Site selection using GIS, Simulated Annealing, AHP and EVAMIX approaches under Fuzzy Set ...

18 Pages·2016·0.78 MB·English

Journal of American Science 2016;12(10) http://www.jofamericanscience.org

Site selection using GIS, Simulated Annealing, AHP and EVAMIX approaches under Fuzzy Set Theory: A
case study of landfill siting in Blantyre City, Malawi.

Rodney Godfrey Tsiko

Department Geoinformatics and Surveying, University of Zimbabwe, P.O Box MP167, Mount Pleasant, Harare,
Zimbabwe.
[email protected]

Abstract: This paper explores the applicability and capability of solving site location problems using an integrated
approach that includes the Markov-chain based Simulated Annealing (SA) algorithm, Geographic Information
System (GIS), fuzzy Analytical Hierarchy Process (AHP) and Evaluation of Mixed Data (EVAMIX). The novelty of
the proposed methodology is its capability to combine the flexibility of fuzzy logic in handling uncertainty with the
simplicity, easy implementation and independence offered by simulated annealing in solving non-linear optimization
problems, thus providing good site location solutions without losing consistency. In addition, the EVAMIX
approach provides complete flexibility in using both quantitative and qualitative criteria. The approach is applied to
a multiple objective decision problem of selecting the best location for a new landfill site, which will serve Blantyre
City, an urban agglomeration in Malawi.
[Tsiko RG. Site selection using GIS, Simulated Annealing, AHP and EVAMIX approaches under Fuzzy Set
Theory: A case study of landfill siting in Blantyre City, Malawi. J Am Sci 2016;12(10):16-33]. ISSN 1545-1003
(print); ISSN 2375-7264 (online). http://www.jofamericanscience.org. 4. doi:10.7537/marsjas121016.04.

Keywords: Simulated Annealing; fuzzy logic; GIS; Analytical Hierarchy Process; EVAMIX

1. Introduction use of both quantitative (cardinal) and qualitative
Decisions regarding site location are often (ordinal) criteria within the same evaluation matrix.
plagued by vagueness, fuzziness, contradictory The algorithm behind EVAMIX maintains the
multiple objectives and a myriad of quantitative and essential characteristics of cardinal and ordinal
qualitative factors. These decisions have an inverse criteria, yet it is designed to eventually combine the
relationship with our ability to define and model them results in a single appraisal score. This unique feature
in a precise way. As the degree of complexity of the gives EVAMIX much greater flexibility and
decision making process increases, the methods and differentiates it from other Multi-Criteria Decision
tools available to resolve them become rare. Hence, Analysis (MCDA) methods such as weighted
there is need for more sophisticated tools to aid summation, range of value method, PROMETHEE II
decision making vis-à-vis multi-objective site location (Brans et al., 1986; Figueira et al., 2005) etc., which
problems, which include some form of uncertainty. are incorrectly applied to ordinal data by treating it as
In this paper, the proposed solving tool is an though they were at a cardinal measurement scale.
integration of simulated annealing (van Laarhoven The approach is applied to a multiple objective
and Aarts, 1987; Kirkpatrick et al., 1983; Cerny, decision problem of selecting the best location for a
1985), geographic information system (GIS) new landfill site, which will serve Blantyre City, an
(Burrough and McDonnell, 1998; Heywood et al., urban agglomeration in Malawi. The example shows
2006; Longley et al., 2010), fuzzy Analytical how decision makers can use this approach to treat
Hierarchy Process (AHP) (Bellman and Zadeh, 1970; landfill siting as a combinatorial optimization problem
Saaty, 1980; Saaty, 1986; Chang, 1996) and within a GIS environment, and use mixed criteria
Evaluation of Mixed Data (EVAMIX) (Voogd, 1982; (qualitative and quantitative) to select the best
Nijkamp et al., 1990; Martel and Matarazzo, 2005; possible landfill site, at the same time accounting for
Hajkowicz and Higgins, 2008). Simulated Annealing the intrinsic imprecision and ambiguity associated
(SA), a heuristic search algorithm, when used in with decision makers when confronted with multi-
conjunction with a GIS, yields optimum and close to criteria data.
optimum site locations that satisfy multiple objective The rest of the paper is organized as follows.
functions or cost criteria within the search domain. Section 2 explores the three dominant themes of this
Fuzzy AHP provides a useful approach to address the study: simulated annealing, fuzzy AHP and EVAMIX.
problems associated with the intrinsic imprecision, In Section 3, a real world case study is presented, the
uncertainty and subjectivity of decision makers when results are discussed and sensitivity analysis is
faced with multi-attribute data. EVAMIX provides a conducted. The paper concludes in Section 4.
way to rank and select the best site location by making
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Journal of American Science 2016;12(10) http://www.jofamericanscience.org

