} \right] \nonumber \\ \end{aligned}$$
(23)

where

$$\begin{aligned} g_{ij} =\left\{ {\begin{array}{l} (n-1)\sum \nolimits _{k=1}^m {r_{ik}^2 ,\quad {i}={j}={1,\ldots ,n}} \\ -\sum \nolimits _{k=1}^m {r_{ik} r_{jk} ,\quad {i,j}={1,\ldots ,n;j}\ne {i}}. \\ \end{array}} \right. \end{aligned}$$
(24)

According to the additive aggregation rule, we have

$$\begin{aligned} \tilde{R}=\sum _{i=1}^n {w_i \tilde{R}_i } =\left( {\sum _{i=1}^n {w_i r_{i1} ,\ldots ,\sum _{i=1}^n {w_i } r_{im} } } \right) \end{aligned}$$
(25)

where \(w_{i}\) is the relative weight of the ith fuzzy opinion, \(i = 1,. . .,\hbox {n}\). The weights satisfy the following condition

$$\begin{aligned} \sum _{i=1}^n {w_i =1} \end{aligned}$$
(26)

which can be expressed in vector form as

$$\begin{aligned} \hbox {e}^{\mathrm{T}} \hbox {W} = 1 \end{aligned}$$
(27)

Note that the weights determined by Eq. (22) do not reflect the relative importance assigned to the experts for their fuzzy opinions. In this study, the LSDM approach was adopted so as to obtain the aggregated fuzzy values of FRs. Next, using Eqs. (21-24) we have obtained the optimal weights and optimal aggregation values of FRs as shown in Table 5.

Table 5 Aggregated judgements for FRs
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Having identified the FRs, the DMs evaluated five continuous fluid bed tea dryers with respect to decision criteria. The DMs’ ratings are demonstrated in Table 6. Afterwards, DMs’ judgements for alternatives are aggregated via LSDM approach. The results are given in Table 7 below. The next step in the WFAD phase is the calculation of the information contents of the alternatives for each criterion using Eq.(7). The computed information contents are represented in Table 8. According to the total unweighted information content values, the alternative \(\hbox {A}_{3}\), which has the minimum information content value (16.314), has emerged as the optimal continuous fluid bed dryer for the tea plant. The information content becomes infinitive (this means the elimination of that alternative) if a certain range is above or below its design range. Thus, \(\hbox {A}_{2}\) and \(\hbox {A}_{5}\) have been eliminated since they have infinite values. The overall ranking of remaining alternatives from best to worst is as \(\hbox {A}_{3} > \hbox {A}_{4 }> \hbox {A}_{1}\). Next, utilizing the normalized weights computed in Fuzzy SMART phase the weighted information contents of the alternatives has been calculated through Eq.(8). The results are presented in Table 9. The alternative that has the minimum information content value is the best alternative. Therefore, the overall ranking of alternatives is as \(\hbox {A}_{3} > \hbox {A}_{1 }> \hbox {A}_{4}\). As seen, this ranking order differs from that of the unweighted FAD approach.

Table 6 DMs’ judgements for alternatives with respect to criteria
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Table 7 Aggregated matrix of DMs’ judgements for alternatives
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Table 8 Information contents of alternatives
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Table 9 Weighted information contents of alternatives
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Discussion and conclusion

In this paper, hybrid combination of fuzzy SMART and WFAD has been introduced in order to select a proper continuous fluid bed tea dryer for a Turkish tea plant. By applying the proposed hybrid method it is aimed to exploit the advantageous attributes of each method. A project team consisting of five experts determined the decision criteria and narrowed down the number of machine alternatives in the initial phase of the application. Afterwards, the weights of the evaluation criteria were calculated via fuzzy SMART in the second phase. Since it does not require much computational effort and incorporates a wide variety of quantitative and qualitative criteria the SMART has been successfully applied to various MCDM problems. However, as crisp SMART is inadequate when handling with the vagueness of the decision making process fuzzy SMART is proposed for our case. The computational procedure of the fuzzy SMART is much faster than the other existing MCDM approaches as it uses simplified ranking fuzzy numbers rather than the other fuzzy ranking methods (Chen and Hwang 1992).

In the last phase of the application, WFAD was exploited to ranking machine alternatives in terms of their overall performance. AD approach takes into account the design range of each criterion specified by the designer (decision maker). Hence, the alternative meeting the design ranges (FRs) is the optimal choice in AD approach whereas the alternative satisfying the criteria at their best levels is selected in most of the other MCDM methods. The AD approach differs from many other existing MCDM methods by eliminating an alternative which does not meet the design range of any criterion (Kahraman and Cebi 2009). In our case, the application results indicate that three out of five alternatives manage to satisfy the expectation levels of the DMs. The other two alternatives (\(\hbox {A}_{2}\) and \(\hbox {A}_{5})\) have been eliminated since they do not satisfy the whole FRs. Another advantage of AD approach over other methods is that it provides flexibility to DMs to establishing the design ranges of the evaluation criteria. For example, the designer may not want to satisfy the criterion the capacity at its best level because of its cost. This flexibility is not involved in many other methods such as AHP, TOPSIS and scoring models etc. Crisp AD model cannot be used with incomplete information in fuzzy environments since the expression of decision variables by crisp numbers would be ill defined. Thus, WFAD approach is proposed in this study. When performing WFAD phase, ratings of both FRs and alternatives were assessed by five DMs. Therefore, individual fuzzy opinions were required to be aggregated in order to set up a group consensus. The LSDM developed by Wang and Parkan (2006) used for deriving group consensus. The proposed LSDM has the advantage of computational simplicity as it eliminates the need to convey time-consuming iterative procedures that are typical of the existing aggregation approaches. In addition, it can be exploited to aggregate any form of fuzzy numbers. This feature considerably widens the range of aggregation problems where it is viable.

The rejection of an alternative which does not meet a design range may seem a dis advantage aspect of crisp and fuzzy AD approaches. It is not possible to compare alternatives which have ‘infinitive’ values and sometimes there would not be any alternative satisfying the entire FRs. However, DMs can assign a numerical value instead of ‘infinitive’ so as to make possible the selection of an alternative which meets all other criteria successfully except the criterion having an ‘infinitive’ value (Kulak and Kahraman 2005a). As a consequence, it is revealed that the proposed hybrid methodology is a robust decision support tool for ranking machine alternatives in the presence of multiple conflicting criteria in a fuzzy environment. The proposed model is expected to provide additional contributions and decision support to the managers in tea industry. Moreover, the presented methodology is flexible and can be used to cope with other MCDM problems. For further research, other multiple criteria evaluation models such as ANP, TOPSIS, VIKOR and ELECTRE etc. can be applied to the same case and the results can be compared.