Estimates of USA Primary Pollutants Sources in million tons per year (Million Tons/Year). [Adapted from Thomas
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
Note: Edited in March 2021
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 191 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 261 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
Note: Edited in March 2021
\n'}],latestNews:[{slug:"intechopen-partners-with-ehs-for-digital-advertising-representation-20210416",title:"IntechOpen Partners with EHS for Digital Advertising Representation"},{slug:"intechopen-signs-new-contract-with-cepiec-china-for-distribution-of-open-access-books-20210319",title:"IntechOpen Signs New Contract with CEPIEC, China for Distribution of Open Access Books"},{slug:"150-million-downloads-and-counting-20210316",title:"150 Million Downloads and Counting"},{slug:"intechopen-secures-indefinite-content-preservation-with-clockss-20210309",title:"IntechOpen Secures Indefinite Content Preservation with CLOCKSS"},{slug:"intechopen-expands-to-all-global-amazon-channels-with-full-catalog-of-books-20210308",title:"IntechOpen Expands to All Global Amazon Channels with Full Catalog of Books"},{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"}]},book:{item:{type:"book",id:"3749",leadTitle:null,fullTitle:"Evolutionary Computation",title:"Evolutionary Computation",subtitle:null,reviewType:"peer-reviewed",abstract:"This book presents several recent advances on Evolutionary Computation, specially evolution-based\r\noptimization methods and hybrid algorithms for several applications, from optimization and learning to pattern recognition and bioinformatics. \r\n\r\nThis book also presents new algorithms based on several analogies and metafores, where one of them is based on philosophy, specifically on the philosophy of praxis and dialectics. In this book it is also presented interesting applications on bioinformatics, specially the use of particle swarms to discover gene expression patterns in DNA microarrays. Therefore, this book features representative work on the field of evolutionary computation and applied sciences. \r\n\r\nThe intended audience is graduate, undergraduate, researchers, and anyone who wishes to become familiar with the latest research work on this field.",isbn:null,printIsbn:"978-953-307-008-7",pdfIsbn:"978-953-51-5913-1",doi:"10.5772/234",price:159,priceEur:175,priceUsd:205,slug:"evolutionary-computation",numberOfPages:584,isOpenForSubmission:!1,isInWos:1,hash:"413cf10893f15955d121a4934493da16",bookSignature:"Wellington Pinheiro dos Santos",publishedDate:"October 1st 2009",coverURL:"https://cdn.intechopen.com/books/images_new/3749.jpg",numberOfDownloads:63677,numberOfWosCitations:80,numberOfCrossrefCitations:61,numberOfDimensionsCitations:115,hasAltmetrics:0,numberOfTotalCitations:256,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"May 16th 2013",dateEndSecondStepPublish:"June 6th 2013",dateEndThirdStepPublish:"September 10th 2013",dateEndFourthStepPublish:"December 9th 2013",dateEndFifthStepPublish:"January 8th 2014",currentStepOfPublishingProcess:5,indexedIn:"1,2,3,4,5,6,7",editedByType:"Edited by",kuFlag:!1,editors:[{id:"125844",title:"Prof.",name:"Wellington",middleName:"Pinheiro Dos",surname:"Santos",slug:"wellington-santos",fullName:"Wellington Santos",profilePictureURL:"https://mts.intechopen.com/storage/users/125844/images/4878_n.jpg",biography:"Wellington Pinheiro dos Santos holds a degree in Electrical and Electronics Engineering (2001) and MS in Electrical Engineering (2003) from the Federal University of Pernambuco, and Ph.D. in Electrical Engineering from the Federal University of Campina Grande (2009). 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He also has experience in the area of Computer Science, with an emphasis on Graphic Processing (Graphics), acting on the following themes: digital image processing, pattern recognition, computer vision, evolutionary computation, numerical methods of optimization, computational intelligence, computer graphics, virtual reality, game design and applications of Computing and Engineering in Medicine and Biology. 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Several studies have been carried out on the quantification of pollutants and analyzing their consequences on public health. It has been estimated that each year between 250 and 300 million tons of air pollutants enter the atmosphere above the United States of American [Dara, 2004; Onianwa, 2001; Stephen and Spencer, 1992]. Tropospheric pollution causes degradation of crops, forests, aquatic systems, structural materials, and human health. It was reported recently, that NOx air pollution is becoming a far reaching threat to USA National Parks and Wilderness Areas as these areas are suffering from harmful effects of oxides of nitrogen pollution [EDFS, 2003]. It has also been confirmed that NOx contributes to ground – level ozone (smog) pollution which can cause serious respiratory problems, especially young children and the elderly, as well as healthy adults that are active outdoors. Furthermore, the same report confirmed worsening ozone concentration in nearly all the national parks over the last ten years [EDFS, 2003]. Towards this, an assessment of new vehicles emission certification standards was carried out in metropolitan area of Mexico city and the results show that light duty gasoline vehicles account for most carbon (II) oxide and NOx emissions [Schifter
During a five-day period marked by temperature inversion and fog in London in 1952, between 3,500 and 4,000 deaths in excess of normal occurred with 1.3 ppm SO2 level recorded [O’Neill, 1993; ACGIH, 1991]. SO2 is oxidized to SO3 in the atmospheric air by photolytic and catalytic processes involving ozone, oxides of nitrogen (NOx), and hydrocarbon (HC), giving rise to the formation of photochemical smog [Dara, 2004].
In contrast to NO2, SO2 is deleterious to plant life [Haagen-Smith, 1952; Molski and Dmuchowski, 1986]. Air pollution causes the decline in Eastern Europe [Nihlgard, 1985; Schutt and Cowling, 1985]. Area with extensive large-scale forest decline correlate with the areas where SO2 concentration were elevated [ECE, 1984; Molski and Dmuchowski, 1986], therefore If the concentration of SO2 is higher along with other gaseous pollutants (NOx, O3, HC, organic and inorganic peroxides etc) in the troposphere and continues to accumulate over time, the overall concentration can have a negative effect on health, vegetation and structures [Abdul Raheem, 2007; Abdul Raheem et al., 2009a].
Surface concentration of ozone has been reported to be on the increase in the last decades in northern hemisphere [Vautard et al., 2007] and in Southeast Asia [Jaffe, 1999]. A rise of about 26% in the ambient concentration of ozone has been reported in Taiwan between 1994 and 2003 [Chou et al., 2006]. High levels of ambient air ozone can cause serious damage to health. The health hazards include shortness of breath, nausea, eye and throat irritation, and lung damage [Menezes and Shively, 2001]. Identification of air pollution source characteristics is an important step in the development of regional air quality control strategies. Receptor modeling, using measurements of pollutant concentrations at one or more sample sites, is often a reliable way to provide information regarding source regions for air pollution [Watson, 1984]. One of such receptor-modeling technique is principal component analysis (PCA) [Einax and Geiss, 1997; Jackson, 1991; Norman, 1987]. This is often combined with multiple linear regression (MLR), principal component regression (PCR), and partial least square regression, which have been demonstrated as powerful tools for handling several environmental problems, especially source apportioning [Otto, 1999; Timm, 1985; Vogt, 1989].
