Hesitant fuzzy axiomatic design technique: A case for humanitarian relief

Abstract

Decision makers can be irresolute evaluating an alternative according to a criterion. In such cases, it is best to consider the hesitancy rather than force the decision maker to make a clear evaluation. This paper focused on how to use a fuzzy axiomatic design technique considering hesitancy. Especially, it was discussed whether triangular hesitant fuzzy numbers defined in the literature were suitable for fuzzy axiomatic design technique. Also the paper contained a humanitarian relief project in terms of the post-natural disaster temporary housing location selection problem. When a natural disaster occurs, temporary housing should be established in order to provide the life safety of natural disaster victims and support them. The main objective of this problem was to determine the best location that would meet requirements of the disaster victims as soon as possible and smoothly. In the case study of this paper, the best temporary housing location was determined in the event of a natural disaster in a city which locates on the North Anatolian Fault Line. The fuzzy axiomatic design technique was used considering hesitancy to make a choice between candidate locations in the City.

Keywords

Multi-criteria decision-making decision-making under hesitancy hesitant fuzzy axiomatic design humanitarian relief temporary housing location

1 Introduction

Numerous multi-criteria decision-making problems are encountered throughout life. In the most part of these problems, decision criteria are not homogenous and deterministic. Often it is necessary to make the best decision by taking into consideration the fuzzy or heterogeneous criteria and evaluating them with one expert or a group of experts. There are various studies in the literature. Chen et al. [1] surveyed the most studies on group decision making problems in heterogeneous structure and classified them as indirect, direct and optimization based approach.

In addition to the fuzziness of multi-criteria decision-making problems are encountered in real life, there may also be cases where the decision maker is sometimes hesitant in making the evaluation.In such cases, it is best to obtain a final decision without forcing the decision maker(s) to make a certain evaluation, but also the hesitancy should be taken into account. Therefore, Torra [2] proposed hesitant fuzzy sets that allows to represent the situation in which different membership functions are considered possible. Afterwards, Rodriguez et al. [3] introduced the concept of hesitant fuzzy linguistic terms that will provide a linguistic elicitation and again Rodriguez et al. [4] presented a multi-criteria linguistic decision-making model in which experts provide their assessments using hesitant fuzzy linguistic term sets. Dong et al. [5] explained a connection between linguistic hierarchy and numerical scale model. Also they proposed a novel computing with words methodology which can be constructs the hesitant fuzzy linguistic term set based on unbalanced linguistic term sets. Recently, various multi-criteria decision-making studies under hesitancy was presented to the literature such as: assessment of bioenergy production technologies by Khishtandar et al. [6], evaluation the eco-tourism environmental carrying capacity by Liu and Wang [7], a personnel selection model by He and Wu [8], green supplier selection by Tang [9], a credit risk evaluation of enterprises by He et al. [10], real estate investment models by Du and Gao [11], a supplier selection model by Fahmi et al. [12], evaluation of alternative-fuel vehicles by Yavuz et al. [13], evaluation of electrical power system safety by Liu [14], emergency response capability assessment of emergency supply chain coordination mechanism by Chen and Song [15], etc.

One of the main motivations of our work is that Sakarya City, is located on the North Anatolian Fault Line and the problem of determining the most suitable temporary housing location as a result of major earthquakes in the past. Finding the best location for unexpected events are the most important agenda of both academician and decision makers in the government. Therefore, this study proposes an approach to determine the best temporary housing location more appropriately in case of a natural disaster for disaster victims. After a natural disaster, disaster victims need emergency shelter firstly for a short period of time and temporary shelter for a few weeks. Before the permanent housing, temporary housing is planned for more than six months to three years. “This way, even in a temporary location, temporary housing is extremely important to promote the return to normalcy in a chaotic and uncertain situation after a disaster, being a necessary step in reconstruction programs” [16]. In the literature there are a few studies about temporary housing problem which handled by decision-making techniques. Hosseini et al. [17] presented a new model in order to supports decision makers selecting a more adequate and sustainable type of temporary housing units. Proposed constructive selection model applied on the four different temporary housing units from the Bam earthquake in 2003. El-Anwar et al. [18] proposed a multi-objective weighted linear model which considers possibly conflicting objectives, namely: minimizing negative socioeconomic and environmental impacts and total public expenditures, maximizing safety. They handled as a case study of temporary housing arrangements after the 1994 Northridge Earthquake. Before the constructive temporary housing planning, determination of the most appropriate location for temporary housing is much more important. Because an inappropriate location (in terms of topography, infrastructure, etc.) can bring with various difficulties or unsolvable constructive problems. Although the problem of determining the temporary housing location is very important, it has not been addressed before and it as a serious shortcoming.

