An Index of Well-Being Based on Maqāṣid al-Sharīʿah and Fuzzy Set Theory

Abstract

This paper presents an index for measuring the well-being of a Muslim community based on Maqāṣid al-Sharīʿah. Its unique feature is that it makes use of fuzzy set theory (Zadeh 1965; Zimmermann 2001). Indeed, numerical estimation of Maqāṣid status may be difficult, especially the Maqāṣid-related intangibles like Dīn and Nafs. The fuzzy set theory allows the use of linguistic variables, which accommodate well the imprecision in human judgment and evaluation of actions and situations. The proposed approach solves the problem of aggregation of numerous values or sub-indexes that are measured in different scales into a single index, which is a serious problem that scholars encounter when constructing an overall index. This contribution opens a new direction of research in a quantitative approach to Maqāṣid al-Sharīʿah theory and in Islamic Economics, in general.

1 Introduction

The idea of expressing the objectives of Islamic, Law (al-Sharīʿah), the Maqāṣid al-Sharīʿah, goes back to works of early prominent Muslim scholars Al-Juwayni (1979), Al-Ghazali (1970) and Al-Shatibi (1302H). However, in the time of these scholars and later, no research was undertaken to explore the potential of Maqāṣid as a tool for the measurement of well-being of a society, while in the West, with the introduction of quantitative methods into human sciences, many indexes were developed to measure the well-being of a society such as the gross domestic production (GDP), Human Development Index (HDI), etc. See OCDE (1982) for more details. Most of these indexes are not comprehensive and do not include the social and psychological dimensions of human life. Moreover, they could not predict the devastating financial and economic crises that occurred in the last decade and before. Therefore, they came under severe critics Stiglitz (2009). In the last decade, Chapra (2009), referring to the work of Ibn Khaldun, pointed out that Maqāṣid al-Sharīʿah could be used as the components of a socio-economic system. This work revived the interest of Muslim scholars to explore the theoretical and practical potential of Maqāṣid al-Sharīʿah in social sciences. However, Chapra did not go beyond the theoretical formulation. Rusita et al. (2013) extended the work of Chapra by Sekaran’s operationalization method to include elements and dimensions, but did not reach the measurement stage, which is the essence of operationalization. In 2009, Larbani and Mustafa formulated an operational model for decision making based on Maqāṣid Al-Sharīʿah for resource allocation. This work opened a new direction of research called, the quantitative operationalization of Maqāṣid al-Sharīʿah. Recently, Bachelor (2013) has introduced an Islamic rating index of well-being for Muslim countries, and Anto (2011) has formulated an Islamic Human Development Index (I-HDI). Both indexes are based on Maqāṣid al-Sharīʿah and are measurable. Bachelor’s index lacks comprehensiveness For instance, he has omitted financial indicators that are essential for financial and economic crisis prevention. Moreover, he did not purify some indexes as the GDP from non-Islamic content. As for the Anto’s index, it is also not comprehensive and relies more on conventional indexes and suffers the same drawbacks as Bachelor’s.

As Maqāṣid al-Sharīʿah are related to all aspects of human life, including material and non-material or intangible, the construction of an overall index for socio-economic well-being would involve two difficulties: The first is the measurement of different variables or indicators that are used to explain or evaluate each of the five Maqāṣid. The second difficulty is the aggregation of the different measurements of each Maqsad into one value than the aggregation of the resulting individual measurements of all the five Maqāṣid into a unique value, that is, the overall index. As for the first difficulty, many indicators are measurable as the GDP, the birth rate, public debt, etc. However, there are also many indicators, especially, those related to the Maqāṣid of Dīn and Nafs as behavior, happiness, imān, etc. that are very difficult to measure. As for the second difficulty, even if one succeeds to measure all the parameters, he/she has to face another problem, the aggregation of measurement made at different scales and dimensions, including material and immaterial ones, into a unique meaningful value that would reflect the aggregated values.

This paper is a further contribution to the operationalization of Maqāṣid direction of research initiated in Larbani and Mustafa (2009). Indeed, we introduce an index of well-being of a nation based on Maqāṣid al-Sharīʿah and fuzzy sets. The use of fuzzy sets considerably reduces the difficulty of measuring immaterial indicators, eliminates the problem of aggregation of measurement of different scales and dimensions, and easy to compute.

