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8/12/2019 A Multi-criteria Decision Making Approach for Location Planning
1/12
Mathematical and Computer Modelling 53 (2011) 98109
Contents lists available atScienceDirect
Mathematical and Computer Modelling
journal homepage:www.elsevier.com/locate/mcm
A multi-criteria decision making approach for location planning forurban distribution centers under uncertainty
Anjali Awasthi a,, S.S. Chauhan b, S.K. Goyal b
a CIISE, Concordia University, Montreal, Canadab Decision Sciences, JMSB, Concordia University, Montreal, Canada
a r t i c l e i n f o
Article history:
Received 9 December 2009
Received in revised form 25 July 2010
Accepted 26 July 2010
Keywords:
Location planning
Urban distribution center
Fuzzy TOPSIS
Multi-criteria decision making
a b s t r a c t
Location planning for urban distribution centers is vital in saving distribution costs and
minimizing traffic congestion arising from goods movement in urban areas. In this paper,
we present a multi-criteria decision making approach for location planning for urban
distribution centers under uncertainty. The proposed approach involves identification of
potential locations, selection of evaluation criteria, use of fuzzy theory to quantify criteria
values under uncertainty and application of fuzzy TOPSIS to evaluate and select the best
location for implementing an urban distribution center. Sensitivity analysis is performed
to determine the influence of criteria weights on location planning decisions for urban
distribution centers.
The strength of the proposed work is the ability to deal with uncertainty arising due
to a lack of real data in location planning for new urban distribution centers. The proposed
approach canbe practically applied by logisticsoperators in deciding on thelocationof new
distribution centers considering the sustainable freight regulations proposed by municipaladministrations. A numerical application is provided to illustrate the approach.
2010 Elsevier Ltd. All rights reserved.
1. Introduction
Location planning for urban distribution centers is vital in minimizing traffic congestion arising from goods movementin urban areas. In recent years, transport activity has grown tremendously and this has undoubtedly affected the traveland living conditions in urban areas. Considering this growth in the number of urban freight movements and theirnegative impacts on city residents and the environment, municipal administrations are implementing sustainable freightregulations like restricted delivery timing, dedicated delivery zones, congestion charging etc. With the implementationof these regulations, the logistics operators are facing new challenges in location planning for distribution centers. For
example, if distribution centers are located close to customer locations, then it will increase traffic congestion in the urbanareas. If they are located far from customer locations, then the distribution costs for the operators will be very high. Underthese circumstances, it is clear that the location planning for distribution centers in urban areas is a complex decision thatinvolves consideration of multiple criteria like maximum customer coverage, minimum distribution costs, least impacts oncity residents and the environment, and conformance to freight regulations of the city.
In this paper, we are focusing on a multi-criteria decision making approach for location planning for urban distributioncenters under uncertainty. We will use fuzzy theory to model decision making parameters. Fuzzy theory was introducedby Zadeh[1] to deal with uncertainty and vagueness in decision making. It is used to model systems that are difficult todefine precisely [2,3]. In such systems, parameters are defined using linguistic terms instead of exact numerical values [4].
Corresponding author. Tel.: +1 514 848 2424x5622; fax: +1 514 848 3171.E-mail addresses:[email protected](A. Awasthi),[email protected](S.S. Chauhan),[email protected](S.K. Goyal).
0895-7177/$ see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2010.07.023
http://dx.doi.org/10.1016/j.mcm.2010.07.023http://www.elsevier.com/locate/mcmhttp://www.elsevier.com/locate/mcmmailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.mcm.2010.07.023http://dx.doi.org/10.1016/j.mcm.2010.07.023mailto:[email protected]:[email protected]:[email protected]://www.elsevier.com/locate/mcmhttp://www.elsevier.com/locate/mcmhttp://dx.doi.org/10.1016/j.mcm.2010.07.0238/12/2019 A Multi-criteria Decision Making Approach for Location Planning
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A. Awasthi et al. / Mathematical and Computer Modelling 53 (2011) 981 09 99
For example, the impact of motor traffic on city residents can be expressed as high, very high, low etc. using linguistic terms.Then, conversion scales are applied to transform the linguistic scales into fuzzy numbers.
