A Mathematical Modeling Approach for Developing a Social Index for Standard of Living in Colombo Municipal Council.H.K.D.S.M.Perera and T.N.WithanachchiDepartment of Mathematics, University of Colombo

ABSTRACTThe Quality of Living index (QOL) is the social index defined to measure the Standard of Living. QOL is a multidimensional concept, which has many distinct domains. We illustrate that the various measures of well-being are highly sensitive to domains which are considered in the construction of comparative indices of QOL. Further we indicate how the measurable inputs are aggregated and weighted to arrive at the composite measures of QOL. Establishing a QOL index requires combinations of different domains which could be qualitative or quantitative. Thus using Fuzzy Set Theory allows capturing the expert knowledge of multi domains of QOL to efficiently generate a single index. Our focus is scoped to the governing area of Colombo Municipal Council. The success of this model serves as a basis for application to rank the Zones of Colombo 1 to 15.

Key words: Quality of Living, Fuzzy Set Theory, Index

CHAPTER 01 - INTRODUCTION QOL has been an object of many studies in different kinds of research classifications such as Community, Education, Environment, Health, Life satisfactions, Safety etc. The notion of QOL has been discussed from different perspectives and interpretations. Expressions like "good city", "good place to live" and "good quality of living" involve conceptual perspectives that, frequently, vary from person to person, from place to place and along the time (Altnok, Kozaman, & engezer). The concept of QOL as used in this project has domains as Housing, Environment, Services and Recreation.While the concept of QOL varies according to the recognized values, it also varies with the wealth of societies as well as with the pass of time. QOL appears as a multidimensional concept. It is defined with respect to a variety of quantitative and qualitative criteria that may change with societies and cultures. Arbitrary is one of the major concepts in describing QOL. Therefore fuzzy theory which was incepted in 1965 by Zadeh provides an approach to deal with arbitrary and uncertain concept. Fuzzy theory comes with understanding that every vague phenomenon can be expressed in membership (Abdullah, 2011). Thus QOL associates and implements the Fuzzy theory in order to approach a mathematical model. The construction of our composite indices for QOL via using the fuzzy theory follows two main steps. The first one concerns the definition of the membership function to each declared variables for each domain. While the membership function can take several formulations, we consider the Fuzzy Analytic Hierarchy Process (FAHP) as defined originally to weight each domain [ (Lelli, 2001) and (Baliamoune, 2003)]. This study is based on the geographical area governed by Colombo municipal council i.e. Colombo 1-15. CHAPTER 02 - RESEARCH METHODOLOGY 2.1 Introduction This chapter aims to set out the methodological framework which is going to use to achieve the results of the study. It describes in detail the research methodology which is going to use in completing this research including the research approach and procedures use for data collection as well as data analysis.2.2 Fuzzy SetA fuzzy set is defined by = {(x, A(x)): xA, A(x)[0,1] }.In the pair (x, A(x)), the first element x belong to the classical set A, the second element A(x), belong to the interval [0, 1], called Membership function (Assarudeen & Gani, 2012).2.3 Triangular Fuzzy NumberAssarudeen & Gani (2012) mentioned that it is a fuzzy number represented with three points as follows: = (a1,a2,a3). This representation is interpreted as membership functions and holds the following conditionsi. a1 to a2 is increasing functionii. a2 to a3 is decreasing functioniii. a1a2a3Figure 1: Triangular Fuzzy Number

= 2.4 Fuzzy Analytic Hierarchy ProcessThe fuzzy AHP technique can be viewed as an advanced analytical method developed from the traditional AHP. Generally, it is impossible to reflect the decision makers uncertain preferences through crisp values (Hasin & Kabir, 2011). Therefore, FAHP is proposed to relieve the uncertainness of AHP method, where the fuzzy comparisons ratios are used. There are the several procedures to attain the priorities in FAHP. Changs extent analysis is utilized in this research to determine the weights of main domains. The first step in this method is to use triangular fuzzy numbers for pairwise comparison by means of FAHP scale, and the next step is to use extent analysis method to obtain priority weights by using synthetic extent values. The fuzzy evaluation matrix of the criteria was constructed through the pairwise comparison of different attributes relevant to the overall objective using the linguistic variables and triangular fuzzy numbers. 2.5 Extent Analysis MethodIn this study, Changs extent analysis method on fuzzy AHP, therefore triangular fuzzy numbers (TFN) are used. Triangular fuzzy numbers are represented as l/m, m/u, (or (l, m, u) in which l, m and u refer to, respectively, the lower value, modal value and upper value (Chang D. Y., 1996) . Let , be an object set and a goal set respectively. Then each object is taken and extent analysis for each goal is performed respectively. Therefore, m extent analysis values for each object can be obtained, with the following signs: where all are TFNs. The steps of Changs extent analysis (Chang D. Y., 1992) can be detailed as follows (Beskese, Bozbura, & Kahraman, 2007): Step 1: The value of fuzzy synthetic extent with respect to the object is defined

