# A Possibilistic Approach for Aggregating Customer Opinions in Product Development

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## Abstract

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## 1. Introduction

## 2. Preliminaries

#### 2.1. State of a Product Feature

#### 2.2. Numerical and Linguistic Truth-Value or Degree of Belief of a Feature

_{i},S

_{j}) be a proposition of the form F

_{i}is S

_{j}where F

_{i}is the i-th feature of a product and S

_{j}∈ State. Let T be a process as follows:

_{i},S

_{j}), i = 1,2,…, j = 1,...,4, DoB(.) ∈ [0,1]. This means that each proposition p(F

_{i},S

_{j}) has a truth-value (or DoB) in the interval [0,1], and the process denoted as T determines it.

_{k}is a state drawn from State, and h is a hedge called “more or less” or “somewhat.”

_{i}be sedan (a feature of a car), i.e., F

_{i}= sedan. Using the states defined in State, the following four propositions can be considered: p

_{1}(sedan, must-be feature), p

_{2}(sedan, should-be feature), p

_{3}(sedan, could-be feature), and p

_{4}(sedan, unreliable feature). A numerical value that lies in the interval [0, 1] can be assigned to each proposition subjectively or following a computation approach as its truth-value or DoB. Let, for instance, DoBs of the propositions be DoB(sedan, must-be feature) = 0.2, DoB(sedan, should-be feature) = 0.7, DoB(sedan, could-be feature) = 0.95, and DoB(sedan, unreliable feature) = 0.05. Linguistically, DoB = 0.2 means that “it is quite false that sedan is a must-be feature of a car,” i.e., DoB = 0.2 refers to a linguistic truth-value “quite false.” Similarly, DoB = 0.7 means that “it is somewhat true that sedan is a should-be feature of a car,” i.e., DoB = 0.7 refers to a linguistic truth-value “somewhat true.” Similarly, DoB = 0.95 means that “it is mostly true that sedan is a could-be feature of a car,” i.e., DoB = 0.95 refers to a linguistic truth-value “mostly true.” Finally, DoB = 0.05 means that “it is mostly false that the opinions obtained on the car feature called sedan is unreliable,” i.e., DoB = 0.05 refers to a linguistic truth-value “mostly false.”

_{i},.)) is given as

_{i}, i = 1, 2, ... can be defined. In this study, a set of seven linguistic truth-values are considered that are given by the membership functions (or DoBs) of the seven fuzzy numbers [16,19,20,21,22] labeled “mostly false (mf),” “quite false (qf),” “somewhat false (sf),” “neither true nor false (tf),” “somewhat true (st),” “quite true (qt),” and “mostly true (mt).” The membership functions are illustrated in Figure 1.

## 3. Logical Aggregation Process

#### 3.1. The Kano Model

_{i}into one of the following types: Class = {One-dimensional (O), Attractive (A), Must-be (M), Indifferent (I), Reverse (R), Questionable (Q)}. As seen from Figure 2, a feature is considered Must-be if its absence produces absolute dissatisfaction, and its presence does not increase the satisfaction. A feature is considered One-dimensional if its fulfillment helps increase the satisfaction and vice versa. A feature is considered Attractive if it leads to a greater satisfaction but is not expected to be in the product. A feature is considered Indifferent if its presence or absence does not contribute to the customers’ satisfaction. A feature is considered Reverse if its presence causes dissatisfaction and vice versa [10,19,26]. To know whether F

_{i}is one of the classes drawn from Class, a respondent needs to answer two questions. One of the questions deals with the scenario that refers to F

_{i}being present in the product, and the other deals with the scenario that refers to F

_{i}being not present in the product. The respondent needs to choose an answer drawn from Answer = {Like, Must-be, Neutral, Live-with, Dislike} for both questions [16,18]. The relationship between the two-answer and classification is listed in Table 1 [16,18]. Note the row and column in Table 1 marked by dark colors that refer to the answer called Neutral for both cases (Present and Not Present).

#### 3.2. Probability-Possibility Transformation

_{i}in terms of O, A, M, I, R, and Q. A relative frequency denoted as f

_{r}(F

_{i},C

_{k}) of a feature F

_{i}in terms of C

_{k}∈ {O, A, M, I, R, Q} is not the truth-value or DoB of the proposition “F

_{i}is C

_{k}”. It is possible to determine the DoB using the information of the relative frequency. To do this, the probability-possibility consistency principle can be used [19,20,21,22,23,24,25]. The probability-possibility consistency principle implies that the degree of possibility (or degree of belief) is always greater than or equal to the degree of probability, i.e., what is probable must be possible with a higher or equal degree of possibility, prob(.) ≤ π(.). The degree of possibility π(.) is, in fact, the Degree of Belief (DoB) or truth-value of a proposition [20,21,22]. The degree of probability, prob(.), is difficult to determine and in most real-life cases, the relative frequencies are taken as an estimation of the degree of probability. Based on this contemplation, the DoB of C

_{k}is given as

_{r}(F

_{i},C

_{k}) denotes the relative frequency of the classification C

_{k}for the feature F

_{i}. Since max(f

_{r}(F

_{i},C

_{k}) | ∀k = 1, …, 6) ≤ 1, DoB(F

_{i},C

_{k}) ≥ f

_{r}(F

_{i},C

_{k}) (≈prob(F

_{i},C

_{k})), i.e., the probability-possibility consistency principle holds if the Equation (4) is used. Other formulations of probability-possibility transformation are not considered in this study.