The problem of location planning for urban distribution centers can be classified as a special case of the more generalfacility location problems. The facility location problem usually involves a set of locations (alternatives) which are evaluatedagainst a set of weighted criteria independent from each other. The alternative that performs best with respect to all criteriais chosen for implementation. The distinct feature in location planning for urban distribution centers is the consideration ofinterests of other stakeholders like city residents, municipal administrators etc. The goal is not only to minimize distribution
costs but also to conform to sustainable freight regulations of the city and create least negative effects on city residentsand their environment. Several approaches have been reported in the literature for solving the facility location problems[58]. Agrawal[9] present a hybrid Taguchi-immune approach to optimize an integrated supply chain design problem withmultiple shipping. Sun et al. [10]present a bilevel programming model for the location of logistics distribution centers.Multi-criteria facility location models have been investigated by researchers in [1114].
The most commonly used approaches can be classified as continuous location models, network location models andinteger programming models. In continuous location models, every point on the plane is a candidate for facility locationand a suitable distance metric is used for selecting the locations. In network location models, distances are computed asshortest paths in a graph. Nodes represent demand points and potential facility sites correspond to a subset of the nodesand to points on arcs. The integer programming models start with a given set of potential facility sites and use integerprogramming to identify best locations for facilities. The objectives may be either of the minsum type or of the minmax type.Minsum models are designed to minimize average distances while minmax models have to minimize maximum distances.A detailed overview of facility location models for distribution network design can be found in[8].
It is worth pointing out that most of the above papers study the location problem under a certain environment, thatis, the parameters in the problem are fixed numbers and known in advance. In reality, often the parameters cannot beobtained with certainty. To deal with such situations, fuzzy theory is used [ 15,16]. Application of fuzzy theory for locationplanning for facilities has been investigated by researchers in[1722]. Liang and Wang [20] developed an algorithm forfacility site selection based on fuzzy theory and hierarchical structure analysis. Chen [21] developed a fuzzy multi-attributedecision making approach for the distribution center location selection problem. Chu [ 23]developed a fuzzy TOPSIS modelto solve the facility location selection problem under group decision making. Kahraman et al. [19] used four fuzzy multi-attribute group decision makingapproaches in evaluating facility locations. Chou et al.[22] presented a fuzzy simple additiveweighting system under group decision making for facility location selection with objective and subjective attributes. Leeand Lin [24] presented a fuzzy quantified SWOT procedure for environmental evaluation of an international distributioncenter.
The rest of the paper is organized as follows. In Section2,we present the preliminaries of fuzzy set theory. Section3presents the proposed framework for multi-criteria location planning for urban distribution centers under a fuzzy
environment. In Section4,we present a numerical application to illustrate the proposed approach. Finally, the conclusionsand future work are presented in Section5.
2. Fuzzy set theory
Some related definitions of fuzzy set theory adapted from [1,3,25,26]are presented as follows.
Definition 1. A fuzzy set a in a universe of discourseX is characterized by a membership function a(x)that maps eachelementxinXto a real number in the interval [0, 1]. The function value a(x)is termed the grade of membership ofxin a[27]. The nearer the value ofa(x)to unity, the higher the grade of membership ofxin a.
Definition 2. A triangular fuzzy number (Fig. 1)is represented as a triplet a= (a1, a2, a3). The membership functiona(x)of triangular fuzzy number ais given by
a(x)=
0, x a1,xa1
a2 a1, a1 x a2,
a3 x
a3 a2, a2 x a3,
0, x> a3
(1)
wherea1, a2, a3are real numbers and a1 < a2 < a3. The value ofxat a2 gives the maximal grade ofa(x), i.e.,a(x) = 1;it is the most probable value of the evaluation data. The value ofxata1gives the minimal grade ofa(x), i.e.,a(x) = 0; itis the least probable value of the evaluation data. Constants a1and a3are the lower and upper bounds of the available areafor the evaluation data. These constants reflect the fuzziness of the evaluation data [ 20]. The narrower the interval [a1, a3],the lower the fuzziness of the evaluation data.