In recent years, certain statistical techniques that incorporate the influence of meteorological variables have been applied to asses the trend in ozone levels in the ambient air [Bakken et al., 1997]. One common approach is the use of a parametric regression model to link some characteristics of ozone, such as the mean level of ozone to meteorological variables. Other scientists have equally used PCA to pattern the spatial and temporal variations of ozone and to identify the important factors influencing ozone concentration [Klaus et al., 2001; Lengyel et al., 2004; Pissimanis et al., 2000]. Different subregions have, however, been identified where ozone concentration exhibited characteristic spatial and temporal patterns based on the differences arising from the interaction of their respective meteorological conditions with anthropogenic effects [Alvarez et al., 2000]. More specifically, Bloomfield et al., (1996) established a nonlinear regression model for hourly ozone data in the Chicago area in which meteorological variables, seasonality, and a trend term were all implicated. Cox and Chu (1992), on their part, proposed a model for the daily maxima of hourly ozone concentrations based on the Weibull distribution in which the scale parameter is allowed to vary as a function of meteorological conditions. On the other hand, Menezes and Shively (2001) used a multivariate approach to estimate the long-term trend in the extreme values of tropospheric ozone in Houston and Texas. They found that there is a downward trend in the probability of an exceedance followed by a relative flat trend. Shively and Sager (1999) extended the work of Cox and Chu (1992) as well as Bloomfield et al., (1996) by using nonparametric regression models to model ozone. The use of multivariate methods was further supported by Lengyel et al., (2004) who analyzed air quality data of which the hidden structure was uncovered by factor analysis and modeled ozone concentration using MLR, PLS, and PCR. While PCA is a very useful tool for selection of properties and different qualities processes leading to a linear model of the data, MLR and PCR or PLS can predict ozone concentration with an error below 2, 5, and 1ppb levels, respectively.
The aim of the present investigation was therefore to analyze environmental data gathered on the daily monitoring of ambient ozone, oxides of nitrogen, and sulfur (IV) oxide at five monitoring sites in Lagos and four monitoring sites in Ilorin, Nigeria. The data were collected between early morning and late evening of the day, and they covered both dry and wet seasons from years 2003 to 2006. The study also included establishment of prediction models on the influence of meteorological parameters on the seasonal variation of concentration of gaseous pollutants in the two Nigerian cities and comparison of measured and modeled concentration values of ozone [Abdul Raheem et al., 2009b].
Air is all around us, odourless, colourless and essential to all life on earth as it acts as a gaseous blanket, protecting the earth from dangerous cosmic radiation from outer space. It helps in sustaining life on earth by screening the dangerous ultraviolet (UV) radiations (< 300nm) from the sun and transmitting only radiations in the range 300nm to 2500nm, comprising of near UV, visible and near infrared (IR) radiations and radio waves (0.01 to 4 x 105nm) [Smart, 1998]. The atmosphere also plays a vital role in maintaining the heat balance on the earth by absorbing the IR radiation received from the sun and re-emitted by the earth. In fact, it is this phenomenon, called “the greenhouse effect”, which keeps the earth warm enough to sustain life on the earth. Yet, the air is actually a combination of gaseous elements that have a remarkable uniformity in terms of their contribution to the totality of life. Thus, oxygen supports life on earth; nitrogen is an essential macro - nutrient for plants; and carbon (IV) oxide is essential for photosynthetic activity of plants. Moreover, atmosphere is a carrier of water from the ocean to land, which is so vital for the hydrological cycle. Any major disturbance in the composition of the atmosphere resulting from anthropogenic activities may lead to disastrous consequences or may even endanger the survival of life on earth [Dara, 2004]. The constituent elements are primarily nitrogen and oxygen, with a small amount of argon. Below 100km, the three main gaseous elements, which account for about 99.9% of the total atmosphere, are N2, O2 and Ar and they have concentration by volume of 78%, 21%, and 0.93% of respectively [Stanley, 1975]. The presence of trace amounts of other gases would account for the remaining 0.07%. These remaining trace gases exist in small quantities and they are measured in terms of a mixing ratio. This ratio is defined as the number of molecules of the trace gas divided by the total number of molecules present in the volumes sampled. For example, ozone (O3), carbon (IV) oxide (CO2), oxides of nitrogen (NO2 + NO) as NOx and chloro fluoro carbons (CFCs) are measured in parts per million by volume (ppmv), parts per billion by volume (ppbv) as well as microgram per cubic meter (µgm-3), [Dale, 1976]
Air pollution has been defined [World Bank, 1978] as ‘the presence in the outdoor atmosphere of one or more contaminants such as dust, fumes, gas, mist, odour, smoke, or vapour in such quantity, characteristics and duration as to make them actually or potentially injurious to human, plant or animal life, or property, or which unreasonably interferes with the comfortable enjoyment of life and property’. Pollution on the whole is caused principally by human activities, though it can also be a natural process. Air pollution arises from people’s economic and domestic activities like modern agriculture which requires agrochemicals. Industrial activities are responsible for wide range of pollution. Thermal power station, burning fossil fuel and moving vehicles emit harmful pollutants like sulphur (IV) oxide, nitrogen (II) oxide and carbon (IV) oxide. Some of these emitted gases have been responsible for acid – rain, global warming and malfunctioning of human / animal’s haemoglobin [Stanley, 1975]. Other causes arising from human activities include inappropriate solid waste disposal, gas flaring and oil exploration. Air pollution can also arise from natural causes such as volcanic eruption, whirlwind, earthquake, decay of vegetation, pollen dispersal, as well as forest fire ignition by lightning.
These are substances introduced into the environment in an amount sufficient to cause adverse measurable effects on human beings, animals, plant, vegetation or materials. Pollutants are referred to as primary pollutants, if they exert the harmful effects in the original form in which they enter the atmosphere e.g. CO, NOx, HCs, SOx, particulate matter and so on. On the other hand, secondary pollutants are products of chemical reactions, among primary pollutants are ozone, hydrogen peroxide, peroxyacetylnitate (PAN) and peroxybenzoyl nitrate (PBN). Classification of pollutants can also be according to chemical compositions i.e. organic or inorganic pollutants or according to the state of matter i.e. gaseous or particulate pollutants. Air pollution is basically made up of three components and these are source of pollutants, the transporting medium, which is air and target or receptor which could be man, animal, plant and structural facility.