The rest of the paper is organized as follows: Section 2 presents the Maqāṣid al-Sharīʿah used in this work. Section 3 presents briefly the fuzzy sets. Section 4 introduces the overall well-being index. Section 5 concludes the paper.

2 Maqāṣid al-Sharīʿah

Many definitions of Maqāṣid al-Sharīʿah have been introduced by prominent early Muslim Scholars as Al-Ghazali (1970), Al-Shatibi (1302), and Al-Juwayni (1979). Some contemporary scholars have also made a contribution to the theory of Maqāṣid al-Sharīʿah as Ibn Ashur (1998), Atya (2003), Al-Raysuni (2006) and Najjar (2006), including the introduction of new Maqāṣid as freedom, environmental protection, etc. Although the scholars differ on a set of Maqāṣid to be considered and on their ordering, they all agree that the ultimate goal of Sharīʿah is serving the interests of all human beings (al-masālih) and protecting them from harm (al-mafāsid).

Al-Ghazali (d. 1111 CE) definition of Maqāṣid emphasizes the sharī’ah concern for safeguarding the five objectives as follows (cited in Dusuki and Bouheraoua 2011):

The very objective of the Sharīʿah is to promote the well-being of the people, which lies in safeguarding their faith (dīn), their lives (Nafs), their intellect (‘aql), their posterity (nasl) and their wealth (māl). Whatever ensures the safeguarding of these five serves public interest and is desirable, and whatever hurts them is against public interest and its removal is desirable.

For the sake of simplicity of presentation, in this paper, we stick to the basic five Maqāṣid introduced by Al-Ghazali. However, the developed index can be adapted to any number of Maqāṣid. Another reason for the use of Al-Ghazali’s five Maqāṣid is to keep the mathematical model as simple as possible. Moreover, almost all the aspects of human life can be related directly or indirectly to the five Maqāṣid. For instance, justice can be related to the Maqsad of Dīn. Environmental protection can be related to Nafs because deterioration of the environment ultimately leads to deterioration of Nafs. It can also be related to the Maqsad of Māl because the resources necessary to develop wealth through production come from the natural environment. In this paper, we do not use the ordering of Maqāṣid as given above by Al-Ghazali; rather, we consider all the Maqāṣid as having the same importance or weights. Indeed, the importance of a Maqsad at some point in time depends on the current situation and attention allocation which generate a prioritization of Maqāṣid and actions. The priority will be given to the Maqsad that is most affected or threatened.

3 Fuzzy Sets

In this section, we give a brief introduction to fuzzy sets. Fuzzy sets will be used in the next section to measure the indicators related to each Maqsad and the overall well-being index. We shall first give the formal definition of a fuzzy set.

Definition 3.1.

Let X be a universal set. A fuzzy set A in X is characterized by a membership function $\mu_{A} \left( x \right)$ which associates with each point in X a real number in the interval [0, 1], with the value $\mu_{A} \left( x \right)$ at x representing the “grade of membership” of x in A.

The nearer the value of $\mu_{A} \left( x \right)$ is to 1, the higher is the grade of membership of x in A. When $\mu_{A} \left( x \right) = 1$, we say that x belongs fully or completely to A and when $\mu_{A} \left( x \right) = 0$, we say that x is not in A. When A is a set in ordinary sense of the term, its membership function $\mu_{A} \left( x \right)$ can take only two values 0 or 1, with $\mu_{A} \left( x \right) = 1$ or 0 according as x does or does not belong to A.

The set $\left\{ {x \in X/\mu_{A} \left( x \right) > 0} \right\}$ is called the support of the fuzzy set A. The set Kernel $\left( A \right) = \left\{ { x \in X/\mu_{A} \left( x \right) = 1} \right\}$ is the core of the fuzzy set A.

Fuzzy sets have been introduced by Zadeh (1965). In the last two decades, they have been applied extensively and successfully in almost all areas of human activity as industry, medicine, management, economics, etc. For more details about fuzzy sets, we refer the reader to Zimmermann (2001).

Example 3.1

Let us consider the set of “young people.” This set is not a set in the ordinary sense as its shape is not precisely known. For instance, a 20-year-old person is definitely young, a 70-year-old person is definitely old, but it is difficult to classify a 38-year-old person as young or not young. The fuzzy set “young people” can be characterized by the following membership function (Fig. 1).

