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Optimal Selection of a Landfill Disposal Site Using a Modified Fuzzy Utility Approach

Optimal Selection of a Landfill Disposal Site Using a Modified Fuzzy Utility Approach Fuzzy Inf. Eng. (2012) 3: 313-338 DOI 10.1007/s12543-012-0118-9 ORIGINAL ARTICLE Optimal Selection of a Landfill Disposal Site Using a Modified Fuzzy Utility Approach Ajit Pratap Singh · Subodh Kant Dubey Received: 10 June 2011/ Revised: 16 May 2012/ Accepted: 5 August 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract The present paper develops an integrated fuzzy based model to select an optimal landfill site among the given alternative sites by using the concept of fuzzy- utility method and multi-nomial logit theory. The suitability of different landfill sites are evaluated based on some important criteria involved in the process such as acces- sibility and transportation; environmental, geological and climatic conditions; socio- economic conditions; land use pattern; and safety at the selected site. These criteria are assessed qualitatively by the decision makers based on their relative degree of importance. The importance weights and ratings of each criterion have been defined in the form of triplets of triangular fuzzy numbers by taking opinion of the decision makers. The corresponding triplets of ratings of each site are used to derive the utility value of the alternative sites. A multi-nomial logit model has been applied to calcu- late the probability of selection of each alternative site which can help policy makers to take appropriate decisions. Finally, the proposed methodology has been applied to allocate suitable landfill sites for disposing off municipal solid waste for Pilani town which is located in Jhunjhunu district of Rajasthan. The results evaluated by the modified fuzzy utility are also compared to the outputs of a direct method which is basically based on certain linguistic aggregation operators for group decision making. Computational results clearly demonstrate that the results obtained by the proposed method are coinciding very well and prepares a basis to adopt an overall strategy for selecting appropriate landfill site for proper solid waste disposal and its management. Keywords Decision support system· Uncertainty· Fuzzy utility value· Solid waste management · Landfill allocation 1. Introduction Ajit Pratap Singh ()· Subodh Kant Dubey Civil Engineering Department, BITS Pilani, Rajasthan 333031, India email: apsbits@gmail.com 314 Ajit Pratap Singh· Subodh Kant Dubey (2012) The solid waste management and its proper disposal have now become an impor- tant issue worldwide, be it the developed countries where the quantity and kind of hazardous wastes, and the lack of disposal sites have caused a greater concern or the developing nations where the growth in both size and concentration of the population, combined with a general lack of public awareness, have made the problem of solid waste a critical public issue [1, 2]. The rapid growth in population and economic de- velopment has resulted in a significant increase in municipal solid waste generation in India. The estimated solid waste generation ranges from 100 grams per capita per day in small towns, 300-400 grams per capita per day in medium size cities and about 500 grams per capita per day in large cities. As per the available trend, the amount of waste generated per capita is estimated to increase at a rate of 1%− 1.33% annually in India. The above projections clearly reflect how the problems of solid waste and its management are complex and critical with respect to both small towns as well as large cities. The solid waste management deals with a series of actions to reduce wastes, re- cover resources from these wastes, and/or dispose the remaining wastes in an envi- ronmentally acceptable way which cannot be eliminated or recovered due to some technical or economic reasons. Thus there are three basic alternatives for handling and disposal of solid wastes: (i) direct disposal of unprocessed waste in a landfill, (ii) processing of waste followed by land disposal, and (iii) processing of waste to recover resources with subsequent disposal of the residues. Though the first choice is usually the cheapest one, landfill space is becoming harder to find, causing costs to increase sharply in populated areas. The second al- ternative of processing of wastes prior to land disposal reduces the volume of wastes. It has definite advantages of reducing ultimate disposal cost which is generally a function of volume of wastes. The third alternative is fast becoming the favorite of environmentalists and solid waste management experts. It consists of the processes that recover energy or materials from solid waste and leave only a residue for ultimate land disposal. While resource recovery techniques may be costlier than other disposal alternatives, they achieve the goal of resource conservation and are environmentally more acceptable [3]. It is important to note here that whatever alternatives are to be adapted for waste disposal, ultimately a fraction of generated wastes has to be disposed off into a landfill which is generally low-lying areas situated on the outskirts of a town or a city. This practice of solid waste management has been continued worldwide to a significant extent due to its economic and technical feasibility. The existing solid waste man- agement system in many Indian cities appears to be highly inefficient because only primary and secondary collection, transportation and open dumping are practiced, that too in a very non-technical manner [4]. The improper disposal of these wastes through landfilling causes several environmental problems especially in highly pop- ulated areas. Thus the selection and use of landfills has to be optimized by keeping in view of the best of economic, environmental and public health practices. It should be ensured that during siting, design, construction and operation, closure and post closure, a landfill complies with all national, provincial/state, and local government rules, regulations, and permits so that it should not become a threat to the society [5, Fuzzy Inf. Eng. (2012) 3: 313-338 315 6]. Although for four decades, this process has been widely accepted, there have been a number of uncertainties that are not well treated or considered in this traditional way of selection process of landfilling, which may result in misleading outputs. Re- search is still focused on the seeking of a more integrated approach to incorporate all important criteria. One of the critical problems involved in integrating various assessment criteria is regarding uncertainty that arises from different sources, such as error in measurement and/or modeling, imprecision in knowledge of relationships between stressors and receptors, and even ambiguity in the meaning of risk, which are not considered in traditional evaluation of suitability of a landfill. To deal with the indicators’ uncertainties arising during the evaluation process, fuzzy set theory [7-11] appears to be a good complimentary approach. Once the in- dicators are represented by fuzzy sets, there are several fuzzy techniques that can be used to facilitate formulation and calculations of uncertainties associated with these fuzzy indicators. Some of them are fuzzy arithmetic [12], fuzzy linear pro- gramming [13, 14], fuzzy rule-based modeling [15] or fuzzy ranking [16, 17]. Sev- eral studies in that direction have been seen in the literature recently. For example, Singh [18] has clearly demonstrated the application of fuzzy set theory for assess- ing potential for water resource development and its impact in Chittorgarh district of Rajasthan. A decision support system has also been developed to evaluate op- timal landfill sites by Singh and Vidyarthi [19] which reflects dynamic, interactive, and uncertain characteristics of the solid waste management system effectively and provides decision makers a decision tool to choose a municipal solid waste manage- ment strategy. Wenger and Rong [20] used fuzzy set models to compare alternative solutions to environmental problems in comprehensive environmental decision mak- ing process. Smith [21] also presented a method for recognizing uncertainty in the evaluation of discrete transportation options characterized along multiple dimensions based on fuzzy and linguistic variables in the case when the objectives are classified as “qualitative imprecise”. Srivastava and Nema [22] developed a fuzzy parametric programming model for integrated solid waste management under uncertainty to un- derstand the waste allocation under different levels of uncertainty which essentially address the uncertainty in waste generation quantities and the capacities of the waste- management facilities. It is clear from the above literature review that many research studies have been conducted to deal with the representation, analysis and evaluation of suitable landfill site. Moreover, some of these studies have faced serious problems of integrating information from many different sources into an overall evaluation and interpretation. However, the fuzzy approach in ranking of landfills is still having a lot of potential to apply wherein fuzzy uncertainty, multidimensional and spatial characteristics of site can be dealt with a simple and straightforward way. Xu [23, 24] devised important operational laws of linguistic variables and developed a few aggregation operators such as linguistic geometric averaging (LGA) operator, lin- guistic weighted geometric averaging (LWGA) operator, linguistic ordered weighted geometric averaging (LOWGA) operator and linguistic hybrid geometric averaging (LHGA) operator. These operators have been used to aggregate linguistic preference information of decision makers in a systematic and straightforward manner without 316 Ajit Pratap Singh· Subodh Kant Dubey (2012) any loss of information [24, 25]. In this study, authors have developed a methodology to evaluate the feasibility of optimal landfill disposal site for solid waste disposal. The different criteria involved in this process are accessibility and transportation; environmental, geological and cli- matic conditions; socio-economic conditions; land use pattern; and safety of selected site. Finally, the optimum landfill disposal site is evaluated using a modified fuzzy utility approach wherein the uncertainty associated with specifying various attributes are incorporated. The study has also incorporated the opinion of experts at the selec- tion process so that real analysis can be performed. 2. Materials and Methods 2.1. Landfill Site Selection Solid waste disposal in landfills is the most widely used method for disposing of waste and about 80% of the wastes go to landfills. It owes its wide acceptance due to ease of maintenance and management. However, the method followed in many Indian case studies is not even equipped with the modern practices of landfilling. The collected wastes are mainly dumped in low-lying areas which are prone to flood which lead to cause both surface and ground water contamination. In addition to this, birds foraging on garbage dumps pose hazard to aircrafts operating in the areas. The selection of sites for landfills should not only be consistent with local land use conditions and zoning codes but should also be able to protect both ecologically sensitive areas (e.g. flood plains, wetlands) and culturally sensitive areas (e.g. archeological, historical) with the minimum impacts on air and/or water quality, and not to otherwise adversely impact upon public health, safety, welfare, community image as well as aesthetic and political issues. Quite often, problems with respect to siting a new landfill are more political than technical in nature. The general perception of public is that landfills are dumping sites and they do not consider the fact that a landfill is designed, built, and operated according to the latest engineering principles. Many people are reluctant to allow construction of a new landfill in their communities. Moreover, a landfill affects the surrounding environment for a long time. The construction and operation of landfills is often more expensive compared to other types of solid waste treatment. Thus the selection process of waste disposal sites should not only deal with technical aspects but also social and political issues [26-29]. The selection of appropriate landfill site is one of the key elements of municipal solid waste management system [30]. Due to rapid rise in environmental awareness among the public and reduction of availability of urban land, the problem of select- ing appropriate waste disposal sites is becoming challenging and complex. It needs to consider several independent factors concerning land use, socio-economy and hy- drogeology. The U.S. Environmental Protection Agency has prescribed a number of criteria to be considered for selection of a suitable waste disposal site. The parame- ters which are of immense importance include the site’s impact on ground-water and air quality, waste material transport feasibility, effect on property values and com- pensation plans, equity in the choice of sites, impact on community image as well as aesthetic and political issues [2]. However, no single landfill site can satisfy all the parameters due to their complex interrelationships and conflicting nature and hence, Fuzzy Inf. Eng. (2012) 3: 313-338 317 tradeoff between them is very clear [5, 31, 32]. The analysis and the evaluation of appropriate landfill sites are critical for several reasons. For example, priorities and preferences to select a landfill always requires the synthesis of two distinct selection process, namely, (i) a technical screening process which is mainly based on economic, engineering and environmental suitability and (ii) public acceptability and approval process along with political will power. More- over, many studies have been reported for allocating waste disposal sites which essen- tially assist decision makers to decide between alternatives when conflicted criteria are taken into account simultaneously. Some of these include linear programming techniques to optimize the location of a site with respect to operation and mainte- nance costs [33, 34]. Zeiss and Lefsrud [35] developed an analytical framework for waste-facility siting to structure the main elements and connections into a framework to explain stakeholder attitudes and siting outcome. The techniques like geographi- cal information systems (GIS) and analytic hierarchy process (AHP) have also been applied in landfill site selection by several researchers [36-40]. For example, Chang et al [36] presented a fuzzy multi-criteria decision analysis alongside with a geospa- tial analysis for the selection of landfill sites which basically employs a two-stage analysis synergistically to form a spatial decision support system (SDSS) for waste management in a fast-growing urban region of south Texas. Nas et al [41] have also presented a study to evaluate the suitability of a landfill in Cumra County of Konya city by integrating multi-criteria evaluation method with ArcGIS 9.0 as a practical in- strument. Ramu and Kennedy [42] presented a heuristic technique which maximizes service to an existing population by locating the new waste facilities based on dis- tance, cost, and environmental, social and political issues which can also be applied to other facility location problems for quick and feasible solution. Some field studies have also been conducted wherein resistivity imaging and ground penetrating radar (GPR) tools were applied to assess pollution level in the vicinity of a landfill [43]. However, the fuzzy approach in ranking of suitable landfill sites is still having a lot of potential to apply wherein all important criteria such as environmental and hydrological conditions, accessibility, ecological and societal effects etc. can be in- corporated in a more effective manner. A systematic approach for selecting a suitable landfill site using the concepts of modified fuzzy utility approach is proposed in this paper to incorporate uncertainty associated in specifying various attributes which are often imprecisely defined by the decision makers. The identification of landfill sites and their ranking are based on objective evaluation of accessibility, receptor, environ- mental, socio-economic, waste management practice, climatological and geological related attributes. A general listing of all important factors considered for landfill sit- ing is presented in Table 1. The above mentioned factors are generally technical, environmental, economical and socio-political in nature which can be usually categorized as vague and impre- cise. Therefore, these factors can be considered as linguistic variables which can be handled using fuzzy concepts [9, 17, 18, 44, 45] and the site selection process of landfill can be performed accordingly. 