2. Heuristic/Optimization Algorithm have given the method its present use in solving a
Since locating a landfill site in a GIS can be variety of global single objective or multiobjective
classified as a combinatorial optimization problem, a optimization problems not only in mathematics, but
heuristic search algorithm is required for solving the also in other fields such as operations research,
model. Note that various researchers have developed geodesy and GIS.
linear programming models to solve landfill siting or For the purposes of this study a description of the
location allocation problems, but encountered key elements of the SA algorithm are summarised as
limitations related to the spatial area that could be follows:
optimized (Arthur and Nalle, 1997; Aerts et al., 2003; Step 1: Define an objective function or cost
Cova and Church, 2000). This problem arises because f
function in the form of a mathematical expression
each map layer in a GIS by itself can consist of
to describe the relationship of the parameters that will
complex information on attribute values, and spatial
optimize the location siting process. It is the main aim
relationships between attributes. When a large number
of the SA algorithm to find an optimum solution that
of layers are involved in GIS analysis (as is normally
f
the case), relationships between layers, and within either minimizes or maximizes .
themselves have to be considered. This often leads to x D
Step 2: Initialize a starting solution i ,
a large solution space from which linear programming
x
solvers find the spatial area too large to explore. In where i is either a user provided or random
consequence, this exploration is time consuming and, generated initial spatial point solution within a GIS
in worse circumstances, linear programming models search domainD.
can get trapped in local minima. Heuristic algorithms, Step 3: Choose an initial control parameter. This
however, are robust, fast, capable of solving large parameter allows the algorithm to converge to the
combinatorial problems and do not get trapped in optimum solution and every algorithm has one. Since
local minima. Application of such algorithms for SA originates from thermodynamics, its control
landfill sitting or location allocation problems include (T )
parameter is in the form of a temperature 0 value.
simulated annealing, tabu search, genetic algorithms
This needs to be large enough to allow the search to
and greedy growing algorithms (Lockwood and
traverse a large portion of the study area and
Moore, 1993; Murray and Church, 1995; Muttiah et
overcome infeasible regions.
al., 1996; Brookes, 1997; Boston and Bettinger, 1999;
Step 4: Obtain a new spatial location solution
Karaganis and Mimis, 2011). In this study the focus is
within the search domain by perturbing the previous
on using simulated annealing as part of a model for
x
solving a location siting problem. solution location by a small amount, i . The
2.1 Simulated Annealing x
increment, i is random and can be in the form of a
In general, Simulated Annealing (SA) is an
function. Among the many functions available, the
iterative meta-heuristic search algorithm capable of
Gauss distribution density function is one of the most
escaping from local optima. Its ease of
suitable for defining this random movement, and is
implementation, convergence properties and its use of
defined as follows,
hill-climbing moves to escape local optima have made
1x2
it a popular technique over the past three decades. The 1 
P(x) e 22..............................................1()
origins and theoretical development of SA are well
2
reviewed within the literature, therefore will not be

repeated here. However, a good overview of With a zero mean and standard deviation .
simulated annealing’s theoretical development and Step 5: Evaluate the objective/cost function to be
domains of application can be found in (Pincus, 1970; optimized using the new spatial solution. If there is an
Kirkpatrick et al., 1983; Cerncy, 1985; Eglese, 1990; improvement in the value of the objective/cost
Koulamas et al., 1994; Fleischer, 1995). van function (i.e., a decrease if the objective/cost function
Laarhoven and Aarts (1987), Aarts and Korst (1989) is to be minimized, or an increase if the objective/cost
also devote entire books to the subject. Aarts and function is to be maximized), then the new spatial
Lenstra (1997) dedicate a chapter to SA in their book location is accepted. However, if the new solution has
on local search algorithms for discrete optimization neither minimized nor maximized the objective/cost
problems. function and the algorithm becomes trapped into a
Of interest to this research from the reviewed local optimum, the SA algorithm is equipped with
literature is the connection between SA and additional criteria to accept this solution (which is a
mathematical optimization, which was first noted by worse solution) with a certain probability. Pardolas
Pincus (1970), but it was the contributions made by and Romeijn (2002), give a number of alternative
Kirkpatrick et al. (1983) and Cerncy (1985), which ways to overcome this problem, which among others
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Journal of American Science 2016;12(10) http://www.jofamericanscience.org