For a detailed study of fuzzy set theory, please refer to [ 1,3,25,26].
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Fig. 1. Triangular fuzzy numbera.
Fig. 2. Two triangular fuzzy numbers.
Table 1
Linguistic terms for alternative ratings.
Linguistic term Membership function
Very poor (VP) (1, 1, 3)
Poor (P) (1, 3, 5)
Fair (F) (3, 5, 7)
Good (G) (5, 7, 9)
Very good (VG) (7, 9, 9)
Table 2
Linguistic terms for criteria ratings.
Linguistic term Membership function
Very low (1, 1, 3)
Low (1, 3, 5)
Medium (3, 5, 7)
High (5, 7, 9)
Very high (7, 9, 9)
2.1. The distance between fuzzy triangular numbers
Let a = (a1, a2, a3)and b= (b1, b2, b3)be two triangular fuzzy numbers(Fig. 2).
The distance between them is given using the vertex method by
d(a,b) =
1
3[(a1 b1)
2 +(a2 b2)2 +(a3 b3)
2]. (2)
2.2. Linguistic variables
In fuzzy set theory, conversion scales are applied to transform the linguistic terms into fuzzy numbers. In this paper, wewill apply a scale of 19 for rating the criteria and the alternatives. Table 1 presents the linguistic variables and fuzzy ratingsfor the alternatives andTable 2presents the linguistic variables and fuzzy ratings for the criteria.
3. The proposed framework for location planning for urban distribution centers
The proposed framework for location planning for urban distribution centers comprises four steps. These steps arepresented in detail as follows:
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Table 3
Criteria for location selection.
Criteria Definition Criteria type
Accessibility (C1) Access by public and private transport modes to the location Benefit (the more the better)
Security (C2) Security of the location from accidents, theft and vandalism Benefit (the more the better)
Connectivity to multimodal
transport (C3)
Connectivity of the location with other modes of transport,
e.g. highways, railways, seaport, airport etc.
Benefit (the more the better)
Costs (C4) Costs in acquiring land, vehicle resources, drivers, and taxesetc. for the location
Cost (the less the better)
Environmental impact (C5) Impact of location on the environment, for example, air
pollution, noise
Cost (the less the better)
Proximity to customers (C6) Distance of location to customer locations Benefit (the more the better)
Proximity to suppliers (C7) Distance of location to supplier locations Benefit (the more the better)
Resource availability (C8) Availability of raw material and labor resources in the
location
Benefit (the more the better)
Conformance to sustainable
freight regulations (C9)
Ability to conform to sustainable freight regulations imposed
by municipal administrations for e.g. restricted delivery
hours, special delivery zones
Benefit (the more the better)
Possibility of expansion (C10) Ability to increase size to accommodate growing demands Benefit (the more the better)
Quality of service (C11) Ability to assure timely and reliable service to clients Benefit (the more the better)
3.1. Selection of location criteria
Step 1 involves the selection of location criteria for evaluating potential locations for urban distribution centers.These criteria are obtained from literature review, and discussion with transportation experts and members of the citytransportation group. 11 criteria are finally chosen to determine the best location for implementing urban distributioncenters. These criteria are shown inTable 3.
It can be seen inTable 3that criterion C3 and criterion C4 belong to the cost category, that is, the lower the value, themore preferable the alternative for the best location. The remaining criteria are benefit type criteria, that means the higherthe value, the more preferable the alternative for the best location.