2.2. Methodological Framework for Site Selection 318 Ajit Pratap Singh· Subodh Kant Dubey (2012) Table 1: List of factors and their attributes used in selection process of a landfill site. Factors to be considered Important attributes Factors to be considered Important attributes for landfill selection associated with corr- for landfill selection associated with process esponding factors process corresponding factors Accessibility to the site • Distance from the road Geological factors • Soil permeability • Distance from the origin • Depth to bedrock of waste • Seismicity Receptor related factors • Proximity of human hab- Environmental factors • Hydro-geological inve- itation /locality stigation • Drinking water sources • Distance to nearest sur- • Land use designation face water • Agriculture value • Air quality • Public utility facility • Soil quality • Historical / Archeological •Water quality monuments • Safety • Public accessibility Socio-economic factors • Job opportunity Waste management pra- • Waste quantity /day • Health ctices related factors • Life of site Since a decision maker is required to allocate best landfill site for efficient and eco- nomic disposal of solid waste, it is necessary to develop a methodological framework which can integrate all important aspects of siting. The top down heuristic approach for evaluating different alternative sites consists of various steps such as constraint mapping, potential site selection, preliminary survey, site investigation on preferred sites, ranking of landfill sites and final selection of the site which in turn depend upon several indicator parameters as shown in Fig.1. The hierarchical structure developed herein aggregates the effects of each indicator parameter. The process of aggregation continues until the final site is selected based on six important steps as shown in Fig.1. The first and foremost important step deals with the identification of various con- straints which essentially eliminates environmentally unsuitable sites and narrows down the number of sites for further consideration. All the constraints should be recorded while prioritizing different alternative sites. For example, one of the most difficult part in the processes of site selection at the initial stage is to get appropriate land for landfilling. Thus the dearth of appropriate land can be a constraint. Several of such negative aspects may conveniently be listed which can further lead to reveal areas in which landfill sites might be located. The important factors which can be con- sidered to eliminate unsuitable sites from further analysis are transportation of wastes to minimize cost and time, natural conditions and land use pattern, safety of selected site and so on. The next step of selection of landfill sites deals with the identification of maximum number of potential sites which mainly depends on land details (area required for site, land ownership and its current use) and infrastructural facilities (ac- cess to roads, sites of existing/former waste disposal facilities and land designated for industrial use). These provide the basis for highlighting promising sites within the alternative sites remaining after first step of analysis. The preliminary survey is another important process where possible selected sites are further examined to elim- Fuzzy Inf. Eng. (2012) 3: 313-338 319 inate some of the sites which basically fail to meet additional socio-economic and environmental concerns at the site and surrounding areas. The detailed investigations on various factors (e.g. geology, hydrogeology, climatology, land etc.) of each site are then performed which are critical to the success of the siting and design of the landfill. Finally, all short listed sites are ranked based on detailed analysis of their en- vironmental, social and community impacts and the best site is selected with highest total score. Fig. 1 Processes involved in the selection of suitable landfill site 2.3. Development of Modified Fuzzy Utility Approach The proposed methodology develops a multi-criteria approach to integrate various criteria with respect to each alternative site of landfill which are available to a decision 320 Ajit Pratap Singh· Subodh Kant Dubey (2012) maker. It deals with fuzzy utility theory and multi-nomial logit model to incorporate imprecision of the site specific attributes. The fuzzy utility theory is mainly applied to calculate utility value associated with each alternative site for selection of opti- mum site whereas the probabilities of alternatives being selected are calculated using multi-nomial logit model. The importance of weights and ratings of various criteria are considered as linguistic variables. These linguistic variable can be expressed as triangular fuzzy numbers. The step by step procedure of the proposed methodology is explained in subsequent paragraphs. The initial step in the development of the model is to identify the possible alter- native sites along with their attributes/criteria which are responsible for performance evaluation of each alternative. th Let A (i = 1, 2,··· , n) represent the i alternative and C ( j = 1, 2,··· , m) repre- i j th sent the j criteria/attributes with which each alternative performances can be evalu- ated. Therefore, these alternatives and criteria can be listed mathematically as given below: A = {A , A ,··· , A }, (1) i 1 2 n C = {C , C ,··· , C }. (2) j 1 2 m Once the attributes (C ) and alternatives A are identified, the next step is the gen- j i eration of importance weight of each criterion associated for every alternative site. Weights are assigned to assess the importance of each criterion as linguistic variables by the different experts. The importance weight of each attribute/criteria (C ) is ob- th tained by deriving a membership function of the perception of k decision maker (DM ). If w  ( j = 1, 2,··· , m) is the fuzzy importance weight of each attribute C ,it k j j can be expressed mathematically by the following matrix form: w  = w  , w  , w  ,··· , w  . (3) j 1 2 3 m The fuzzy weight of each attribute C can be represented in terms of triangular fuzzy number [(x ,μ (x )), (x ,μ (x )), (x ,μ (x ))] as shown in Fig.2, whereμ (x) I l I II 2 II III 3 III i is membership value of any decision variable x to represent linguistic view of the decision maker mathematically. ሺ࢞ሻ ࢞ ࢞ ࢞ ࢞ ࡵ  ࡵࡵ ࡵࡵࡵ Fig. 2 Triangular fuzzy number with its triplets and membership values ࣆ Fuzzy Inf. Eng. (2012) 3: 313-338 321 The membership functions are derived in the form of triplets of triangular fuzzy numbers, i.e., [μ (x ),μ (x ),μ (x )] for any given values [x , x , x ] of the deci- l I 2 II 3 III I II III sion variables. Thus the fuzzy weights of each criterion can be expressed as w  = w  , w  , w  . The membership function for triangular fuzzy numbers can be cal- jI jII jIII culated by the following formulation: ⎪ 0, for x < x x− x ⎪ I , for x ≤ x ≤ x , ⎪ I II x − x II I μ (x) = (4) i ⎪ ⎪ x − x III ⎪ , for x ≤ x ≤ x , II III ⎪ x − x III II 0, for x > x . III The third step is to evaluate final weights for each criterion. The fuzzy weight of any th j criteria can be calculated as 1 2 3 k w + w + w +···+ w j j j j w  = , (5) k th th where k is the number of experts and w is the weight assigned by k expert for j criteria. The fourth step is to convert final triangular fuzzy weights into a crisp value by using Yager’s unit interval method [46] as given in Equation (6): Max F(A) = M(α)dα, (6) L + U α α where M(α) = and L and U are the lower alpha cut and the upper alpha α α cut respectively. The fifth step is to define the ideal or reference alternative site (S ) using the crisp weights calculated in pervious step, which can be expressed as S = {C , j(x)}, where j = 1, 2,··· , m and j(x) = membership value of corresponding attributes for ideal case. The sixth step deals with the formulation of linguistic variables for assigning rat- ings of each alternative with respect to all available criteria. A pairwise comparison matrix is constructed by selecting available alternatives for a landfill site under differ- ent criteria. The linguistic terms of the pairwise comparisons are assigned by asking importance degree of each criteria using fuzzy approach. The pairwise comparison th th matrix for evaluating i alternative with respect to j criteria can be expressed in the 322 Ajit Pratap Singh· Subodh Kant Dubey (2012) following matrix form: ⎡ ⎤ ⎢ ⎥ r  r  r ···  r ⎢ 11 12 13 1m⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ r  r  r ···  r ⎢ 21 22 23 2m⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ r  r  r ···  r ⎢ ⎥ 31 32 33 3m R = ⎢ ⎥ , (7) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . . . . ⎥ ⎢ ⎥ ⎢ . . . . . ⎥ ⎢ . ⎥ . . . . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ r  r  r ···  r n1 n2 n3 nm where  r is the fuzzy rating of alternative A (i = 1, 2,··· , n) with respect to cri- ij i terion C and is derived in the form of triplets of triangular fuzzy numbers, i.e., th a = (a , a , a ). The membership value of rating of i alternative with respect ij I II III th to j criteria ( r ) is also derived to represent linguistic view of the decision maker ij mathematically. The ratings ( r ) for each alternative is also expressed as triangular ij fuzzy numbers and the final rating of each alternative is calculated as given below: 1 2 3 k r + r + r +···+ r ij ij ij ij r = , (8) ij k th th where r is the rating for i alternative site with respect to j criteria as assigned by ij th k expert. The seventh step is the formulation of utility matrix, elements of which is repre- sented in the form of μ(x), r , where μ(x) is the membership value and  r is the ij ij th fuzzy rating of any alternative A with respect to the j criteria. In some cases, more than one membership value will be available for  r . In such cases, the maximum ij value should be assigned as suggested by Wang and Mendel [47]. The eighth step is to calculate fuzzy utility value (U ) of each alternative (A ) with respect to ideal alternative (S ) based on the decision maker’s perception, which can be calculated by using Equation (9): U = μ (x)⊗ μ(x), r = μ , r , (9) j ij m ij whereμ = μ (x)∩μ(x). m j After evaluating associated utility of each site as mentioned above, it is necessary to determine a set that can maximize the utility of each alternative site. It can be obtained by deriving the power set of U with respect to each alternative. The max- imizing set can be expressed by deriving its membership value with respect to each element of normal utility set as given by Equation (10): U = μ, r , (10) ij where μ = ,  r is the corresponding rating value of that element,  r is the Max Max supremum of union of normal utility set (i.e.,  r = sup∪ r ) and n is the degree of Max ij confidence that decision makers have on their judgment. If decision makers are more confident about their judgment, value of n can be varied from 2 to 4. In case they are not confident, it will vary backward from 1 to 0.1. In this analysis, the value of n has been taken as 1 to represent the normal situation. Fuzzy Inf. Eng. (2012) 3: 313-338 323 The next step is to form an optimum set (U ) for each alternative which can be N M evaluated by the intersection of U and U obtained in previous two steps, and can i i be expressed as O N M U = U ∩ U . (11) i i i Once optimum set is derived for each alternative (A ), the utility value or degree of satisfaction (DoS) of each alternative can be computed by deriving supreme member- ship value for each of them from their optimum set which is expressed by Equation (11), u = DoS (A ) = sup U  r ,  r ∈ U . (12) i n ij ij Subsequently, alternatives can be ranked on the basis of utility values. Addition- ally, the probability of the alternatives being selected by different decision makers can also be obtained by the multi-nomial logit model after evaluating degree of satis- faction which can finally be expressed as P(A ) =  , (13) where U is the utility of alternatives which can be directly expressed in terms of the degree of satisfaction. The series of steps adopted in present methodology to select an optimal landfill site among the given alternative sites by integrating the concept of fuzzy-utility method and multi-nomial logit model are shown in Fig.3. 2.4. Study Area The proposed methodology has been applied for evaluation of the best landfill site among the possible landfill sites to dispose the solid wastes of Vidya Vihar, Pilani, Rajasthan, India. As a policy maker, Nagarpalika of Vidya Vihar Pilani formed a committee of three experts of solid waste management namely DM , DM and DM 1 2 3 to take their opinions. Keeping with the site selection procedure described earlier under ‘development of modified fuzzy utility approach’, three probable landfill sites A , A , and A were shortlisted for further evaluation. A set of 13 criteria was consid- 1 2 3 ered to select the best landfill site. The criteria were decided on the basis of literature review pertaining to site selection, experts from waste management and it was also decided taking in view the factors contributing to the pollution pathways. The criteria considered in this application are depth to groundwater (C ), life of site (C ), soil 1 2 permeability (C ), distance to nearest drinking water (C ), population within 500 me- 3 4 ters (C ), distance from collection point (C ), air quality (C ), health (C ), land use 5 6 7 8 zoning (C ), type of road (C ), public acceptability (C ), odor (C ), public utility 9 10 11 12 facility within 2 km (C ) which are also described in Table 4 with their qualitative weights assigned by each of the three experts. 