New Product Design Using Chebyshev’s Inequality Based Interval-Valued Intuitionistic Z-Fuzzy QFD Method

1Introduction

With each passing day, customers’ expectations of the product that they are planning to purchase are increasing. Today, manufacturers and service providers must meet customer demands at the maximum level in order to be successful and maintain their continuity. Their competitive advantage depends on the aesthetic success of the product they offer for sale as well as the technical features. Customers generally expect the product to be affordable, durable, easy to use and appealing to the eye. However, it is difficult, even impossible sometimes, for the producers to meet all these demands at the same time due to economical and timewise limitations. Companies must first prioritize customer needs in order to determine the best product they can produce using their competencies and the maximum customer demands they can respond to. One of the most used methods for this purpose is Quality Function Deployment (QFD).

House of Quality (HOQ) is a special and mostly used part of QFD which is named for its shape that reminds of a house with a roof on top. A classical HOQ consists of some parts in matrix form such as customer demands (CDs), customer evaluations (CEs) of those demands, technical descriptors (TDs), relationship matrix between CDs and TDs, and correlation matrix among TDs. In some recent studies, new matrices are added eligibly to the common parts such as technical difficulty and direction of improvement of TDs, and competitive analysis for both CDs and TDs. The HOQ matrices are generally constructed by an effort of a team of experts and multiple customers. Since humans tend to express their thoughts and ideas linguistically rather than exact and precise numbers, this brings vagueness and impreciseness to the design and development process. To overcome this obstacle and deal with complex problems more realistically, the fuzzy set theory has been applied successfully for decades.

The fuzzy set theory was introduced in the literature by Zadeh (1965) as ordinary fuzzy sets which are represented by an x value and its membership degree. Later, in 1986, intuitionistic fuzzy sets (IFSs) have been developed as a generalization of Zadeh’s ordinary fuzzy sets by Atanassov (1986) which involve the degrees of membership and non-membership together with experts’ hesitancies for an x value. Later, neutrosophic sets are introduced in the literature by Smarandache (1998) which consist of three components truthiness, indeterminacy, and falsity where these components can be assigned independently. Pythagorean fuzzy sets are developed by Yager (2013) and allowed the squared sum of the membership and non-membership degrees to be at most one. Picture fuzzy sets (PiFS) have been developed by Cuong (2015) in order to define a fuzzy set by membership, non-membership, and hesitancy degrees so that their squared sum is at most equal to one. As an extension of PiFs, Kutlu Gündoğdu and Kahraman (2019) developed the spherical fuzzy sets that the squared sum of three components (membership, non-membership, and hesitancy degrees) to be between zero and one. One of the latest extensions of intuitionistic fuzzy sets is circular intuitionistic fuzzy sets developed by Atanassov (2020). They add the uncertainty of the membership and non-membership degrees by defining a circle with radius “r” for these values.

In this paper IVIFSs are employed in the proposed QFD method taking into consideration the reliability of the assigned IVIF numbers. The reliability in this method is handled by Z-fuzzy numbers developed by Zadeh (2011). Z-fuzzy number is an ordered pair of fuzzy numbers where the first component is a real-valued uncertain variable as a restriction on the values. The second component is a measure of reliability for the first component. Z- fuzzy numbers are used to make computations with fuzzy numbers which are not totally reliable. A Z-fuzzy number can represent the information about an uncertain variable, whose first component represents a value of the variable, and the second component represents an idea of uncertainty or probability. In other words, the second component shows how sure the decision maker is with the first component (Yaakob and Gegov, 2015). Chebyshev’s inequality is employed to calculate the maximum probability to determine the expected values of lower and upper bounds of the IVIF number in the first component. Thus, we obtain more realistic and objective results compared to classical Z-fuzzy approaches.

The advantage of our study and its contribution to the literature can be explained as follows. In most of the Z-fuzzy number studies, sufficient details on how to construct the reliability function are not presented. This study scientifically explains how to create the reliability function and integrate it into the restriction function with the help of Chebyshev’s theory. Obtaining the extreme values in IVIF numbers through the integration of reliability factor is realized by using probability theory. Therefore, this paper offers a very different Z-fuzzy number idea from Zadeh’s classical Z-fuzzy proposal. The advantage of our method is that it presents the QFD approach under intuitionistic fuzziness with all its aspects such as technical difficulty, competitive analysis through CDs and TDs.