Generalized Plot of Atmospheric Concentrations of Species Involved In Smog Formation As A Function of Time of The Day
The various chemical and photochemical reactions taking place in the atmosphere mostly depend upon the temperature, composition, humidity and the intensity of sun light. Thus the ultimate fate of chemical species in the atmosphere depends upon these parameters. Photochemical reactions take place in the atmosphere by the absorption of solar radiations in the UV region. Absorption of photons by chemical species gives rise to electronically excited molecules. These reactions are not possible under normal laboratory conditions except at higher temperature and in the presence of chemical catalysts [Hansen
Generalized Scheme For The Formation Of Photochemical Smog
return to the original state [Dara, 2004]. Any of these transformation pathways may serve as an initiating chemical step or a primary process. The three steps involved in an overall photochemical reaction are Absorption of radiation, Primary reactions and Secondary reactions.
Smoggy atmosphere show characteristics variations with time of the day in levels of NO, NO2, hydrocarbons, aldehydes and oxidants. A generalized plot showing these variations is shown in Figure 1. This shows that shortly after dawn the level of NO in the atmosphere decreases markedly, a decrease which is accompanied by a peak in the concentration of NO2. During the mid – day the levels of aldehydes and oxidants become relatively high, however, the concentration of total hydrocarbons in the atmosphere peaks sharply in the morning, then decreases during the remaining daylight hours. The variations in species concentration shown in the above Figure 1 may be explained by a generalized reaction scheme in Figure 2. This is based on the photochemically initiated reactions which occur in an atmosphere containing oxides of nitrogen, reactive hydrocarbons, and oxygen. The various chemical species that can undergo photo - chemical reactions in the atmosphere include NO2, SO2, HNO3, N2, ketones, H2O2, organic peroxides and several other organic compounds and aerosols; the time – variations of which are explained in a group of overall reactions first proposed by Friedlander nd Seinfeld (1974)
Primary photochemical reaction:
NO2 + hע NO + O
Reactions involving oxygen species
O2 + O + M O3 + M
(M is an energy absorbing third body)
O3 + NO NO2 + O
Production of organic free radicals from hydrocarbons,
RHO + RH R.\n\t\t\t\t
o3 + RH R. + other products
R. is a free radical, which may or may not contain oxygen
Chain propagation, branching and termination
NO + R. NO2 + R1\n\t\t\t
In this case R. contains oxygen and oxidizes NO. It is one of many chain propagation reactions, some of which involve NO
NO2 + R. products (e.g. PAN)
A number of specific reactions are involved in the above overall scheme for the formation of photochemical smog, which is smoke and fog [Thomas
NO2 + hט NO + O
This leads to several reactions involving oxygen and nitrogen oxide species. Examples of such reactions are given below:
O + O2 +M O3 + M
O + NO + M NO2 + M
O + NO2 NO + O2
O3 + NO NO2 + O2
O + NO2 + M NO3 + M
O3 + NO2 NO3 + O2
Pollutant Source | CO | NOx | HC | SOx | Particulate | Total |
Transportation | 111.0 | 11.7 | 19.5 | 1,0 | 0.7 | 143.9 |
Fuel Combustion (Stationary Source) | 0.8 | 10.0 | 0.6 | 26.5 | 6.8 | 44,7 |
Industrial Processes | 11.4 | 0.2 | 5.5 | 6.0 | 13.1 | 36.2. |
Solid wastes disposal | 7,2 | 0.4 | 2.0 | 0.1 | 1.4 | 11.1 |
Miscellaneous | 16.8 | 0.4 | 7.1 | 0.3 | 3.4 | 28.0 |
Total | 147.2 | 22.7 | 34.7 | 33.9 | 25.4 | 263.9 |
Estimates of USA Primary Pollutants Sources in million tons per year (Million Tons/Year). [Adapted from Thomas
The determinations of the concentrations of total oxidants (undertaken as O3), NOX and SO2 in the ambient air were carried out between 2003 and 2006 to cover the two seasons; dry season (November to April) and rainy season (May to October) using standard methods. Total oxidants were determined by buffered potassium iodide solution method proposed by Byers and Saltzman (1958). Determination of oxides of nitrogen concentrations were done using the Intersociety Committee Method of Analysis (1972) which is based on the Griess–Saltzman (1954) colorimetric, azo dye forming reagent while oxides of sulphur were determined by conductivity measurements as proposed by Stanley (1975). The application of these techniques to monitoring required the preparation of the following range of reagents as sampling and absorbing solutions as well as preparation of calibration curves. All the chemicals used in the preparation of sampling and absorbing solutions were of analytical grade. The reagents had been prepared and seen stable for months prior to their use. The validation of all procedures was further confirmed using LaMotte standard air sampler.
The absorbing solution used for trapping ozone was 1% KI buffered at 6.8 ± 0.2. The standard solution and calibration curve were prepared as follows: 4.09ml of standard 0.001M iodine solution was taken and diluted to 100ml with distilled water. 10ml of this solution was taken and further diluted to 100ml with absorbing reagent and this was labelled solution A. 1ml of A was further diluted to 100ml using absorbing reagent, which gives solution B. This provides a calibrated iodine solution equivalent to 1.92μg of ozone per ml [Stanley, 1975 and Dara, 2004] as ozone reacts with iodine ion in neutral buffer solution according to the following equation:
O3 + 3I - + H2O → I3- + 2 OH - + O2\n\t\t\t\t
The tri – iodide ion liberated has an intense yellow colour. The standard solution was always prepared freshly when needed. A series of standard solutions prepared from above were used to obtain calibration curve. The absorbance measurement was carried out at 352nm. The calibration curve is shown in Figure 3 and it has regression value of 0.9972
Total oxidants calibration curve
The set-up of the high volume sampler is given in Figure 4. To follow strict guidelines needed when monitoring for criteria pollutants as discussed earlier, our equipment is validated with LaMotte air sampler (Figure 5) purchased from LaMotte & Company, USA, for capability, repeatability and reliability needed to collect accurate data, and operation of the equipment within our established methods. The impinger used is a big boiling tube of capacity 250cm3 with 42mm diameter. Teflon tubing’s used as delivery tubes along with
Picture of The Sampling train
Lamotte Air- Sampler
glass tubing that serves as inlet for the ambient air sampled. Silicone grease was used to make the set up airtight by it application to all necessary joints. The air was sampled at the rate of 1 dm3 min -1 with absorbing solution fixed at 30 cm3 as found appropriate and used [Abdul Raheem et al., 2009c], after each sampling for one hour, the impinger was carefully removed and the sample transferred quantitatively into the sample bottle for analysis. The impinger was wrapped with aluminum foil to avoid sunrays’ interference. In order to determine pollutant variability over daytime periods, air samples were taken at each location over ten defined sixty - minute periods for any sampling day. The ten sampling periods were spaced equally between approximately 6 am and 6 pm to reflect morning, afternoon and evening. Minimum of 30 samples were collected at each site per week for the pollutants monitored. The time of sixty minutes was found to be optima from the preliminary investigations for the quantitative sampling of these toxic gases within the environment [Abdul Raheem et al., 2009c]. All samplings reported were carried out in triplicates. As the road traffic is the common source of pollution cutting across all sites classes, the average traffic volume were determined for all sampling zones. Traffic count was manually done, counting the vehicles passing on the road for 10 minutes in every hour from which hourly traffic was calculated [Abam and Unachukwu, 2009].