When the selection of temporary housing location problem is handled, it is seen that the group decision making problem has heterogeneous preference structure by the reason of some of the criteria have crisp evaluation value and some of them have fuzzy value. Determining process of evaluation criteria and alternatives also evaluation process were carried out by a group which consisted of experts in this study. At this point, there are many proposed techniques in the literature to make a consensus. For example, Dong et al. [19] handled real-world multi criteria group decision problems which denoted by complex and dynamic, and proposed a resolution framework. Zang et al. [20] presented a novel consensus reaching model for large-scale problems which considered individuals concerns and satisfactions.

Some of the real-life problems such as the selection of temporary housing location problem ad-dressed in this study, is not easy to evaluate certainly for some of the alternatives according to the fuzzy criteria. That is to say, when an alternative is evaluated according to a criterion, in case of hesitancy, then this should definitely be taken into account and the group decision making problem should be handled as fuzzy hesitant. In addition, there are very limited study in the literature about this subject so this is another main motivation for us. One of the outstanding paper was studied by Dong et al. [21]. They focused making a consensus in the hesitant linguistic group decision making problem and proposed an optimization based consensus model which minimized the number of adjusted simple terms. Also Rodriguez and Martinez [22] proposed another consensus model for group decision making problems under hesitancy.

In this study, unlike the others, when the group decision was given in the evaluation process, the hesitancy and fuzziness were not completely removed. Based on this, all fuzzy hesitant evaluations was considered without clarification. Namely, the group, which was consisted of a substructure responsible of the Sakarya Metropolitan Municipality, a geographical information system expert, a delegate of the Chamber of Agriculture and a delegate of the Republic of Turkey Prime Ministry Disaster and Emergency Management Presidency, decided for criteria and alternatives experientially. But when evaluating the alternatives according to the criteria, the Delphi method developed by Dalkey and Helmer [23] was used for making consensus. The evaluation dataconsidered hesitant fuzzy axiomatic design (HFAD) was chosen as a solution technique. Because, although it was proper, has not been used under hesitancy before.

The most important advantage of the HFAD method is that it does not force the decision maker when it is in doubt and allows to develop a solution considering the hesitation. In addition, the most basic knowledge base of this method is also the experience. For this reason, it can also be adapted to some types of problems, especially where the experience is desired to be taken into consideration.

In the second section of this study, it is explained how to use FAD under hesitancy. The case study, which motivated us, was presented in third section. In the fourth section, the triangular fuzzy numbers expressed in leading studies in the literature were discussed in terms of FAD method. Finally, the solution of the temporary housing location selection problem was given in the fifth section.

2 Fuzzy Axiomatic Design (FAD) technique under hesitancy

Axiomatic Design (AD) was developed by Suh [24] as a design theory in order to eliminate undesirable characteristics in a design process and to concentrate only on the required properties necessary for a design decision. AD consisted of two axioms as below [24, 25]:

“Independence axiom: Maintain the independence of functional requirements (FR).”

“Information axiom: Minimize the information content. (I)”

According to the independence axiom, the independence of FRs must be provided and the design parameters are related with FRs in an acceptable design. According to the information axiom, the best design has minimum I.