3.2. Selection of potential locations
Step 2 involves selection of potential locations for implementing urban distribution centers. The decision makers usetheir knowledge, prior experience with the transportation conditions of the city and the presence of sustainable freightregulations in the city to identify candidate locations for implementing urban distribution centers. For example, if certainareas are restricted for delivery by municipal administration, then these areas are barred from being considered as potentiallocations for implementing urban distribution centers. Ideally, the potential locations are those that cater to the interest ofall city stakeholders, that is city residents, logistics operators, municipal administrations etc.
3.3. Locations evaluation using fuzzy TOPSIS
The third step involves evaluation of potential locations against the selected criteria ( Table 3) using the technique calledfuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Situation). The TOPSIS approach chooses the alternativethat is closest to thepositive ideal solution andfarthestfrom thenegative ideal solution.A positive ideal solution is composedof the best performance values for each attribute whereas the negative ideal solution consists of the worst performancevalues. Fuzzy TOPSIS has been applied to facility location problems by researchers in[23,2831]. The various steps of fuzzyTOPSIS are presented as follows:
Step1. Assignment of ratings to the criteria and the alternatives.Let us assume there are J possible candidates called A = {A1,A2, . . . ,Aj} which are to evaluated against m criteria,
C = {C1, C2, . . . , Cm}. The criteria weights are denoted bywi(i = 1, 2, . . . , m). The performance ratings of each decisionmaker Dk(k= 1, 2, . . . , K) for each alternativeAj(j= 1, 2, . . . , n) with respect to criteria Ci(i = 1, 2, . . . , m) are denoted
by Rk =xijk(i = 1, 2, . . . , m; j= 1, 2, . . . , n; k= 1, 2, . . . , K)with membership functionRk (x).
Step2. Compute aggregate fuzzy ratings for the criteria and the alternatives.
If the fuzzy ratings of all decision makers are described as triangular fuzzy numbers Rk = (ak, bk, ck), k = 1, 2, . . . , K,
then the aggregated fuzzy rating is given by R = (a, b, c), k= 1, 2, . . . , K, where
a = mink
{ak}, b=1
K
Kk=1
bk, c=maxk
{ck}. (3)
If the fuzzy rating and importance weight of the kth decision maker are xijk = (aijk, bijk, cijk)and wijk = (wjk1, wjk2, wjk3),i = 1, 2, . . . , m,j = 1, 2, . . . , n, respectively, then the aggregated fuzzy ratings ( xij)of alternatives with respect to each
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criterion are given by xij =(aij, bij, cij)where
aij =mink
{aijk}, bij =1
K
Kk=1
bijk, cij =maxk
{cijk}. (4)
The aggregated fuzzy weights(wij)of each criterion are calculated as wj =(wj1, wj2, wj3)where
wj1 =mink {wjk1}, wj2 =
1
K
Kk=1
wjk2, wj3 =maxk {cjk3}. (5)
Step3. Compute the fuzzy decision matrix.
The fuzzy decision matrix for the alternatives(D)and the criteria( W)is constructed as follows:
D=
C1 C2 Cn
A1 x11 x12 . . . x1nA2 x21 x22 . . . x2nA3 . . . . . . . . . . . .A4 xm1 xm2 . . . xmn
, i = 1, 2, . . . , m; j= 1, 2, . . . , n (6)
W =( w1, w2, . . . , wn). (7)
Step4. Normalize the fuzzy decision matrix.
The raw data are normalized using a linear scale transformation to bring the various criteria scales onto a comparablescale. The normalized fuzzy decision matrix Ris given by
R = [rij]mxn, i = 1, 2, . . . , m; j= 1, 2, . . . , n (8)
where
rij =
aij
cj,
bij
cj,
cij
cj
and cj =max
icij (benefit criteria) (9)
rij =
aj
cij,
aj
bij,
aj
aij
and aj =min
iaij (cost criteria). (10)
Step5. Compute the weighted normalized matrix.