include using a fixed probability (such as 0.2, 0.1, can be summarised in seven steps, which are
0.05 or 0.01), the Tsallis’ Acceptance Rule and Boltz- discussed as follows:
man’s distribution. Step 1: Construct an abevaluation matrix E ,
Step 6: Decrease the temperature during the a b
where is the number of alternatives and is the
search for the optimum solution by making use of a
number of evaluation criteria (quantitative and/or
certain function known as the cooling schedule. There
qualitative). Thus given a set of evaluation criteria
are several theoretical and empirical cooling schedules
j(j1,2,,b)
suggested in the literature, but given a sufficiently and a finite set of alternatives
high initial temperature, T0, some successful cooling i(i 1,2,,a) , the evaluation matrix E will be
functions are as follows: characterized by its ordinal and cardinal components
as follows:
T
T(t) 0 .........................................(2)
e e  e 
1log(t1)  11 12 1a
T(t) T0 .....................................................(3) E e21 e22  e2a
1t    
T(t)Tt,01......................................(4) e e  e 
0  b1 b2 ba
t
Where, , is the number of iterations the algorithm is Then, distinguishing the set of evaluation criteria into
allowed to perform at each temperature two subsets, ordinal and cardinal criteria, we obtain
According to Berne and Baselga (2003), the final E
two distinct evaluation matrices: o(ordinal criteria)
temperature at which to stop the iterations can be
E
stated as a fraction of the initial one, for example, and C(cardinal criteria).
104T Step 2: Standardize the cardinal data set to a
0 , or as the moment when the differences
common unit using linear normalization. Beneficial
x
between two consecutive solutions in terms of i criteria are normalized using Eq. (5), whilst non-
becomes negligible. beneficial criteria are normalized using Eq. (6).
Step 7: Return to step 4 until the finish criteria are x min(x )
r  ij ij (i1,2,,a;j1,2,,b).......................(5)
fulfilled. ij max(x )min(x )
ij ij
2.2 Integrated EVAMIX/ Fuzzy AHP

The EVAMIX approach developed by Voogd
max(x )x
(1982, 1983), and described in Nijkamp et al. (1990), r  ij ij (i1,2,,a;j1,2,,b).......................(6)
ij max(x )min(x )
Martel and Matarazzo (2005) and Hajkowicz and ij ij
Higgins (2008) is an outranking method for evaluating According to Eq. (5) and Eq. (6) in the normalized
alternative options using both quantitative (cardinal) decision matrix, the maximum value will always be 1
and qualitative (ordinal) criteria within the same and the minimum value equal to 0.
evaluation matrix and ranking the options from best to Step 3: Calculate the weights or measure of
worst. The literature shows some applications of importance of each criterion (ordinal or cardinal) with
EVAMIX in the field of multi criteria analysis in respect to the others. This step involves making use of
material selection by Chatterjee et al. (2011), the fuzzy AHP as an integral part of the EVAMIX
industrial environment by Darji and Rao (2013), method. The study concentrates on the fuzzy AHP
spatial ranking of vulnerability proposed by Chung approach introduced by Chang (1996), in which
and Lee (2009), analysis of investments in triangular fuzzy numbers (TFNs) are preferred for the
construction by Ustinovichius et al. (2007) and multi pairwise comparison scale. When using TFNs, an
criteria analysis of small-scale forestry by Jeffreys interval is used to define the decision maker’s
(2004). It is very clear from the above listed judgement. This is in the form of three numbers or
references that, there exist few applications of the parameters, expressed as (l, m,u), where the lowest
EVAMIX method in science, ecological, industrial, l
possible parameter or value is , the middle possible
financial and non-financial units, and rare applications m
parameter or value is and the upper possible
in the related fields of multi objective optimization
u
parameter or value is . The domain of this interval is
and GIS.
described by a triangular membership function, which
As part of a model for solving a site location
can be represented in both math and graph form using
problem, this study adopted EVAMIX for evaluating
Eq. (7) and Figure 1, respectively (Chang, 1996;
alternative landfill site locations using both
Kaufmann and Gupta, 1988) as follows:
quantitative and qualitative multi criteria and rank the

sites from best to worst. From a procedural point of
view, the EVAMIX method used in this present paper
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 0 xl By using TFNs for pairwise comparison of the
 ~
A(x)((uxxl))//((umml)) lmxxmu................................(7) eisv aclrueaatteiodn a ncrdi tiesr ioaf ath feu fzozrym c:o mparison matrix A
 0 xu
 (1,1,1) (l ,m ,u )  (l ,m ,u )
12 12 12 1b 1b 1b
 
A~(a~ ) (l21,m21,u21) (1,1,1) (l2b,m2b,u2b)
pqbb      
 
(lb1,mb1,ub1) (lb2,mb2,ub2)  (1,1,1) 
wherea~ (l ,m ,u )anda~ 1(1/u ,1/m ,1/l ) forp,q1,...,bandpq
pq pq pq pq pq qp qp qp