To obtain , the fuzzy addition operation of m extent analysis values for a particular matrix is performed such as:

And to obtain perform the fuzzy addition operation of values such that

And then inverse of the vector above is computed, such as:

Step 2: The degree of possibility of is defined as

And can be equivalently expressed as follows:

Where d is the ordinate of the highest intersection point D between and as shown in Figure 2.Figure 2: The Intersection between and

To compare and we need both values of Step3: The degree possibility for a convex fuzzy number to be greater than k convex fuzzy numbers can be defined by

Assume that Then the weight vector is given by where are n elementsStep 4: Via normalization, the normalized weight vectors are where is a non-fuzzy number.2.6 Fuzzy OperatorsThe modified algebraic product F of A and B is denoted AB and is defined by

The modified Hamacher operator H defines the intersection of two fuzzy sets A and B by 0

CHAPTER 03 DEVELOPING THE QUALITY OF LIVING INDEX3.1 Identifying the domains that will be used to develop the indexIn the literature, the QOL context encompasses multi domains. Considering the existing approaches main four domains and several sub domains as well as the variables for each domain were determined. The hierarchical structure which was illustrated in Table I was constructed. (Mercer, 2012), (Mittelhammer, Rahman, & Wandschneider, 2003), (Kerce, 1992), (numbeo, 2013),Table 1: The Hierarchical StructureThe Hierarchical Structure of the Multi domains of Quality Of Living

Main DomainsSub DomainsSub Domains

EnvironmentLandPopulation density

Green coverage percentage

ClimateTemperature

DisturbancesAir Quality (CO Density)

ServicesEducationAvailability of schools

EconomicAvailability of banks

Availability of markets/ supermarkets

HealthAvailability of Hospitals

HousingPercentage of slums

Percentage of luxury apartments

RecreationThe percentage of the total area devoted to Parks

Availability of Hotels/Movie Theatre

3.2 Membership FunctionsThe Quality of Living Index will be based on the values taken by following variables under 4 main domains.

8 | PageDepartment of Mathematicst1: Population density in persons per hectaret2: Green coverage percentaget3: Temperature in 0Ct4: CO density in mg/m3t5: Availability of schoolst6: Availability of bankst7: Availability of markets/supermarketst8: Availability of hospitalst9: Percentage of slumst10: Percentage of luxury apartmentst11: The percentage of the total area devoted to Parkst12: Availability of Hotels/Movie Theatre

If a fuzzy set approach is to be used, membership functions have to be defined for all sub domains. The fuzzy set A of the zones with a low level of population density can then be defined by the membership function UA(x,t1). (population matters, 2013)

UA(x,t1)=

The fuzzy set B of the zones with an appropriate level of green coverage percentage can then be defined by the membership function UB(x,). (Tsutsumi, Ishii, & Katayama)

UB(x,) =

The fuzzy set C of the zones with an acceptable level of temperature can then be defined by the membership function UC(x,). (West Midlands Public Health Observatory)UC(x,) =

The fuzzy set D of the zones with an acceptable level of CO density can then be defined by the membership function UD(x,)UD(x,)=

The fuzzy set E of the zones with an appropriate percentage of availability of schools can then be defined by the membership function UE(x,).UE(x,) =

The fuzzy set F of the zones with an appropriate percentage of availability of banks can then be defined by the membership function UF(x,)UF(x,) =The fuzzy set G of the zones with an appropriate percentage of availability of markets/supermarkets can then be defined by the membership function UG(x,)UG(x,) =

The fuzzy set H of the zones with an appropriate percentage of availability of hospitals can then be defined by the membership function UH(x,)UH(x,) =

The fuzzy set I of the zones with an appropriate percentage of slums can then be defined by the membership function UI(x,)UI(x,t9)=

The fuzzy set J of the zones with an appropriate percentage of luxury apartments can then be defined by the membership function UJ(x,)UJ(x,) =

The fuzzy set K of the zones with an appropriate percentage of the total area devoted to parks can then be defined by the membership function UK(x,) (Standards for recreational areas)UK(x,) =

The fuzzy set L of the zones with an appropriate percentage of availability of Hotels/Movie Theatre can then be defined by the membership function UL(x,)UL(x,) =

3.3 Estimation of the weights of main domains using Fuzzy AHP.Membership functions are aggregated using weights of domains which are determined using Fuzzy AHP. Once the model was constructed, a questionnaire form in the appendix A was established to obtain the pair wise comparisons for main domains Environment, Services, Housing and Recreation. In the form, ten people indicated their pairwise comparisons to obtain the weights of the main domain using the linguistic scale (Buyukozkan, Ertay, & Kahraman, 2006) which is presented in Table 2. Table 2: Linguistic Scale for ImportanceLinguistic scaleLinguistic scale Triangular Fuzzy

