Multi-criteria decision making using axiomatic design and hesitant fuzzy linguistic term sets

Abstract

Multi-criteria decision making (MCDM) is a discipline in which several contradictory criteria are evaluated in a decision making process. There are many MCDM techniques described in the literature, and important considerations have been made in this area because the hesitant evaluations of decision makers, regarding both the criteria and the alternatives, have to be taken into consideration. This study focuses on the MCDM problem using axiomatic design (AD), which is one of the "state of the art" and efficient MCDM techniques related to hesitant evaluations, which are defined as hesitant fuzzy linguistic term sets (HFLTS). In this study, AD is used throughout a methodology in which hesitancy is taken into account in the assessments of the decision makers. System and design ranges are determined in light of the hesitant evaluations of the field experts, and alternatives are listed in the case of both the unweighted and weighted fuzzy environments.

Keywords

Multi criteria decision making Axiomatic design Hesitant fuzzy linguistic term set Medical imaging systems

1 Introduction

Multi-criteria decision making (MCDM) involves making an optimal choice that provides the highest degree of fitness among the alternatives characterized in terms of specific properties, and is a common task in our daily lives. MCDM methods are designed to identify preferred alternatives, to classify these alternatives into a small number of categories, and/or to rank them in subjective order of preference.

In classical MCDM, the evaluations of alternatives are well known. However, owing to the inherent uncertainties of human preferences and objects that are blurred and ambiguous, the properties used in decision-making problems are not always expressed in terms of real numbers, and it is more appropriate to employ fuzzy values for these expressions.

Nevertheless, the results obtained when decision-makers cannot make a clear decision about the superiority of alternatives and criteria, may not be trustable. Decision makers may hesitate while deciding between different criteria or alternatives and may need better expressions to express their knowledge. For these reasons, the hesitant fuzzy linguistic term set (HFLTS) is at the forefront. HFLTS describes the situations that permit the membership of an element of a given set having a few different values, which is a useful way to describe and deal with uncertain information in the MCDM process. Therefore, the use of hesitant fuzzy assessments enable the decision makers’ judgments to be more reliable and informative in the decision making process [40]. HFLTS has been used in the literature frequently in recent years, Wang et. al [35] reviewed the developments related to this subject. Dong et al. [8], Xu et al. [38], Tüysüz and Şimşek [33], Liang et al. [24], Xu et al. [37], Donget al. [7], Liao et al. [25], Li et al. [22], Wu et al. [36], Zhou et al. [43], Xu et al. [39], Li et al. [23] and Yuan et al. [42] are also some of the most recent studies.

Axiomatic design (AD) is one of the commonly used techniques of MCDM. The technique and its principles have improved rapidly over the recent years. Kulak et al. [21] presented a literature study in 2010 related to the applications of AD principles from the past 20 years. In recent studies, many AD applications are seen in different areas, such as; project selection [14], ergonomic evaluation [26], knowledge services [5], personnel selection [18], machine scheduling [19], location selection [3], production design and improvement [4 , 34], failure modes analysis [29], maintenance strategy selection [30], supplier selection [2, 6], industrial robot [15], robot arm [1], and material handling equipment selection [16].

In AD, two major axioms are used: independence axiom and information axiom. The independence axiom points out that the independence of functional requirements, which are defined as the minimum number of independent requirements that illustrate the design aims, should always be protected. The information axiom states that, amongst designs that satisfy the independence axiom, the best design is the one that has the minimum information content, where the information content determines the design with the highest probability of success. A multi-criteria information axiom is based on the designs, systems and common ranges, which enable definition of the lower and upper bounds of design targets and the distribution function of the system performance. This provides an important advantage as it does not force the decision maker to define a single numerical design target [12].

The decision problem considered in this study is the selection of the most appropriate ultrasound (US) device at the Department of Pain Medicine in a university hospital in Ankara, Turkey. In this department, interventional pain management procedures and operations are performed on the patients. The university hospital analyzed here is regarded as one of the biggest hospitals in its region, and the department has an average of 150 patient visits per day. The majority of these patients are admitted with complaints such pain due to cancer, back pain, and headache, which may negatively affect their living standards and may cause serious loss to the workforce in the society. Because of such discomforts, interventional pain treatment is administered to 70-80 patients per week. The US is an imaging technology that is often used to diagnose unexplained pain, swelling, and infection. It is also used to provide imaging guidance to needle biopsies or to view and evaluate conditions related to blood flow. Furthermore, the popularity of US devices is increasing as they can be used in outpatient clinic conditions as well as in bedside applications and, have enable synchronous observation of the tissues in the treated area together with the needles and medicines used during the procedure, which consequently decreases side effects, and prevents both the patient and the physician from being exposed to radiation. The use of US devices is of great importance as the areas that these procedures are applied require a high degree of precaution.