#### 3.3. Logical Transformation

_{i},C

_{k}), it is possible to find out the DoBs of the members of State (must be included, should be included, could be included, and unreliable). In doing so, it is important to understand the semantics of the classifications in Table 1 as follows:

## 4. Results and Discussions

- Step 1
- Considering plausible product features (e.g., the features called Sedan, SUV, and Van for a product called car) and a customer needs model (e.g., the Kano model)
- Step 2
- Developing a questionnaire
- Step 3
- Sending the developed questionnaire to certain respondents
- Step 4
- Collecting the respondents’ opinions based on the developed questionnaire
- Step 5
- Performing logical aggregation through the Degree of Belief (DoB) of all features in terms of must-be, should-be, and could-be features
- Step 6
- Ranking the features based on the compliance analysis using the quantity called certainty and requirement compliances denoted as CC and RC, respectively
- Step 7
- Making final decision on the features

#### 4.1. Implementation

#### 4.1.1. Execution of Steps 1–5

#### 4.1.2. Execution of Steps 6–7

_{E}) sets the requirement: X is a must-be included feature, X is a should-be included feature, X is a could-be included feature, X is a somewhat should-be included feature, and X is a must-be or should-be included feature. The user interface of a system, presented elsewhere [26,27] and shown in Figure 7, is used to calculate the information content in terms of (CC, RC). In this system, the information content in terms of (CC, RC) is defined as follows.

_{i},S

_{j}) and DoB(R

_{E}) are needed for calculating the values of CC and RC for each feature F

_{j}∈ {Sedan, SUV, Van}. In doing so, the numerical degrees of beliefs shown in Figure 6 are first converted to their respective linguistic counterparts, as described in Section 2.2. The expected values of the respective linguistic truth-values based on the centriod method (E(.), see Section 2.2) are considered the degrees of belief of the respective features and used while executing Equations (16) and (17). Therefore, the DoBs corresponding to Equations (16) and (17) refer to the expected values of the linguistic counterparts of the DoB shown in Figure 6. For example, consider the feature called SUV. For SUV, the linguistic truth-value of must-be included is mostly true (mt) because its numerical truth-value is equal to 1 (Figure 6), which belongs to the linguistic truth-value called mostly true more than it belongs to other linguistic truth-values, as illustrated in Figure 1. Since the expected value of mostly true (mt) is E(mt) = 0.967 (based on the centriod method), this value is considered the degree DoB(F

_{i}= SUV, S

_{j}= must-be included) of SUV when it is a must-be included feature.

_{E}). The results listed in Table 2 are also plotted in Figure 8. As seen from Figure 8, the variability in the information content of SUV is low compared to those of Sedan and Van. Van exhibits low information content when it is considered could-be included feature. When the requirement is set to must-be included feature or should-be included feature, Sedan’s information content becomes low. The same nature is seen for the feature called SUV. As such, the customers in Bangladesh prefer SUV and Sedan more than they prefer Van. SUV and Sedan must be included in the passenger vehicle population. On the other hand, Van could be included in the passenger vehicle population but not as many as SUV and Sedan. This decision is schematically illustrated in Figure 9.

## 5. Concluding Remarks

## Author Contributions

## Conflicts of Interest

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**Figure 2.**The Kano model [16].

Present (↓) | Not Present | ||||
---|---|---|---|---|---|

Like | Must-be | Neutral | Live-with | Dislike | |

Like | Q | A | A | A | O |

Must-be | R | I | I | I | M |

Neutral | R | I | I | I | M |

Live-with | R | I | I | I | M |

Dislike | R | R | R | R | Q |

Requirement (R_{E}) | X | ||
---|---|---|---|

Sedan | SUV | Van | |

X must be included | (0.533, 0) | (0.383, 0) | (0.216, 0.88) |

X should be included | (0.533, 0.772) | (0.383, 0.32) | (0.216, 0.88) |

X could be included | (0.533, 0.5) | (0.383, 0.32) | (0.216, 0) |

X somewhat should be included | (0.533, 0.435) | (0.383, 0.156) | (0.216, 0.602) |

X must be or should be included | (0.533, 0.0) | (0.383, 0) | (0.216, 0.88) |

- | (CC, RC) |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Rashid, M.M.; Ullah, A.S.
A Possibilistic Approach for Aggregating Customer Opinions in Product Development. *Systems* **2016**, *4*, 17.
https://doi.org/10.3390/systems4020017

**AMA Style**

Rashid MM, Ullah AS.
A Possibilistic Approach for Aggregating Customer Opinions in Product Development. *Systems*. 2016; 4(2):17.
https://doi.org/10.3390/systems4020017

**Chicago/Turabian Style**

Rashid, Md. Mamunur, and AMM Sharif Ullah.
2016. "A Possibilistic Approach for Aggregating Customer Opinions in Product Development" *Systems* 4, no. 2: 17.
https://doi.org/10.3390/systems4020017