3. Results and Discussions 3.1. Selection of Optimum Landfill Site Using Modified Fuzzy Utility Approach 324 Ajit Pratap Singh· Subodh Kant Dubey (2012) $$# $$# $$ $$   # $$ $$   # # $$'$ $$ $$ $$'$ Fig. 3 Important steps of fuzzy based site selection process of a suitable landfill Fuzzy Inf. Eng. (2012) 3: 313-338 325 The formulation developed herein has been applied in a case study for screening potential landfill sites for disposing off municipal solid waste of Pilani town which is one of the educational hubs located in Jhunjhunu district of Rajasthan state in India. All important data with respect to each site have been taken from the earlier work of the first author and they are referred elsewhere [19]. The step by step procedure adopted for the present case study is described as below: 1) Important selection criteria, its attributes and all possible alternatives for landfill site were identified for the given problem. For example, a total of 13 criteria are identified with 3 alternative sites in this case. 2) The linguistic variables for assigning importance weights are classified as pre- sented in Table 2. These variables are then used to assess the importance of each criteria by the experts DM , DM and DM and depicted in Table 3. 1 2 3 Table 2: Important weight of each criterion. Linguistic description Weights with triangular elements Very low (VL) (0.0, 0.1, 0.3) Low (L) (0.1, 0.3, 0.5) Medium (M) (0.3, 0.5, 0.7) High (H) (0.5, 0.7, 0.9) Very high (VH) (0.7, 0.9, 1.0) Table 3: Ratings of each criterion. Linguistic description Ratings with triangular elements Very poor (VP) (0.0, 1.0, 3.0) Poor (P) (1.0, 3.0, 5.0) Fair (F) (3.0, 5.0, 7.0) Good (G) (5.0, 7.0, 9.0) Very good (VG) (7.0, 9.0, 10.0) 3) The fuzzy weight of each criterion has been calculated using Tables 2 and 4 and Equation (5) which is given in Table 5. For example, in Table 4, for the criterion related to soil permeability (C ), there have been different opinions of three experts, i.e., VH, H and VH which correspond to triangular membership value of importance weight as (0.7, 0.9, 1.0), (0.5, 0.7, 0.9), and (0.7, 0.9, 1.0), respectively. Therefore the th fuzzy weight of any j criteria (say related to soil permeability) can be calculated as 0.7+ 0.5+ 0.7 0.9+ 0.7+ 0.9 1.0+ 0.9+ 1.0 w  = , , , i.e., w  = (0.63, 0.83, 0.96). 3 3 3 3 3 Similarly, other values can be calculated and filled in Table 5. 4) According to Table 5 which was initially derived by Singh and Vidyarthi (2008), 326 Ajit Pratap Singh· Subodh Kant Dubey (2012) Table 4: Importance weights of each attributes. S.No. Attributes DM DM DM 1 2 3 1. Depth to groundwater (C)VH H H 2. Life of site (C ) H VH H 3. Soil permeability (C)VH H VH 4. Distance to nearest drinking water (C)VH VH VH 5. Population within 500 meters (C ) H VH M 6. Distance from collection point (C ) H VH H 7. Air quality (C ) H VH VH 8. Health (C)VHHH 9. Land use zoning (C)M M M 10. Type of road (C)M M H 11. Public acceptability (C)M M M 12. Odor (C)VHHH 13. Public utility facility within 2 km (C ) H VH H the fuzzy weights expressed by triplets of triangular fuzzy numbers have been con- verted into a crisp value using Equation (6) of Yager’s unit interval method. These crisp values are also given in Table 5. 5) The ideal or reference alternative site (S ) has now been defined using the crisp weights associated with each criterion as calculated in pervious step according to the decision maker’s perception. Therefore, the ideal site (S ) can be expressed as S = (0.75, C ), (0.75, C ), (0.82, C ), (0.88, C ), (0.69, C ), 1 2 3 4 5 (0.75, C ), (0.82, C ), (0.75, C ), (0.5, C ), (0.56, C ), (14) 6 7 8 9 10 (0.5, C ), (0.75, C ), (0.75, C ) . 11 12 13 6) The linguistic variables for assigning ratings have also been classified as given in Table 3. These variables are used to assess the ratings of each criterion by the experts DM , DM and DM for all three possible alternative sites and are presented in Table 1 2 3 7) The fuzzy utility matrix has been derived using the ratings of each criterion ob- tained from Step 6 and Table 6 with respect to each of the possible alternatives so that the chosen site can be compared with reference to the ideal one. Each element of this matrix is expressed by μ, r , whereμ is the membership function and r is the fuzzy ij ij th rating of any alternative A with respect to any j criteria assigned by the experts and is calculated using Equation (8). For example, in Table 6, for fifth criteria related to population within 500 m (C ), there are different opinions of three experts, i.e., Poor, Fair and Poor for alternative 1 (i.e., for first site) which correspond to triangular membership value rating as (1.0, 3.0, 5.0), (3.0, 5.0, 7.0) and (1.0, 3.0, 5.0) respec- Fuzzy Inf. Eng. (2012) 3: 313-338 327 Table 5: Fuzzy weights of the criteria. S.No. Attributes Fuzzy weights Crisp value using Normalized Yager’s unit weight (w) interval method 1. Depth to groundwater (C)(0.56, 0.76, 0.93 0.75 0.08090615 2. Life of site (C)(0.56, 0.76, 0.93) 0.75 0.08090615 3. Soil permeability (C)(0.63, 0.83, 0.96) 0.82 0.08845739 4. Distance to nearest drinking water (C )(0.70, 0.90, 1.0) 0.88 0.09492988 5. Population within 500 meters (C)(0.50, 0.70, 0.86) 0.69 0.07443366 6. Distance from collection point (C)(0.56, 0.76, 0.93) 0.75 0.08090615 7. Air quality (C)(0.63, 0.83, 0.96) 0.82 0.08845739 8. Health (C)(0.56, 0.76, 0.93) 0.75 0.08090615 9. Land use zoning (C)(0.30, 0.50, 0.70) 0.50 0.05393743 10. Type of road (C)(0.36, 0.56, 0.76) 0.56 0.06040992 11. Public acceptability (C)(0.30, 0.50, 0.70) 0.50 0.05393743 12. Odor (C)(0.56, 0.76, 0.93) 0.75 0.08090615 13. Public utility facility within 2 km(C )(0.56, 0.76, 0.93) 0.75 0.08090615 th tively. Therefore the fuzzy rating of any j criteria (say related to population within 1.0+ 3.0+ 1.0 3.0+ 5.0+ 3.0 5.0+ 7.0+ 5.0 500 m) can be calculated as  r = , , , 3 3 3 i.e.,  r = (1.6, 3.6, 5.6). Thus the elements of the fuzzy utility matrix can be derived by assigning their re- spective membership functions with their ratings. For example, elements with respect th to any j criteria μ, r (say related to population within 500 m) can be expressed as ij 1.0+ 3.0+ 1.0 3.0+ 5.0+ 3.0 5.0+ 7.0+ 5.0 0.7, , 0.7, , 0.7, , 3 3 3 i.e., μ, r = [(0.7, 1.6), (0.7, 3.6), (0.7, 5.6)]. It is important to note that the fuzzy ij th rating of any alternative A with respect to j criteria ( r ) may attain two member- i ij ship values in some cases. In such cases, the highest membership values should be assigned as per Wang and Mendal (1992). For example,  r = 4.3 have two mem- ij bership values 0.65 and 0.35 in linguistic description of fair and poor respectively and therefore, the corresponding membership value with respect to the rating of 4.3 would be 0.65. However, if all the decisions makers have assigned same linguistic rating for any particular attribute, then the membership value would be same as the assigned value given in Table 3. For example, alternative site 1 with respect to criteria 6 (distance from collection point: C ) has been assigned ‘fair’ rating by all the three decision makers. The membership value for this alternative site with respect to C can therefore be represented as [(0, 3), (1, 5), (0, 7)]. There is no need to assign mem- bership value of 1 for 3 and 1 for 7 as every decision maker has given same rating. 328 Ajit Pratap Singh· Subodh Kant Dubey (2012) Table 6: Opinion of three decision makers for ratings of each alternative site. S. No. Attributes Site 1 Site 2 Site 3 DM DM DM DM DM DM DM DM DM 1 2 3 1 2 3 1 2 3 1. Depth to VG VG VG VG VG VG VG VG VG groundwater (C ) 2. Life of GGG F GGGGG site (C ) 3. Soil F F FG GFGFG permeability (C ) 4. Distance to nearest GGGGGG F F F drinking water (C ) 5. Population within PFP VG G VG PP VP 500 meters (C ) 6. Distance from FFFFPFP VP P collection point (C ) 7. Air quality (C) P P P GGGGGG 8. Health (C)P F P VG G G P VP P 9. Land use PFPFF G G F G zoning (C ) 10. Type of road (C) G G G FFFPP VP 11. Public FVP VPF F P P PVP acceptability (C ) 12. Odor (C ) F PVP G G G VP PVP 13. Public utility facility PPPPPP G FF within 2 km (C ) Similarly other entries of this matrix can be obtained as shown in Table 7. th 8) Using Equations (9) and (14) and Table 7, normal utility value of any i alternative with reference to ideal site (U ) are evaluated based on the decision maker’s percep- tion. It is important to note that the lowest membership value should be assigned to each rating value. For example, criteria C is expressed as (0.75, C ) in Equation (14). 1 1 Thus, (0.75, C ) = (0.75, [(0, 7), (1, 9), (0, 10)]) = [(0, 7), (0.75, 9), (0, 10)]. Similarly, other entries can be obtained. Therefore Fuzzy Inf. Eng. (2012) 3: 313-338 329 Table 7: Fuzzy utility for every criteria/attributes of each site/alternative. Attributes Site 1 (A ) Site 2 (A ) Site 3 (A ) 1 2 3 Depth to groundwa- [(0,7), (1,9), (0,10)] [(0,7), (1,9), (0,10)] [(0,7), (1,9), (0,10)] ter (C ) Life of site (C ) [(0,5), (1,7), (0,9)] [(0.65,4.3), (0.65,6.3), (0.65,8.3)] [(0,5), (1,7), (0,9)] Soil permeability [(0,3), (1,5), (0,7)] [(0.65,4.3), (0.65,6.3), (0.65,8.3)] [(0.65,4.3), (0.65,6.3), (0.65,8.3)] (C ) Distance to nearest [(0,5), (1,7), (0,9)] [(0,5), (1,7), (0,9)] [(0,3), (1,5), (0,7)] drinking water (C ) Population within [(0.7,1.6), (0.7,3.6), (0.7,5.6)] [(0.65,6.3), (0.7,7.6), (0.4,9.6)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] 500 meters (C ) Distance from col- [(0,3), (1,5), (0,7)] [(0.65,2.3), (0.65,4.3), (0.65,6.3)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] lection point (C ) Air quality (C ) [(0,1), (1,3), (0,5)] [(0,5), (1,7), (0,9)] [(0,5), (1,7), (0,9)] Health (C ) [(0.7,1.6), (0.7,3.6), (0.7,5.6)] [(0.7,5.6), (0.7,7.6), (0.7,9.3)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] Land use zoning (C ) [(0.7,1.6), (0.7,3.6), (0.7,5.6)] [(0.65,4.3), (0.7,5.6), (0.7, 7.6)] [(0.65,4.3), (0.65,6.3), (0.65,8.3)] Type of road (C ) [(0,5), (1,7), (0,9)] [(0,3), (1,5), (0,7)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] Public acceptability [(1,1), (0.65,2.3), (0.65,4.3)] [(0.65,2.3), (0.65,4.3), (0.65,6.3)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] (C ) Odor (C ) [(0.85,1.3), (1,3), (1,5)] [(0,5), (1,7), (0,9)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] Public utility facility [(0,1), (1,3), (0,5)] [(0,1), (1,3), (0,5)] [(0.7,3.6), (0.7,5.6), (0.7,9.6)] within 2 km (C ) U = [{0.75, [(0, 7), (1, 9), (0, 10)]}, 0.75, [(0, 5), (1, 7), (0, 9)], {0.82, [(0, 3), (1, 5), (0, 7)]}, {0.88, [(0, 5), (1, 7), (0, 9)]},{0.69, [(0.7, 1.6), (0.7, 3.6), (0.7, 5.6)]}, {0.75, [(0, 3), (1, 5), (0, 7)]},{0.82, [(0, 1), (1, 3), (0, 5)]}, {0.75, [(0.7, 1.6), (0.7, 3.6), (0.7, 5.6)]},{0.5, [(0.7, 1.6), (0.7, 3.6), (0.7, 5.6)]},{0.56, [(0, 5), (1, 7), (0, 9)]},{0.5, [(1, 1), (0.65, 2.3), (0.65, 4.3)]},{0.75, [(0.85, 1.3), (1, 3), (1, 5)]},{0.75, [(0, 1), (1, 3), (0, 5)]}]. Which is further simplified as U = {[(0, 7), (0.75, 9), (0, 10)], [(0, 5), (0.75, 7), (0, 9)], [(0, 3), (0.82, 5), (0, 7)], [(0, 5), (0.88, 7), (0, 9)], [(0.69, 1.6), (0.69, 3.6), (0.69, 5.6)], [(0, 3), (0.75, 5), (0, 7)], [(0, 1), (0.82, 3), (0, 5)], [(0.7, 1.6), (0.7, 3.6), (0.7, 5.6)], [(0.5, 1.6), (0.5, 3.6), (0.5, 5.6)], [(0, 5), (0.56, 7), (0, 9)], [(0.5, 1), (0.5, 2.3), (0.5, 4.3)], [(0.75, 1.3), (0.75, 3), (0.75, 5)], [(0, 1), (0.75, 3), (0, 5)]}. As observed in the above analysis, there are different membership values for given 330 Ajit Pratap Singh· Subodh Kant Dubey (2012) r , thus these membership values can be aggregated corresponding to the respective ij r , as shown below: ij U = [[(0+ 0.75+ 0+ 0+ 0.56), 7], [(0.75+ 0+ 0), 9], [0, 10], [(0+ 0.82+ 0+ 0.75+ 0+ 0+ 0.75+ 0), 5], [(0+ 0+ 0.75+ 0.75), 3], (15) [(0.69+ 0.7+ 0.5), 1.6], [(0.69+ 0.7+ 0.5), 3.6], [(0.69+ 0.7+ 0.5), 5.6], [(0+ 0.5+ 0), 1], [0.5, 2.3], [0.5, 4.3], [0.75, 1.3]]. The values of U obtained from the above analysis can further be simplified using algebraic sum method as mentioned below: μ +μ = μ +μ −μ ∗μ , (16) 1 2 1 2 1 2 U = [(0.89, 7), (0.75, 9), (0, 10), (0.99, 5), (0.94, 3), (0.95, 1.6), (0.95, 3.6), (0.95, 5.6), (0.5, 1), (0.5, 2.3), (0.5, 4.3), (0.75, 1.3)]. Similarly, U = (0.95, 7), (0.75, 9), (0, 10), (0.99, 4.3), (0.99, 6.3), (0.88, 8.3), (0.56, 5), (0.95, 7.6), (0.4, 9.6), (0.83, 2.3), (0.85, 5.6), (0.7, 9.3), (0.75, 3), (0, 1)]. Similarly, U = [(0.96, 7), (0.75, 9), (0, 10), (0.88, 5), (1, 4.3), (0.83, 6.3), (0.83, 8.3), (0, 3), (1, 0.6), (0.99, 2.3), (0.7, 3.6), (0.7, 5.6), (0.7, 9.6)]. However, other methods can also be used to get single crisp value which can be referred elsewhere [26]. 9) After evaluating associated utility of each site in Step 8, a set that can maximize the utility of each alternative site is determined using Equation (10). In this analysis, the degree of confidence that decision makers have on their judgment (n) is taken as 1 to represent the normal situation. If Y represents the union of normal utility set obtained in Step 8, then it can be expressed as Y = {7.0, 9.0, 5.0, 3.0, 1.6, 3.6, 5.6, 1.0, 2.3, 4.3, 1.3, 6.3, 8.3, 7.6, 9.6, 9.3, 0.6}. It may be observed here that the rating of 10 is not included in the above calculation as it has zero membership value in each normal utility set. Therefore the superimum of normal utility set,  r = 9.6. Thus the membership value of 7 in the new set will max be 7/9.6 = 0.73, and the maximizing sets can be expressed as U = [(0.73, 7.0), (0.94, 9.0), (0.52, 5.0), (0.31, 3.0), (0.17, 1.6), (0.38, 3.6), (0.58, 5.6), (0.1, 1.0), (0.24, 2.3), (0.45, 4.3), (0.14, 1.3)], U = [(0.73, 7.0), (0.94, 9.0), (0.45, 4.3), (0.66, 6.3), (0.87, 8.3), (0.52, 5.0), (0.79, 7.6), (1.0, 9.6), (0.24, 2.3), (0.58, 5.6), (0.97, 9.3), (0.31, 3.0), (0.1, 1.0)], U = [(0.73, 7.0), (0.94, 9.0), (0.52, 5.0), (0.45, 4.3), (0.66, 6.3), (0.87, 8.3), (0.31, 3.0), (0.1, 0.6), (0.24, 2.3), (0.38, 3.6), (0.58, 5.6), (1.0, 9.6)]. Fuzzy Inf. Eng. (2012) 3: 313-338 331 10) After evaluating maximizing set of each alternative site, it is desirable to form O N M optimum set (U ) which can be evaluated by the intersection of U and U obtained i i i in Steps 8 and 9, and can be expressed as U = [(0.73, 7.0), (0.75, 9.0), (0.52, 5.0), (0.31, 3.0), (0.17, 1.6), (0.38, 3.6), (0.58, 5.6), (0.1, 1.0), (0.24, 2.3), (0.45, 4.3), (0.14, 1.3)], U = [(0.73, 7.0), (0.75, 9.0), (0.45, 4.3), (0.66, 6.3), (0.87, 8.3), (0.52, 5.0), (0.79, 7.6), (0.4, 9.6), (0.24, 2.3), (0.58, 5.6), (0.7, 9.3), (0.31, 3.0), (0.1, 1.0)], U = [(0.73, 7.0), (0.75, 9.0), (0.52, 5.0), (0.45, 4.3), (0.66, 6.3), (0.83, 8.3), (0.0, 3.0), (0.1, 0.6), (0.24, 2.3), (0.38, 3.6), (0.58, 5.6), (0.7, 9.6)]. 11) Once optimum set is derived for each alternative, DoS or utility value of each alternative can be computed by deriving supreme membership value for each alterna- tive from their optimum set. DoS (A ) = SUP(0.73, 0.75, 0.52, 0.31, 0.17, 0.38, 0.58, 0.10, 0.24, 0.45, 0.14) = 0.75, DoS (A ) = SUP(0.73, 0.75, 0.45, 0.66, 0.87, 0.52, 0.79, 0.4, 0.24, 0.58, 0.7, 0.31, 0.0) = 0.87, DoS (A ) = SUP(0.73, 0.75, 0.52, 0.45, 0.66, 0.83, 0.0, 0.1, 0.24, 0.38, 0.58, 0.7) = 0.83. Therefore, A > A > A . 2 3 1 12) Additionally, the probability of the alternatives being selected by the different decision makers is also evaluated using Equation (13) of the multi-nomial logit model as shown below: 0.75 P(A )= = 31.00%, 0.75 0.87 0.83 (e + e + e ) 0.87 P(A )= = 35.00%, 0.75 0.87 0.83 (e + e + e ) 0.83 P(A )= = 33.68%. 