In this study, AD is used for selecting the best US device among various alternatives. Throughout this method, system properties of the alternatives are determined and an attempt is made to minimize the information contents calculated according to the relation between the design and the system properties. With this approach, it is possible to make highly objective evaluations related to the proximity of the current properties of the alternatives to those of the desired characteristics, allowing the decision makers to use more flexible data using ranges instead of single values. Although this approach has become very popular in recent years, to the best of the authors’ knowledge, no application of AD with HFLTS has been found in the literature. In this study, AD is used in a methodology in which hesitancy is taken into account in the assessments of the decision makers. In the proposed method, system ranges and design ranges are determined in the same manner as those of AD applications, but in light of the hesitant evaluations of the field experts. Alternatives are first listed regardless of the weights of the criteria, and then, considering the priorities of these factors, the alternatives are sorted using the criteria weights, which are again determined according to the hesitant evaluations. Finally, the two results are compared and, related observations and analyses are given.

The remainder of the paper is organized as follows; Section 2 introduces some of the basic concepts of AD and HFLTS, briefly. Section 3 presents the application of the proposed algorithm in the decision problem of ultrasound device selection. Finally, Section 4 offers conclusions and recommendations for future studies.

2 Preliminaries

This section introduces the basic ideas and definitions about AD and HFLTS. In essence, this section is divided into four parts: (1) AD, (2) Fuzzy AD, (3) Weighted fuzzy AD and (4) HFLTS.

2.1 Axiomatic Design (AD)

The AD is a design method proposed by Suh [31] to establish a scientific technique to improve design activities, by providing a theoretical foundation grounded on logical and rational thought processes and tools [32]. Some of the most important aspects of AD principles are that they allow researchers to not only determine the best alternative or choice from a certain set of criteria but also to choose the best suitable alternative for the decision makers [13].

In any design situation, the probability of success is determined by information; it is given by what the designer wishes to achieve in terms of tolerance (i.e. design range) and what the system can deliver (i.e. system range). Information is expressed by I_i information content and it is related to the actualization probability of given functional requirements (FR). For a given FR, I_i value is calculated as follows: $I_{i} = {log}_{2} \frac{1}{p_{i}}$ (1) Here, p_i expresses the probability to reach FR_i. In the design process, the probability of reaching the determined target is defined based in tolerance values with what the designer wants to reach (design range) and system capabilities (system range). As shown in Fig. 1, the intersection of design range and system range occurs as the common range where theacceptable solution is found.

Fig.1

Design range, system range, common range and probability density function (p.d.f.) of a functional requirement (FR)

For an FR with normal probability density function, p_i value is calculated as follows: $p_{i} = \frac{Common range}{System range}$ (2) Considering the expectation level as design range and the current situation as system range, information content is calculated as follows: $I_{i} = {log}_{2} (\frac{System range}{Common range})$ (3) Following these calculations; the final solution is obtained by ranking the information contents in descending order.

2.2 Fuzzy Axiomatic Design

MCDM techniques in the literature are generally available when the data are specific, while fuzzy multi-criteria AD approach is available if the data are not definitive. Real numbers are used to express the exact specified data, however, when the data are not specific and are expressed by linguistic variables instead of numerical values, this data must be transformed into a numeric form depending on a certain rule base. The fuzzy set theory is an important tool that can be used at this stage.

In AD, the system and design ranges of the functional requirements can not cannot always be expressed by certain intervals. Above or below a certain value can be approximated and these values can be represented by triangular or trapezoidal fuzzy numbers. In the case of a fuzzy AD, when the range values are given linguistically, the triangular or trapezoidal fuzzy membership functions are used when the probability density function is specified. Therefore, the common area is the intersection area between the fuzzy areas of the system and design ranges. Thus, the information content is calculated by Equation 4: $I_{i} = {log}_{2} (\frac{Fuzzy area of system range}{Common range})$ (4)

2.3 Weighted Fuzzy Axiomatic Design

In subsection 1, the weights of the criteria are expected to be equal. However, this is not a realistic approach for real-life applications. In case each criterion has a specific weight value (w_j), in addition to the previously defined equations, the following one is used to calculate the information content and this approach is named as the weighted axiomatic design [20]. $I = {\begin{matrix} [{log}_{2} (\frac{1}{p_{ij}})]^{\frac{1}{w_{j}}}, & 0 \leq I_{ij} < 1 \\ [{log}_{2} (\frac{1}{p_{ij}})]^{w_{j}}, & I_{ij} > 1 \\ w_{j} & I_{ij} = 1 \end{matrix}}$ (5)

2.4 Hesitant Fuzzy Linguistic Term Sets (HFLTS)

In this study, hesitant fuzzy linguistic term sets (HFLTS), which were originally proposed by Rodríguez et al. [27], are used. Rodríguez et al. [27] introduced the concept of HFLTS to provide a linguistic and computational basis to increase the richness of elicitation based on the fuzzy linguistic approach and the use of context-free grammar using comparative terms. They proposed a single-criterion hesitant linguistic group decision-making model to achieve the aims in their related work.