The rest of this study is organized as follows. Section 2 presents a literature review on fuzzy QFD (F-QFD). Section 3 gives the preliminaries for intuitionistic Z-fuzzy numbers based on Chebyshev’s inequality. Section 4 develops the intuitionistic Z-fuzzy QFD method based on Chebyshev’s inequality. Section 5 illustrates the application of the proposed model on a new hand sanitizer design and development. Section 6 concludes the paper with discussions and future directions.

2Literature Review

A literature review on F-QFD based on Scopus database gives a list of 185 publications. Figure 1 shows the distribution of the F-QFD publications with respect to years.

Fig. 1

Distribution of the F-QFD publications with respect to years.

Fig. 2

Document type distributions of F-QFD publications.

Fig. 3

Document type distributions of F-QFD publications.

After the first study on F-QFD was published in 1998, the highest publication rate was attained in 2019 with 25 studies.

As given in Fig. 2, most of the F-QFD studies are in article form which is followed by conference papers and book chapters.

F-QFD has been applied to many subject areas. Figure 3 shows the frequencies of these publications. Engineering, computer science, and business, management and accounting are the most frequently applied subjects, respectively.

Some representative F-QFD studies are presented in Table 1 together with the type of fuzzy sets used, integrated methods, and application areas.

We can conclude at the end of the literature review that TFNs are used more than other types of fuzzy numbers. The most integrated methods with F-QFD are AHP, ANP, TOPSIS, FMEA, and DM, respectively. The most used extensions of ordinary fuzzy sets with F-QFD are IFNs, HFNs, T2FNs and SFNs, respectively. The application areas of F-QFD are quite different from delivery drone design to choosing the ideal gas fuel at wastewater treatment plants. A focused application area of F-QFD is not observed in this comprehensive literature review.

3Chebyshev’s Inequality Based IV-Intuitionistic Z-Fuzzy Numbers

In this section, we first present the preliminaries of single-valued intuitionistic fuzzy (SVIF) and IVIF sets with some of their arithmetic operations. Then, ordinary Z-fuzzy numbers are introduced. And finally, Chebyshev’s inequality-based interval-valued intuitionistic Z-fuzzy numbers are developed.

3.2Classical Z-Fuzzy Numbers

A Z-fuzzy number is defined by Zadeh (2011) as an ordered pair of fuzzy numbers, (A˜,R˜) which includes a restriction function A˜ and a reliability function R˜ representing the reliability level of the restriction function. If a fuzzy number is not totally reliable, Z-fuzzy numbers can provide a systematic approach to increase the reliability of that fuzzy number.

A Z-fuzzy number can be defined as in Fig. 4.

Fig. 4

A Z-fuzzy number.

Definition 7.

The expected value of a fuzzy set is calculated as in Eq. (12) (Zadeh, 2011):

(12)

EA˜(x)=∫xxμA˜(x)dx,

where A˜ is defined as A˜={⟨x,μA˜(x)⟩|x∈X}, and μA˜:X→[0,1].

Definition 8.

Consider a Z-fuzzy number Z=(A˜,R˜), which is described as in Fig. 4. Let A˜={⟨x,μA˜(x)⟩|μ(x)∈[0,1]} and R˜={⟨x,μR˜(x)⟩|μ(x)∈[0,1]} (Zadeh, 2011).

The triangular fuzzy reliability function can be converted into a classical number by Eq. (13):

(13)

α=∫xμR˜(x)dx∫μR˜(x)dx.

Then, the result of Eq. (13) is integrated with the trapezoidal fuzzy restriction function as in Eq. (14):

(14)

Z˜α={⟨x,μA˜α(x)⟩|μA˜α(x)=αμA˜(x),μ(x)∈[0,1]}.

After applying Eq. (14), the Z-fuzzy number becomes a single ordinary fuzzy number as in Fig. 5.

Fig. 5

Z-fuzzy number converted into a single ordinary fuzzy number.

In the next section, ordinary Z-fuzzy numbers will be extended by a new approach using Chebyshev’s inequality. In this approach, reliability component of the Z-fuzzy number is calculated more objectively based on Chebyshev’s probability terms.