Fig. 1

(Source Authors)

Membership function of the fuzzy set “young people”

A 30-year-old person has a membership grade or degree of 0.8 to the fuzzy set “young people,” while a 70-year-old person has a very small membership grade of 0.1. This reflects the imprecision in human judgment and the power of fuzzy sets to express it.

A fuzzy set can also express imprecise values. For instance, the body temperature of a healthy person should be around 37. Being “around 37” can be represented by a fuzzy set, the membership of which is (Fig. 2).

Fig. 2

(Source Authors)

Membership function of the fuzzy set “around 37”

When temperature, t, of a person is between [36.8, 37.8], the membership is $\mu_{{{\text{Around}}\,37}} \left( x \right) = 1$. When the temperature is below 36.8 or above 37.8, the membership declines sharply. When a person’s temperature is below 36 or above 39, the membership grade is 0, meaning that the person may be sick. The membership function is

$$\begin{aligned} \hfill \\ \mu_{{{\text{Around}}\,37}} \left( x \right) = \left\{ {\begin{array}{*{20}c} 0 & {if} & {0 \le t \le 1.2} \\ {\frac{t - 36}{36.8 - 36}} & {if} & {36 < t < 36.8} \\ 1 & {if} & {36.8 < t < 37.8} \\ {\frac{39 - t}{39 - 37.8}} & {if} & {37.8 \le t < 39} \\ 0 & {if} & {39 \le t} \\ \end{array} } \right. \hfill \\ \end{aligned}$$

(1)

Similarly, an economic or social indicator that should remain around some target value like the birth rate or inflation can be represented by a fuzzy set. In the next section, we will see how fuzzy sets can be used to construct a comprehensive well-being index based on Maqāṣid al-Sharīʿah.

Definition 3.2

Zadeh (1975) A linguistic variable is a variable whose values are words or sentences of a natural or artificial language.

For example, age is a linguistic variable if its values are linguistic rather than numerical, e.g. young, not young, very young, quite young, old, not old, very old, etc. rather than 20, 21, 23, etc. A fuzzy set is used to express the compatibility of numerical values with linguistic values. This is illustrated in Example 4.1. We will use a special linguistic variable to evaluate states of Maqāṣid in the next section.

4 Maqāṣid al-Sharīʿah and Fuzzy Sets Based Index of Well-Being

In this section, we introduce a Maqāṣid and fuzzy set-based index of well-being. The index uses the five Maqāṣid as defined by Al-Ghazali as the main indicators of the well-being. As the Maqāṣid are too general to be measured or estimated or evaluated, each Maqsad will be considered as a dependent variable of some independent variables related to it. For instance, public debt and poverty levels may be considered as independent variables related to the Maqsad of Māl (Fig. 3).

Fig. 3

(Source Authors)

Structure of Māl Index

For the ease of presentation, denote the five Maqāṣid, Dīn, Nafs, ‘Aql, Posterity, and Māl by $M_{1} ,M_{2} ,M_{3} ,M_{4,} \,{\text{and}}\,M_{5}$ respectively. For each Maqsad $\varvec{M}_{\varvec{i}} , \, i = 1,2, \ldots 5$, let $x_{i,j} \,j = 1,2, \ldots n_{i} ,$ be the related independent variables, where $n_{i}$ is their number. Graphically, this can be represented by the following figure (Fig. 4).

Fig. 4

(Source Authors)

Maqsad $M_{i}$ and its independent variables

To each independent variable $x_{i,j} \,j = 1,2,..,n_{i} ,$ and Maqsad $\varvec{M}_{\varvec{i}} ,i = 1,2, \ldots ,5$, we associate a fuzzy set $\check{x}_{i,j}$ defined on the possible states of the variable. Here, we have five cases:

Case 1

The independent variable $x_{i,j}$ is numerically measurable and the more the better, then one can easily construct the membership function of the fuzzy set $\check{x}_{i,j}$. For instance, if $x_{i,j}$ is the GDP of the country, the support of the fuzzy set $\check{x}_{i,j}$ is the interval [0, MaxGDP], where MaxGDP is a maximum or target value for GDP. Then the membership function $\mu_{i,j} \left( x \right)$ is defined by

$$\mu_{i,j} \left( x \right) = \frac{x}{\text{MaxGDP}}$$

where x is the observed GDP value. The form of this membership function is as follows.