Before the AD-HFLTS integrated methodology is presented, some basic concepts related to HFLTS are given. In this study, linguistic term sets, which are proposed by Yavuz et al. [41], are used together with the context-free grammar G_H of Rodríguez et al. [27]. Assuming S = {s₀, …, s_g} be a linguistic term set, the elements of G_H = {V_N, V_T, I, P} are defined as follows:

= {〈primary term〉, 〈composite term〉, 〈unary relation〉, 〈binary relation〉, 〈conjunction〉}

= {lower than, greater than, between, at least, at most, and, s₀, s₁, …, s_g}

∈V_N

= {I : : = 〈primary term〉 | 〈composite term〉

〈composite term〉 : : = 〈unary relation〉〈primary term〉 | 〈binary relation〉〈primary term〉〈conjunction〉〈primary term〉

〈primary term〉 : : = s₀|s₁| … |s_g

〈unary relation〉 : : = lower than | greater than | at least | at most

〈binary relation〉 : : = between

〈conjunction〉 : : = and }.

Assuming that E_GH is a function that converts linguistic expressions into HFLTS, linguistic expressions are converted into HFLTSs as follows [27]:

(s_i)] = {s_i|S_i ∈ S}

(at most s_i)] = {s_j|s_j ∈ S and s_j ≤ s_i}

(lower than s_i)] = {s_j|s_j ∈ S and s_j < s_i}

(at least s_i)] = {s_j|s_j ∈ S and s_j ≥ s_i}

(greater than s_i)] = {s_j|s_j ∈ S and s_j > s_i}

(between s_i and s_j)] = {s_k|s_k ∈ S and s_i ≤ s_k ≤ s_j}

The envelope of an HFLTS, env (H_S), is a linguistic interval whose limits are obtained by its maximum value and minimum value:

env (H_{S}) = [H_{S^{-}}, H_{S^{+}}], H_{S^{-}} \leq H_{S^{+}},

(6) where H_{S
^-} = min(s_i) = s_j, s_i ∈ H_S and s_i ≥ s_j ∀ i and H_{S
⁺} = max(s_i) = s_j, s_i ∈ H_S and s_i ≤ s_j ∀ i.

3 Fuzzy AD-HFLTS Integrated Approach

In this study, a fuzzy AD-HFLTS integrated approach is used for device selection. In the proposed solution, in accordance with the AD methodology, design and system ranges related to US device selection are determined first. From these determined ranges, information contents are calculated using the common ranges for each alternative, and the alternatives are listed on the basis of these contents.

Expert opinions are inevitable during this work. Rodríguez et al. [28] stated that in order the use of hesitant information to be useful for real world decision problems, any decision-making problem including hesitant fuzzy information should state what causes the doubt or the hesitancy in the problem framework. In this study, hesitancy arises due to the ambivalent evaluations of experts focusing on the selection of a specific device. As mentioned in previous sections, HFLTS is used in these evaluations, since there may be instability in the opinions of the experts, and HFLTS can help ensure consistency and increase the clarity of expert opinions.

Throughout the proposed method, the basic steps of AD are followed. However, as the effect of hesitancy is taken into account, the following steps are applied while determining the design and system ranges of alternatives.

Step 1. Gather the performance scores for alternatives with respect to the criteria using the context-free grammar of Rodríguez et al. [27].

Step 2. Transfer the evaluations into intervals using the transformation function E_{G
_H}

Step 3. Calculate the pessimistic and optimistic preference relations (P^- and P⁺) using 2-tuple operations.

In the first method, assuming that all the criteria are of equal importance, weights are not taken into account, whereas in the second one, decision makers’ hesitant evaluations are used to obtain criteria weights that are calculated using the above-mentioned steps for the criteria and sub-criteria, and then normalizing the midpoints of the intervals.

3.1 Application of Fuzzy AD - HFLTS Integrated Approach

The department requires a US device that is low-cost, high quality, easy to use, and has satisfactory technical properties. After a real-world decision-making investigation with a group of field experts, composed of a purchasing department director (PD), two radiology physicians (RP ♯1 and ♯2), and two pain medicine physicians (PP ♯1 and ♯2), a consensus on the definitions of the criteria, alternatives, and the functional requirements is achieved.

The decision making criteria used for selection of the most appropriate US device are determined as purchase cost, after-sales cost, image quality, data management, system properties, a variety of probes, ease of operation, and ease of mobility. These decision-making criteria are determined to capture important aspects of US device selection for pain medicine operations. These eight criteria are categorized into three main groups (i.e., cost, quality, and usability) as can be seen from Fig. 2. Throughout this study, experts from different fields are consulted, even though it is unlikely that physicians will have precise quantitative “cost” data; for example, it is not possible for the purchaser to have a definite opinion of “usability”. In order to prevent the loss of information related to the evaluation of qualitative criteria, fuzzy logic must be employed throughout the solution procedure.

Fig.2

Hierarchical organization of the decision-making criteria

Taking a closer look at these criteria, “purchase cost” is probably the most easy-to-understand among all the others. “After-sale cost”, needs to be paid attention to as it may generate a considerable amount of money in case of need. As can be noticed by a careful reader, these two criteria can be defined by scalar representations. However, as physicians may not have exact knowledge of “cost” values, yet a coherent view is desired in this study, related data that are obtained from the field experts are used as linguistic values.