The number of comparisons for each matrix is
b(b1)/2 b
, where is the total number of
evaluation criteria.
 Compute the normalized value of row sums (i.e.
~
S
Figure 1: Fuzzy Triangular Number the fuzzy synthetic extent, j ) for the fuzzy
~
comparison matrix A by making use of fuzzy
arithmetic operations from Eq. (8-10) such that:
In order to utilize TFNs in a fuzzy environment, 1
their basic algebraic operations should be understood. ~ b ~  b b ~ 
Sj   ajk   ajk ..............................................(11)
An extensive discussion on these operations can be k1 j1k1 
found in Chang (1996), Kaufmann and Gupta (1988)
and Zadeh (1965). However, for the purposes of this b
~
research only three operations are illustrated. Consider  ajk
~ ~ To obtain, k1 , the fuzzy addition operation
M M
that 1 and 2 are two TFNs were (Eq. (8)) is applied to the fuzzy judgement
~ ~
M (l ,m ,u ) M (l ,m ,u ) matrices, such that,
1 1 1 1 and 2 2 2 2 . The three

basic operations used in this study are:
b b b b
M~ M~ (l l ,m m ,u u )..............................(8) k1a~jk (k1ljk,k1mjk,k1ujk).............................(12)
1 2 1 2 1 2 1 2
~ ~
M M (ll ,mm ,uu )............................................(9)
1 2 12 1 2 1 2 The second part of Eq. (11) is obtained by
M~ 1 1 , 1 ,1...........................................................(10) applying the fuzzy addition operation (Eq. (8)) to
1 u1 m1 l1 column values in the matrix obtained from Eq.
(12), followed by using Eq. (10) to compute the
Using these three algebraic operations the inverse of the resulting vector such that,
process of calculating the weights of the evaluation  
criteria is as follows:  b b ~ 1  1 1 1 
 Caiomn storfu ctth teh em fuatzrzixy AisH Pto coemlupciadraistoe na mn atorridxe. rT hoef j1k1ajk b b ujk,b b mjk,b b ljk................................(13)
preference given a number of evaluation criteria. j1k1 j1k1 j1k1 
Central to this is a series of pairwise comparisons,
indicating the relative preferences between pairs  Using the normalized TFNs of two criteria
of evaluation criteria. Since decision makers use obtained from Eq. (11) determine the degree of
linguistic terms or verbal judgements during possibility of one criteria fuzzy number’s being
pairwise comparison, a measurement scale is greater than or equal to the other criteria fuzzy
~ ~
required to convert these pairwise comparisons S S
into fuzzy numbers. One such scale showing the number’s j1 j2 (see Figure 2). This can be
proposed TFNs, membership functions and represented by Eq. (14) and Figure 2.
matching linguistic variables or verbal
judgements was provided by Saaty (1980) and is
shown in Table 1.

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Journal of American Science 2016;12(10) http://www.jofamericanscience.org

Table 1: Proposed TFN, linguistic variables and membership functions
Saaty’s scale
of relative Definition Membership function Domain (l,m,u) Linguistic
TFNs scale variables
importance
Just equal (1.0,1.0,1.0) Just equal
Least
1 Equal importance A(x)(3x)/(31) 1 x3 (1.0,1.0,3.0) importance
Moderate A(x)(x1)/(31) 1x3 Moderate
3 importance of one (1.0,3.0,5.0)
over another A(x)(5x)/(53) 3x5 importance
5 Essential or strong A(x)(x3)/(53) 3x5 (3.0,5.0,7.0) Essential
importance A(x)(7x)/(75) 5x7 importance
7 Demonstrated A(x)(x5)/(75) 5x7 (5.0,7.0,9.0) Demonstrate
importance A(x)(9x)/(97) 7 x9 importance
Extreme
9 Extreme importance A(x)(x7)/(97) 7 x9 (7.0,9.0,9.0) importance
If an activity has
one of the above
numbers (e.g., 3)
Reciprocals compared with a
Reciprocals of above;
of above second activity, then
non-zero the second activity A1 (1/u ,1/m ,1/l )
1 1 1 1
numbers has the reciprocal
value (i.e., 1/3)
when compared to
the first.
Adapted from Saaty (1980)

 1 if mj1mj2

l u
~ ~  j2 j1
V(Sj1Sj2)(m u )(m l ) if lj2uj1..............(14)
 j1 j1 j2 j2
 0 otherwise