Case 2

The variable $x_{i,j}$ is measurable, but it is such that the lesser the better like public debt, one may use a different formula for the membership function of the fuzzy set $\check{x}_{i,j}$ as follows:

$$\mu_{i,j} \left( x \right) = 1 - \frac{x}{\text{MaxDebt}}$$

The support of the fuzzy set $\check{x}_{i,j}$ is [0, MaxDebt], where Maxdebt is the maximum possible for the debt beyond which the collapse of the government is almost a certainty.

Case 3

The variable $x_{i,j}$ is measurable, but it is such that it should be around some desirable value. For instance, the birth rate should be around some value, let say 2.5%, for a high birth rate may be unsustainable, while a low birth rate many lead to population aging. Then a fuzzy set $\check{x}_{i,j}$, being around 2.5% with a membership function of trapezoidal type as depicted in Fig. 2, is the most appropriate. The corresponding membership function is similar to (1) with appropriate thresholds.

Case 4

When it is not possible to numerically measure the variable $x_{i,j}$, one may use a linguistic variable to express its state as the variable taking the following values Extremely Bad (EB), Very Bad (VB), Bad (B), Moderate (M), Good (G), Very Good (VG), and Excellent (E). For ease of presentation, we denote the states by $S_{1} , S_{2} , \ldots ,S_{7}$ respectively. Then a fuzzy set has to be constructed based on these states as follows. For each state, we associate a membership value with the interval [0,1], $\mu_{i,j} \left( {S_{p} } \right),\,p = 1,2, \ldots ,7$. Then for any state S in between two states $S_{p}$ and $S_{p + 1} ,\,p \le 6,\,\mu_{i,j} \left( S \right)$ is defined by $\mu_{i,j} \left( S \right) = \frac{{\mu_{i,j} \left( {S_{p + 1} } \right) + \mu_{i,j} \left( {S_{p} } \right)}}{2}$, for any state S in beyond or better than the state $S_{7} ,\,\mu_{i,j} \left( S \right)$ is defined by $\mu_{i,j} \left( S \right) = \frac{{1 + \mu_{i,j} \left( {S_{7} } \right)}}{2}$ and for any state S below the state $S_{1} ,\,\mu_{i,j} \left( S \right)$ is defined by $\mu_{i,j} \left( S \right) = \mu_{i,j} \left( {S_{1} } \right)/2$.

The function $\mu_{i,j} \left( S \right)$ can assume any other types of non-decreasing function between the states $S_{1} ,S_{2} , \ldots ,S_{6}$. For illustration, assume that

$\mu_{i,j} \left( {S_{p} } \right) = 0.1p,\,p = 1, 2, \ldots ,6$ and $\mu_{i,j} \left( {S_{7} } \right) = 0.8$.

The membership function will look as follows.

Case 5

The decision maker wants to evaluate some numerically measurable independent variable or indicator $x_{i,j}$, by a linguistic variable. For example, the decision maker wants to evaluate the health care system by the ratio of the number of doctors per 1000 citizens. He/she may set the highest ratio in the world as a benchmark, then evaluate the ratio of his/her country using the linguistic variable of the Case 4. Then use the membership function of the form presented in Fig. 5 to incorporate this health care evaluation in the Maqsad Nafs evaluation.

Fig. 5

(Source Authors)

Membership function for a linguistic variable

Assume now that the fuzzy sets $\check{x}_{i,j} \,,j = 1,2,..,n_{i}$ of all the independent variables $x_{i,j} ,j = 1,2, \ldots ,n_{i} ,$ related to Maqsad $\varvec{M}_{\varvec{i}}$, then the overall individual index of the Maqsad $\varvec{M}_{\varvec{i}} \varvec{ }$ is defined by

$$\pi_{i} \left( {\left( {z_{i,j} } \right)_{{\varvec{i},\,\varvec{j} = 1,2, \ldots \varvec{n}_{\varvec{i}} }} } \right) = \mathop \prod \limits_{j = 1}^{{n_{i} }} \mu_{i,j} \left( {z_{i,j} } \right)$$

(2)

where $z_{i,j}$ is a numerical value if the independent variable $x_{i,j}$ is measurable or a state S if the variable $x_{i,j}$ is not measurable. That is the index $\pi_{i}$ of the Maqsad $M_{i}$ is the product of all the membership functions of the fuzzy sets of its independent variables.