The first one among the quality related criteria is “image quality”, which is an important issue allowing experts to see the needle and anatomy in every patient, even in the obese. This avoids improper needle placementand increases confidence. “Data management” includes the documentation and connectivity possibilities. “System properties” define the specific properties of the device, including the frame rate, which is defined as the speed of frames generated by the device and effects the quality of image, screen properties, the technology used in the screen, the usability of the touchscreen and the screen size of the device, the number of digital channels that corresponds to the processing power that may cause poor image quality in case of low number of channels, etc.

The third group is related to the “usability” of the device by the experts. The first criterion is the “variety of probes”, which is important in terms of the differentiation of the applications. Different types of US probes have different properties, such as maximum display depth, which is related to image structures deep in the body and is important especially in obese patients and for deep tissues, and a feature for supporting high frequency probes, which helps to obtain better quality images in superficial tissues. This differentiation can cause significant differences during the operations and can be an important reason for preference. In the context of “ease of operation”, ergonomic design for ease of use in challenging environments and configurable presets including the device’s ability to obtain more ideal images for each operation using the already adjusted settings, quick opening-closing, auto-optimization, delivering contrast resolution, optimized spectral modes, automatic adjustments of the auto-depth according to the corresponding frequency, and focal zone position, for clarity throughout the near and far field are included. These properties make the system easy to use without many adjustments, which helps the expert focus on the patient rather than a complicated control panel and provide timely and effective pain blocking. “Ease of mobility” is mostly related to the portability and the weight of the device. This property is beneficial in facilitating access to patients in different clinics or operating rooms, whereas lighter devices are preferred for bedside procedures.

Alternatives of the study are also determined together with the field experts. Five different devices commonly used in many different pain centers have been considered as alternatives for evaluation. As the brands of these alternatives cannot be given due to ethical and legal obligations, they are named as Alternatives 1 to 5 throughout the solution procedure.

While constructing the design and system ranges with respect to the features of the devices, decision-makers are required to perform their evaluations using the semantics and syntax of the linguistic term set S₁. $S_{1} = {\begin{matrix} very poor (VP), \\ poor (P), \\ fair (F), \\ good (G), \\ very good (VG) . \end{matrix}}$ (7) As is mentioned by Dong et al. [9], there are different models based on linguistic 2-tuples which are used to identify the term sets as if they are uniformly and symmetrically distributed, or not. In this study, the first class of distribution is taken into account, in which the term sets are uniformly and symmetrically distributed. These notations respond the values given in the scale in Table 1.

Table 1

The scale for the linguistic terms

VP	P	F	G	VG
0	1	2	3	4

The design ranges for selecting the best US device, showing the FRs that should be satisfied by an ultrasound device, are defined according to the opinions and experience of the experts. Performance scores with respect to the criteria are given in Table 2.

Table 2

Linguistic evaluations of experts with respect to the design range

		P	RP ♯1	RP ♯2	PP ♯1	PP ♯2
C	PC	At least (G)	At least (G)	At least (F)	At least (G)	At least (G)
	AC	At least (G)	At least (F)	At least (F)	At least (G)	At least (G)
Q	IQ	At least (G)	At least (G)	At least (G)	Is (VG)	At least (G)
	DM	At least (F)	At least (F)	At least (G)	At least (G)	At least (F)
	SP	At least (F)	Is (VG)	At least (G)	At least (G)	At least (G)
U	VP	At least (G)	At least (G)	At least (G)	At least (G)	At least (G)
	EO	At least (F)	At least (G)	At least (G)	Is (VG)	At least (G)
	EM	At least (F)	At least (F)	At least (F)	At least (G)	At least (G)

Then these evaluations are converted into intervals using the transformation function E_{G
_H}. The obtained envelops are presented in Table 3.

Table 3

Obtained envelops for HFLTS with respect to the design range

		P	RP ♯1	RP ♯2	PP ♯1	PP ♯2
C	PC	[G,VG]	[G,VG]	[F,VG]	[G,VG]	[G,VG]
	AC	[G,VG]	[F,VG]	[F,VG]	[G,VG]	[G,VG]
Q	IQ	[G,VG]	[G,VG]	[G,VG]	[VG,VG]	[G,VG]
	DM	[F,VG]	[F,VG]	[G,VG]	[G,VG]	[F,VG]
	SP	[F,VG]	[VG,VG]	[G,VG]	[G,VG]	[G,VG]
U	VP	[G,VG]	[G,VG]	[G,VG]	[G,VG]	[G,VG]
	EO	[F,VG]	[G,VG]	[G,VG]	[VG,VG]	[G,VG]
	EM	[F,VG]	[F,VG]	[F,VG]	[G,VG]	[G,VG]

Assigning the scale in Table 1 to the linguistic terms, P^- and P⁺ values are calculated. For example, the pessimistic and optimistic values for “purchase cost” (PC) are calculated as follows:

$\begin{matrix} P_{PC}^{-} & = & \begin{matrix} \frac{1}{5} (Δ^{- 1} (G, 3) + Δ^{- 1} (G, 3) \\ + Δ^{- 1} (F, 2) + Δ^{- 1} (G, 3) \\ + Δ^{- 1} (G, 3)) \end{matrix} \\ = & \frac{1}{5} (3 + 3 + 2 + 3 + 3) \\ = & 2.80 \end{matrix}$ (8) $\begin{matrix} P_{PC}^{+} & = & \begin{matrix} \frac{1}{5} (Δ^{- 1} (VG, 4) + Δ^{- 1} (VG, 4) \\ + Δ^{- 1} (VG, 4) + Δ^{- 1} (VG, 4) \\ + Δ^{- 1} (VG, 4)) \end{matrix} \\ = & \frac{1}{5} (4 + 4 + 4 + 4 + 4) \\ = & 4.00 \end{matrix}$ (9) Obtained design ranges are given in Table 4.

Table 4

Design ranges

		Pessimistic	Optimistic	Interval Utilities
C	PC	2.80	4.00	[2.80, 4.00]
	AC	2.60	4.00	[2.60, 4.00]
Q	IQ	3.20	4.00	[3.20, 4.00]
	DM	2.40	4.00	[2.40, 4.00]
	SP	3.00	4.00	[3.00, 4.00]
U	VP	3.00	4.00	[3.00, 4.00]
	EO	3.00	4.00	[3.00, 4.00]
	EM	2.40	4.00	[2.40, 4.00]

Face-to-face interviews are conducted with the relevant field experts about the selection of US device for the department and five US device alternatives are determined. System ranges for alternative US devices are identified according to the opinions and experiences of the experts again. Performance scores are gathered from the experts and transferred into intervals. Obtained system range values with respect to the alternative devices are given in Table 5.

Table 5

System ranges

		Alternative 1	Alternative 2	Alternative 3	Alternative 4	Alternative 5
C	PC	[2.40, 3.00]	[2.20, 3.00]	[2.60, 3.20]	[2.60, 3.40]	[3.00, 3.80]
	AC	[3.00, 3.60]	[2.40, 3.20]	[3.40, 4.00]	[2.40, 3.00]	[2.00, 2.80]
Q	IQ	[3.00, 3.80]	[3.40, 4.00]	[3.00, 3.60]	[2.80, 3.40]	[2.40, 3.40]
	DM	[3.20, 3.80]	[2.60, 3.40]	[2.20, 3.00]	[2.00, 2.60]	[2.20, 2.60]
	SP	[3.40, 3.80]	[2.80, 3.60]	[2.60, 3.40]	[2.60, 3.20]	[2.40, 3.20]
U	VP	[3.20, 3.80]	[2.80, 3.60]	[2.80, 3.80]	[2.60, 3.20]	[1.40, 2.00]
	EO	[2.80, 3.40]	[2.60, 3.40]	[3.40, 3.80]	[2.80, 3.20]	[2.60, 3.20]
	EM	[2.60, 3.40]	[2.00, 2.80]	[3.00, 3.80]	[1.80, 2.60]	[2.20, 3.00]

Arriving at the last step of the solution procedure, information contents for all of the alternatives are calculated and presented in Table 6. According to these results, Alternative 1, having the minimum information content, emerges as the most appropriate alternative.

Table 6

Information contents for alternatives

	C		Q			U			∑
	PC	AC	IQ	DM	SP	VP	EO	EM
Alternative 1	1.585	0.000	0.415	0.000	0.000	0.000	0.585	0.000	2.585^*
Alternative 2	2.000	0.415	0.000	0.000	0.415	0.415	1.000	1.000	5.245
Alternative 3	0.585	0.000	0.585	0.415	1.000	0.322	0.000	0.000	2.907
Alternative 4	0.415	0.585	1.585	1.585	1.585	1.585	1.000	2.000	10.340
Alternative 5	0.000	2.000	2.322	1.000	2.000	∞	1.585	0.415	∞

3.2 Application of the Weighted Fuzzy AD HFLTS Integrated Approach

When the criteria used in the decision making phase are not equal in importance and the significance levels are different from each other, weighted fuzzy AD is preferred as an approach to use. In this approach, first, weights of the criteria are determined.

As is mentioned by Dong et al. [11], there are several approaches to obtain the attribute weights that can be classified into three categories: the subjective approach, the objective approach and the integrated approach, in which the decision maker is assumed to be honest, and aims to obtain “best” attribute weights to get a ranking of alternatives. In this study, the weights are obtained by asking the decision makers to make pairwise comparisons for criteria and sub-criteria. The semantics and syntax of the linguistic term set S₂ are defined in the same manner as in Yavuz et al. [41] together with the hesitant linguistic term sets that are previously defined.