~ ~
S (l ,m ,u )and S (l ,m ,u )
Where, j1 j1 j1 j1 j2 j2 j2 j2

Figure 2: Degree of possibility

 Extend the basic principles from above and
determine the degree of possibility of one
criterion fuzzy numbers being greater than all the
(b1) * * * T
other criteria fuzzy numbers. This can be w(d (A ),d (A ),...,d (A )) ,
1 2 b
defined as follows, * ~ ~
~ ~ ~ ~ ~ where d (A )min V(S S | j1,...,b,).
V(S  S ,S ,...,S V[(S  S ) b j jb
j j1 j2 jb j j1
........................(16)
~ ~
and (Sj  Sj2)and ... The normalized weight vectors for each fuzzy
~
~ ~ comparison matrix, A , at each level of the
and (S  S )]....................(15)
j jb hierarchy is then determined by normalizing the
By taking the minimum values in the degree of weight vector, w . This is known as de-
possibility sets created from Eq. (15), it is possible to fuzzification and involves dividing each value in
w
determine a weight vector, , as the weight vector, w , by their total sum as
follows,
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Journal of American Science 2016;12(10) http://www.jofamericanscience.org

~ ~
were used to represent uncertainty in the
V(S S |j1,...b,)
b jb
W  ,i1,..b..........(.1.7) judgment matrix, the FCR values obtained from
i b ~ ~ Eq. (22) will be in the form of a set with 3
V(S S |j1,...b,;jk)
k j values. As such, determine the FCR as a
k1
preference ratio, such that it is defined as the
 Determine whether the process of assigning
ith
weights was consistent by calculating the Fuzzy percentage of the fuzzy number within a set
Consistency Ratio (FCR). For this research the being the most preferred. This ratio according to
algorithm used to calculate the FCR is that Modarres and Sadi-Nezhad (2001) can be
proposed by Modarres et al. (2010), which is | |
R(i) i ................(23)
based on the preference ration concept. The
||
expressed as , where
algorithm is as follows:
~ 
 Define a fuzzy matrix h , such that, i and  are values in FCR set obtained from
~ ~ Eq. (22). Usually a CR of 0.1 or less is
hjk  wj ajk.................(18) w
, where j considered as acceptable, and it reflects an
jth informed judgement attributable to the
is the weight for the criterion for knowledge of the analysts regarding the problem
~
j1,2,...,b and ajk are the TFN’s in the under study.
Step 4: Use the criteria and their associated
fuzzy judgement matrix.
W
jth weights ( j ) to compare the alternative options by
 Sum the values of each row of the
~ computing the dominance scores of each alternative
h
matrix , that is, (i,i')
b b option pair, . For all ordinal criteria use Eq.
~s  h~ ...........(19) (24), otherwise use Eq. (25) for all cardinal criteria.
j jk
j1 k1 1
~  c
 F~or ~sj1tob calculate j such that ii' jO{Wjsgnr(ijri'j)}c ,c1,3,5....................................2.(4)
j
  .....................(20)
j w
j 1
 Calculate the Consistency Index (CI) as    C
1(~ b) ii' WJsgnrijri'j c ........................................(25)
b j jC 
CI  ...........(21)
follows: b1 . The 1 if rij ri'j
smaller the CI, the smaller the deviation from sgn(r r ) 0 if r r .
ij i'j ij i'j
the consistency. 
1 if r r
 Obtain the random index (RI) for the number Where  ij i'j
of criteria from Table 2. C
and is a subset of criteria with a cardinal

measurement scale.
Table 2: Random Indices
Step 5: Calculate standardized dominance scores
Order of Matrix Random Index
for all ordinal and cardinal criteria. The dominance
2 0
 
3 0.58 scores ii' and ii' are standardized into the same
4 0.90 measurement unit in order to make them comparable.
5 1.12 If Z is a standardization function, then the
6 1.24 standardized dominance measures for all ordinal
7 1.32 ( ) (d )
8 1.41 ii' and cardinal ii' criteria have the following
expressions:
9 1.45
10 1.51 ii' Z(ii') and dii' Z(ii')
The standardized dominance scores can be
 Calculate the FCR using, obtained using three different approaches, i.e., (a)
CI subtractive summation, (b) subtracted shifted interval,
FCR .............(22)
RI . However, since TFNs and (c) additive interval technique. For this study, the
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Journal of American Science 2016;12(10) http://www.jofamericanscience.org

additive interval technique will be used and is
represented as follows:
  
( ) ii' ...................................(26)
ii'  
 

()
Where is the highest (lowest) ordinal
i,i'
dominance score for the alternative pair, ( ).
  