Suh defines the I as a function depending on the intersection of system and design range as Fig. 1 and calculates according to Equation (1). I of all alternatives are calculated for each criteria. Finally all the calculated information contents are summed up in terms of each alternative. Total I of the alternatives is compared and minimum one is determined as the best alternative. $I = {log}_{2} (\frac{System Range}{Common Range})$ (1)

Fig.1

Design, system and common ranges.

The design range can be defined as lower and upper bounds for target value and the system range can be defined as lower and upper bounds of the system performance. Also the common range is where the system and design range intersect. The common range shows how well the system performance meets the design performance. Also FR can be a continuous random variable and the design range can be expressed as a probability as in Fig. 2. Then I is calculated according to Equation (2) where p(FR) is probability density function of FR. $I = \int_{lower bound design range}^{upper bound design range} p (FR) dFR$ (2)

Fig.2

Design, system and common ranges for if FR is a continuous random variable.

Kulak and Kahraman [26] improved AD as a FAD to be able to consider fuzzy data, especially on multi-criteria decision-making problems. They define the I as a function depending on intersection of system and design area alike Suh as Fig. 3 and calculated according to Equation (3). The best alternative is determined likewise AD. $I = {log}_{2} (\frac{System Area}{Common Area})$ (3)

Fig.3

Design, system and common areas.

The concepts in FAD are almost same to axiomatic design, except fuzzy one can take triangular or trapezoidal values, so it has fields instead of range.

Decision makers can sometimes be instable when evaluating the alternatives considering criteria. This instability can affect the final decision. It would be more appropriate to adapt the decision support mechanism to this situation. For example, let S = {very low, low, medium, high, very high} and assume that the distance of all the linguistic variables is equal to each other. Let the design area is ‘at least S3’. If the decision maker hesitate between S3 and S4 evaluating an alternative according a criterion, then this hesitancy should be considered. The situation illustrated in the Fig. 4.

Fig.4

Difference between certain and hesitant evaluation.

So, if the decision maker hesitates evaluating an alternative, he/she should not be forced to make a certain decision and should consider this hesitancy in order to decide most appropriately.

3 Determining temporary housing location for disaster victims in Sakarya City

Sakarya City is the 1st degree seismic zone because it is located on the North Anatolian Fault Line. It was heavily influenced by 1943 Hendek, 1957 Abant, 1967 Akyazı, 1967 Mudurnu, 1999 Gölcük, 1999 Düzce earthquakes in the past. In addition to the earthquakes, rivers such as Sakarya River, Mudurnu River and Dariçayiri Stream sometimes cause floods. Sudden soaker and snow melting cause floods, too. In this context, when the natural disaster potential of Sakarya is taken into account, it is important to determine the ideal temporary housing location for the victims. Sapanca, Hendek, Arifiye, Geyve and Kaynarca were chosen as options for the temporary housing location. The criteria considered in the selection process were inspired by the related studies [16 , 27]:

Closeness to city center (CCC)

Closeness to healthcare organization (CHO)

Closeness to springs (CS)

Field slope (FS)

Closeness to fault line (CFL)

Geographical area (GA)

Vegetation (V)

Sewage infrastructure (SI)

Expansion compatibility (EC)

Electrical infrastructure (EI)

Climate conditions (CC)

According to these criteria, design ranges/areas of each criteria and system ranges/areas of each alternatives were determined and shown in Table 1. All the design ranges/areas and system ranges/areas were determined by an expert group which was detailed in the introduction section.