3.3Chebyshev’s Inequality Based IV-Intuitionistic Z-Fuzzy Numbers

Chebyshev’s inequality provides the maximum probability between two points with a given mean and variance as illustrated in Fig. 6 when the distribution of the considered data is not known. Let’s assume that μ=E(X)∈R and σ=sd(X)∈(0,∞), where X is a random variable.

Chebyshev’s inequality is given in Eq. (15):

(15)

P(|X−μ|⩾kσ)⩽1k2,k>0,

where k determines the distance from the population mean as in Fig. 6.

Fig. 6

Chebyshev’s inequality.

Assume that n number of linguistic evaluations is given as A˜={E1,E2,…,En}, each is represented by an interval-valued intuitionistic fuzzy number. Let the arithmetic mean of the lower and upper values of the membership degrees be μx‾L and μx‾U, respectively. Similarly, let the lower and upper values of non-membership degrees be vx‾L, and vx‾U, respectively. Then let the standard deviation of the lower and upper values of the membership degrees be μσL and μσU, respectively, whereas let the lower and upper values of non-membership degrees be vσL, and vσU, respectively.

Next operation is to find k value in Eq. (15) in a way that the maximum reliability Rmax of the lower and upper values of membership and non-membership degrees is obtained. In this operation the k value must satisfy that x‾−kS=0 and/or x‾+kS=1. Then maximum reliability is calculated by Rmax=1−1/k2 for each lower and upper values of membership and non-membership degrees to be RmaxμL, RmaxμU, RmaxvL, and RmaxvU, respectively. Thus, the maximum reliability level becomes maximum between RmaxμL and RmaxμU and between RmaxvL and RmaxvU. Then the expected value of the IVIF number is obtained by Eqs. (16)–(19):

(16)

E[μL]=[(x‾μL−kμLSμL)×RmaxμL,(x‾μL+kμLSμL)×RmaxμL]=[μLL,μLU],

(17)

E[μU]=[(x‾μU−kμUSμU)×RmaxμL,(x‾μU+kμUSμU)×RmaxμL]=[μUL,μUU],

(18)

E[vL]=[(x‾vL−kvLSvL)×RmaxvL,(x‾vL+kvLSμL)×RmaxvL]=[vLL,vLU],

(19)

E[vU]=[(x‾vU−kvUSvU)×RmaxvL,(x‾vU+kvUSvU)×RmaxvL]=[vUL,vUU].

The IVIF number ([E[μL],E[μU]],[E[vL],E[vU]]) is converted to a SVIF number by Eq. (20) for membership interval and Eq. (21) for non-membership interval, respectively.

(20)

D([E[μL],E[μU]])=E[μLL]+E[μLU]+(1−E[vLL])+(1−E[vLU])+E[μLL]×E[μLU]−(1−E[vLL])×(1−E[vLU])4,

(21)

D([E[vL],E[vU]])=E[μUL]+E[μUU]+(1−E[vUL])+(1−E[vUU])+E[μUL]×E[μUU]−(1−E[vUL])×(1−E[vUU])4.

Thus, SVIF number (D(E[μ]),D(E[v])) is obtained.

4Intuitionistic Z-Fuzzy QFD Based on Chebyshev’s Inequality

In this section, we present our novel Chebyshev’s inequality based intuitionistic Z-fuzzy QFD approach. The proposed approach requires the number of experts to be ne and the number of customers to be nc that we interviewed. The steps of the proposed approach are composed of two phases and 10 steps in total, each is presented in detail below. The phase of customer demands (CDs) and technical descriptors (TDs) relation analysis and the phase of competitive analysis are the two main phases of the approach.

Phase 1 – CD&TD Relation Analysis

Step 1: Let nc number of customers define the linguistic CDs and assign the linguistic customer evaluations using the scale in Table 2. The total number of CDs is T. Then, translate the linguistic customer evaluations into IVIF values by using Table 2 and aggregate by using Eqs. (20)–(21). Here, customers’ weights (wc) can be assigned differently. This is realized by Eqs. (22)–(25) which require the weighted mean and the weighted standard deviation of the assigned customer evaluations, respectively. This is applied for each element of T number of CDs. Please note that after the aggregation operations, the IVIF values are turned into SVIF values which is to decrease the vagueness.