0.75 0.87 0.83 (e + e + e ) 3.2. Linguistic Aggregation Operators Based Approach for Selection of Landfill Sites A direct method [23, 24] based on linguistic aggregation operators for group decision making with linguistic preference relations was also performed to compare the results evaluated by the proposed method. The data with respect to rating of each alternative site under the given criteria have been reorganized in appropriate format as per the requirement of the methodology suggested by Xu (2004). The data was analyzed by LGA and LWGA operators for all three landfill sites. The salient steps applicable to use these operators for identifying best alternative landfill site under 13 available criteria by aggregating opinion of all three experts are described below: 332 Ajit Pratap Singh· Subodh Kant Dubey (2012) • Based on the opinion of the DMs as given in Tables 3 and 6 earlier, the com- parative ratings of all three possible alternative sites under each criterion are evaluated. For example, DM has assigned fair, fair and poor ratings under criteria 6 (distance from collection point: C ) for alternative sites 1, 2 and 3 respectively. It clearly indicates that alternative sites 1 and 2 with respect to criterion C are equally good whereas these two sites are slightly good in com- parison with site 3. It should be noted that the ratings of a site under a given criteria can be assigned from VG to VP depending on the opinion of DMs. • Set S = {s} is defined by representing possible linguistic variable s (i = i i 1, 2··· , t) so that the elements of it are uniformly distributed on a prescribed scale in a totally ordered manner. The details of it can be referred elsewhere [23]. In this study, a total of nine linguistic variables have been taken to rep- resent set S = s (i = 1, 2,··· , 9), where S = s = extremely poor, s = very i 1 2 poor, s = poor, s = slightly poor, s = equally good (fair), s = slightly good, 3 4 5 6 s = good, s = very good, s = extremely good. 7 8 9 • The three alternative landfill sites should be compared with each other under each of the criteria using the linguistic terms of set S as suggested by Xu [23, 24]. The linguistic preference relation matrix (R) is derived for all the three alternative sites under each criterion based on the assessment of each decision maker. For example, all 3 alternative sites have been rated ‘VG’ by DM under criterion 1 (i.e., depth to groundwater: C as given in Table 6) and therefore they are equally good (s ) when compared to each other. Thus the linguistic preference relation matrix (R) dealing with alternative sites 1, 2 and 3 under criteria C as assessed by DM can be represented as follows: 1 1 • Similarly, all 13 possible linguistic preference relation matrices can be con- structed under each one of the thirteen criteria based on the opinion of each of the decision makers. Thus there will be 39 matrices in total for all 3 decision makers. th • Using LGA operator [23], the preference degree Z of i alternative site over th th all other alternative sites under j criteria as assessed by the k decision maker can be expressed as j j j j j j j n−1 Z = LGA r , r ,··· , r = r ⊗ r ⊗···⊗ r , k k k k k k k i i1 i2 in i1 i2 in where ⊗ is fuzzy multiplication operator. For example, the preference degree Z for alternative site 1 over other two alternative sites under criteria 1 (C )of 1 1 DM can be expressed as 1 1 1 1 1 1 1 3−1 0.5 Z = LGA r , r , r = r ⊗ r ⊗ r = (s ⊗ s ) , 1 1 1 1 1 1 1 5 5 1 11 12 13 11 12 13 0.5 0.5 = (s ) = s = s . 25 5 • Similarly, preference information Z of all decision makers (k = 1, 2, 3) for all three alternative sites (i = 1, 2, 3) under all 13 criteria ( j = 1, 2,··· , 13) can be derived as shown in Table 9. Fuzzy Inf. Eng. (2012) 3: 313-338 333 Table 8: Linguistic preference relation matrix for criteria 1. Site 1 Site 2 Site 3 Site 1 - s s 5 5 Site 2 s - s 5 5 Site 3 s s - 5 5 Table 9: Preferences for all alternative sites by the DM , DM and DM . 1 2 3 Criterion ( j) C C C C C C C C C C C C C 1 2 3 4 5 6 7 8 9 10 11 12 13 Z s s s s s s s s s s s s s 1 5 5.48 4 5.48 3.16 5.48 3 3.16 3.46 6.48 5.48 5.29 3.87 Z s s s s s s s s s s s s s 1 5 4 4.90 5.48 8 5.48 5.92 8 4.90 4.90 5.48 6.93 3.87 Z s s s s s s s s s s s s s 1 5 5.48 5.48 4 3.16 4 5.92 3.16 6.48 3.46 4 2.45 7 Z s s s s s s s s s s s s s 2 5 5 4.47 5.48 4.90 6.48 3 5.29 5 6.48 3.46 3.87 4.47 Z s s s s s s s s s s s s s 2 5 5 6 5.48 6.48 4.90 5.92 6.92 5 4.90 6.48 7 4.47 Z s s s s s s s s s s s s s 2 5 5 4.47 4 3.46 3.46 5.92 2.45 5 3.46 4.90 3.87 6 Z s s s s s s s s s s s s s 3 5 5 4.47 5.48 3.46 5.48 3 3.87 3 6.93 4.47 3.16 4.47 Z s s s s s s s s s s s s s 3 5 5 4.47 5.48 8.48 5.48 5.92 7 5.92 5.29 6 8 4.47 Z s s s s s s s s s s s s s 3 5 5 6 4 2 4 5.92 3.87 5.92 2.45 4.47 3.16 6 • Using LWGA operator [23], the overall preference degree ( Z )of ith alter- k i native landfill site can be evaluated by aggregating evaluation of each landfill site against all criteria made by each of the DM [23-24]. This is expressed as follows: Z = LWGA (S , S ,··· , S ) (17) k i w α1 α2 αn w w w 1 2 n = (S ) ⊗ (S ) ⊗···⊗ (S ) . α1 α2 αn • Using Tables 5 and 9, for example, the overall preference degree Z of first 1 1 alternative site over all other alternative sites as assessed by DM can be ex- pressed as w w w w w w 1 2 3 4 5 13 Z = (S ) ⊗ (S ) ⊗ (S ) ⊗ (S ) ⊗ (S ) ···⊗ (S ) 1 1 5 5.48 4 5.48 3.16 3.87 = S . 4.40 334 Ajit Pratap Singh· Subodh Kant Dubey (2012) Similarly, other values can be calculated as Z = S , Z = S , Z = S , Z = S , 1 2 5.48 1 3 4.39 2 1 4.73 2 2 5.61 Z = S , Z = S , Z = S , Z = S . 2 3 4.26 3 1 4.32 3 2 5.74 3 3 4.26 • Using LGA operator again, the final preference degree Z of ith alternative site over all other alternative sites can be expressed as 1/3 Z = LGA ( Z, Z, Z ) = ( Z ⊗ Z ⊗ Z ) . i 1 i 2 i 3 i 1 i 2 i 3 i Thus the final preference degree of site 1, 2 and 3 can be expressed as Z = S , Z = S , and Z = S respectively. Therefore alternative site A > 4.47 2 5.60 3 4.80 2 A > A . The ranking values of all 3 landfill sites exhibit that landfill Site 2 has 3 1 the highest potential in the site selection process. It clearly demonstrates that these results are coinciding with the modified fuzzy utility approach proposed in this paper. The results obtained by the proposed method also coincide with other studies [5, 19]. However, the modified fuzzy utility approach presented here integrates the con- cept of fuzzy-utility method with multi-nomial logit theory to evaluate the ranking order of all possible landfill sites. It also evaluates the probability of selection of each alternative site which prepares a basis to adopt an overall strategy for selecting appro- priate landfill site for proper solid waste disposal and its management. The modified fuzzy utility approach proposed herein is simple, robust and capable of incorporating uncertainty along with flexible policies of solid waste management without any loss of information. It can be observed that both the methods discussed here not only allow decision makers to determine the ranking order of all alternatives but can also indicate the de- gree of preference of each alternative. It may be desirable in many situations where one may be willing to use next optimal alternative due to certain practical constraints. Therefore, they are more suitable and effective in dealing with subjective judgments in fuzzy environment. Both of the methods can successfully select the best alternative landfill site when the states of the system and/or utilities associated with alternatives are known in terms of a linguistic variable without any loss of original information. The selection and usefulness of different methods may also depend on data availabil- ity at the landfill sites, which may be available in different formats for its analysis. Moreover, the number of alternative sites and criteria involved in the process will also be the deciding factors to choose the most appropriate method. 4. Conclusion In this paper, a modified fuzzy utility method is used to evaluate the choice of landfill site for dumping of solid waste under different alternatives with a number of criteria which takes into account parameters related to accessibility and transportation; en- vironmental, geological and climatic condition; socio-economic conditions; land use pattern; and safety at the selected site. An ideal site is framed based on the perception of decision maker. All the alternative sites are evaluated in terms of their utility value Fuzzy Inf. Eng. (2012) 3: 313-338 335 with reference to an ideal site. The optimum utility value of each site is calculated to choose the best site among the given sites. In this methodology, individual utility value is calculated with reference to ideal site rather than inter utility value. It gives a better view of judging the alternatives, since degree of fulfillment of each alternative is known whereas in previous paper [19] competing degree of fulfillment was known (that is how much superior one alternative is with reference to other). Multi-nomial Logit model is used to calculate the probability of acceptance of each alternative, which is helpful in decision making process. Alternative site (A ) has 35.0% prob- ability of acceptance (chance that 35.0% decision makers will agree that site (A ) ) has should be chosen as a first landfill dumping site), whereas alternative site (A 33.68%. A stepwise and objective method to evaluate probability of acceptance is of great advantage which determines the ranking order of each site. Ranking order is given for all the alternatives and the best one is chosen which depends upon several fac- tors as mentioned in the results. The study demonstrates that alternative landfill site 2 is the best among the available three sites for the disposal of solid wastes. How- ever, if alternative site 2 is not available due to some constraints, decision maker can adopt alternative site 3 which is of second highest order of ranking with the prob- ability of acceptance of 33.68%. The methodology applied in this paper is easy to use and understand and clearly screen out the landfill alternatives which are not ap- propriate under the given set of criteria. The methodology presented in this paper is comparatively easier and more comprehensive in comparison with the earlier works of Singh and Vidyarthi [19] who initially developed fuzzy preference relation matrix and computed non-dominated degree of each alternative site for assessing the suit- able landfill. At the end of the analysis, most suitable landfill site is identified for a case study which satisfies all important requirements of the landfill as mentioned initially in Fig.1. The case study clearly shows that the utility value of alternative site 2 (i.e., A ) is 0.87 which is nearest to the reference value of 1 and hence it is the most suitable landfill site among all three possible sites. Due to higher utility value, the chosen landfill site for solid waste disposal will have the least impact on surrounding environment and therefore will be the most acceptable. The results obtained by the proposed method also coincide with other studies [5, 19, 23, 24]. It can be observed that proposed method not only allows decision mak- ers to determine the ranking order of all alternatives but can also indicate the degree of preference of each alternative. It may be desirable in many situations where one may be willing to use next optimal alternative due to certain practical constraints. Therefore, they are more suitable and effective in dealing with subjective judgments in fuzzy environment. However, a number of extensions and applications of this model may be possi- ble by taking opinion of more number of decision makers and by collecting detailed data pertaining to the factors such as geology, hydrogeology, climatology, land use pattern and socio-economic characteristics towards the protection of overall environ- ment. A more practical implementation of the model can be achieved by integrating geographical information system and multi-criteria evaluation techniques. 336 Ajit Pratap Singh· Subodh Kant Dubey (2012) Acknowledgements Writers are deeply grateful to Birla Institute of Technology and Science Pilani for providing necessary facilities to carry out this research work. Authors are also thank- ful to Central Pollution Control Board (CPCB) of India for its useful publication on hazardous waste management series and all other references cited in the text. 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Hazardous Waste Management Series, Central Pollution Control Board, New Delhi 6. Hanrahan D, Srivastava S, Ramakrishna A S (2006) Improving management of municipal solid waste in India: overview and challenges. World Bank Report, South Asia Region, Environment and Social Development Unit 7. Zadeh L A (1965) Fuzzy sets. Information and Control 8: 338-353 8. Zadeh L A (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3-28 9. Klir G J, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall Inc. Englewood Cliffs, N.J. 1-94, 390-416 10. Sakawa M (1993) Fuzzy sets and interactive multi-objective optimization. Plenum Press, New York 11. Singh A P, Ghosh S K, Sharma P (2007) Water quality management of a stretch of river Yamuna: an interactive fuzzy multi-objective approach. Water Resources Management 21(2): 515-532. DOI 10.1007/s11269-006-9028-0 12. Kaufman A, Gupta M M (1991) Introduction to fuzzy arithmetic, theory and application. Van Nos- trand Reinhold, New York 13. Liang T F (2008) Fuzzy multi-objective production/distribution planning decisions with multi-product and multi-time period in a supply chain. Computers and Industrial Engineering 55(3): 676-694 14. Pramanik S, Roy T K (2008) Multi-objective transportation model with fuzzy parameters: priority based fuzzy goal programming approach. Journal of Transportation Systems Engineering and Infor- mation Technology 8(3): 40-48 15. Badossy A, Duckstein L (1995) Fuzzy rule-based modeling with applications to geophysical, biolog- ical and engineering systems. CRC Press, Boca Raton, Florida 16. Chen H, Chang N (2000) Prediction analysis of solid waste generation based on grey fuzzy dynamic modelling. Resources, Conservation and Recycling 29(1-2): 1-18 17. Chen C T (2001) A fuzzy approach to select the location of the distribution center. Fuzzy Sets and Systems 118: 65-73 18. 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Information Sciences 24(2): 143- 161 47. Wang L X, Mendel J M (1992) Generating fuzzy rules by learning from examples. IEEE Transactions on Systems, Man, and Cybernetics 22(6): 1414-1427 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Optimal Selection of a Landfill Disposal Site Using a Modified Fuzzy Utility Approach

Optimal Selection of a Landfill Disposal Site Using a Modified Fuzzy Utility Approach

Abstract

AbstractThe present paper develops an integrated fuzzy based model to select an optimal landfill site among the given alternative sites by using the concept of fuzzy-utility method and multi-nomial logit theory. The suitability of different landfill sites are evaluated based on some important criteria involved in the process such as accessibility and transportation; environmental, geological and climatic conditions; socioeconomic conditions; land use pattern; and safety at the selected site....