Table 1
Design ranges/areas of each criteria and system ranges/areas of each alternatives

Criteria Design Ranges/ Alternatives

Areas Sapanca Hendek Arifiye Geyve Kaynarca

CCC (km) ≤45 15–30 28–42 8–13 44–51 32–48

CHO (km) ≤10 8–20 6–17 4–8 4–9 5–16

CS (km) ≤15 2–4 3–9 6–11 8–17 14–25

FS (%) ≥ ≤15 7 5 4 8 12

CFL (km) ≥20 30–37 40–48 20–22 38–52 41–55

GA At worst Mountainside Lowland Lowland Valley and Mountainside

Valley and Lowland Lowland and Lowland

V At least Good Between Good and Good At most Good Good Good

Very Good

SI At least Mean At least Mean At most Mean At least Good Between Mean At most Good

and Good

EC At least Mean At most Mean Excellent Poor At least Good At most Mean

EI At least Mean At least Good At least Good At least Good Mean Mean

CC At least Mean Between Mean At most Mean Between Mean At most Good Excellent

and Good and Good

Criteria	Design Ranges/	Alternatives
CCC (km)	≤45	15–30	28–42	8–13	44–51	32–48
CHO (km)	≤10	8–20	6–17	4–8	4–9	5–16
CS (km)	≤15	2–4	3–9	6–11	8–17	14–25
FS (%) ≥	≤15	7	5	4	8	12
CFL (km)	≥20	30–37	40–48	20–22	38–52	41–55
GA	At worst	Mountainside	Lowland	Lowland	Valley and	Mountainside
	Valley	and Lowland			Lowland	and Lowland
V	At least Good	Between Good and	Good	At most Good	Good	Good
		Very Good
SI	At least Mean	At least Mean	At most Mean	At least Good	Between Mean	At most Good
					and Good
EC	At least Mean	At most Mean	Excellent	Poor	At least Good	At most Mean
EI	At least Mean	At least Good	At least Good	At least Good	Mean	Mean
CC	At least Mean	Between Mean	At most Mean	Between Mean	At most Good	Excellent
		and Good		and Good

For example the closest and farthermost point of Sapanca is 15 km and 30 km far from the city center respectively. So, system and design ranges can be drawn as Fig. 5 and information content of Sapanca in terms of closeness to city center criterion calculated as Equation (4).

Fig.5

System and design ranges of closeness to city center criterion for Sapanca.

$\begin{matrix} I_{CCC} & = & {log}_{2} (\frac{System Range}{Common Range}) \\ = & {log}_{2} (\frac{15}{15}) = 0 \end{matrix}$ (4)

Another illustrative example, Sapanca is a both mountainside and lowland place. The triangular fuzzy expressions and design area of the graphical area criterion is shown in Fig. 6. Also system and design areas can be drawn as Fig. 7 and information content of Sapanca in terms of the geographical area criterion calculated as Equation (5). $I_{GA} = {log}_{2} (\frac{1.5}{0.667}) = 1.17$ (5)

Fig.6

The triangular fuzzy expressions and design area of the graphical area criterion.

Fig.7

The design and system areas of the graphical area criterion for Sapanca.

Another illustrative example, Sapanca is evaluated between good and very good in terms of the vegetation. The design areas of the vegetation criterion are shown in Fig. 8. Also the same design areas were determined for sewage infrastructure, electrical infrastructure, expansion compatibility and climate conditions criteria. System and design areas can be drawn as Fig. 9 and information content of Sapanca in terms of vegetation criterion calculated as Equation (6). $I_{v} = {log}_{2} (\frac{1.5}{0.993}) = 0.596$ (6)

Fig.8

The triangular fuzzy expressions of the vegetation, sewage and electrical infrastructure, expansion compatibility and climate conditions criteria.

Fig.9

System and design areas of the vegetation criterion for Sapanca.

The information content of each criteria of all alternatives were calculated as in the illustrative examples. Then all the calculated information contents were summed up in terms of each alternatives and all the results were shown in Table 2.

Table 2

Total information contents of each alternatives

Criteria	Sapanca	Hendek	Arifiye	Geyve	Kaynarca
I _CCC	0.000	0.000	0.000	2.807	0.000
I _CHO	2.585	1.459	0.000	0.000	2.585
I _CS	0.000	0.000	0.000	0.363	0.000
I _FS	3.539	3.248	3.125	3.716	3.539
I _CFL	0.000	0.000	0.000	0.000	0.000
I _GA	1.169	0.000	0.000	0.000	1.169
I _V	0.596	0.973	2.029	0.973	0.596
I _SI	0.000	1.609	0.000	0.874	0.000
I _EC	1.609	0.000	2.414	0.000	1.609
I _EI	0.000	0.000	0.000	1.322	0.000
I _CC	0.874	1.609	0.874	1.202	0.874
Total	10.372	8.898	8.442	11.257	10.372

According to the Table 2, Arifiye is the best location for temporary housing. Also information content value of Hendek is pretty close to the information content value of Arifiye.