Step 2: Let the ne number of experts define the TDs. The total number of TDs is S. Then translate their linguistic assessments for the CD-TD relationship matrix into IVIF numbers by using Table 2. Experts’ weights (we) can be assigned differently depending on our trust in their experiences. Next, aggregate each IVIF relation to a SVIF number by using Eqs. (20)–(21). Eqs. (26)–(29) are used to calculate the weighted mean and the weighted standard deviation of the assigned relations, respectively. This is applied for each element of S number of TDs. Please note that after the aggregation operations, the IVIF values are turned into SVIF values which is to decrease the vagueness.

Step 3: Let the experts determine the level of technical difficulty of the TDs by using the scale given in Table 2. The weights of the experts are accepted to be the same as Step 2 and similar calculations are applied to find the aggregated SVIF values for each TDs’ technical difficulty as in Step 2.

Step 4: Construct the correlation matrix among TDs based on the IVIF scale presented in Table 3. In this matrix two types of correlations are considered: positive and negative. Positive correlations and negative correlations are indicated by PC and NC, respectively. PC means that two TDs move to the same direction whereas NC means that two TDs move to the opposite directions whenever the value of one of these two TDs is changed. When there exists no correlation, the cell includes no linguistic value in the correlation matrix. The differences between PCs and NCs are obtained by Eq. (31).

Step 5: Obtain the Chebyshev’s inequality-based absolute priority degree (AP˜C) for each TD as in Eq. (30):

Relative technical difficulty (RTDF˜C) in Eq. (30) is calculated as in Eq. (32):

Fuzzy relative absolute priority (RAP˜ijC) values are found by Eq. (33):

Step 6: Rank the TDs regarding their RAP˜ijC values. The highest RAP˜ijC shows the TD with the highest priority for the product developers to consider in the new product design and development phase.

Phase 2 – Competitive Analysis

Step 7: Determine the customers’ linguistic assessments for the competitive analysis through CDs assigned by nc number of customers using the IVIF scale given in Table 2. To locate the position of our company among the competitors whose number is y, the customer assessments should be first aggregated with regarding the corresponding CDs. Next, the distances between our company and other companies (D˜O−CℓCD) are calculated by using Eq. (34):

κO−CellCD in Eq. (32) is defined as in Eq. (35):

Step 8: Find the linguistic customer assessments of the competitive analysis through TDs assigned by ne number of experts using the IVIF scale given in Table 2. To locate the position of our company among the competitors, the expert assessments should be first aggregated with regarding the corresponding TDj. Next, the distances between our company and other companies (D˜O−CℓTD) are calculated by using Eq. (37):

κO−CℓTD in Eq. (37) is defined as in Eq. (38):

Step 9: Calculate our company’s combined performance rating score (CPR˜) to locate the position of our firm among the competitors regarding engineering assessments and customer ratings together as in Eq. (40):

Step 10: Find the location of our company relative to the other competitive firms as in Fig. 7. Larger positive distance between our company and Cℓ indicates that our company is in a more advantageous position than Cℓ. At the other negative side, bigger distance between our company and Cℓ indicates that our company is in a more disadvantageous position than Cℓ. The relative location of our company is determined by the indicators in Table 4.

Fig. 7

Scale to indicate the position of our company.

5Application: Hand Sanitizer Design and Development

COVID-19 is a contagious disease, first identified in China, in December 2019 and has since spread worldwide, leading to an ongoing pandemic. Centres for Disease Control and Prevention recommend washing the hands with soap and water for at least 20 seconds to prevent the spread of the virus and minimize the risk of getting infected. However, in many cases especially at public places, they are mostly not available. In such situations, hand sanitizers with at least 60% of alcohol are the most suggested solutions. Hand sanitizers (Fig. 8) are generally liquid, gel or foam form of agents applied on the hands to remove viruses/bacteria/microorganisms.

Fig. 8

Hand sanitizer representation.

In this section an application on hand sanitizer design and development will be presented in steps to illustrate the proposed novel intuitionistic Z-fuzzy QFD approach based on Chebyshev’s inequality.