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Fuzzy Inf. Eng. (2012) 3: 313-338 DOI 10.1007/s12543-012-0118-9 ORIGINAL ARTICLE Optimal Selection of a Landfill Disposal Site Using a Modified Fuzzy Utility Approach Ajit Pratap Singh · Subodh Kant Dubey Received: 10 June 2011/ Revised: 16 May 2012/ Accepted: 5 August 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract The present paper develops an integrated fuzzy based model to select an optimal landfill site among the given alternative sites by using the concept of fuzzy- utility method and multi-nomial logit theory. The suitability of different landfill sites are evaluated based on some important criteria involved in the process such as acces- sibility and transportation; environmental, geological and climatic conditions; socio- economic conditions; land use pattern; and safety at the selected site. These criteria are assessed qualitatively by the decision makers based on their relative degree of importance. The importance weights and ratings of each criterion have been defined in the form of triplets of triangular fuzzy numbers by taking opinion of the decision makers. The corresponding triplets of ratings of each site are used to derive the utility value of the alternative sites. A multi-nomial logit model has been applied to calcu- late the probability of selection of each alternative site which can help policy makers to take appropriate decisions. Finally, the proposed methodology has been applied to allocate suitable landfill sites for disposing off municipal solid waste for Pilani town which is located in Jhunjhunu district of Rajasthan. The results evaluated by the modified fuzzy utility are also compared to the outputs of a direct method which is basically based on certain linguistic aggregation operators for group decision making. Computational results clearly demonstrate that the results obtained by the proposed method are coinciding very well and prepares a basis to adopt an overall strategy for selecting appropriate landfill site for proper solid waste disposal and its management. Keywords Decision support system· Uncertainty· Fuzzy utility value· Solid waste management · Landfill allocation 1. Introduction Ajit Pratap Singh ()· Subodh Kant Dubey Civil Engineering Department, BITS Pilani, Rajasthan 333031, India email: apsbits@gmail.com 314 Ajit Pratap Singh· Subodh Kant Dubey (2012) The solid waste management and its proper disposal have now become an impor- tant issue worldwide, be it the developed countries where the quantity and kind of hazardous wastes, and the lack of disposal sites have caused a greater concern or the developing nations where the growth in both size and concentration of the population, combined with a general lack of public awareness, have made the problem of solid waste a critical public issue [1, 2]. The rapid growth in population and economic de- velopment has resulted in a significant increase in municipal solid waste generation in India. The estimated solid waste generation ranges from 100 grams per capita per day in small towns, 300-400 grams per capita per day in medium size cities and about 500 grams per capita per day in large cities. As per the available trend, the amount of waste generated per capita is estimated to increase at a rate of 1%− 1.33% annually in India. The above projections clearly reflect how the problems of solid waste and its management are complex and critical with respect to both small towns as well as large cities. The solid waste management deals with a series of actions to reduce wastes, re- cover resources from these wastes, and/or dispose the remaining wastes in an envi- ronmentally acceptable way which cannot be eliminated or recovered due to some technical or economic reasons. Thus there are three basic alternatives for handling and disposal of solid wastes: (i) direct disposal of unprocessed waste in a landfill, (ii) processing of waste followed by land disposal, and (iii) processing of waste to recover resources with subsequent disposal of the residues. Though the first choice is usually the cheapest one, landfill space is becoming harder to find, causing costs to increase sharply in populated areas. The second al- ternative of processing of wastes prior to land disposal reduces the volume of wastes. It has definite advantages of reducing ultimate disposal cost which is generally a function of volume of wastes. The third alternative is fast becoming the favorite of environmentalists and solid waste management experts. It consists of the processes that recover energy or materials from solid waste and leave only a residue for ultimate land disposal. While resource recovery techniques may be costlier than other disposal alternatives, they achieve the goal of resource conservation and are environmentally more acceptable [3]. It is important to note here that whatever alternatives are to be adapted for waste disposal, ultimately a fraction of generated wastes has to be disposed off into a landfill which is generally low-lying areas situated on the outskirts of a town or a city. This practice of solid waste management has been continued worldwide to a significant extent due to its economic and technical feasibility. The existing solid waste man- agement system in many Indian cities appears to be highly inefficient because only primary and secondary collection, transportation and open dumping are practiced, that too in a very non-technical manner [4]. The improper disposal of these wastes through landfilling causes several environmental problems especially in highly pop- ulated areas. Thus the selection and use of landfills has to be optimized by keeping in view of the best of economic, environmental and public health practices. It should be ensured that during siting, design, construction and operation, closure and post closure, a landfill complies with all national, provincial/state, and local government rules, regulations, and permits so that it should not become a threat to the society [5, Fuzzy Inf. Eng. (2012) 3: 313-338 315 6]. Although for four decades, this process has been widely accepted, there have been a number of uncertainties that are not well treated or considered in this traditional way of selection process of landfilling, which may result in misleading outputs. Re- search is still focused on the seeking of a more integrated approach to incorporate all important criteria. One of the critical problems involved in integrating various assessment criteria is regarding uncertainty that arises from different sources, such as error in measurement and/or modeling, imprecision in knowledge of relationships between stressors and receptors, and even ambiguity in the meaning of risk, which are not considered in traditional evaluation of suitability of a landfill. To deal with the indicators’ uncertainties arising during the evaluation process, fuzzy set theory [7-11] appears to be a good complimentary approach. Once the in- dicators are represented by fuzzy sets, there are several fuzzy techniques that can be used to facilitate formulation and calculations of uncertainties associated with these fuzzy indicators. Some of them are fuzzy arithmetic [12], fuzzy linear pro- gramming [13, 14], fuzzy rule-based modeling [15] or fuzzy ranking [16, 17]. Sev- eral studies in that direction have been seen in the literature recently. For example, Singh [18] has clearly demonstrated the application of fuzzy set theory for assess- ing potential for water resource development and its impact in Chittorgarh district of Rajasthan. A decision support system has also been developed to evaluate op- timal landfill sites by Singh and Vidyarthi [19] which reflects dynamic, interactive, and uncertain characteristics of the solid waste management system effectively and provides decision makers a decision tool to choose a municipal solid waste manage- ment strategy. Wenger and Rong [20] used fuzzy set models to compare alternative solutions to environmental problems in comprehensive environmental decision mak- ing process. Smith [21] also presented a method for recognizing uncertainty in the evaluation of discrete transportation options characterized along multiple dimensions based on fuzzy and linguistic variables in the case when the objectives are classified as “qualitative imprecise”. Srivastava and Nema [22] developed a fuzzy parametric programming model for integrated solid waste management under uncertainty to un- derstand the waste allocation under different levels of uncertainty which essentially address the uncertainty in waste generation quantities and the capacities of the waste- management facilities. It is clear from the above literature review that many research studies have been conducted to deal with the representation, analysis and evaluation of suitable landfill site. Moreover, some of these studies have faced serious problems of integrating information from many different sources into an overall evaluation and interpretation. However, the fuzzy approach in ranking of landfills is still having a lot of potential to apply wherein fuzzy uncertainty, multidimensional and spatial characteristics of site can be dealt with a simple and straightforward way. Xu [23, 24] devised important operational laws of linguistic variables and developed a few aggregation operators such as linguistic geometric averaging (LGA) operator, lin- guistic weighted geometric averaging (LWGA) operator, linguistic ordered weighted geometric averaging (LOWGA) operator and linguistic hybrid geometric averaging (LHGA) operator. These operators have been used to aggregate linguistic preference information of decision makers in a systematic and straightforward manner without 316 Ajit Pratap Singh· Subodh Kant Dubey (2012) any loss of information [24, 25]. In this study, authors have developed a methodology to evaluate the feasibility of optimal landfill disposal site for solid waste disposal. The different criteria involved in this process are accessibility and transportation; environmental, geological and cli- matic conditions; socio-economic conditions; land use pattern; and safety of selected site. Finally, the optimum landfill disposal site is evaluated using a modified fuzzy utility approach wherein the uncertainty associated with specifying various attributes are incorporated. The study has also incorporated the opinion of experts at the selec- tion process so that real analysis can be performed. 2. Materials and Methods 2.1. Landfill Site Selection Solid waste disposal in landfills is the most widely used method for disposing of waste and about 80% of the wastes go to landfills. It owes its wide acceptance due to ease of maintenance and management. However, the method followed in many Indian case studies is not even equipped with the modern practices of landfilling. The collected wastes are mainly dumped in low-lying areas which are prone to flood which lead to cause both surface and ground water contamination. In addition to this, birds foraging on garbage dumps pose hazard to aircrafts operating in the areas. The selection of sites for landfills should not only be consistent with local land use conditions and zoning codes but should also be able to protect both ecologically sensitive areas (e.g. flood plains, wetlands) and culturally sensitive areas (e.g. archeological, historical) with the minimum impacts on air and/or water quality, and not to otherwise adversely impact upon public health, safety, welfare, community image as well as aesthetic and political issues. Quite often, problems with respect to siting a new landfill are more political than technical in nature. The general perception of public is that landfills are dumping sites and they do not consider the fact that a landfill is designed, built, and operated according to the latest engineering principles. Many people are reluctant to allow construction of a new landfill in their communities. Moreover, a landfill affects the surrounding environment for a long time. The construction and operation of landfills is often more expensive compared to other types of solid waste treatment. Thus the selection process of waste disposal sites should not only deal with technical aspects but also social and political issues [26-29]. The selection of appropriate landfill site is one of the key elements of municipal solid waste management system [30]. Due to rapid rise in environmental awareness among the public and reduction of availability of urban land, the problem of select- ing appropriate waste disposal sites is becoming challenging and complex. It needs to consider several independent factors concerning land use, socio-economy and hy- drogeology. The U.S. Environmental Protection Agency has prescribed a number of criteria to be considered for selection of a suitable waste disposal site. The parame- ters which are of immense importance include the site’s impact on ground-water and air quality, waste material transport feasibility, effect on property values and com- pensation plans, equity in the choice of sites, impact on community image as well as aesthetic and political issues [2]. However, no single landfill site can satisfy all the parameters due to their complex interrelationships and conflicting nature and hence, Fuzzy Inf. Eng. (2012) 3: 313-338 317 tradeoff between them is very clear [5, 31, 32]. The analysis and the evaluation of appropriate landfill sites are critical for several reasons. For example, priorities and preferences to select a landfill always requires the synthesis of two distinct selection process, namely, (i) a technical screening process which is mainly based on economic, engineering and environmental suitability and (ii) public acceptability and approval process along with political will power. More- over, many studies have been reported for allocating waste disposal sites which essen- tially assist decision makers to decide between alternatives when conflicted criteria are taken into account simultaneously. Some of these include linear programming techniques to optimize the location of a site with respect to operation and mainte- nance costs [33, 34]. Zeiss and Lefsrud [35] developed an analytical framework for waste-facility siting to structure the main elements and connections into a framework to explain stakeholder attitudes and siting outcome. The techniques like geographi- cal information systems (GIS) and analytic hierarchy process (AHP) have also been applied in landfill site selection by several researchers [36-40]. For example, Chang et al [36] presented a fuzzy multi-criteria decision analysis alongside with a geospa- tial analysis for the selection of landfill sites which basically employs a two-stage analysis synergistically to form a spatial decision support system (SDSS) for waste management in a fast-growing urban region of south Texas. Nas et al [41] have also presented a study to evaluate the suitability of a landfill in Cumra County of Konya city by integrating multi-criteria evaluation method with ArcGIS 9.0 as a practical in- strument. Ramu and Kennedy [42] presented a heuristic technique which maximizes service to an existing population by locating the new waste facilities based on dis- tance, cost, and environmental, social and political issues which can also be applied to other facility location problems for quick and feasible solution. Some field studies have also been conducted wherein resistivity imaging and ground penetrating radar (GPR) tools were applied to assess pollution level in the vicinity of a landfill [43]. However, the fuzzy approach in ranking of suitable landfill sites is still having a lot of potential to apply wherein all important criteria such as environmental and hydrological conditions, accessibility, ecological and societal effects etc. can be in- corporated in a more effective manner. A systematic approach for selecting a suitable landfill site using the concepts of modified fuzzy utility approach is proposed in this paper to incorporate uncertainty associated in specifying various attributes which are often imprecisely defined by the decision makers. The identification of landfill sites and their ranking are based on objective evaluation of accessibility, receptor, environ- mental, socio-economic, waste management practice, climatological and geological related attributes. A general listing of all important factors considered for landfill sit- ing is presented in Table 1. The above mentioned factors are generally technical, environmental, economical and socio-political in nature which can be usually categorized as vague and impre- cise. Therefore, these factors can be considered as linguistic variables which can be handled using fuzzy concepts [9, 17, 18, 44, 45] and the site selection process of landfill can be performed accordingly. 