Now we are able to formulate an overall index for the well-being of a society by aggregating the individual indexes of the five Maqāṣid, $\pi_{i} ,i = 1,2, \ldots ,5$ as follows:

$$\pi \left( {\pi_{1} ,\pi_{2} , \ldots ,\pi_{5} } \right) = \frac{{\mathop \sum \nolimits_{{\varvec{i} = 1}}^{5} \pi_{i} (\left( {z_{i,j} } \right)_{{\varvec{i},\,j = 1,2, \ldots n_{i} }} }}{5}$$

(3)

In other words, the overall index of well-being is the average of the indexes $\pi_{i} ,i = 1,2, \ldots ,5$ of the five Maqāṣid, $M_{i} , i = 1,2, \ldots ,5.$ As the index $\pi$ is a number in the interval [0, 1], one can use the fuzzy membership function in Fig. 5 of the linguistic variable introduced in the Case 4 to give a linguistic evaluation of it. This value would help decision makers have an idea about how good is the obtained value $\pi$. One needs to locate the index value in the vertical axis then find the corresponding linguistic evaluation in the horizontal axis. Other forms of aggregation may be used. One of them is the product of the five Maqāṣid indexes $\pi_{i} ,i = 1,2, \ldots , 5$, but the index $\pi$ will be small as a result of multiplication of numbers smaller than 1.

Let us now explain the computation process of the introduced overall well-being index in steps.

• Step 1. Identify the independent variables $x_{i,j} ,j = 1,2, \ldots , n_{i}$ of each Maqsad $M_{i} ,i = 1,2, \ldots ,5$.

• Step 2. Construct the fuzzy sets $\check{x}_{i,j}$ of each of the independent variable $x_{i,j} ,j = 1,2, \ldots ,n_{i} ,$ associated with each Maqsad $\varvec{M}_{\varvec{i}} ,i = 1,2, \ldots ,5$. The membership function can be constructed according to the Cases 1–5 presented above.

• Step 3. Measure the variables that are measurable and determine or assign values to linguistic variables, and as a result we get the values $\left( {z_{i,j} } \right)_{{\varvec{i} = 1,2, \ldots 5,\,j = 1,2, \ldots n_{i} }}$.

• Step 4. Compute the value of the membership functions $\mu_{i,j} \left( {z_{i,j} } \right)$ of the fuzzy set $\check{x}_{i,j}$ of each of the independent variable $x_{i,j} ,j = 1,2, \ldots ,n_{i} ,$ associated with each Maqsad $\varvec{M}_{\varvec{i}} ,i = 1,2, \ldots ,5$. As a result we get the values $\mu_{i,j} \left( {z_{i,j} } \right),i = 1,2, \ldots ,5,j = 1,2, \ldots ,n_{i}$.

• Step 5. Compute the individual index $\pi_{i}$ of each Maqsad $\varvec{M}_{\varvec{i}} ,i = 1,2, \ldots ,5$ by Formula (2). As a result we get the five indexes $\pi_{i} ,i = 1,2, \ldots ,5$.

• Step 6. Compute the overall well-being index $\pi$ using Formula (3).

Example 4.1.

For illustration purpose, we give the following hypothetical example. Assume that for some country C the following data are available.

• For Dīn Maqsad

  $x_{1,1} -$ The ratio of people who went to Hajj to the quota allowed is 0.7 in the current year.

  $x_{1,2} -$ The result of a survey on the religiosity of Imaams indicating that the level of religiosity is moderate among people on an ordinal scale such as Extremely Bad (EB), Very Bad (VB), Bad (B), Moderate (M), Good (G), Very Good (VG), and Excellent (E).

• For Nafs Maqsad

  $x_{2,1} -$ The ratio of number of doctors to 1000 citizens is 1.5/1000 = 0.0015.

• For Maqsad ‘Aql

  $x_{3,1} -$ The proportion of students who left school before university level in the current year is 10%.

  $x_{3,2} -$ The literacy level is 90%.

• For Maqsad Nasl

  $x_{4,1} -$ Birth rate is 1.8%.

  $x_{4,2} -$ Mortality rate at birth per 100 is 7% (a ratio of 0.07).

• For Māl Maqsad

  $x_{5,1} -$ The growth rate is 3.7%.