The daily minimum and maximum temperatures were between 23 C and 36.5 C, throughout the sampling period.
A freshly prepared absorbing solution serves as sample reference or blank solution in order to take care of any impurities during preparation. Absorbance of samples for total oxidants was measured at 352 nm with UV / Visible spectrophotometer. The concentration was read out in μg / 10ml from the reference plot of which one of the examples is shown in Figure 3. The concentrations were converted to μgm-3 or ppm or ppb using appropriate conversion factor.
for 1 μgm-3 of ozone, the ppb value will be
QUANTITY | CONTENTS | CODE |
2 × 120 Ml | Total oxidants reagent #1 | 7740-J |
30Ml | Total oxidants reagent #2 | 7741-G |
30Ml | Total oxidants reagent #3 | 7742-G |
3 | Test Tubes, 5mL, w/ caps | 0230 |
1 | Total oxidants in Air Comparator | 7739 |
LaMotte total oxidants in air test kit code 7738. Total oxidants in air calibration chart** [LaMotte 6.05] Comparator index number. ** Values in ppm.
10 mL of reagent #1 was put into impinging tube, followed by 2 drops of reagent #2 added and swirled to mix then 2 drops of reagent #3 added and also swirled to mix. The impinging apparatus was connected to intake of the sampling pump as shown in Figure 5 such that the long tube was immersed in the absorbing solution. The impinging tube was covered with foil to protect it from light while sampling. The flow meter of sampling apparatus was adjusted to collect air at 1.0 Lm-1 rate. The sampling continued until 15 minutes when a measurable pink colour developed. The impinging tube was disconnected from the pumping apparatus and the contents poured into a clean test tube (0230). The test tube was later inserted into the total oxidants in air comparator (7739) and the sample colour was matched with an index value. The index value was recorded and the calibration chat was used to convert the index readings into concentration of the pollutant in the atmosphere in parts per million.
Time(min) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
5 | 0.14 | 0.36 | 0.72 | 1.08 | 1.44 | 2.88 | 4.32 | 5.76 |
10 | 0.07 | 0.18 | 0.36 | 0.54 | 0.72 | 1.44 | 2.16 | 2.88 |
15 | 0.05 | 0.12 | 0.24 | 0.36 | 0.48 | 0.96 | 1.44 | 1.72 |
Total oxidants in air calibration chart** [LaMotte 6.05] Comparator index number. ** Values in ppm.
The absorbing solution used for trapping NOx was Saltzman solution which is an azo dye forming reagent.
The standard solution and calibration curve were prepared as follows: 2.16g of sodium di - oxo nitrate (III), NaNO2 was dissolved in 1000 cm3 volumetric flask and the solution labeled A. 1ml of solution A was measured out into 100 ml volumetric flask and the solution made up to the mark. This solution of concentration 0.0216 gL-1 was labeled B. 1ml of B was added to 100 ml volumetric flask and distilled water added to the mark to give 0.000216 gL-1 solution C. 10ml of solution C was added to 100ml volumetric flask and filled to the mark with distilled water to give solution of concentration 0.0000216 gL-1 labeled D. Further dilutions of the last two solutions C and D were used for calibration plot. As the standardization was based on the empirical observation that 0.72 mole of NaNO2 produces the same colour as 1mole of NO2 [Hesketh, 1972]. In other words, 1ml of the 0.000216 gL-1 working standard which contains 0.216μg of NaNO2 should be equivalent to 0.2 μg of NO2. Series of standard solutions prepared in 10 ml volumetric flasks from solutions C and D above were allowed to stay for 15 minutes for colour development and the spectra run at 550 nm to obtain a set of absorbance value which were recorded against known concentrations. The formation of red azo dye of which the absorbance is picked at 550 nm can be explained according to the equation in Figure 6 However, a plot of absorbance against concentration in μg / 10 ml was made, a straight line graph obtained with regression value of 0.9962 as shown in Figure 7.
Equation showing the formation of azo dye
The procedure for sampling is as given above
Oxides of nitrogen (nox) calibration curve
The absorbing solution serves as sample reference or blank solution in order to take care of any impurities during preparation. Absorbance of samples for oxides of nitrogen was measured at 550nm with UV / Visible spectrophotometer. The concentration was read out in μg / 10ml from the reference plot of which an example is shown in figure 7. The concentrations were converted to μgm-3 or ppm or ppb of which the conversion factors are explained hereafter.
for 1 μgm-3 of NOx as NO2, the ppb value will be
QUANTITY | CONTENTS | CODE |
2 × 120 mL | Nitrogen (Iv) oxide reagent #1 Absorbing solution | 7684-J |
30 mL | Nitrogen (Iv) oxide reagent #2 | 7685-G |
10g | Nitrogen (Iv) oxide reagent #3 powder | 7688-D |
2 | Test tubes,10mL, glass, w/caps | 0822 |
1 | Spoon, 0.005g, plastic | 0696 |
1 | Pipet, droping, plastic | 0352 |
1 | Nitroge (IV) oxide in air comparator | 7689 |
1 | Tubing | 23609 |
1 | Pipet | 30410 |
1 | Needle | 27336-01 |
LaMotte nitrogen (IV) oxide in air test kit code 7690
10mL of reagent #1 i.e. absorbing reagent was poured into the impinging tube, a gas bubbler impinger (0934). The impinging apparatus was connected to the intake of air sampling pump and the long tube was immersed in the absorbing solution. The special adaptor was attached to the intake of the pump to sample at 0.2Lm-1 while the sampling was done for 20 minutes when a measurable amount of nitrogen (IV) oxide was absorbed. At the end of the sampling period the contents of the impinging tube was poured into test tube (0822). The pipette (0352) was used to add a drop of reagent #2, the test tube capped and mixed after which the 0.05g spoon was used to add 0.05g of reagent #3. The test tube capped and the solution left for 10 minutes for colour development after which the test tube was placed into comparator (7689) and the sample colour matched to index of colour standards. The index number which gave the proper colour matched was recorded and the calibration chart used to convert the index read to concentration of nitrogen (IV) oxide in ppm.