Absolutely more important(5/2,3,7/2)

Very strongly more important(2,5/2,3)

Strongly more important(3/2,2,5/2)

Weakly more important(1,3/2,2)

Equally important(1/2,1,3/2)

Just equal(1,1,1)

Table 3: Aggregated Comparison Matrix of the Attributes for AHP ModelAttributesHousingServicesEnvironmentRecreation

Housing(1,1,1)(1.80,2.20,2.70)(1.65,2.15,2.65)(1.15,1.65,2.15)

Services(0.378,0.470,0.625)(1,1,1)(1.633,2.090,2.55)(1.55,2.05,2.55)

Environment(0.379,0.470,0.619)(0.537,0.670,0.864)(1,1,1)(1.491,1.856,2.230)

Recreation(0.487,0.658,1.051)(0.399,0.504,0.685)(0.937,1.136,1.364)(1,1,1)

Using the tables 2 & 3, the values of fuzzy synthetic extent with respect to each main domain is calculated as follows:

The fuzzy values are compared and the following values are obtained:The vector of priority of weight is found by normalization:

The result indicates that housing is the most important main domain with the priority of 0.64 for the quality of living. The lowest priority belongs to the domain recreation in the decision makers judgments.3.4 Estimation of the Quality of Living Indices.

The effects of sub domains and variables under each domain are assumed to be same and the weights of main domains mentioned above are used to aggregate the membership functions. Considering the power of operators as , , and p=0.5 following Quality of Living Indices are calculated using Modified Algebraic Product Operator and Modified Hamacher Operator.Table 4 Quality of Living IndicesZones of ColomboUsing Modified Algebraic ProductUsing Modified Hamacher Operator

QOL IndexRankingQOL IndexRanking

Colombo 010.0472140.074814

Colombo 020.4427050.523505

Colombo 030.4184060.438906

Colombo 040.5871030.604003

Colombo 050.3511100.399507

Colombo 060.2364120.290612

Colombo 070.7705020.849801

Colombo 080.8073010.814702

Colombo 090.4169070.375210

Colombo 100.5157040.592604

Colombo 110.0192150.028015

Colombo 120.2971110.358211

Colombo 130.3675080.393709

Colombo 140.3624090.398508

Colombo 150.1767130.246913

CHAPTER 04 - DISCUSSIONThe two methods which were used to aggregate the membership values, Modified Algebraic Product and Modified Hamacher Operator provided adequate results for the study. Both methods gave somewhat same results but Modified Hamacher Operator provided that the zone with the best quality of living in Colombo city is Colombo 07 and Modified Algebraic Product Operator provided that the zone with the best quality of living in Colombo city is Colombo 08. The people in Colombo 07 and Colombo 08 enjoy the comfort of living conditions. The zones with the lowest ratings were Colombo 01 and Colombo 11. Those areas are commercially important but proven not the best living environments.As each domain has not the same importance, using dilation operator the algebraic product and Hamacher operator have been modified. Thus they allow for compensation and maximum interaction, since the effect of one domain fully impacts the others. The results reveal the modified Hamacher operator allows for cumulative effects, interactions and recompenses between the domains rather than the modified algebraic product. Therefore, Modified Hamacher operator proved to be a convenient method in developing the indices.According to the weights using Fuzzy AHP, Housing has higher priority than other domains. The lowest priority is for the domain, Recreation. Thus the responses of the decision makers have a high impact on the final weights of the domains. Therefore decision makers should identify their importance of domains properly. The main domains of this study were limited to four domains; Housing, Environment, Services and Recreation. Thus an expansion of the scope could be used to enhance the accuracy and sensitivity of the index.Figure 3: Quality of living indices of Colombo 01 to 15

The final output map in Figure 3 indicates that the areas with the high quality of living as well as the low quality of livingCHAPTER 05 - CONCLUSIONThis study compared neighborhoods in Colombo as from Colombo one to fifteen to rank them according to the quality of living in them. Both qualitative and quantitative measurements were taken to consideration to approach a solution in sort of a mathematical model to develop a social Index i.e. QOL. Fuzzy Theory has been repeated proposed as an effective technique to deal with arbitrary and uncertain. Fuzzy AHP technique was used to synthesize the opinion of the decision makers to identify the weight of each domain. Modified Hamacher operator proved to be a convenient method in developing the indices. Housing is the most important domain for quality of living. The study proven that the zones with the best quality of living in Colombo city are Colombo 07 and Colombo 08. Further the index can be extended to compare Quality of Living in Districts or provincial wise to ensure the quality of living in the country as well.