4 Discussion

One of the most cited studies in the literature, Rodriguez et al. [4] defined hesitant fuzzy linguistic term sets for decision-making as in Fig. 10.

Fig.10

Commonly held hesitant fuzzy linguistic term sets.

For example, according to this definition we should handle ‘at least S3’ equal to greater than Si as in Fig. 11.

Fig.11

Illustration of ‘at least S3’ according to the commonly held fuzzy linguistic term sets.

But then, the intersection of S3 and S4 as system areas with at least S3 as design area will be same under assumption that the distance of all the linguistic variables is equal to each other. So the information content of these two evaluations will be same too as in Fig. 12.

Fig.12

Illustration of ‘S3 ∩ at least S3’ and ‘S4 ∩ at least S3’.

In fact, intersection of S₄ must be greater than S₃ with the design area, which determined as ‘at least S₃’. In this direction, let S = {very low, low, medium, high, very high} and assume that all the distance of all the linguistic variables is equal to each other. Fuzzy and hesitant fuzzy expressions can be convert to triangular fuzzy numbers as shown in Fig. 13.

Fig.13

Triangular fuzzy numbers of fuzzy and hesitant fuzzy linguistic expression for HFAD technique.

5 Conclusion

In multi-criteria decision-making problems, we are often confronted with hesitancy in real life when we are evaluating. In such cases, often the decision makers are expected to make a certain evaluation. It is necessary to take account of the hesitancy as well as the fuzziness. There are various studies in the related literature. However, there is no previous study on the FAD technique under hesitancy, which is one of the important and practical multi-criteria decision-making techniques. In this study, there are descriptions of how the FAD technique can be used under hesitancy, and how the hesitant fuzzy linguistic expressions accepted in the literature should be implemented in this technique. In addition, the proposed HFAD technique has been applied in a humanitarian relief project, which has not been addressed previously in the field of multi-criteria decision-making. In Sakarya City, which is located on the North Anatolian Fault Line and experienced serious earthquakes, the post-natural disaster temporary housing location selection problem was addressed and a selection was made by using the HFAD technique suggested in the determined alternatives. According to these criteria, Arifiye was determined as a temporary housing location for disaster victims.

Also in the future works, HFAD can be used for the heterogeneous decision making problems such as selection problems, assignment problems, sequencing problems, etc. that is not possible to make definite evaluations or is convenient hesitation considering criteria.

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Criteria	Design Ranges/	Alternatives
	Areas	Sapanca	Hendek	Arifiye	Geyve	Kaynarca
CCC (km)	≤45	15–30	28–42	8–13	44–51	32–48
CHO (km)	≤10	8–20	6–17	4–8	4–9	5–16
CS (km)	≤15	2–4	3–9	6–11	8–17	14–25
FS (%) ≥	≤15	7	5	4	8	12
CFL (km)	≥20	30–37	40–48	20–22	38–52	41–55
GA	At worst	Mountainside	Lowland	Lowland	Valley and	Mountainside
	Valley	and Lowland			Lowland	and Lowland
V	At least Good	Between Good and	Good	At most Good	Good	Good
		Very Good
SI	At least Mean	At least Mean	At most Mean	At least Good	Between Mean	At most Good
					and Good
EC	At least Mean	At most Mean	Excellent	Poor	At least Good	At most Mean
EI	At least Mean	At least Good	At least Good	At least Good	Mean	Mean
CC	At least Mean	Between Mean	At most Mean	Between Mean	At most Good	Excellent
		and Good		and Good