Comparator index number | ||||||||
Time (min) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
1 | 0.00 | 2.8 | 7.0 | 14.0 | 21.0 | 28.0 | 42.0 | 56.0 |
5 | 0.00 | 0.56 | 1.40 | 2.80 | 4.20 | 5.60 | 8.40 | 11.20 |
10 | 0.00 | 0.28 | 0.70 | 1.40 | 2.10 | 2.80 | 4.20 | 5.60 |
15 | 0.00 | 0.19 | 0.47 | 0.93 | 1.40 | 1.87 | 2.80 | 3.74 |
20 | 0.00 | 0.14 | 0.35 | 0.70 | 1.05 | 1.40 | 2.10 | 2.80 |
Nitrogen (IV) oxide in air calibration chart
The absorbing solution used for trapping SO2 was 0.3M H2O2 solution buffered at pH 5 ± 0.2.
The standard solution and calibration curve were prepared as follows:
0.1M H2SO4 was used as parent standard solution. All other lower concentrations were prepared from serial dilution of 0.1M H2SO4. 0.1M H2SO4 was standardized by titration against Na2CO3 using methyl orange as indicator. The conductivity measurement of each of the concentrations of H2SO4 (0.001 – 0.01M) obtained from serial dilution were taken, using Hanna Instrument EC 214 conductivity model. A graph of conductivity values in Siemens per centimeter (Scm-1) against concentrations of H2SO4 in mol dm-3 was plotted. The data gave a straight line which passes through the origin with regression value of 0.9874. The calibration curve so obtained is shown in Figure 8. This was used as a working curve for the determination of SO2 during the analysis of samples.
The procedure for sampling others remained except the flow rate that was increased to 2 Lmin-1 for optimization purpose [Abdul Raheem et al., 2009c].
Sulphur (iv) oxide calibration curve
Conductivity measurements were undertaken using the Hanna Instrument Model E 214 conductivity meter.From the sample and reference solutions 20 cm3 volume was measured respectively into a liquid sample holder test tube of Hanna model conductivity meter. The concentrations in mol dm-3 of H2SO4 formed from SO2 of the samples were read out from the reference plot (Fig.8), the concentrations obtained in mol dm-3 were converted to parts per million or parts per billion or microgram per cubic meter (ppm or ppb or μgm-3) as shown below using appropriate conversion factor. Equation of reaction for formation of H2SO4 from SO2 is shown below:
for 1 moldm-3 of SO2, ppm value will be
in μgm-3, the value becomes:
QUANTITY | CONTENTS | CODE |
2 × 250 mL | Sulphur (IV) oxide absorbing solution | 7804-K |
15g | Sulphur (IV) oxide reagent #1 | 7693-E |
30mL | Sodim hydroxide, 1.0 N | 4004PS-G |
60mL | Sulphur (IV) oxide passive bubbler indicator | 7805-H |
2 | Pipets, 1.0mL, plastic | 0354 |
2 | Test tubes, 5 mL, plastic, w/caps | 0230 |
2 | Test tubes, Hester, w/caps | 0204 |
1 | Spoon, 0.25g | 0695 |
1 | Dispenser caps | 0693 |
1 | Sulphur (IV) oxide passive bubbler comparator | 7746 |
LaMotte sulphur (IV) oxide in air test kit code 7714
10mL of Sulphur (IV) oxide absorbing solution was added to impinging tube and connected to the impinging apparatus as shown in Figure 5. The long tube was immersed into the absorbing solution. Sampling was done at 1.0 Lpm for 60 minutes or 90 minutes. The impinging apparatus was covered with foil to protect it from light. At the end of the sampling time the small test tube (0230) was filled to the line with the sample and 0.25g spoon was used to add a level measured of Sulphur (IV) oxide reagent #1. The test tube containing the mixture was capped and vigorously shaken to dissolve the powder. A 1 mL pipette was used to add 1mL sodium hydroxide, 1.0N, to the same small test tube, capped and inverted several times to mix. The other 1mL pipette was also used to add 2mL (2 measures) of Sulphur (IV) oxide passive bubbler indicator (7805) to a large test tube (0204). The contents of the small test tube were poured into the large test tube containing the indicator. Immediately the tube capped and inverted six times, holding the cap firmly in place with the index finger. After waiting for 15 minutes, the test tube was placed into the Sulphur (IV) oxide passive bubbler comparator (7746). The sample colour matched with the standard colour and the index number read and recorded from the comparator. The index number was converted to concentration in ppm using the calibration chart provided.
Comparator index number | ||||||||
Time (min) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
10 | 0.00 | 0.19 | 0.29 | 0.38 | 0.48 | 0.57 | 0.67 | 0.76 |
30 | 0.00 | 0.06 | 0.10 | 0.13 | 0.16 | 0.19 | 0.22 | 0.25 |
60 | 0.00 | 0.03 | 0.05 | 0.06 | 0.08 | 0.10 | 0.11 | 0.13 |
90 | 0.00 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 |
Sulphur (IV) oxide in air calibration chart.** Values in ppm.
The impinger was well rinsed with distilled water and properly wrapped with foil paper before each use. The tubing’s and corks in the sampling train were checked before and during sampling, in case they had become slackened, however silicone grease was used to increase the pressure by making them air tight.
The absorbing reagents were always prepared freshly ahead of sampling for the solution to stabilize. They were stored in amber coloured bottles and refrigerated because of light interference. They were always allowed to thaw and assume the 25 C temperature before use.
Lengthy contact with air by the absorbing reagent was avoided during both preparation and use to prevent absorption of the oxides. The absorbance of the reagent blank was deducted from that of the samples where the machine could not be adjusted to zero to avoid matrix error, especially with the conductivity meter.
For the nitrogen oxides determination, a gas bubbler impinger (fritted gas bubbler) was used instead of a general purpose impinger as absorption tube. The general purpose impinger has been reported to give low absorption efficiency with oxides of nitrogen [ICMA, 1972; Onianwa et al., 2001; Saltzman, 1954]. However the results were corrected and correlated with the fritted bubbler as well as standardized absorbing solution imported from LaMotte and Company, USA.
Greatest accuracy has been reported to be achieved by standardizing the sampling train with accurately known gas sample in a precision flow dilution system like a permeation tube [Dara, 2004]. Due to lack of the apparatus necessary for the standardization of the train, the actual collection efficiency is not known. However with the use of LaMotte sampling pump with inbuilt flow meter and standardized reagents, we recorded high collection efficiency at sites with increase concentrations of samples.
