DDSS2006

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• DDSS2006

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Motivation I
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• 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.

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Motivation II
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• 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.

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Contents
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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.
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Residential location
Undesirable facility
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Distribution of Endurance Distance
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Fig.1 Relationship between the endurance rate
and distance to a facility
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Assumption for Distribution of Endurance Distance
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where a and m are scale and shape parameters.
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Survey Concerning Endurance Distance
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• Estimation of parameters for endurance rate
  function

Data concerning the endurance rate P(x)
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?
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Questionnaire Survey in Chengdu City
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Fig.2 The location of Chengdu City
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Case Study Area-object of Survey
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Fig.3 Object area of the research
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The Result of Survey
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Fig.4 Percentage by age
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Endurance Distance and Endurance Rate
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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
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Average Endurance Distance Classified by
Attribute
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Fig.7 Average endurance distance classified
by age for garbage transfer stations
Fig.8 Average endurance distance classified by
age for final waste disposal facilities
Fig.9 Average endurance distance classified by sex
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Estimated Parameters of Endurance Rate Function
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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
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Fig.10 The residential endurance rate for
garbage transfer stations
Fig.11 The residential endurance rate for final
waste disposal facilities
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NON-PARAMETRIC DISTRIBUTION METHOD
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In matrix form, non-linear models are given by
the formula y f(X, ß) e, where y is an
n-by-1 vector of responses, f is a function of
ß and X, ß is a m-by-1 vector of coefficients, X
is the n-by-m design matrix for the model, e 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.
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Distribution Fitting for Non-parametric
Distribution
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(7)
where, y(x) is probability distribution function
Table 2. The coefficients of equation (7) and
goodness of fit
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Cumulative Distribution Function Calculation
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where erf(.) is the error function
(8)
Table 3. The coefficients of equation (8) (with
95 confidence bounds)
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Fig.12 Distribution of the endurance rate h(?)
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3-PARAMETER LOGLOGISTIC DISTRIBUTION METHOD
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• Analysis of Data

Fig.13 A plot of the original data to 12km
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Flipping the Data
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Fig.14. Data flipping
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Flipping the Data
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(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
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Distribution Analysis of NewData
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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
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The Resulting Values for the Parameters the
Goodness of Fit
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a1.183 b0.399 c0.296
Fig.16 Result of parameter estimation and test
of goodness of distribution (Where, C1 means
NewData shown in Figure 13)
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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 -8 to 0.
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(14)
Fig.17 Distribution of the endurance rate sr(z)
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CONCLUSIONS
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• Regarding undesirable facilities, we defined
  residential endurance distance and endurance
  rate, modeled the relationship between facilitys
  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 its 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 theres 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 both kinds 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.

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Thank you for your attention!
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? ? ? ?
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Slide 3

• Whats the meaning of the highest effect?
• What are the meaning of fixed points?
• What means a series of fixed points?

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Slide 3

• Why need consider residential awareness in this
  study?

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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
  parameters changing. 2. The distribution
  function is completely integrabel, which make it
  possiblefor next step of parameter estimation.

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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 theres 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.

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What means

• In matrix form, non-linear models are given by
  the formula
• y f(X, ß) e,
• where y is an n-by-1 vector of responses,
• f is a function of ß and X, ß is a m-by-1 vector
  of coefficients,
• X is the n-by-m design matrix for the model,
• e 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.

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Table 2

• If the value of goodness of fit can be gain
  at the step of distribution fitting?

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The procedure of selecting equation (11) by
employing MINITAB
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Explaining the following paragraph

• The function j(x) in equation (13) exists for
  values xlt0, 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 -8 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.