$S_{2} = {\begin{matrix} no importance (N), \\ very low importance (VL), \\ low importance (L), \\ medium importance (M), \\ high importance (H), \\ very high importance (VH), \\ absolute importance (A) \end{matrix}}$ (10)

Field experts’ evaluations of the main criteria with respect to the goal are converted to envelops for HFLTS using the same manner in the previous section and related pessimistic and optimistic collective preferences are obtained and given in Table 7. Criteria weights are obtained by normalizing the midpoints of these intervals. These values are given in Table 8, representing the relative importance of each of the main criteria.

Table 7

Pessimistic and optimistic collective preferences

Criteria	C	Q	U	C	Q	U
C	-	1.20	1.40	-	2.40	2.60
Q	3.60	-	2.00	4.80	-	3.20
U	3.40	2.80	-	4.60	4.00	-

Table 8

Linguistic intervals and weights

Criteria	Avg.	Avg.	Interval	Midpoints	Weights
	pessimistic	optimistic	utilities
C	1.30	2.50	[1.30, 2.50]	1.90	0.211
Q	2.80	4.00	[2.80, 4.00]	3.40	0.378
U	3.10	4.30	[3.10, 4.30]	3.70	0.411

Table 9

Criteria and sub-criteria weights

Criteria		Criteria Weight	Sub-Criteria Weight	Sub-Criteria Global Weight
C		0.211
	PC		0.633	0.134
	AC		0.367	0.077
Q		0.378
	IQ		0.439	0.166
	DM		0.172	0.065
	SP		0.389	0.147
U		0.411
	VP		0.400	0.164
	EO		0.422	0.174
	EM		0.178	0.073

In a similar fashion, same steps are repeated for all of the 8 sub-criteria with respect to each of the three main criteria and the criteria and sub-criteria weights given in Table 9 are obtained.

After determining the criteria weights, information contents for the criteria are calculated using Equation 5 and the results are given in Table 10.

Table 10

Weighted information contents for alternatives

	C		Q			U			∑
	PC	AC	IQ	DM	SP	VP	EO	EM
Alternative 1	1.064	0.000	0.005	0.000	0.000	0.000	0.046	0.000	1.114
Alternative 2	1.097	0.000	0.000	0.000	0.003	0.005	0.174	0.073	1.351
Alternative 3	0.018	0.000	0.039	0.000	0.147	0.001	0.000	0.000	0.205^*
Alternative 4	0.001	0.001	1.079	1.030	1.070	1.079	0.174	1.052	5.486
Alternative 5	0.000	1.055	1.150	0.065	1.107	∞	1.083	0.000	∞

From the results in Table 10, Alternative 3, which has the minimum sum of the information content value, is determined as the best alternative. The results obtained with fuzzy AD and weighted fuzzy AD are summarized in Table 11. When the results from these tables are examined, it is seen that the ranking has changed. Based on this result, it can be said that, considering the weights of the criteria has an important and correcting effect on the result.

Table 11

Results obtained by Fuzzy AD and Weighted Fuzzy AD

Alternatives	Fuzzy AD	Weighted Fuzzy AD
Alternative 1	1^st priority	2^nd priority
Alternative 2	3^rd priority	3^rd priority
Alternative 3	2^nd priority	1^st priority
Alternative 4	4^th priority	4^th priority
Alternative 5	Eliminated	Eliminated

Taking a closer look at the data in Tables 6 and 10, it is possible to obtain secondary rankings of alternatives to be made on a criteria basis. Using such an examination, for example, according to the results obtained with fuzzy AD, although Alternative 3 comes into prominence in the “cost” and “usability” criteria, Alternative 1 is chosen to have the first priority in the overall evaluation. In addition to this, Alternative 5 is eliminated and ranked last, as it is not included in the design range for the “usability” criterion, irrespective of the other criteria. Similarly, looking at the results obtained with weighted fuzzy AD, Alternatives 4 and 2 have the smallest information content values in the “cost” and “quality” criteria, respectively. However, Alternative 3 has first priority in the overall evaluation.

When the ranking results are evaluated together with the experts; it is declared that systematic, logical, and consistent results have been achieved. These evaluations make it possible to see the effectiveness of AD HFLTS better.

4 Conclusion

This study focuses on the MCDM problems arising in our daily lives. In order to solve such problems, one needs the opinions of experts working in related areas. Furthermore, because of the fact that these experts may hesitate in deciding among different criteria or alternatives and may need better ways to express their knowledge, the opportunity to use HFLTS is proposed throughout their evaluations. As it may be better to choose the best suitable alternative instead of determining the best alternative to be chosen from a certain set of criteria, AD is used throughout the solution procedure.

First, the decision-making criteria are determined together with the field experts. Then the system and design ranges are composed in light of the hesitant evaluations. Following the steps of AD, the information for all of the alternatives are calculated and are listed regardless of the weights of the criteria. Then, the experts are asked to evaluate the criteria and related weights are obtained. In the second solution, the alternatives are sorted using these criteria weights.