(d ) ii' ...............................(27)
ii'  
 

()
Where is the highest (lowest) cardinal
i,i'
dominance score for the alternative pair, ( ).
W
Step 6: Assuming that weights j have
quantitative properties, the overall dominance
D
measure ii' for each pair of alternatives landfill sites
Figure 3: Study Area
i,i'
( ) is:
3.2 State of solid waste management in Blantyre
D w  w d ..............................(28)
ii' O ii' C ii; City
w According to Matope (2000), it is estimated that
Where O is the sum of the weights for the ordinal
more than a decade ago the city of Blantyre generated

(wO  Wj) w waste totalling 0.37 kg per capita per day, the
criteria jO and C is the sum of the
equivalent of 192 tons a day, three-quarters of which
(w  W ) was classed as domestic waste. By the year 2006, 540
C j
weights for the cardinal criteria jC . metric tons were being generated in the city as solid
This overall dominance score reflects the degree to waste. More recently, the average amount of waste
a a being produced is 0.9 kg per capita per day, which
which option i dominates option i' for the given
amounts to 647 metric tons per day – 81 per cent of
set of criteria and weights.
which is organic and biodegradable (Matope, 2000;
Step 7: For each option, its final appraisal
Berman, 2011). A breakdown of the waste according
score is given by,
to the city’s different economic backgrounds is shown
 a D 1 in Table 3. According to UN-HABITAT (2011), only
S  ii ......................................(29) a third of the solid waste produced is collected and
i  i Dii disposed properly.
D D 1
Where ii ii . The higher the appraisal score
Table 3: Breakdown of waste by economic
the better is the performance of the option. Thus, the
background
best option is the one with the highest appraisal score.
Component of Waste Fraction (% by mass)
3.0 Empirical study of landfill site selection
the waste High Middle Low
3.1 Study Area
stream Income Income Income
The study is applied in Blantyre City, which is
Paper 30 20 7
the second largest urban center in Malawi and is
Plastics 30 20 8
located 311km south east on the national capital,
Glass 18 10 2
Lilongwe (Figure 3).
Metals 5 3 2
Situated at an altitude of approximately 1,150
Textiles 4 3 1
meters above sea level, Blantyre City covers an area
Garden waste 1 1 0
of over 220 square kilometers. According to the
National Statistical Office the population of Blantyre Builder’s 1.5 1 0
City was estimated at 661,256 in 2008, with an rubble
average population density of 3,006 people per square Ash 0.5 20 25
kilometer. This accounted for a 5.1 per cent share on Putrescibles 10 28 55
the national population. However, the population is
projected to rise to 1,068,681 by 2015.
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Journal of American Science 2016;12(10) http://www.jofamericanscience.org