This is already discussed extensively in Abdul Raheem, 2007 and Abdul Raheem et al., 2009a,b,c. Typical tables are shown to show the typical measurements concentration results and the meteorological data
Start of sampling | End of sampling | OX (ppb) | NOx (ppb) | SO2 (ppb) | RELHUM (%) | WND ms-1 | DWND (oC) | AIRTEMP (oC) | Sun ExpWm-2 |
6.30am | 7.30am | 29.08 ±11.73 | 1.47 | 7.83 | 78.17 | 27.60 | 144.60 | 22.70 | -1.55 |
7.45am | 8.45am | 29.72 ±10.5 | 3.44 | 6.54 | 71.67 | 36.30 | 156.40 | 23.20 | 0.51 |
9.00am | 10.0am | 29.71 ±5.57 | 0.43 | 4.17 | 57.30 | 44.60 | 156.50 | 27.90 | 8.63 |
10.15am | 11.15am | 33.11 ±5.51 | 1.67 | 4.42 | 53.30 | 42.00 | 160.50 | 29.80 | 12.61 |
11.30am | 12.30pm | 46.69 ±7.49 | 1.73 | 6.27 | 42.00 | 42.30 | 153.20 | 31.50 | 15.36 |
12.45pm | 1.45pm | 69.94 ±15.45 | 1.04 | 7.36 | 38.67 | 43.40 | 154.00 | 32.80 | 16.09 |
2.00pm | 3.00pm | 35.55 ±11.21 | 2.46 | 8.84 | 35.50 | 41.60 | 160.00 | 34.30 | 13.39 |
3.15pm | 4.15pm | 21.44 ±6.31 | 2.46 | 7.62 | 37.17 | 39.40 | 167.90 | 33.80 | 10.16 |
4.30pm | 5.30pm | 17.62 ±3.13 | 2.69 | 9.52 | 39.00 | 39.30 | 178.00 | 33.00 | 5.66 |
5.45pm | 6.45pm | 11.56 ±2.19 | 2.91 | 9.11 | 42.67 | 37.60 | 176.70 | 31.30 | 0.86 |
Dry season environmental data for Ilorin
Start of sampling | End of sampling | OX (ppb) | NOx (ppb) | SO2 (ppb) | RELHUM (%) | WND ms-1 | DWND (oC) | AIRTEMP (oC) | Sun Exp Wm-2 |
6.30am | 7.30am | 14.26 | 12.40 | 10.72 | 90.58 | 3.07 | 107.02 | 25.99 | -1.65 |
7.45am | 8.45am | 22.92 | 5.89 | 7.20 | 87.58 | 4.67 | 156.38 | 26.38 | 2.80 |
9.00am | 10.0am | 28.95 | 5.39 | 11.15 | 73.75 | 7.09 | 189.02 | 29.05 | 8.90 |
10.15am | 10.0am | 46.86 | 5.66 | 14.82 | 67.92 | 7.35 | 182.63 | 30.25 | 11.00 |
11.30am | 12.30pm | 43.21 | 6.41 | 10.51 | 63.50 | 8.76 | 170.67 | 31.30 | 12.06 |
12.45pm | 1.45pm | 85.31 | 5.68 | 12.74 | 60.33 | 10.11 | 159.55 | 32.00 | 17.30 |
2.00pm | 3.00pm | 73.77 | 6.45 | 16.62 | 60.08 | 10.36 | 155.00 | 31.98 | 15.10 |
3.15pm | 4.15pm | 26.06 | 6.84 | 15.47 | 62.67 | 10.94 | 163.79 | 31.37 | 13.20 |
4.30pm | 5.30pm | 12.23 | 5.72 | 16.48 | 67.00 | 10.21 | 165.22 | 30.38 | 10.70 |
5.45pm | 6.45pm | 8.58 | 6.90 | 19.21 | 72.75 | 8.99 | 166.64 | 29.20 | 3.30 |
Dry season environmental data for Lagos
This is showing typical results of statistical modeled analysis of Ilorin and Lagos during dry season MLR with backward selection in stepwise mode (without intercept) results in the following equation:
OXILO = 6.092 x SO2 + 0.657 x RHUM – 2.653 x ATEMP + 4.385 x SUNEXP
Where R = 0.981, F (4, 6) = 38.389, p < 0.000
This shows that only four of the variables are found to be significant for retention in the model.
MLR using backward selection in stepwise mode (without intercept) results in the following equation:
OXLag = 1.679 x ATEMP + 5.622 x SUNEXP – 8.079 x WND
where, R = 0.961, F (3, 7) = 27.874, p < 0.000
MLR shows that only three of the variables are significant for retention in the model.
A table Comparing the ozone measured concentration with calculated results from MLR model equations
ILORIN | LAGOS | |||
RAIN | DRY | RAIN | DRY | |
MEASURED | 21.86 ± 2.47 | 32.44 ± 5.13 | 9.87 ± 0.99 | 36.22 ± 5.76 |
MODELED | 16.12 ± 1.86 | 44.32 ± 4.25 | 9.89 ± 0.82 | 36.29 ± 3.87 |
MLR equation modeled results for ozone compared with monitored results for the two cities of interest during rainy and dry seasons (ppb)
The direction and spatial extent of transport and the relative contribution of transported ozone and precursors to individual downwind areas are highly variable. A number of factors influence site to site differences in ozone concentrations, including sources of precursor’s emissions and meteorological conditions.
Data analysis also reveals that NOx and SO2 as well as volatile organic compounds contribute to ozone formation and this is in accordance with other researchers [Winer
On the whole the chemometric multivariate analysis results confirmed our experimental results and unfold the fact that meteorological influence plays a major role in the atmospheric chemistry of ozone.
Finally, these results and analysis suggested that ozone acting in concert with other pollutants need to be recognized as important health and ecosystem related air quality concern in Nigeria. Based on increasing evidence on regional transport of ozone all over the world, there is need for recognition that ground – level ozone would be an appropriate issue to be considered by the Nigerian government. In particular, a proactive measure has to be formulated towards reducing NOx and SO2 and by consequence O3 in Nigeria.
A well-founded risk assessment is a mandatory stage in the effective implementation of almost any project. This chapter discusses credit risk assessments. Credit risk assessment includes determining whether or not to give a loan to a borrower and what is the probability of bankruptcy or inability to service a loan due to financial problems. Despite the fact that during this process the lender gets a lot of information from the borrower, there is no unambiguous rule for decision-making.
\nThere are various models for assessing these risks. Before the 2008 financial crisis, emphasis was placed on developing models for assessing the financial stability of borrowers. But when, after this crisis, many companies faced a significant risk of bankruptcy, the vector of model developers took the direction to developing effective forecast models.
\nExisting forecasting models can be classified into two main groups: statistical and theoretical. A description of these models can be found in the literature (see, e.g., [1]). However, in a number of cases, these models are unacceptable. Basically, they are not suitable for assessing credit risks for corporations in developing countries, as well as for assessing the risks of lending to investment projects.
\nLet us dwell very briefly on the reasons for the unsuitability of these models for corporations in developing countries (a more detailed analysis can be found, e.g., in [2]). For statistical models:
An insufficient defaulting history does not provide a relevant background for assessment of the credit risk.
Problems arise with classification process too: a borrower may have so-called partial default status.
For theoretical models:
A potential problem that could arise when applying this model is that majority of companies’ stock is not traded on a stock exchange. In this case, a rapid evaluation of the market value of the assets is difficult.
