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A Comparison Study about the Allocation Problem of Undesirable Facilities Based on Residential Awareness – A Case Study on Waste Disposal Facility in ChengDu City, Sichuan China – K. Zhou, A. Kondo, A.Cartagena Gordillo 1 and K. Watanabe
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A Comparison Study about the Allocation Problem of Undesirable Facilities Based on Residential Awareness • – A Case Study on Waste Disposal Facility in • ChengDu City, Sichuan China – • K. Zhou, A. Kondo, A.Cartagena Gordillo1 and K. Watanabe • The University of Tokushima, Tokushima, Japan 1Yokohama National University, Yokohama, Japan DDSS2006
2 Motivation I • In the existing facility location theory, there are many studies concerned with location modeling of facilities that put more importance on nearness. But there are also some facilities that give undesirable feeling to residents. Location problems for those kind of facilities require new methodologies with corresponding solutions. • Thereupon, in this research, our purpose is to consider the location problem of undesirable facilities • Waste disposal facilities are appointed for analysis. Specifically, we choose garbage transfer stations and final disposal facilities as research objects due to their high level of variety.
3 Motivation II • As we review the methodologies for location problems of undesirable facility, we found that the most popular way of handling undesirability for a single facility is to minimize the highest effect on a series of fixed points applying the principle of locating the undesirable facilities as far as possible from all sensitive places. • Therefore, in the existing literature we can appreciate that physical magnitudes, such as distance or time, were mainly used as important parameters on the study of facility location. However, the psychology element of the facility users was not given enough attention. • Regarding those characteristic, our objective in this research is to analyze location problems of undesirable facilities by using a model based on probability theory, which considers residential awareness.
4 Modeling Stochastic Methods Weibull Non-Parametric 3-Parameter Loglogistic Regression Analysis Matlab Minitab Endurance Rate Function Parameters Estimation Conclusions Contents
5 X>W (X: distance to a facility) W Endurance distance Definition of Endurance Distance & Endurance Rate For purely undesirable facilities, we can consider that residents hope the undesirable facility can be located farther than a certain distance, which means the residents can endure the location of the undesirable facility if the facility is located farther than that distance. Then the minimum of this desired distance can be defined as endurance distance, which is expressed here as w. And, when an undesirable facility is located at a certain distance, the rate of residents who could endure the facility location is defined as endurance rate, which is expressed here as P(x) in this research. ● Undesirable facility Residential location
6 Distribution of Endurance Distance Fig.1 Relationship between the endurance rate and distance to a facility
7 Y c mX Assumption for Distribution of Endurance Distance where α and m are scale and shape parameters.
8 Survey Concerning Endurance Distance Data concerning the endurance rate P(x) Estimation of parameters for endurance rate function Carry out a questionnaire survey toward the residents in object area Questionnaire Survey At least how far should a waste facility be located to your home?
9 Questionnaire Survey in Chengdu City Fig.2 The location of Chengdu City
10 Case Study Area-object of Survey Fig.3 Object area of the research
11 The Result of Survey Fig.4 Percentage by age
12 Endurance Distance and Endurance Rate Fig.5 Distance to garbage transfer stations and corresponding residential endurance rate Fig.6 Distance to final waste disposal facilities and corresponding residential endurance rate
13 Average Endurance Distance Classified by Attribute Fig.7 Average endurance distance classified by age for garbagetransfer stations Fig.8 Average endurance distance classified by age for final waste disposal facilities Fig.9 Average endurance distance classified by sex
14 Estimated Parameters of Endurance Rate Function Table 1. Result of parameters estimation for the endurance rate model where the numbers between parentheses represent the value of t. R2 is determination Coefficient
15 Fig.10 The residential endurance rate for garbage transfer stations Fig.11 The residential endurance rate for final waste disposal facilities
16 NON-PARAMETRIC DISTRIBUTION METHOD In matrix form, non-linear models are given by the formula: y = f(X, β)+ ε, where y is an n-by-1 vector of responses, f is a function of β and X, β is a m-by-1 vector of coefficients, X is the n-by-m design matrix for the model, ε is an n-by-1 vector of errors, n is the number of data and m is the number of coefficients. The fitting process was automated, employing the commercial software Fitting Toolbox from Matlab.