Blantyre City Council (BCC) Health and general waste from households, industry, hospitals
Engineering Department, which is responsible for and educational institutions, however, there are
solid waste management as mandated by the Local sightings of restricted materials such as hazardous
Government Act of 1980 contributes to this escalating (batteries, paint etc.) and medical waste being dumped
problem by offering selective waste collection at the site illegally. To create room waste is piled up
services. Waste collection services are not in heaps then eventually spread level then covered in
consistently made available to all areas, with high- earth before being compacted at the site. This is done
income areas getting more services than low-income twice a year by a private company hired by the BCC.
areas. This is a common practice by local authorities The process is neither sustainable nor cost-effective as
in developing countries to allocate resources to areas the BCC spends an estimated MK 10 million (US35,
with higher tax yields. In high-income areas collection 335) just to bulldoze the Mzedi dump site each year.
of waste is carried out six days a week and in the As no impervious soil is placed on the ground before
city’s busier markets it is done every day. There are compaction, leachate from the waste is more likely to
also a number of private entities which provide waste percolate into the ground and eventually contaminate
collection services within the city. These include the city’s groundwater systems. The BCC also often
private trash collectors who focus on commercial or resorts to burning the waste to curb the nuisance
industrial companies. In addition, the privately owned produced by flying litter and to create room for more
Malawi Housing Corporation focuses on collecting waste. At least 150 scavengers (mostly women and
waste from high-income residential areas. In other children) who visit the site on a daily basis run the
residential areas and market places, the BCC has skips risk of contracting respiratory diseases as they inhale
in place, but collection is irregular. In peri-urban the smoke or are exposed continuously to the strong
areas, informal settlements and Traditional Housing foul odours emanating from the decomposing waste.
Areas (THAs), regular solid waste collection is non- The dumpsite also creates ideal conditions for vectors
existent, partly due to the shortage of access roads. such as mosquitos to thrive. During the rainy season,
Consequently, household solid waste is often nearby streams transport the malaria vectors in the
indiscriminately discarded or dumped in the streets, city’s water system thus an upsurge of Malaria.
pits, or drains. In light of the above, the Mzedi dump site has
The situation is compounded by the frequent fuel outlived its lifespan and is dilapidated. It is also a long
shortages and breaking down or shortage of distance from the city center, influencing the cost
equipment. According the BCC, the city owns 100 involved in collecting and disposing of waste. The
skips, of which 58 are hired out to private companies BCC has determined that a second site for building a
while the rest are placed in markets and unplanned proper landfill is needed and should be located closer
areas. This leaves 42 skips to service the city markets, to the city. However, they do not expect it to be
THAs and informal sectors. As a result, only 125 operational before 2015. In addition, the continuous
tonnes of waste is collected from these areas meaning increase in population and subsequent waste
a further 475 tonnes of waste go uncollected by the generation within Blantyre City calls for an urgent
council. In addition, poor maintenance and misuse determination of the best location for a new landfill.
such as setting garbage loaded skips on fire are 3.3 Implementation and Results
contributing to their deteriorated status. With an The combined methodology of the study in
average working life of 3 years for a skip, poor shown in Figure 4. After determining the problem
replacement ratio is compounding this situation. area, landfill siting was treated as a combinatorial
Furthermore, the 14 refuse trucks and 3 tractors used optimization problem. As such, simulated annealing
to collect these skips frequently break down due to was used to identify optimum sites for locating a
poor maintenance, which is not helped by the fact that landfill. For ranking and selecting the alternative
the majority of them are overused and have not been landfill sites an integrated EVAMIX/ Fuzzy AHP
replaced in the past decade, thus have succumbed to approach was employed. The study was implemented
natural wear and tear. under the setting of fuzzy set theory to accommodate
As of 1993, the official disposal site for solid the inherent uncertainty or ambiguity associated with
waste from Blantyre City is located on the slopes of decision makers when faced with complex multi-
Mzedi Mountain along the eastern boundary of the attribute decision making problems such as landfill
city (UN-HABITAT, 2011). The site is more of an siting.
open dump than a proper landfill. The site is not 3.3.1 Optimizing landfill siting
fenced or walled and lacks a leachate or gas In this study, the process of finding alternative
management system, which are basic requirements of sites for locating a landfill in Blantyre City was
a landfill. On a monthly basis it receives roughly treated as a combinatorial optimization model with all
3,000 tonnes of waste. The majority of the waste is the main elements of a complex problem, that is,
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Journal of American Science 2016;12(10) http://www.jofamericanscience.org

multiple objectives, multi-facilities, and non-linearity. x x x
landfills i and j, and (2) a new landfill i and an
The optimization problem was approached as a
x
continuous model and the alternative landfill sites existing landfill g.
were considered to be entering an existing network.
Having defined the objective functions a pseudo-
Formally, the problem was formulated by assuming
computer code of the simulated annealing algorithm
D R2 used for this research was developed as shown in
that is the region of interest (i.e., the study
Figure 5. The approach was implemented by tight
x ,x ,...,x D x ,x ,...,x
area) and 1 2 n , where 1 2 n are coupling, with the SA algorithm for continuous non-
n
the spatial locations of landfill sites. As discussed in linear optimization being executed within a
step 1 of section 2.1, an objective function or cost Geographical Information System (GIS) environment
function in the form of a mathematical expression is where with the usage of the associated spatial
required to describe the relationship of the parameters information, the objective functions were evaluated.
that will optimize the landfill siting process. Within Python was used as the scripting language in
the context of this research, the objective functions combination with ArcPy, a new site package in
were modelled by assuming that the aim was to ArcGIS 10 that builds on (and is a successor) to the
maximize the distance between, (i) new and the successful arcgisscripting module. Using ArcPy it was
existing Mzedi disposal site, and (ii) new landfill sites possible to access and work with numerous Python
only. In that respect, two objective functions modules from different sources but not necessarily
represented by Eq. (30) and Eq. (31), were defined. part of ArcGIS 10.
As a distance function the Euclidean distance was
used as comparisons by Love et al. (1998) and
Apparicio et al. (2008) have shown that it is close to
reality than the Manhattan or other distance functions.