The fact that the method used in the study is based on the difference between the system and the design ranges reveals an important difference between it and other MCDM techniques. On the basis of the results obtained from the sample problem, it can be seen that, when using AD, an alternative with a better ranking sequence for several criteria might be lower in the overall evaluation. Owing to this important feature of the method, there occurs a difference between the rankings obtained on the basis of the criteria and the final results obtained in both of the cases where the weights are taken into consideration or not. It is concluded that the final results are systematic, logical, and consistent when the calculated rankings are interpreted in consultation with the field experts. From this judgment, the effectiveness of the integrated method is once more visible.

Rodríguez et al. [28] stated the importance of the identification of real world decision situations in which the hesitancy produces uncertainty. As is mentioned before, obtained rankings are found to be better accepted than the solutions obtained with other type of information modeling by the field experts. Regarding the necessity of being more creative than simple and straight-away extensions of previous models for using hesitant information [28]; to the best of the authors’ knowledge, this is the first study to use fuzzy AD together with HFLTS and to apply this procedure to multi-criteria comparisons of a medical imaging system at a university hospital. Further research can be performed in different sectors as well. Besides, regarding to the recent progress in HFLTSs; strategic weight manipulation [11], dynamically changing sets of attributes and/or alternatives [10] can also be integrated into the solution approach that is proposed in this study.

References

Bahadir

M.C.

and Satoglu

S.I.

, A novel robot arm selection methodology based on axiomatic design principles, International Journal of Advanced Manufacturing Technology 71(9) (2014), 2043–2057.

Büyüközkan

and Göçer

, Application of a new combined intuitionistic fuzzy MCDM approach based on axiomatic design methodology for the supplier selection problem, Applied Soft Computing 52 (2017), 1222–1238.

Cebi

, Ilbahar

and Atasoy

, A fuzzy information axiom based method to determine the optimal location for a biomass power plant: A case study in Aegean Region of Turkey, Energy 116 (2016), 894–907.

Chakraborty

, Mondal

and Mukherjee

, Analysis of product design characteristics for remanufacturing using Fuzzy AHP and Axiomatic Design, Journal of Engineering Design 28(5) (2016), 338–368.

Chen

, Li

, Fan

Z.P.

, Zhou

and Zhang

, Matching demanders and suppliers in knowledge service: A method based on fuzzy axiomatic design, Information Sciences 346-347 (2016), 130–145.

Deng

and Jiang

, An evidential axiomatic design approach for decision making using the evaluation of belief structure satisfaction to uncertain target values, International Journal Of Intelligent Systems 33 (2018), 15–32.

Dong

, Yuan

and Wan

, Extended VIKOR method for multiple criteria decision-making with linguistic hesitant fuzzy information, Computers and Industrial Engineering 112 (2018), 305–319.

Dong

, Chen

and Herrera

, Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making, Information Sciences 297 (2015), 95–117.

Dong

, Li

C.-C.

and Herrera

, Connecting the linguistic hierarchy and the numerical scale for the 2-tuple linguistic model and its use to deal with hesitant unbalanced linguistic information, Information Sciences 367-368 (2016), 259–278.

10.

Dong

, Zhang

and Herrera-Viedma

, Consensus reaching model in the complex and dynamic MAGDM problem, Knowle dge-Base d Systems 106 (2016), 206–219.

11.

Dong

, Liu

, Liang

, Chiclana

and Herrera-Viedma

, Strategic weight manipulation in multiple attribute decision making, Omega 75 (2018), 154–164.

12.

Kahraman

, Onar

S.C.

, Cebi

and Oztaysi

, Extension of information axiom from ordinary to intuitionistic fuzzy sets: An application to search algorithm selection, Computers and Industrial Engineering 105 (2017), 348–361.

13.

Kannan

, Govindan

and Rajendran

, Fuzzy axiomatic design approach based green supplier selection: A case study from Singapore, Journal of Cleaner Production 96 (2015), 194–208.

14.

Khandekar

A.V.

, Antuchevic~ienė

and Chakraborty

, Small hydro-power plant project selection using fuzzy axiomatic design principles, Technological and Economic Development of Economy 21(5) (2015), 756–772.

15.

Khandekar

A.V.

and Chakraborty

, Selection of industrial robot using axiomatic design principles in fuzzy environment, Decision Science Letters 4(2) (2015), 181–192.

16.

Khandekar

A.V.

and Chakraborty

, Selection of material handling equipment using fuzzy axiomatic design principles, Informatica 26(2) (2015), 259–282.

17.

Khandekar

A.V.

and Chakraborty

, Application of fuzzy axiomatic design principles for selection of non-traditional machining processes, The International Journal of Advanced Manufacturing Technology 83(1) (2016), 529–543.

18.

Khandekar

A.V.

and Chakraborty

, Personnel selection using fuzzy axiomatic design principles, Verslas: Teorija ir Praktika / Business: Theory and Practice 17(3) (2016), 251–260.

19.

Kir

and Yazgan

, A sequence dependent single machine scheduling problem with fuzzy axiomatic design for the penalty costs, Computers and Industrial Engineering 92 (2016), 95–104.

20.

Kulak

and Kahraman

, Fuzzy multi-attribute selection among transportation companies using axiomatic design and analytic hierarchy process, Information Sciences 170 (2005), 191–210.