2.2. Methodological Framework for Site Selection 318 Ajit Pratap Singh· Subodh Kant Dubey (2012) Table 1: List of factors and their attributes used in selection process of a landfill site. Factors to be considered Important attributes Factors to be considered Important attributes for landfill selection associated with corr- for landfill selection associated with process esponding factors process corresponding factors Accessibility to the site • Distance from the road Geological factors • Soil permeability • Distance from the origin • Depth to bedrock of waste • Seismicity Receptor related factors • Proximity of human hab- Environmental factors • Hydro-geological inve- itation /locality stigation • Drinking water sources • Distance to nearest sur- • Land use designation face water • Agriculture value • Air quality • Public utility facility • Soil quality • Historical / Archeological •Water quality monuments • Safety • Public accessibility Socio-economic factors • Job opportunity Waste management pra- • Waste quantity /day • Health ctices related factors • Life of site Since a decision maker is required to allocate best landfill site for efficient and eco- nomic disposal of solid waste, it is necessary to develop a methodological framework which can integrate all important aspects of siting. The top down heuristic approach for evaluating different alternative sites consists of various steps such as constraint mapping, potential site selection, preliminary survey, site investigation on preferred sites, ranking of landfill sites and final selection of the site which in turn depend upon several indicator parameters as shown in Fig.1. The hierarchical structure developed herein aggregates the effects of each indicator parameter. The process of aggregation continues until the final site is selected based on six important steps as shown in Fig.1. The first and foremost important step deals with the identification of various con- straints which essentially eliminates environmentally unsuitable sites and narrows down the number of sites for further consideration. All the constraints should be recorded while prioritizing different alternative sites. For example, one of the most difficult part in the processes of site selection at the initial stage is to get appropriate land for landfilling. Thus the dearth of appropriate land can be a constraint. Several of such negative aspects may conveniently be listed which can further lead to reveal areas in which landfill sites might be located. The important factors which can be con- sidered to eliminate unsuitable sites from further analysis are transportation of wastes to minimize cost and time, natural conditions and land use pattern, safety of selected site and so on. The next step of selection of landfill sites deals with the identification of maximum number of potential sites which mainly depends on land details (area required for site, land ownership and its current use) and infrastructural facilities (ac- cess to roads, sites of existing/former waste disposal facilities and land designated for industrial use). These provide the basis for highlighting promising sites within the alternative sites remaining after first step of analysis. The preliminary survey is another important process where possible selected sites are further examined to elim- Fuzzy Inf. Eng. (2012) 3: 313-338 319 inate some of the sites which basically fail to meet additional socio-economic and environmental concerns at the site and surrounding areas. The detailed investigations on various factors (e.g. geology, hydrogeology, climatology, land etc.) of each site are then performed which are critical to the success of the siting and design of the landfill. Finally, all short listed sites are ranked based on detailed analysis of their en- vironmental, social and community impacts and the best site is selected with highest total score. Fig. 1 Processes involved in the selection of suitable landfill site 2.3. Development of Modified Fuzzy Utility Approach The proposed methodology develops a multi-criteria approach to integrate various criteria with respect to each alternative site of landfill which are available to a decision 320 Ajit Pratap Singh· Subodh Kant Dubey (2012) maker. It deals with fuzzy utility theory and multi-nomial logit model to incorporate imprecision of the site specific attributes. The fuzzy utility theory is mainly applied to calculate utility value associated with each alternative site for selection of opti- mum site whereas the probabilities of alternatives being selected are calculated using multi-nomial logit model. The importance of weights and ratings of various criteria are considered as linguistic variables. These linguistic variable can be expressed as triangular fuzzy numbers. The step by step procedure of the proposed methodology is explained in subsequent paragraphs. The initial step in the development of the model is to identify the possible alter- native sites along with their attributes/criteria which are responsible for performance evaluation of each alternative. th Let A (i = 1, 2,··· , n) represent the i alternative and C ( j = 1, 2,··· , m) repre- i j th sent the j criteria/attributes with which each alternative performances can be evalu- ated. Therefore, these alternatives and criteria can be listed mathematically as given below: A = {A , A ,··· , A }, (1) i 1 2 n C = {C , C ,··· , C }. (2) j 1 2 m Once the attributes (C ) and alternatives A are identified, the next step is the gen- j i eration of importance weight of each criterion associated for every alternative site. Weights are assigned to assess the importance of each criterion as linguistic variables by the different experts. The importance weight of each attribute/criteria (C ) is ob- th tained by deriving a membership function of the perception of k decision maker (DM ). If w  ( j = 1, 2,··· , m) is the fuzzy importance weight of each attribute C ,it k j j can be expressed mathematically by the following matrix form: w  = w  , w  , w  ,··· , w  . (3) j 1 2 3 m The fuzzy weight of each attribute C can be represented in terms of triangular fuzzy number [(x ,μ (x )), (x ,μ (x )), (x ,μ (x ))] as shown in Fig.2, whereμ (x) I l I II 2 II III 3 III i is membership value of any decision variable x to represent linguistic view of the decision maker mathematically. ሺ࢞ሻ ࢞ ࢞ ࢞ ࢞ ࡵ  ࡵࡵ ࡵࡵࡵ Fig. 2 Triangular fuzzy number with its triplets and membership values ࣆ Fuzzy Inf. Eng. (2012) 3: 313-338 321 The membership functions are derived in the form of triplets of triangular fuzzy numbers, i.e., [μ (x ),μ (x ),μ (x )] for any given values [x , x , x ] of the deci- l I 2 II 3 III I II III sion variables. Thus the fuzzy weights of each criterion can be expressed as w  = w  , w  , w  . The membership function for triangular fuzzy numbers can be cal- jI jII jIII culated by the following formulation: ⎪ 0, for x < x x− x ⎪ I , for x ≤ x ≤ x , ⎪ I II x − x II I μ (x) = (4) i ⎪ ⎪ x − x III ⎪ , for x ≤ x ≤ x , II III ⎪ x − x III II 0, for x > x . III The third step is to evaluate final weights for each criterion. The fuzzy weight of any th j criteria can be calculated as 1 2 3 k w + w + w +···+ w j j j j w  = , (5) k th th where k is the number of experts and w is the weight assigned by k expert for j criteria. The fourth step is to convert final triangular fuzzy weights into a crisp value by using Yager’s unit interval method [46] as given in Equation (6): Max F(A) = M(α)dα, (6) L + U α α where M(α) = and L and U are the lower alpha cut and the upper alpha α α cut respectively. The fifth step is to define the ideal or reference alternative site (S ) using the crisp weights calculated in pervious step, which can be expressed as S = {C , j(x)}, where j = 1, 2,··· , m and j(x) = membership value of corresponding attributes for ideal case. The sixth step deals with the formulation of linguistic variables for assigning rat- ings of each alternative with respect to all available criteria. A pairwise comparison matrix is constructed by selecting available alternatives for a landfill site under differ- ent criteria. The linguistic terms of the pairwise comparisons are assigned by asking importance degree of each criteria using fuzzy approach. The pairwise comparison th th matrix for evaluating i alternative with respect to j criteria can be expressed in the 322 Ajit Pratap Singh· Subodh Kant Dubey (2012) following matrix form: ⎡ ⎤ ⎢ ⎥ r  r  r ···  r ⎢ 11 12 13 1m⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ r  r  r ···  r ⎢ 21 22 23 2m⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ r  r  r ···  r ⎢ ⎥ 31 32 33 3m R = ⎢ ⎥ , (7) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . . . . ⎥ ⎢ ⎥ ⎢ . . . . . ⎥ ⎢ . ⎥ . . . . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ r  r  r ···  r n1 n2 n3 nm where  r is the fuzzy rating of alternative A (i = 1, 2,··· , n) with respect to cri- ij i terion C and is derived in the form of triplets of triangular fuzzy numbers, i.e., th a = (a , a , a ). The membership value of rating of i alternative with respect ij I II III th to j criteria ( r ) is also derived to represent linguistic view of the decision maker ij mathematically. The ratings ( r ) for each alternative is also expressed as triangular ij fuzzy numbers and the final rating of each alternative is calculated as given below: 1 2 3 k r + r + r +···+ r ij ij ij ij r = , (8) ij k th th where r is the rating for i alternative site with respect to j criteria as assigned by ij th k expert. The seventh step is the formulation of utility matrix, elements of which is repre- sented in the form of μ(x), r , where μ(x) is the membership value and  r is the ij ij th fuzzy rating of any alternative A with respect to the j criteria. In some cases, more than one membership value will be available for  r . In such cases, the maximum ij value should be assigned as suggested by Wang and Mendel [47]. The eighth step is to calculate fuzzy utility value (U ) of each alternative (A ) with respect to ideal alternative (S ) based on the decision maker’s perception, which can be calculated by using Equation (9): U = μ (x)⊗ μ(x), r = μ , r , (9) j ij m ij whereμ = μ (x)∩μ(x). m j After evaluating associated utility of each site as mentioned above, it is necessary to determine a set that can maximize the utility of each alternative site. It can be obtained by deriving the power set of U with respect to each alternative. The max- imizing set can be expressed by deriving its membership value with respect to each element of normal utility set as given by Equation (10): U = μ, r , (10) ij where μ = ,  r is the corresponding rating value of that element,  r is the Max Max supremum of union of normal utility set (i.e.,  r = sup∪ r ) and n is the degree of Max ij confidence that decision makers have on their judgment. If decision makers are more confident about their judgment, value of n can be varied from 2 to 4. In case they are not confident, it will vary backward from 1 to 0.1. In this analysis, the value of n has been taken as 1 to represent the normal situation. Fuzzy Inf. Eng. (2012) 3: 313-338 323 The next step is to form an optimum set (U ) for each alternative which can be N M evaluated by the intersection of U and U obtained in previous two steps, and can i i be expressed as O N M U = U ∩ U . (11) i i i Once optimum set is derived for each alternative (A ), the utility value or degree of satisfaction (DoS) of each alternative can be computed by deriving supreme member- ship value for each of them from their optimum set which is expressed by Equation (11), u = DoS (A ) = sup U  r ,  r ∈ U . (12) i n ij ij Subsequently, alternatives can be ranked on the basis of utility values. Addition- ally, the probability of the alternatives being selected by different decision makers can also be obtained by the multi-nomial logit model after evaluating degree of satis- faction which can finally be expressed as P(A ) =  , (13) where U is the utility of alternatives which can be directly expressed in terms of the degree of satisfaction. The series of steps adopted in present methodology to select an optimal landfill site among the given alternative sites by integrating the concept of fuzzy-utility method and multi-nomial logit model are shown in Fig.3. 