Figure 5: Simulated Annealing Algorithm

Figure 4: The Proposed Landfill Siting Model

Line 2 of the pseudo-computer code initializes a
2
max x x dx (30) starting point xD , where x represents the x, y
i j
S
coordinates of a landfill site within the search space or
m n 2 domain D, which is the study area. In this case it was
max x x dx (31)
i g user provided as the centroid of Blantyre City. Line 3
S
g1 i1 (T )
chooses a high enough temperature 0 , such that
x
m almost any trial landfill location is accepted. Here,
Where, is the number of existing landfill sites (in
n an initial temperature of 100,000 was considered to be
this study its one, the Mzedi landfill), is the number
high enough for the SA algorithm to converge to the
x x x x optimal location of a landfill site. At each temperature
i j i g
of new landfill sites, , and are the
N
Euclidean distances between; (1) any two new the SA algorithm in executed T times (see Line 4)
using Eq. (32) as follows:

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Journal of American Science 2016;12(10) http://www.jofamericanscience.org

N (10vm) and 10(vm).......................(32) 
T Where, v, is the number of variables and m is  1 1 if 0
the number of constraints. Eq. (32) is a by-product of Tsalli'sAcceptanceRule[1(1q) ](1q) if 0and(1q)  1 (34)
the heuristic rule (Corana et al. (1987). It should be  AT AT
noted that for each landfill site, the problem variables  0 if 0and(1q)AT1
are two, one for each coordinate. Line 6 terminates the qisknownastheTsalli'sparametearndstartsfrom2anddecreaseesxponentially
algorithm if the current temperature, T , is small to1asTdecrease.s
enough (i.e., less or equal than 0.0001). Line 8
generates a random trial landfill site, x, within the Finally, Line 16 reduces T after looping NT
(x) x x
neighbourhood, , from the current location in times of generating and accepting them with
search space D . The neighbour landfill sites were probability given by the Tsallis’ rule. T was reduced
generated through the use of polar coordinates that T T
using a geometric cooling schedule, , where
guarantee the necessary randomness of the selection 
is a constant known as the cooling rate and is
x
process for the new locations considered. Thus, if smaller than 1 (typically between 0.5 and 0.99). At
represents a shift in the location of a site, then highT , any trial landfill site was accepted with high
x cos [0,2] probabilities, allowing the search to traverse a large
1 1 1
portion of the study area and overcome infeasible
x sincos [0,]
2 1 2 2 regions. As T was gradually reduced, its acceptance

probability decreased, and at very low temperatures
x sinsin cos  [0,] the algorithm behaved like a local search.
n1 1 n2 n1 n1
At the end of the SA algorithm one candidate
x sinsin sin sin [0,]
n 1 n2 n1 n n landfill site was generated. However, in order to give
Where
decision makers more candidate sites to choose from,
 1 if t  100 the SA algorithm was run again six more times thus
 2 if 100  t  150 generating six more alternative sites. During each run,

 a random spatial location within the study area was
   3 if 150  t  250
used as the starting point for the algorithm. As a

5 if 250  t  350 result, the SA algorithm provided a good set of seven

15 if t  350 (7) alternative candidate landfill sites (denoted as L1,
L , L, L , L, L and L respectively) for further
2 3 4 5 6 7
t is the number of iterations
evaluation based on criteria that had not been
The new landfill site, in the neighbourhood of the considered early on in the research and are difficult to
previous location is then given by: account for by a numerical scheme. Figure 6 shows
x xx the location of the alternative sites in relation to the
(x,,x)(x ,,x )(x ,,x ) (33) transport network, water sources, land cover type and
1 n 1 n 1 n
Line 9 calculates the difference between the new waste generation centers as well as the existing Mzedi
and previous cost/objective functions. In this case it’s disposal site. Because a high temperature of 100,000
the Euclidean distances between, (1) new and already degrees Celsius was used to initialize the search area,
existing Mzedi landfill site and (2) new landfill sites. it allowed the SA algorithm to traverse a large portion
 of Blantyre City and overcome infeasible regions
These are represented by variable . Line 11 checks
hence the location of these sites is such that they are
if the new location has maximized or minimized the
fairly distributed around the study area. The sites are
objective functions shown in Eq. (30) and Eq. (31).
also located such that the Euclidean distances between
This location is now accepted as the new site for the
them maximizes the objective functions set out in Eq.
landfill. If the new location has neither minimized nor
(30) and Eq. (31).
maximized the objective functions (failure to move to
3.3.2 Selecting and ranking landfill sites
a smaller energy state) and the algorithm becomes
For identifying the selection criteria to rank and
trapped into a local optimum, Lines 13 and 14 accepts
select the seven candidate sites, a number of
this location (which gives a worse solution) with a
qualitative and quantitative factors affecting the
certain probability, which in this study is given by the
location evaluation procedure were considered from a
Tsallis’ Acceptance Rule as in Eq. (34). This helps the
long list of criteria discussed in Zeiss and Lefsrud
SA algorithm to escape from local minima and be able
(1995), Wang et al. (2009), Siddiqui et al. (1996),
to provide a global optimum solution.
Sener et al. (2006) and Kontos et al. (2003). The
following seven criteria were adopted: land cover
25