  $x_{5,2} -$ The ratio of public debt to GDP is 0.7.

• Step 1. It is straightforward.

• Steps 2 and 3. Construction of the fuzzy sets corresponding to the indicators. First, we categorize the indicators into (i) the more the better and not requiring benchmark value, which includes $x_{1,1} ,\,x_{3,2}$ (ii) the more the better and requiring a benchmark value, which includes $x_{2,1}$ and $x_{5,1}$ (iii) the lesser the better requiring benchmark or tolerance level, which includes $x_{3,1} ,x_{4,2} \,{\text{and}}\,x_{3,2}$, (iv) required to be around some target value, which includes $x_{4,1}$, and (v) linguistic indictor, which includes $x_{1,2}$.

• i. For the indicators $x_{1,1} ,\,x_{3,2}$, the membership function of the corresponding fuzzy sets can be constructed by using the formulas

$$\mu_{1,1} \left( {x_{1,1} } \right) = x_{1,1} \,{\text{and}}\,\mu_{3,2} \left( {x_{3,2} } \right) = x_{3,2} ,\,{\text{respectively}}.$$

(4)

• ii. The indicators $x_{2,1}$ and $x_{5,1}$ require some benchmark value for 100% does not make any practical sense. Let us assume that 6/1000 = 0.006 is the benchmark value for $x_{2,1}$ and 12% for $x_{5,1}$, then the membership functions of the corresponding fuzzy sets are

$$\mu_{2,1} \left( {x_{2,1} } \right) = \frac{{x_{2,1} }}{0.006}\,{\text{and}}\,\mu_{5,1} \left( {x_{5,1} } \right) = \frac{{x_{5,1} }}{0.12},\,{\text{respectively}}$$

(5)

For example, the membership function $\mu_{5,1} \left( {x_{5,1} } \right)$ has the following shape

• iii. The indicators $x_{3,1} ,\,x_{4,2} \,,{\text{and}}\,x_{5,2}$ are of the type: the lesser the better. They all require some tolerance level or maximum value because 100% or more does not make practical sense. Let us assume that for $x_{3,1}$, 40% is the tolerance level; for $x_{4,2}$ the tolerance level is 15%; and for $x_{5,2}$ the tolerance level or ceiling is 120%. Therefore, the membership functions of the corresponding fuzzy sets are (Fig. 6).

  Fig. 6

  

  (Source Authors)

  Membership functions of the corresponding fuzzy sets

$$\begin{aligned} & \mu _{{3,1}} \left( {x_{{3,1}} } \right) = 1 - \frac{{x_{{3,1}} }}{{0.4}},\mu _{{4,2}} \left( {x_{{4,2}} } \right) = 1 - \frac{{x_{{4,2}} }}{{0.15}}\quad {\text{and}} \\ & \quad \mu _{{5,2}} \left( {x_{{5,2}} } \right) = 1 - \frac{{x_{{5,2}} }}{{1.2}}, \\ \end{aligned}$$

(6)

respectively. For example, the membership function $\mu_{5,2} \left( {x_{5,2} } \right)$ has the shape (Fig. 7).

Fig. 7

(Source Authors)

Membership function shape

• iv. The indicator $x_{4,1}$ is desired to be around some target value. Let us assume that the birth rate is desired to be around 2.5% to avoid both the aging of the population and its unsustainable increase. We define the membership function of the associated fuzzy set “around 25” similarly to the membership of the fuzzy set “around 37 degrees” in Fig. 7, (Fig. 8).

  Fig. 8

  

  (Source Authors)

  Membership function of the associated fuzzy set “around 25”

The decision maker has to specify the meaning of “being around 2.5.” For instance, a birth rate of 3.8 and above or 1.2 and below are not accepted. Any rate between 2.2 and 2.8 implies that the objective of being around 2.5 is reached. The analytical form of the membership function is as follows:

$$\mu_{4,1} \left( x \right) = \mu_{{{\text{Around}} 2.5}} \left( x \right) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {if\quad 0 \le x \le 1.2} \hfill \\ {\frac{x - 1.2}{2.2 - 1.2}} \hfill & {if\quad 1.2 < x < 2.2} \hfill \\ 1 \hfill & {if\quad 2.2 < x < 2.8} \hfill \\ {\frac{3.8 - x}{3.8 - 2.8}} \hfill & {if\quad 2.8 \le x < 3.8} \hfill \\ 0 \hfill & {if\quad 3.8 \le x} \hfill \\ \end{array} } \right.$$

(7)

• v. The indicator $x_{1,2}$ is linguistic and takes values in the ordinal scale Extremely Bad (EB), Very Bad (VB), Bad (B), Moderate (M), Good (G), Very Good (VG), and Excellent (E). The decision maker may choose the membership function $\mu_{1,2} \left( {x_{1,2} } \right)$ of the corresponding fuzzy set as described in Case 4 above and depicted in Fig. 5.