2.4. Study Area The proposed methodology has been applied for evaluation of the best landfill site among the possible landfill sites to dispose the solid wastes of Vidya Vihar, Pilani, Rajasthan, India. As a policy maker, Nagarpalika of Vidya Vihar Pilani formed a committee of three experts of solid waste management namely DM , DM and DM 1 2 3 to take their opinions. Keeping with the site selection procedure described earlier under ‘development of modified fuzzy utility approach’, three probable landfill sites A , A , and A were shortlisted for further evaluation. A set of 13 criteria was consid- 1 2 3 ered to select the best landfill site. The criteria were decided on the basis of literature review pertaining to site selection, experts from waste management and it was also decided taking in view the factors contributing to the pollution pathways. The criteria considered in this application are depth to groundwater (C ), life of site (C ), soil 1 2 permeability (C ), distance to nearest drinking water (C ), population within 500 me- 3 4 ters (C ), distance from collection point (C ), air quality (C ), health (C ), land use 5 6 7 8 zoning (C ), type of road (C ), public acceptability (C ), odor (C ), public utility 9 10 11 12 facility within 2 km (C ) which are also described in Table 4 with their qualitative weights assigned by each of the three experts. 3. Results and Discussions 3.1. Selection of Optimum Landfill Site Using Modified Fuzzy Utility Approach 324 Ajit Pratap Singh· Subodh Kant Dubey (2012) $$# $$# $$ $$   # $$ $$   # # $$'$ $$ $$ $$'$ Fig. 3 Important steps of fuzzy based site selection process of a suitable landfill Fuzzy Inf. Eng. (2012) 3: 313-338 325 The formulation developed herein has been applied in a case study for screening potential landfill sites for disposing off municipal solid waste of Pilani town which is one of the educational hubs located in Jhunjhunu district of Rajasthan state in India. All important data with respect to each site have been taken from the earlier work of the first author and they are referred elsewhere [19]. The step by step procedure adopted for the present case study is described as below: 1) Important selection criteria, its attributes and all possible alternatives for landfill site were identified for the given problem. For example, a total of 13 criteria are identified with 3 alternative sites in this case. 2) The linguistic variables for assigning importance weights are classified as pre- sented in Table 2. These variables are then used to assess the importance of each criteria by the experts DM , DM and DM and depicted in Table 3. 1 2 3 Table 2: Important weight of each criterion. Linguistic description Weights with triangular elements Very low (VL) (0.0, 0.1, 0.3) Low (L) (0.1, 0.3, 0.5) Medium (M) (0.3, 0.5, 0.7) High (H) (0.5, 0.7, 0.9) Very high (VH) (0.7, 0.9, 1.0) Table 3: Ratings of each criterion. Linguistic description Ratings with triangular elements Very poor (VP) (0.0, 1.0, 3.0) Poor (P) (1.0, 3.0, 5.0) Fair (F) (3.0, 5.0, 7.0) Good (G) (5.0, 7.0, 9.0) Very good (VG) (7.0, 9.0, 10.0) 3) The fuzzy weight of each criterion has been calculated using Tables 2 and 4 and Equation (5) which is given in Table 5. For example, in Table 4, for the criterion related to soil permeability (C ), there have been different opinions of three experts, i.e., VH, H and VH which correspond to triangular membership value of importance weight as (0.7, 0.9, 1.0), (0.5, 0.7, 0.9), and (0.7, 0.9, 1.0), respectively. Therefore the th fuzzy weight of any j criteria (say related to soil permeability) can be calculated as 0.7+ 0.5+ 0.7 0.9+ 0.7+ 0.9 1.0+ 0.9+ 1.0 w  = , , , i.e., w  = (0.63, 0.83, 0.96). 3 3 3 3 3 Similarly, other values can be calculated and filled in Table 5. 4) According to Table 5 which was initially derived by Singh and Vidyarthi (2008), 326 Ajit Pratap Singh· Subodh Kant Dubey (2012) Table 4: Importance weights of each attributes. S.No. Attributes DM DM DM 1 2 3 1. Depth to groundwater (C)VH H H 2. Life of site (C ) H VH H 3. Soil permeability (C)VH H VH 4. Distance to nearest drinking water (C)VH VH VH 5. Population within 500 meters (C ) H VH M 6. Distance from collection point (C ) H VH H 7. Air quality (C ) H VH VH 8. Health (C)VHHH 9. Land use zoning (C)M M M 10. Type of road (C)M M H 11. Public acceptability (C)M M M 12. Odor (C)VHHH 13. Public utility facility within 2 km (C ) H VH H the fuzzy weights expressed by triplets of triangular fuzzy numbers have been con- verted into a crisp value using Equation (6) of Yager’s unit interval method. These crisp values are also given in Table 5. 5) The ideal or reference alternative site (S ) has now been defined using the crisp weights associated with each criterion as calculated in pervious step according to the decision maker’s perception. Therefore, the ideal site (S ) can be expressed as S = (0.75, C ), (0.75, C ), (0.82, C ), (0.88, C ), (0.69, C ), 1 2 3 4 5 (0.75, C ), (0.82, C ), (0.75, C ), (0.5, C ), (0.56, C ), (14) 6 7 8 9 10 (0.5, C ), (0.75, C ), (0.75, C ) . 11 12 13 6) The linguistic variables for assigning ratings have also been classified as given in Table 3. These variables are used to assess the ratings of each criterion by the experts DM , DM and DM for all three possible alternative sites and are presented in Table 1 2 3 7) The fuzzy utility matrix has been derived using the ratings of each criterion ob- tained from Step 6 and Table 6 with respect to each of the possible alternatives so that the chosen site can be compared with reference to the ideal one. Each element of this matrix is expressed by μ, r , whereμ is the membership function and r is the fuzzy ij ij th rating of any alternative A with respect to any j criteria assigned by the experts and is calculated using Equation (8). For example, in Table 6, for fifth criteria related to population within 500 m (C ), there are different opinions of three experts, i.e., Poor, Fair and Poor for alternative 1 (i.e., for first site) which correspond to triangular membership value rating as (1.0, 3.0, 5.0), (3.0, 5.0, 7.0) and (1.0, 3.0, 5.0) respec- Fuzzy Inf. Eng. (2012) 3: 313-338 327 Table 5: Fuzzy weights of the criteria. S.No. Attributes Fuzzy weights Crisp value using Normalized Yager’s unit weight (w) interval method 1. Depth to groundwater (C)(0.56, 0.76, 0.93 0.75 0.08090615 2. Life of site (C)(0.56, 0.76, 0.93) 0.75 0.08090615 3. Soil permeability (C)(0.63, 0.83, 0.96) 0.82 0.08845739 4. Distance to nearest drinking water (C )(0.70, 0.90, 1.0) 0.88 0.09492988 5. Population within 500 meters (C)(0.50, 0.70, 0.86) 0.69 0.07443366 6. Distance from collection point (C)(0.56, 0.76, 0.93) 0.75 0.08090615 7. Air quality (C)(0.63, 0.83, 0.96) 0.82 0.08845739 8. Health (C)(0.56, 0.76, 0.93) 0.75 0.08090615 9. Land use zoning (C)(0.30, 0.50, 0.70) 0.50 0.05393743 10. Type of road (C)(0.36, 0.56, 0.76) 0.56 0.06040992 11. Public acceptability (C)(0.30, 0.50, 0.70) 0.50 0.05393743 12. Odor (C)(0.56, 0.76, 0.93) 0.75 0.08090615 13. Public utility facility within 2 km(C )(0.56, 0.76, 0.93) 0.75 0.08090615 th tively. Therefore the fuzzy rating of any j criteria (say related to population within 1.0+ 3.0+ 1.0 3.0+ 5.0+ 3.0 5.0+ 7.0+ 5.0 500 m) can be calculated as  r = , , , 3 3 3 i.e.,  r = (1.6, 3.6, 5.6). Thus the elements of the fuzzy utility matrix can be derived by assigning their re- spective membership functions with their ratings. For example, elements with respect th to any j criteria μ, r (say related to population within 500 m) can be expressed as ij 1.0+ 3.0+ 1.0 3.0+ 5.0+ 3.0 5.0+ 7.0+ 5.0 0.7, , 0.7, , 0.7, , 3 3 3 i.e., μ, r = [(0.7, 1.6), (0.7, 3.6), (0.7, 5.6)]. It is important to note that the fuzzy ij th rating of any alternative A with respect to j criteria ( r ) may attain two member- i ij ship values in some cases. In such cases, the highest membership values should be assigned as per Wang and Mendal (1992). For example,  r = 4.3 have two mem- ij bership values 0.65 and 0.35 in linguistic description of fair and poor respectively and therefore, the corresponding membership value with respect to the rating of 4.3 would be 0.65. However, if all the decisions makers have assigned same linguistic rating for any particular attribute, then the membership value would be same as the assigned value given in Table 3. For example, alternative site 1 with respect to criteria 6 (distance from collection point: C ) has been assigned ‘fair’ rating by all the three decision makers. The membership value for this alternative site with respect to C can therefore be represented as [(0, 3), (1, 5), (0, 7)]. There is no need to assign mem- bership value of 1 for 3 and 1 for 7 as every decision maker has given same rating. 328 Ajit Pratap Singh· Subodh Kant Dubey (2012) Table 6: Opinion of three decision makers for ratings of each alternative site. S. No. Attributes Site 1 Site 2 Site 3 DM DM DM DM DM DM DM DM DM 1 2 3 1 2 3 1 2 3 1. Depth to VG VG VG VG VG VG VG VG VG groundwater (C ) 2. Life of GGG F GGGGG site (C ) 3. Soil F F FG GFGFG permeability (C ) 4. Distance to nearest GGGGGG F F F drinking water (C ) 5. Population within PFP VG G VG PP VP 500 meters (C ) 6. Distance from FFFFPFP VP P collection point (C ) 7. Air quality (C) P P P GGGGGG 8. Health (C)P F P VG G G P VP P 9. Land use PFPFF G G F G zoning (C ) 10. Type of road (C) G G G FFFPP VP 11. Public FVP VPF F P P PVP acceptability (C ) 12. Odor (C ) F PVP G G G VP PVP 13. Public utility facility PPPPPP G FF within 2 km (C ) Similarly other entries of this matrix can be obtained as shown in Table 7. th 8) Using Equations (9) and (14) and Table 7, normal utility value of any i alternative with reference to ideal site (U ) are evaluated based on the decision maker’s percep- tion. It is important to note that the lowest membership value should be assigned to each rating value. For example, criteria C is expressed as (0.75, C ) in Equation (14). 1 1 Thus, (0.75, C ) = (0.75, [(0, 7), (1, 9), (0, 10)]) = [(0, 7), (0.75, 9), (0, 10)]. Similarly, other entries can be obtained. Therefore Fuzzy Inf. Eng. (2012) 3: 313-338 329 Table 7: Fuzzy utility for every criteria/attributes of each site/alternative. Attributes Site 1 (A ) Site 2 (A ) Site 3 (A ) 1 2 3 Depth to groundwa- [(0,7), (1,9), (0,10)] [(0,7), (1,9), (0,10)] [(0,7), (1,9), (0,10)] ter (C ) Life of site (C ) [(0,5), (1,7), (0,9)] [(0.65,4.3), (0.65,6.3), (0.65,8.3)] [(0,5), (1,7), (0,9)] Soil permeability [(0,3), (1,5), (0,7)] [(0.65,4.3), (0.65,6.3), (0.65,8.3)] [(0.65,4.3), (0.65,6.3), (0.65,8.3)] (C ) Distance to nearest [(0,5), (1,7), (0,9)] [(0,5), (1,7), (0,9)] [(0,3), (1,5), (0,7)] drinking water (C ) Population within [(0.7,1.6), (0.7,3.6), (0.7,5.6)] [(0.65,6.3), (0.7,7.6), (0.4,9.6)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] 500 meters (C ) Distance from col- [(0,3), (1,5), (0,7)] [(0.65,2.3), (0.65,4.3), (0.65,6.3)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] lection point (C ) Air quality (C ) [(0,1), (1,3), (0,5)] [(0,5), (1,7), (0,9)] [(0,5), (1,7), (0,9)] Health (C ) [(0.7,1.6), (0.7,3.6), (0.7,5.6)] [(0.7,5.6), (0.7,7.6), (0.7,9.3)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] Land use zoning (C ) [(0.7,1.6), (0.7,3.6), (0.7,5.6)] [(0.65,4.3), (0.7,5.6), (0.7, 7.6)] [(0.65,4.3), (0.65,6.3), (0.65,8.3)] Type of road (C ) [(0,5), (1,7), (0,9)] [(0,3), (1,5), (0,7)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] Public acceptability [(1,1), (0.65,2.3), (0.65,4.3)] [(0.65,2.3), (0.65,4.3), (0.65,6.3)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] (C ) Odor (C ) [(0.85,1.3), (1,3), (1,5)] [(0,5), (1,7), (0,9)] [(0.6,0.6), (0.65,2.3), (0.65,4.3)] Public utility facility [(0,1), (1,3), (0,5)] [(0,1), (1,3), (0,5)] [(0.7,3.6), (0.7,5.6), (0.7,9.6)] within 2 km (C ) U = [{0.75, [(0, 7), (1, 9), (0, 10)]}, 0.75, [(0, 5), (1, 7), (0, 9)], {0.82, [(0, 3), (1, 5), (0, 7)]}, {0.88, [(0, 5), (1, 7), (0, 9)]},{0.69, [(0.7, 1.6), (0.7, 3.6), (0.7, 5.6)]}, {0.75, [(0, 3), (1, 5), (0, 7)]},{0.82, [(0, 1), (1, 3), (0, 5)]}, {0.75, [(0.7, 1.6), (0.7, 3.6), (0.7, 5.6)]},{0.5, [(0.7, 1.6), (0.7, 3.6), (0.7, 5.6)]},{0.56, [(0, 5), (1, 7), (0, 9)]},{0.5, [(1, 1), (0.65, 2.3), (0.65, 4.3)]},{0.75, [(0.85, 1.3), (1, 3), (1, 5)]},{0.75, [(0, 1), (1, 3), (0, 5)]}]. Which is further simplified as U = {[(0, 7), (0.75, 9), (0, 10)], [(0, 5), (0.75, 7), (0, 9)], [(0, 3), (0.82, 5), (0, 7)], [(0, 5), (0.88, 7), (0, 9)], [(0.69, 1.6), (0.69, 3.6), (0.69, 5.6)], [(0, 3), (0.75, 5), (0, 7)], [(0, 1), (0.82, 3), (0, 5)], [(0.7, 1.6), (0.7, 3.6), (0.7, 5.6)], [(0.5, 1.6), (0.5, 3.6), (0.5, 5.6)], [(0, 5), (0.56, 7), (0, 9)], [(0.5, 1), (0.5, 2.3), (0.5, 4.3)], [(0.75, 1.3), (0.75, 3), (0.75, 5)], [(0, 1), (0.75, 3), (0, 5)]}. As observed in the above analysis, there are different membership values for given 330 Ajit Pratap Singh· Subodh Kant Dubey (2012) r , thus these membership values can be aggregated corresponding to the respective ij r , as shown below: ij U = [[(0+ 0.75+ 0+ 0+ 0.56), 7], [(0.75+ 0+ 0), 9], [0, 10], [(0+ 0.82+ 0+ 0.75+ 0+ 0+ 0.75+ 0), 5], [(0+ 0+ 0.75+ 0.75), 3], (15) [(0.69+ 0.7+ 0.5), 1.6], [(0.69+ 0.7+ 0.5), 3.6], [(0.69+ 0.7+ 0.5), 5.6], [(0+ 0.5+ 0), 1], [0.5, 2.3], [0.5, 4.3], [0.75, 1.3]]. The values of U obtained from the above analysis can further be simplified using algebraic sum method as mentioned below: μ +μ = μ +μ −μ ∗μ , (16) 1 2 1 2 1 2 U = [(0.89, 7), (0.75, 9), (0, 10), (0.99, 5), (0.94, 3), (0.95, 1.6), (0.95, 3.6), (0.95, 5.6), (0.5, 1), (0.5, 2.3), (0.5, 4.3), (0.75, 1.3)]. Similarly, U = (0.95, 7), (0.75, 9), (0, 10), (0.99, 4.3), (0.99, 6.3), (0.88, 8.3), (0.56, 5), (0.95, 7.6), (0.4, 9.6), (0.83, 2.3), (0.85, 5.6), (0.7, 9.3), (0.75, 3), (0, 1)]. Similarly, U = [(0.96, 7), (0.75, 9), (0, 10), (0.88, 5), (1, 4.3), (0.83, 6.3), (0.83, 8.3), (0, 3), (1, 0.6), (0.99, 2.3), (0.7, 3.6), (0.7, 5.6), (0.7, 9.6)]. However, other methods can also be used to get single crisp value which can be referred elsewhere [26]. 