17 Distribution Fitting for Non-parametric Distribution (7) where, y(x) is probability distribution function Table 2.The coefficients of equation (7) and goodness of fit
18 Cumulative Distribution Function Calculation where erf(.) is the error function (8) Table 3. The coefficients of equation (8) (with 95% confidence bounds)
19 Fig.12 Distribution of the endurance rate h(λ)
20 3-PARAMETER LOGLOGISTIC DISTRIBUTION METHOD Analysis of Data Fig.13 A plot of the original data to 12km
21 Flipping the Data Fig.14. Data flipping
22 Flipping the Data (9) Where f(x) is the distribution in Figure 9 , g(x) is the distribution in Figure 10. (10) Where h(x) is the distribution function for New Data, j(x) is the distribution function for original data. Fig.15 NewData
23 Distribution Analysis of NewData Employing the estimation method of Least Squares, a 3-Parameter Loglogistic distribution function was found as: (11) where, a = Location parameter, b = Scale parameter, c = Threshold parameter
24 The Resulting Values for the Parameters & the Goodness of Fit a=1.183 b=0.399 c=0.296 Fig.16 Result of parameter estimation and test of goodness of distribution(Where, C1 means NewData shown in Figure 13)
25 The modified Loglogistic function is: (12) Finally, the function of the endurance rate model is: (13) where z is a value between 0 and 12, 0.04 is the integral of the resulting function from -∞ to 0.
26 (14) Fig.17 Distribution of the endurance rate sr(z)
27 CONCLUSIONS • Regarding undesirable facilities, we defined residential endurance distance and endurance rate, modeled the relationship between facility’s location and the endurance rate. • From the questionnaire survey carried out in Chengdu City, we could dissect the distribution of residential endurance distance for garbage transfer stations and final waste disposal facility. Using the endurance rate model, we indicated it’s possible to propose waste facilities’ location from the viewpoint of residents. • Based on different probability distribution functions, we proposed three models for estimating the residential endurance rate and make a comparison study. Based on those models, we found there’s no big difference between the results when residential endurance rate according to facility location is 80%, the calculation results of suitable distance for garbage transfer stations are all around 10km. • From the comparison study, we found that the advantage of the model employing Weibull distribution is its simplicity; it has only 2 parameters and can be used for the bothkinds of facilities though the accuracy was not good enough. The Non-parametric one described a better modelling even though a lot of parameters were needed for describing the detail of the data. As computer technology is developed today, we consider that this method can be used for any kind of situation as a numerical analysis model. Based on a parametric distribution function, we also found a model by analysing and flipping the data as explained above. For this case, the model using Loglogistic distribution function is a new experiment with good modelling characteristics.
Slide 3 • What’s the meaning of “the highest effect”? • What are the meaning of “fixed points”? • What means “a series of fixed points”?
Slide 3 • Why need consider residential awareness in this study?
Why & how could find Weibull During the proceeding, we found Weibull distribution has some interesting characters as following: 1. It is a distribution with good elasticity. The shape changes following shape parameter’s changing. 2. The distribution function is completely integrabel, which make it possiblefor next step of parameter estimation.
Contents of the survey For getting the data, a survey on residential awareness about undesirable facilities was carried out in Feb. 2004. The area object of survey is shown in Figure 3. The question was: At least how far should a waste facility be located to your home? According to the endurance distance, a few alternatives were given in advance. Then respondents choose their desired endurance distance from the alternatives or a certain number they considered adequate. The choices, for garbage transfer stations, were from 1km to 10km, for final waste disposal facilities, were from 5km to 30km. For both facilities there was the option: “If there’s no endurance distance you considered, please write down a distance you can endure”. Data analysis was based on the endurance distance which residents chose or wrote. A simple explanation concerning present condition of waste disposal in Chengdu was given before the questions.
What means In matrix form, non-linear models are given by the formula: y = f(X, β)+ ε, where y is an n-by-1 vector of responses, f is a function of β and X, β is a m-by-1 vector of coefficients, X is the n-by-m design matrix for the model, ε is an n-by-1 vector of errors, n is the number of data and m is the number of coefficients. The fitting process was automated, employing the commercial software Fitting Toolbox from Matlab.
Table 2 If the value of “goodness of fit” can be gain at the step of distribution fitting?
The procedure of selecting equation (11) by employing MINITAB
Explaining the following paragraph The function j(x) in equation (13) exists for values x<0, which is unreal for the processed data, then an adjusting value of 0.04 is included in equation (13) which corresponds to the integral of j(x) from -∞ to 0. For this reason, the endurance rate never reaches 100%. A corrective coefficient can be applied to the equation of the endurance rate sr(z). Then it becomes equation (14). The corrected function newsr(z) is illustrated in Figure 12.