It is very important to note that companies whose shares are not traded on the stock exchange are not so rare in developed countries.
\nLet us consider in more detail the credit risks when financing investment projects. It is known that an investment project involves planning over time of three main cash flows: investment, current (operating) expenses, and income.
\nWhen implementing an investment project, the investor never has a comprehensive risk assessment, since frequent changes in the dynamically developing world cannot be foreseen. Therefore, there is an unforeseen circumstance not taken into account by the project (e.g., a catastrophe), which nevertheless happened and disrupt the investment process. At the same time, the investor must be as informed as possible in order to assess the risk of his investment decisions both at the stage of project development and during the investment process itself. In addition, it is important to keep in mind that prices and volumes of products sold, as well as cash values for materials, raw materials, and other goods and services, in the future can radically differ from their expected values at the time of planning the investment project.
\nThus, the incompleteness and uncertainty of the information significantly affects the effectiveness of the investment project and often poses insurmountable risks. Therefore, a project considered to be profitable, in fact, may be losing. This may occur due to the risk of deviation of the values of the design parameters from the actual values or due to the complete neglect of any factors.
\nBased on the foregoing and world practice, models for assessing any object of study, including risks, are suitable only if there is a sufficiently large statistical base (general population). Consequently, these methods do not result in cases under uncertainty. What to do in the absence of such statistics? The accumulated experience shows that the only way out in this case is to use expert estimates. Thus, we come to the process of group decision-making. Decision-making processes are used in quite a variety of applications. The inherent property of these processes is to represent the transformation of individual opinions of experts to the resulting one.
\nFirst you need to perform the parameterization process, i.e., identify parameters that experts should evaluate. Denote the set of selected parameters by \n
In 1965, a professor at California University (Berkeley) Lotfi A. Zadeh published a paper “Fuzzy Sets” that gave a birth to the modeling of human intellectual activity and allowed for new interpretations of some mathematical theories. According to classical mathematics, an object either belongs to some set or not, so the characteristic function of an ordinary set is defined as {0, 1} (if an element does not belong to the set, then 0, and if it does, then 1).
\nLotfi A. Zadeh was the first to propose a generalization of the range of values of the membership function {0, 1} to the closed interval [0; 1]. Thus, the value of the membership function can be any real number, starting from zero and ending with unity [3]. Such sets were called “fuzzy” sets. By gradually developing the proposed approach, Zadeh introduced the concept of a fuzzy linguistic variable, which was able to model mathematically linguistic variables [4]. For example, Zadeh made it possible to express mathematically the following linguistic notions: “childhood, young, middle-aged, old.” Zadeh also introduced the concept of fuzzy relations and basic operations on them.
\nAs noted above, in our study we cannot do without expert evaluations. Since subjectivity, vagueness, and imprecision influence the assessments of experts, the use of fuzzy set theory seems to be an effective tool for our research work (see, e.g., [5, 6]).
\nIn [2], to assess credit risks, we used precisely fuzzy relations. In the present work, we propose to use trapezoidal fuzzy numbers as a form of presentation of experts’ estimates. The rationale for this choice will be given in the third section.
\nThe chapter consists of six sections. The second section includes all the necessary information to understand the material. The theoretical basis of the offered approach is laid out, and some theoretical results are given. In the third section, the approach to assessment of credit risk is introduced and discussed. In particular, the rationale for the choice of fuzzy trapezoidal numbers as a form for the presentation of experts’ estimates is given. The fourth section looks at the algorithm for realization of the proposed approach. The fifth section contains toy example of the practical application of the introduced approach. The sixth and the final section summarize the chapter.
\nIn conclusion, we note that when reading a chapter, the reader is not required to have knowledge of higher mathematics, but only elementary knowledge of arithmetic, algebra, and geometry. Nevertheless, we will try to explain meaningfully mathematical symbols and concepts that may be unfamiliar to the reader.
\nIn the introduction, we gave a substantive description of a fuzzy set; now we give its mathematical description.
\nHere
An important special case of fuzzy sets is fuzzy numbers. A
In this chapter we will deal with trapezoidal fuzzy numbers. Almost all the results given in this and third sections are, with minor modifications, taken from [7]; therefore we will refrain from further citation.
\nWe denote trapezoidal fuzzy numbers in
We denote by \n
The determinations of some operations on trapezoidal fuzzy numbers are given below.
\nIt is known that fuzzy maximum and minimum of two trapezoidal fuzzy numbers is defined as follows [8]:
\nHence it follows that the above definition is equivalent to those given in the literature (see, e.g., [9, 10]):
\nNow we are going to introduce a metric on Ψ(
We say that the function
and
\nThe isotone valuation
Ψ(
Let us clarify the meaning of this definition. A representative of the given finite collection of trapezoidal fuzzy numbers is a trapezoidal fuzzy number such that the sum of the distances between it and all members of this collection is minimal.
\nFor an accommodation of posterior theoretical constructions, we need to introduce a concept of
Suppose we have the finite collection of trapezoidal fuzzy numbers:
\n\n | \n | \n\n | \n\n | \n\n | \n
---|---|---|---|---|
\n\n | \n7 | \n7.5 | \n8 | \n8.8 | \n
\n\n | \n6 | \n6.1 | \n7.7 | \n9 | \n
\n\n | \n1 | \n3 | \n5 | \n9.5 | \n
\n\n | \n7.6 | \n7.9 | \n8.1 | \n8.9 | \n
Compare with it the following finite collection of trapezoidal fuzzy numbers:
\n\n | \n | \n\n | \n\n | \n\n | \n
---|---|---|---|---|
\n\n | \n1 | \n3 | \n5 | \n8.8 | \n
\n\n | \n6 | \n6.1 | \n7.7 | \n8.9 | \n
\n\n | \n7 | \n7.5 | \n8 | \n9 | \n
\n\n | \n7.6 | \n7.9 | \n8.1 | \n9.5 | \n
We see that the matching columns in both tables consist of equal sets; at the same time the elements of the sets in the second table form nondecreasing sequences. By the regulation of the finite collection of trapezoidal fuzzy numbers \n
Due to this definition and Eq. (9), it is obvious that the equality
\nholds in the metric space for any \n
It is obvious that the regulation represents a finite collection of nested trapezoidal fuzzy numbers: \n
The following theorem yields a formal definition of a representative.
\nIt follows from the theorem that when the number of members in a finite collection of trapezoidal fuzzy numbers is even, a representative can take on an infinite number of values. Now we introduce the specific aggregation operator that uniquely identifies the representative (here and further on, expression [
First of all, it is necessary to parameterize the risk assessment process, i.e., identify those parameters that to one degree or another affect credit risks. Such parameters may be credit history of the borrower, revenue, provision, market share, etc. The number and characteristics of risk assessment parameters are determined by an experienced lender manager.