To determine the CDs for hand sanitizer, a questionnaire was designed to ask their expectations from this product. This questionnaire was distributed to the e-mail addresses of the customers of one of the largest markets in İstanbul. The total number of the customers was 2078 and 219 of them replied. Based on these responses, the following CDs from a hand sanitizer product were determined: Easy storage, compact package, nice smell, fast absorption and/or drying, moisturizing formula, aesthetic design, powerful formula, environmentally friendly and cruelty free, easy and convenient use, and no hard chemicals. After determining these CDs from the customers, we gathered a small focus group to interview and discuss with them the importance degrees of these CDs. Then we asked a chemical cleaning supplies producer in İstanbul how these CDs can be met by which TDs. The producer firm determined the following TDs: Active ingredients, hazardous ingredients, colour, fragrance, package design, and compliance with laws. The relations between these CDs and TDs can be seen in Table 8.

Now the steps of the proposed intuitionistic Z-fuzzy QFD approach based on Chebyshev’s inequality will be given in details in the following.

Phase 1 – CD&TD Relation Analysis

Step 1: Linguistic CDs are defined, and linguistic customer evaluations are assigned by three customers using the scale in Table 2. Customers’ weights are assigned to be wc1=3, wc2=2, and wc3=1, based on the scale in Table 5. Then, the linguistic customer evaluations are translated into IVIF numbers by using Table 2 and aggregated by using Eqs. (20)–(21). The linguistic CDs and corresponding evaluations are given in Table 6 with their aggregated SVIF representations. These are calculated based on the weighted mean and the weighted standard deviation of the assigned customer evaluations by using Eqs. (22)–(25). Please note that after the aggregation operations, the IVIF numbers are turned into SVIF numbers which is to decrease the vagueness.

To have a better understanding with the calculations, a sample calculation is given in Table 7 showing the aggregation operation for the customer demand “Easy Storage, Compact Package” evaluated by three customers.

Step 2: TDs are defined by three experts where their weights are we1=1, we2=2, and we3=1 depending on the scale given in Table 5. Then their linguistic assessments for the CD-TD relationship matrix are translated into IVIF numbers by using Table 2. Later, each IVIF relation is aggregated to a SVIF number by using Eqs. (20)–(21). These are calculated based on the weighted mean and the weighted standard deviation of the values in the relationship matrix by using Eqs. (26)–(29). Table 8 presents this linguistic relationship matrix between CDs and TDs, and their aggregated SVIF correspondences.

To have a better understanding with the calculations, a sample calculation is given in Table 9 showing the aggregation operation for the relation between the CD “Nice Smell” and the TD “Active Ingredients” evaluated by three experts.

Step 3: The level of technical difficulty of the TDs are determined by using the scale given in Table 2 by the three experts. The weights are accepted to be the same as in Step 2 and similar calculations are applied to find the aggregated SVIF numbers for each TDs’ technical difficulty. Table 10 shows the linguistic technical difficulty of each TD and their corresponding aggregated SVIF value.

Step 4: The linguistic correlation matrix among TDs is constructed by the experts as given in Fig. 9 by using the scale given in Table 2. In this way the directions of the correlations which can be positive or negative have been determined. These directions of improvements are represented with “+” and “−” signs to show whether the TD is needed to be increased or decreased, respectively. In Fig. 9, each cell shows three assessments from three experts. The blank cells in Fig. 9 indicate no correlation between the considered two TDs.

Step 5: We obtained the Chebyshev’s inequality based absolute priority degrees for each TD by using Eq. (30) as given in Table 11.

Fig. 9

Linguistic and SVIF correlation matrices.

To better explain this step, a sample calculation is given below for TD “active ingredients”.

First, we multiplied each SVIF customer evaluation value with the corresponding cell in the relation matrix for TD “active ingredients” by using Eq. (4) and then summed these values up by using Eq. (3). Results are shown in Table 12. We added up each SVIF value separately to the summation of the previous ones by applying Eq. (3) successively. The summation result is found to be (0.68, 0.01). Next, we defuzzified this value with Eq. (7) and the result is found as 0.76, where 0.76=1−0.012−0.68−0.01.

Next, to find the correlation correction factor for TD “active ingredients”, first we defuzzified the SVIF correlation values. Then applied Eq. (31) as (4/5)×(0.53+0.53+0.563−0.61)=−0.06, where ncc1=4, S=6. Then, we defuzzified all the SVIF technical difficulty values of TDs and divided the technical difficulty of TD “active ingredients” to all technical difficulty’s summation as 0.63/(0.63+0.65+0.37+0.49+0.42+0.56)=0.20. This gives us the relative technical difficulty of “active ingredients”, given in Eq. (32).