21.

Kulak

, Cebi

and Kahraman

, Applications of axiomatic design principles: A literature review, Expert Systems Applications 37(9) (2010), 6705–6717.

22.

C.-C.

, Rodríguez

R.M.

, Martínez

, Dong

and Herrera

, Consistency of hesitant fuzzy linguistic preference relations: An interval consistency index, Information Sciences 432 (2018), 347–361.

23.

C.-C.

, Rodríguez

R.M.

, Martínez

, Dong

and Herrera

, Personalized individual semantics based on consistency in hesitant linguistic group decision making with comparative linguistic expressions, Knowledge-Based Systems 145 (2018), 156–165.

24.

Liang

, Wang

and Zhang

, Projection-based PROMETHEE methods based on hesitant fuzzy linguistic term sets, International Journal of Fuzzy Systems (2017), 1–14.

25.

Liao

H.C.

, Yang

L.Y.

and Xu

Z.S.

, Two new approaches based on ELECTRE II to solve the multiple criteriadecision making problems with hesitant fuzzy linguistic term sets, Applied Soft Computing 63 (2018), 223–234.

26.

Maldonado-Macías

, García-Alcaraz

, Reyes

R.M.

and Hernández

, Application of a fuzzy axiomatic design methodology for ergonomic compatibility evaluation on the selection of plastic molding machines: A case study, Procedia Manufacturing 3 (2015), 5769–5776.

27.

Rodríguez

R.M.

, Martínez

and Herrera

, Hesitant fuzzy linguistic term sets for decision making, IEEE Transactions on Fuzzy Systems 20(1) (2012), 109–119.

28.

Rodríguez

R.M.

, Bedregal

, Bustince

, Dong

Y.C.

, Farhadinia

, Kahraman

, Martínez

, Torra

, Xu

Y.J.

, Xu

Z.S.

and Herrera

, A position and perspective analysis of hesitant fuzzy sets on information fusion in decision making. Towards high quality progress, Information Fusion 29 (2016), 89–97.

29.

Seiti

H.R.

, Behnampour

, Imani

D.M.

and Houshmand

, Failure modes and effects analysis under fuzzy environment using fuzzy axiomatic design approach, International Journal of Research in Industrial Engineering 6(1) (2017), 51–68.

30.

Seiti

H.R.

, Hafezalkotob

and Fattahi

, Extending a pessimistic-optimistic fuzzy information axiom basedapproach considering acceptable risk: Application in the selection ofmaintenance strategy, Applied Soft Computing 67 (2018), 895–909.

31.

Suh

N.P.

The principles of design, in: Oxford University Press, New York, NY,1990.

32.

Suh

N.P.

Axiomatic design: Advances and applications 2001–, Oxford University Press New York, NY.

33.

Tüysüz

and Simşek

, A hesitant fuzzy linguistic term setsbased AHP approach for analyzing the performance evaluation factors: An application to cargo sector, Complex Intelligent Systems 3 (2017), 167–175.

34.

Vinodh

, Kamala

and Jayakrishna

, Application of fuzzy axiomatic design methodology for selection of design alternatives, Journal of Engineering, Design and Technology 13(1) (2015), 2–22.

35.

Wang

, Zeshui

and Zeng

X.-J.

, Hesitant fuzzy linguistic term sets for linguistic decision making: Current developments, issues and challenges, Information Fusion 43 (2018), 1–12.

36.

, Xu

, Ren

and Liao

, Hesitant fuzzy linguistic projection model to multi-criteria decision making for hospital decision support systems, Computers and Industrial Engineering 115 (2018), 449–458.

37.

, Cabrerizo

F.J.

and Herrera-Viedma

, A consensus model for hesitant fuzzy preference relations and itsapplication in water allocation management, Applied Soft Computing 58 (2017), 265–284.

38.

, Chen

, Rodríguez

R.M.

, Herrera

and Wang

, Deriving the priority weights from incomplete hesitant fuzzy preference relations in group decision making, Knowledge- Based Systems 99 (2016), 71–78.

39.

, Li

and Wen

, Missing values estimation and consensus building for incomplete hesitant fuzzy preference relations with multiplicative consistency, International Journal of Computational Intelligence Systems 11 (2018), 101–119.

40.

and Zhang

, Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information, Knowledge-Based Systems 52 (2013), 53–64.

41.

Yavuz

, Oztaysi

, Onar

S.C.

and Kahraman

, Multicriteria evaluation of alternative-fuel vehicles via a hierarchical hesitant fuzzy linguistic model, Expert Systems with Applications 42 (2015), 2834–2848.

42.

Yuan

, Li

, Liu

and Li

, Linguistic hesitant fuzzy multi-criterion decision-making for renewable energy: A case study in Jilin, Journal of Cleaner Production 172 (2018), 3201–3214.

43.

Zhou

, Wang

and Zhang

, Multi-criteria decision making approaches based on distance measures for linguistic hesitant fuzzy sets, Journal of the Operational Research Society 69(5) (2018), 661–675.