\nAs already mentioned in the introduction, there is always uncertainty in forecasting the values of the parameters for assessing credit risks, and, unfortunately, this fact cannot be completely avoided. An effective way out of this situation is to attract experts whose estimates are based on experience and intuition.
\nA very important point is the determination of a form for the submission of expert assessments. Here we present expert estimates in the form of trapezoidal fuzzy numbers. Let us justify our choice.
\nThe expert has the opportunity to outline the following intervals (Figure 1):
[a, b]—Where the parameter takes positive values, increasing from 0 to 1
[
[c, d]—Where the parameter takes positive values, decreasing from 1 to 0
General form of trapezoidal fuzzy number.
If, according to the expert’s opinion, the parameter takes the maximum value at a single point, then
General form of triangular fuzzy number.
If the expert is confident that the parameter reaches its maximum value in the interval [
Segment of straight line y = 1.
If in an extreme case the expert considers that the maximum value of the parameter is reached at a single point, and at all other points the values are zero, then
Thus, the membership function of a trapezoidal fuzzy number is common to the cases considered and can easily be reduced to particular cases.
\nNow we need to aggregate expert assessments for each parameter and obtain the result of group decision-making.
\nSuppose a group of experts estimates the rating of an alternative under some given criterion. Though the experts are professionals of the same level, their subjective estimates may be essentially different. The problem consists in processing these estimates so that a consensus could be found. In constructing any kind of aggregation method under group decision-making, the key task is to determine the well-justified weights of importance for each expert.
\nLet us consider the finite collection of trapezoidal fuzzy numbers formed by experts’ estimates. To our mind, the representative of this collection, i.e., a trapezoidal fuzzy number such that the sum of distances between it and all other members of the given finite collection is minimal, is of particular interest. A representative can be regarded as a kind of group consensus, but in that case the degrees of experts’ importance are neglected. A representative is something like a standard for the members of the considered collection. As the weights of physical bodies are measured by comparing them with the Paris standard kilogram, it seems natural for us to determine experts’ weights of importance depending on how close experts’ estimates are to a representative.
\nThus, the main idea of the proposed method reduces to the following. The weight of importance for each expert is determined by a function inversely proportional to the distance between his estimate and the representative of the finite collection of all experts’ estimates, i.e., the smaller the distance between an expert’s estimate and the representative, the larger the weight of his importance.
\nLet \n
By the above reasoning, the weights and the final result of aggregation can be defined as follows:
\nand
\nIt is obvious that \n
In the following proposition and its corollaries, the properties and values of the aggregation result for special cases are established.
\n\n\n
\n
Let
The
To assess the values of the parameters according to the credit risk criterion, we find a representative of each column of the matrix. As a result, we obtain the following sequence: \n
Now we present an algorithm for finding a representative of finite collection of trapezoidal fuzzy numbers.
\nStep 0:
Step 1:
Step 2:
Step 3:
If at least one Δj = 0 then \n
If Δj > 0 for all j then compute ωj by Eq. (15) and obtain the final result by Eq. (16).
For what follows, we need to determine a scale that can “measure” the opinions of experts regarding the risk of bankruptcy of the borrower. We use the general approach described in [11].
\nIn almost any field, you can get a rating scale using the following principles:
Define a list of characteristics by which the concept (object) is evaluated.
Find polar characteristics in this list and form a polar scale.
At the poles, determine to what extent the concept possesses this characteristic (in the original, specific numerical closed intervals were used).
The collection of ratings on the scale was called the
In the introduction, we mentioned the concept of a linguistic variable. This concept plays an important role in our study. Let us introduce the linguistic variable “degree of credit risk”:
\nwhere:
\n\n
\n
\n
\n
\n
Therefore, by constructing the linguistic variable
We construct the membership function of profile
Conformity between the percentage scale and the trapezoidal fuzzy numbers.
numbers
To transform the coordinate system of the percentage scale to the coordinate system for expert estimates, the following mappings should be performed \n
It is easy to see that the coefficient of proportionality between the abscissas of the primary and the new coordinate system is
\nThereby, the coordinates of the original trapezoidal fuzzy numbers will change as follows:
\nSo, condition (c) is also satisfied.
\nWe continue the description of the implementation of the proposed approach. Based on Algorithm 1, we find the value of the representative of the finite collection of the trapezoidal fuzzy numbers for each parameter. Consider a representative calculated for the
In general, if
Here \n
Let us summarize the above as a generalized algorithm. So, we have the following input:
Step 0:
Step 1:
Step 2:
Step 3:
If the condition is met then the level of risk is acceptable;
If the condition is not met then the level of risk is unacceptable.
Here we give a toy example that will allow the reader to understand the essence of the proposed approach. To begin the practical implementation of our approach, it is necessary to determine the specific values of the isotone valuation
Without loss of generality, we consider the process of determining the degree of risk for one parameter evaluated by three experts. For any other parameter, the procedure described below is similar. As noted above, the parameterization of the risk assessment process is carried out by the lender manager. Suppose that for evaluation the parameter “
We ask three experts to evaluate the parameter
We follow Algorithm 1, detailing it along the way.
\nStep 0:
Step 1:
Step 2:
Step 3:
Now we form the percentage scale of the linguistic variable
\n
\n
\n
\n
\n
So,
Graphical expression of the degree of risk in percent.
So, we have the following input: three experts’ estimates of the parameter out of the components of the linguistic variable
We follow Algorithm 2, detailing it along the way.
\nStep 0:
Step 1:
Step 2:
Step 3:
the condition is satisfied and the level of risk is acceptable.
The presented work aims to propose a new approach for an assessment of the credit risks under uncertainty. The novelty of the proposed approach is the use of trapezoidal fuzzy numbers, which makes it possible to adequately form and process the experts’ estimates. An important fact is that the proposed approach takes into account the degrees of experts’ importance.
\nThe main results of the work are as follows:
A brief analysis of existing models is carried out, and the feasibility of creating a new approach is justified.
The rationale for the presentation of experts’ assessments of the credit risk in the form of trapezoidal fuzzy numbers is given.
The linguistic variable “degree of credit risk” is formed.
A polar percentage and coordinate scales of trapezoidal fuzzy numbers with a gradation of assigned levels are defined. The formalization of the mapping of the percentage scale to the coordinate scale is given.
The criteria for an assessment of the credit risks are generated.
Generalized algorithms for implementing the proposed approach are constructed.
A toy example which illustrates the practical application of the proposed approach is provided.
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I am also a member of the team in charge for the supervision of Ph.D. students in the fields of development of silicon based planar waveguide sensor devices, study of inelastic electron tunnelling in planar tunnelling nanostructures for sensing applications and development of organotellurium(IV) compounds for semiconductor applications. I am a specialist in data analysis techniques and nanosurface structure. 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