• Step 4. Now we are able to compute the values of all the nine membership function for all the indicators based on the given data:

For Maqsad Dīn

$\mu_{1,1} \left( {0.7} \right) = 0.7$, see Formula (4), $\mu_{1,2} \left( {\text{Moderate}} \right) = 0.4$ (based on Fig. 4).

For Maqsad Nafs

$\mu_{2,1} \left( {0.0015} \right) = \frac{0.0015}{0.006} = 0.25$, see Formula (5).

For Maqsad ‘Aql

$\mu_{3,1} \left( {0.1} \right) = 1 - \frac{0.1}{0.4} = 0.75$, see Formula (6), $\mu_{3,2} \left( {0.9} \right) = 0.9$, see Formula (4).

For Maqsad Nasl

$\mu_{4,1} \left( {1.8} \right) = 0.6$, see Formula (7), $\mu_{4,2} \left( {0.07} \right) = 1 - \frac{0.07}{0.15}$ = 0.41, see Formula (6).

For Maqsad Māl

$\mu_{5,1} \left( {0.037} \right) = \frac{0.037}{0.12} = 0.308$, see Formula (5), $\mu_{5,2} \left( {0.7} \right) = 1 - \frac{0.7}{1.2} = 0.58$, see Formula (6).

• Step 5. Computation of indexes of individual Maqāṣid

For Maqsad Dīn

$$\pi_{1} = 0.7 \times 0.4 = 0.28$$

For Maqsad Nafs

$$\pi_{2} = 0.25$$

For Maqsad ‘Aql

$$\pi_{3} = 0.75\, \times \,0.9 = 0.675$$

For Maqsad Nasl

$$\pi_{4} = 0.6\, \times \,0.41 = 0.246$$

For Maqsad Māl

$$\pi_{5} = 0.308\, \times \,0.58 = 0.178$$

• Step 6. Computation of the overall well-being index.

$$\pi = \left( {0.28 + 0.35 + 0.675 + 0.246 + 0.178} \right)/5 = 0.346$$

Evaluating the overall well-being index using the fuzzy set with the membership function in Fig. 5 that is associated with the linguistic variable whose values are Extremely Bad (EB), Very Bad (VB), Bad (B), Moderate (M), Good (G), Very Good (VG), and Excellent (E), we obtain that the considered country is between Bad and Moderate statuses. One can also notice that individually, Maqāṣid Dīn and Nafs are at bad status. This means that the country has to develop a strategy to improve all the Maqāṣid with more resource allocation to Maqāṣid Dīn and Nafs.

Remark 4.1

The introduced overall well-being index has the following unique features.

• 1. It can incorporate variables or indicators that are not measurable when it is possible to use a linguistic variable to express them. Therefore, it can operationally incorporate more indicators or variables than the existing indexes.

• 2. It can incorporate indicators that are assumed to be around some target value (limited excess and/or slack are allowed) and indicators that should remain in some fuzzy range.

• 3. It can aggregate indicators that are measured in different scales or dimensions via fuzzy sets.

• 4. It computes individual index for each Maqsad, which allows a decision maker to analyze the status of each Maqsad individually.

• 5. It is operational in the sense that it is easy to compute once real data are available and can incorporate as many variables or indicators as one wants.

5 Conclusion

In this paper, we have introduced an overall well-being index of a nation based on Maqāṣid al-Sharīʿah. Its unique feature is that it uses fuzzy sets for aggregating variables of different dimensions or scales and allows the incorporation of non-measurable indicators when they can be expressed by linguistic variables. We have shown how to construct a fuzzy set associated with each independent variable or indicator to make the index operational and easy to compute. We intend to use this index to evaluate the overall well-being of Malaysia in the nearer future; we are at the stage of data collection. Later, we apply the index to OIC countries.