9) After evaluating associated utility of each site in Step 8, a set that can maximize the utility of each alternative site is determined using Equation (10). In this analysis, the degree of confidence that decision makers have on their judgment (n) is taken as 1 to represent the normal situation. If Y represents the union of normal utility set obtained in Step 8, then it can be expressed as Y = {7.0, 9.0, 5.0, 3.0, 1.6, 3.6, 5.6, 1.0, 2.3, 4.3, 1.3, 6.3, 8.3, 7.6, 9.6, 9.3, 0.6}. It may be observed here that the rating of 10 is not included in the above calculation as it has zero membership value in each normal utility set. Therefore the superimum of normal utility set,  r = 9.6. Thus the membership value of 7 in the new set will max be 7/9.6 = 0.73, and the maximizing sets can be expressed as U = [(0.73, 7.0), (0.94, 9.0), (0.52, 5.0), (0.31, 3.0), (0.17, 1.6), (0.38, 3.6), (0.58, 5.6), (0.1, 1.0), (0.24, 2.3), (0.45, 4.3), (0.14, 1.3)], U = [(0.73, 7.0), (0.94, 9.0), (0.45, 4.3), (0.66, 6.3), (0.87, 8.3), (0.52, 5.0), (0.79, 7.6), (1.0, 9.6), (0.24, 2.3), (0.58, 5.6), (0.97, 9.3), (0.31, 3.0), (0.1, 1.0)], U = [(0.73, 7.0), (0.94, 9.0), (0.52, 5.0), (0.45, 4.3), (0.66, 6.3), (0.87, 8.3), (0.31, 3.0), (0.1, 0.6), (0.24, 2.3), (0.38, 3.6), (0.58, 5.6), (1.0, 9.6)]. Fuzzy Inf. Eng. (2012) 3: 313-338 331 10) After evaluating maximizing set of each alternative site, it is desirable to form O N M optimum set (U ) which can be evaluated by the intersection of U and U obtained i i i in Steps 8 and 9, and can be expressed as U = [(0.73, 7.0), (0.75, 9.0), (0.52, 5.0), (0.31, 3.0), (0.17, 1.6), (0.38, 3.6), (0.58, 5.6), (0.1, 1.0), (0.24, 2.3), (0.45, 4.3), (0.14, 1.3)], U = [(0.73, 7.0), (0.75, 9.0), (0.45, 4.3), (0.66, 6.3), (0.87, 8.3), (0.52, 5.0), (0.79, 7.6), (0.4, 9.6), (0.24, 2.3), (0.58, 5.6), (0.7, 9.3), (0.31, 3.0), (0.1, 1.0)], U = [(0.73, 7.0), (0.75, 9.0), (0.52, 5.0), (0.45, 4.3), (0.66, 6.3), (0.83, 8.3), (0.0, 3.0), (0.1, 0.6), (0.24, 2.3), (0.38, 3.6), (0.58, 5.6), (0.7, 9.6)]. 11) Once optimum set is derived for each alternative, DoS or utility value of each alternative can be computed by deriving supreme membership value for each alterna- tive from their optimum set. DoS (A ) = SUP(0.73, 0.75, 0.52, 0.31, 0.17, 0.38, 0.58, 0.10, 0.24, 0.45, 0.14) = 0.75, DoS (A ) = SUP(0.73, 0.75, 0.45, 0.66, 0.87, 0.52, 0.79, 0.4, 0.24, 0.58, 0.7, 0.31, 0.0) = 0.87, DoS (A ) = SUP(0.73, 0.75, 0.52, 0.45, 0.66, 0.83, 0.0, 0.1, 0.24, 0.38, 0.58, 0.7) = 0.83. Therefore, A > A > A . 2 3 1 12) Additionally, the probability of the alternatives being selected by the different decision makers is also evaluated using Equation (13) of the multi-nomial logit model as shown below: 0.75 P(A )= = 31.00%, 0.75 0.87 0.83 (e + e + e ) 0.87 P(A )= = 35.00%, 0.75 0.87 0.83 (e + e + e ) 0.83 P(A )= = 33.68%. 0.75 0.87 0.83 (e + e + e ) 3.2. Linguistic Aggregation Operators Based Approach for Selection of Landfill Sites A direct method [23, 24] based on linguistic aggregation operators for group decision making with linguistic preference relations was also performed to compare the results evaluated by the proposed method. The data with respect to rating of each alternative site under the given criteria have been reorganized in appropriate format as per the requirement of the methodology suggested by Xu (2004). The data was analyzed by LGA and LWGA operators for all three landfill sites. The salient steps applicable to use these operators for identifying best alternative landfill site under 13 available criteria by aggregating opinion of all three experts are described below: 332 Ajit Pratap Singh· Subodh Kant Dubey (2012) • Based on the opinion of the DMs as given in Tables 3 and 6 earlier, the com- parative ratings of all three possible alternative sites under each criterion are evaluated. For example, DM has assigned fair, fair and poor ratings under criteria 6 (distance from collection point: C ) for alternative sites 1, 2 and 3 respectively. It clearly indicates that alternative sites 1 and 2 with respect to criterion C are equally good whereas these two sites are slightly good in com- parison with site 3. It should be noted that the ratings of a site under a given criteria can be assigned from VG to VP depending on the opinion of DMs. • Set S = {s} is defined by representing possible linguistic variable s (i = i i 1, 2··· , t) so that the elements of it are uniformly distributed on a prescribed scale in a totally ordered manner. The details of it can be referred elsewhere [23]. In this study, a total of nine linguistic variables have been taken to rep- resent set S = s (i = 1, 2,··· , 9), where S = s = extremely poor, s = very i 1 2 poor, s = poor, s = slightly poor, s = equally good (fair), s = slightly good, 3 4 5 6 s = good, s = very good, s = extremely good. 7 8 9 • The three alternative landfill sites should be compared with each other under each of the criteria using the linguistic terms of set S as suggested by Xu [23, 24]. The linguistic preference relation matrix (R) is derived for all the three alternative sites under each criterion based on the assessment of each decision maker. For example, all 3 alternative sites have been rated ‘VG’ by DM under criterion 1 (i.e., depth to groundwater: C as given in Table 6) and therefore they are equally good (s ) when compared to each other. Thus the linguistic preference relation matrix (R) dealing with alternative sites 1, 2 and 3 under criteria C as assessed by DM can be represented as follows: 1 1 • Similarly, all 13 possible linguistic preference relation matrices can be con- structed under each one of the thirteen criteria based on the opinion of each of the decision makers. Thus there will be 39 matrices in total for all 3 decision makers. th • Using LGA operator [23], the preference degree Z of i alternative site over th th all other alternative sites under j criteria as assessed by the k decision maker can be expressed as j j j j j j j n−1 Z = LGA r , r ,··· , r = r ⊗ r ⊗···⊗ r , k k k k k k k i i1 i2 in i1 i2 in where ⊗ is fuzzy multiplication operator. For example, the preference degree Z for alternative site 1 over other two alternative sites under criteria 1 (C )of 1 1 DM can be expressed as 1 1 1 1 1 1 1 3−1 0.5 Z = LGA r , r , r = r ⊗ r ⊗ r = (s ⊗ s ) , 1 1 1 1 1 1 1 5 5 1 11 12 13 11 12 13 0.5 0.5 = (s ) = s = s . 25 5 • Similarly, preference information Z of all decision makers (k = 1, 2, 3) for all three alternative sites (i = 1, 2, 3) under all 13 criteria ( j = 1, 2,··· , 13) can be derived as shown in Table 9. Fuzzy Inf. Eng. (2012) 3: 313-338 333 Table 8: Linguistic preference relation matrix for criteria 1. Site 1 Site 2 Site 3 Site 1 - s s 5 5 Site 2 s - s 5 5 Site 3 s s - 5 5 Table 9: Preferences for all alternative sites by the DM , DM and DM . 1 2 3 Criterion ( j) C C C C C C C C C C C C C 1 2 3 4 5 6 7 8 9 10 11 12 13 Z s s s s s s s s s s s s s 1 5 5.48 4 5.48 3.16 5.48 3 3.16 3.46 6.48 5.48 5.29 3.87 Z s s s s s s s s s s s s s 1 5 4 4.90 5.48 8 5.48 5.92 8 4.90 4.90 5.48 6.93 3.87 Z s s s s s s s s s s s s s 1 5 5.48 5.48 4 3.16 4 5.92 3.16 6.48 3.46 4 2.45 7 Z s s s s s s s s s s s s s 2 5 5 4.47 5.48 4.90 6.48 3 5.29 5 6.48 3.46 3.87 4.47 Z s s s s s s s s s s s s s 2 5 5 6 5.48 6.48 4.90 5.92 6.92 5 4.90 6.48 7 4.47 Z s s s s s s s s s s s s s 2 5 5 4.47 4 3.46 3.46 5.92 2.45 5 3.46 4.90 3.87 6 Z s s s s s s s s s s s s s 3 5 5 4.47 5.48 3.46 5.48 3 3.87 3 6.93 4.47 3.16 4.47 Z s s s s s s s s s s s s s 3 5 5 4.47 5.48 8.48 5.48 5.92 7 5.92 5.29 6 8 4.47 Z s s s s s s s s s s s s s 3 5 5 6 4 2 4 5.92 3.87 5.92 2.45 4.47 3.16 6 • Using LWGA operator [23], the overall preference degree ( Z )of ith alter- k i native landfill site can be evaluated by aggregating evaluation of each landfill site against all criteria made by each of the DM [23-24]. This is expressed as follows: Z = LWGA (S , S ,··· , S ) (17) k i w α1 α2 αn w w w 1 2 n = (S ) ⊗ (S ) ⊗···⊗ (S ) . α1 α2 αn • Using Tables 5 and 9, for example, the overall preference degree Z of first 1 1 alternative site over all other alternative sites as assessed by DM can be ex- pressed as w w w w w w 1 2 3 4 5 13 Z = (S ) ⊗ (S ) ⊗ (S ) ⊗ (S ) ⊗ (S ) ···⊗ (S ) 1 1 5 5.48 4 5.48 3.16 3.87 = S . 4.40 334 Ajit Pratap Singh· Subodh Kant Dubey (2012) Similarly, other values can be calculated as Z = S , Z = S , Z = S , Z = S , 1 2 5.48 1 3 4.39 2 1 4.73 2 2 5.61 Z = S , Z = S , Z = S , Z = S . 2 3 4.26 3 1 4.32 3 2 5.74 3 3 4.26 • Using LGA operator again, the final preference degree Z of ith alternative site over all other alternative sites can be expressed as 1/3 Z = LGA ( Z, Z, Z ) = ( Z ⊗ Z ⊗ Z ) . i 1 i 2 i 3 i 1 i 2 i 3 i Thus the final preference degree of site 1, 2 and 3 can be expressed as Z = S , Z = S , and Z = S respectively. Therefore alternative site A > 4.47 2 5.60 3 4.80 2 A > A . The ranking values of all 3 landfill sites exhibit that landfill Site 2 has 3 1 the highest potential in the site selection process. It clearly demonstrates that these results are coinciding with the modified fuzzy utility approach proposed in this paper. The results obtained by the proposed method also coincide with other studies [5, 19]. However, the modified fuzzy utility approach presented here integrates the con- cept of fuzzy-utility method with multi-nomial logit theory to evaluate the ranking order of all possible landfill sites. It also evaluates the probability of selection of each alternative site which prepares a basis to adopt an overall strategy for selecting appro- priate landfill site for proper solid waste disposal and its management. The modified fuzzy utility approach proposed herein is simple, robust and capable of incorporating uncertainty along with flexible policies of solid waste management without any loss of information. It can be observed that both the methods discussed here not only allow decision makers to determine the ranking order of all alternatives but can also indicate the de- gree of preference of each alternative. It may be desirable in many situations where one may be willing to use next optimal alternative due to certain practical constraints. Therefore, they are more suitable and effective in dealing with subjective judgments in fuzzy environment. Both of the methods can successfully select the best alternative landfill site when the states of the system and/or utilities associated with alternatives are known in terms of a linguistic variable without any loss of original information. The selection and usefulness of different methods may also depend on data availabil- ity at the landfill sites, which may be available in different formats for its analysis. Moreover, the number of alternative sites and criteria involved in the process will also be the deciding factors to choose the most appropriate method. 4. Conclusion In this paper, a modified fuzzy utility method is used to evaluate the choice of landfill site for dumping of solid waste under different alternatives with a number of criteria which takes into account parameters related to accessibility and transportation; en- vironmental, geological and climatic condition; socio-economic conditions; land use pattern; and safety at the selected site. An ideal site is framed based on the perception of decision maker. All the alternative sites are evaluated in terms of their utility value Fuzzy Inf. Eng. (2012) 3: 313-338 335 with reference to an ideal site. The optimum utility value of each site is calculated to choose the best site among the given sites. In this methodology, individual utility value is calculated with reference to ideal site rather than inter utility value. It gives a better view of judging the alternatives, since degree of fulfillment of each alternative is known whereas in previous paper [19] competing degree of fulfillment was known (that is how much superior one alternative is with reference to other). Multi-nomial Logit model is used to calculate the probability of acceptance of each alternative, which is helpful in decision making process. Alternative site (A ) has 35.0% prob- ability of acceptance (chance that 35.0% decision makers will agree that site (A ) ) has should be chosen as a first landfill dumping site), whereas alternative site (A 33.68%. A stepwise and objective method to evaluate probability of acceptance is of great advantage which determines the ranking order of each site. Ranking order is given for all the alternatives and the best one is chosen which depends upon several fac- tors as mentioned in the results. The study demonstrates that alternative landfill site 2 is the best among the available three sites for the disposal of solid wastes. How- ever, if alternative site 2 is not available due to some constraints, decision maker can adopt alternative site 3 which is of second highest order of ranking with the prob- ability of acceptance of 33.68%. The methodology applied in this paper is easy to use and understand and clearly screen out the landfill alternatives which are not ap- propriate under the given set of criteria. The methodology presented in this paper is comparatively easier and more comprehensive in comparison with the earlier works of Singh and Vidyarthi [19] who initially developed fuzzy preference relation matrix and computed non-dominated degree of each alternative site for assessing the suit- able landfill. At the end of the analysis, most suitable landfill site is identified for a case study which satisfies all important requirements of the landfill as mentioned initially in Fig.1. The case study clearly shows that the utility value of alternative site 2 (i.e., A ) is 0.87 which is nearest to the reference value of 1 and hence it is the most suitable landfill site among all three possible sites. Due to higher utility value, the chosen landfill site for solid waste disposal will have the least impact on surrounding environment and therefore will be the most acceptable. The results obtained by the proposed method also coincide with other studies [5, 19, 23, 24]. It can be observed that proposed method not only allows decision mak- ers to determine the ranking order of all alternatives but can also indicate the degree of preference of each alternative. It may be desirable in many situations where one may be willing to use next optimal alternative due to certain practical constraints. Therefore, they are more suitable and effective in dealing with subjective judgments in fuzzy environment. However, a number of extensions and applications of this model may be possi- ble by taking opinion of more number of decision makers and by collecting detailed data pertaining to the factors such as geology, hydrogeology, climatology, land use pattern and socio-economic characteristics towards the protection of overall environ- ment. A more practical implementation of the model can be achieved by integrating geographical information system and multi-criteria evaluation techniques. 336 Ajit Pratap Singh· Subodh Kant Dubey (2012) Acknowledgements Writers are deeply grateful to Birla Institute of Technology and Science Pilani for providing necessary facilities to carry out this research work. Authors are also thank- ful to Central Pollution Control Board (CPCB) of India for its useful publication on hazardous waste management series and all other references cited in the text. 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Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Sep 1, 2012

Keywords: Decision support system; Uncertainty; Fuzzy utility value; Solid waste management; Landfill allocation

References