The weighted normalized matrix Vfor criteria is computed by multiplying the weights ( wj)of evaluation criteria withthe normalized fuzzy decision matrixrij:
V = [vij]mxn, i = 1, 2, . . . , m;j= 1, 2, . . . , n where vij =rij(.)wj. (11)
Step6. Compute the fuzzy ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS).The FPIS and FNIS of the alternatives are computed as follows:
A =(v1 ,v2 , . . . , v
n ) where v
j =max
i{vij3}, i = 1, 2 . . . , m; j= 1, 2, . . . , n (12)
A =(v1,v2, . . . , vn)where v
j =mini
{vij1}, i = 1, 2 . . . , m; j= 1, 2, . . . , n. (13)
Step7. Compute the distance of each alternative from FPIS and FNIS.The distance(di, d
i )of each weighted alternativei = 1, 2, . . . , mfrom the FPIS and the FNIS is computed as follows:
di =
nj=1
dv (vij,v
j), i = 1, 2, . . . , m (14)
di =
nj=1
dv (vij,v
j ), i = 1, 2, . . . , m (15)
wheredv (a,b)is the distance measurement between two fuzzy numbers aand b.
Step8. Compute the closeness coefficient(CCi)of each alternative.The closeness coefficient CCirepresents the distances to the fuzzy positive ideal solution (A
) and the fuzzynegative idealsolution (A) simultaneously. The closeness coefficient of each alternative is calculated as
CCi =di
d
i
+d
i
, i = 1, 2, . . . , m. (16)
Step9. Rank the alternatives.
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A2A3
A1
Highway
Distribution Center location
Customer locations
Restricted delivery zones
Fig. 3. Potential locations for the urban distribution center.
Rank the alternatives according to the closeness coefficient(CCi)in decreasing order and select the alternative with thehighest closeness coefficient for final implementation. The best alternative is closest to the FPIS and farthest from the FNIS.
3.4. Sensitivity analysis
Sensitivity analysis addresses the question How sensitive is the overall decision to small changes in the individualweights assigned during the pairwise comparison process? This question can be answered by slightly varying the values ofthe weights and observing the effects on the decision. This is useful in situations where uncertainties exist in the definitionof the importance for different factors. In our case, we will conduct sensitivity analysis in order to see the importance ofcriteria weights in deciding the best location for implementing the urban distribution center.
4. Numerical illustration
Let us assume that a logistics company is interested in implementing a new urban distribution center. A committee ofthree decision makers D1, D2 and D3 is formed to select the best selection. The alternatives available for location selectionare A1, A2 and A3(Fig. 3).
It can be seen that location A1 is situated outside the city close to a highway while locations A2 and A3 are located insidethe city. Location A2 is situated on the outskirts inside the city close to highways and to the customer locations whereaslocation A3 is situated in the city center far from highways.
The criteria used for evaluation of locations for urban distribution centers are same as those presented in Table 3, that is:Accessibility (C1), Security (C2), Connectivity to multimodal transport (C3), Costs (C4), Environmental impact (C5), Proximityto customers (C6), Proximity to suppliers (C7), Resource availability (C8), Conformance to sustainable freight regulations(C9), Possibility of expansion (C10), and Quality of service (C11). Among these, Criteria C3 and C4 are the cost categorycriteria (the less the better) while the remaining ones are the benefit category (the more the better) criteria.
The committee provided linguistic assessments for the eleven criteria using rating scales given in Table 1 and to the three
alternatives (locations) for each of the 11 location criteria using rating scales ofTable 2. Tables 4 and 5 present the linguisticassessments for the criteria and the alternatives.
Then, the aggregated fuzzy weights (wij) for each criterion are calculated using Eq. (5).For example, for criterion C1Accessibility, the aggregated fuzzy weight is given by wj =(wj1, wj2, wj3)where
wj1 =mink
{7, 7, 7}, wj2 =1
3
3k=1
(9+9+9), wj3 =maxk
{9, 9, 9} = wj =(7, 9, 9).
Likewise, we compute the aggregate weights for the remaining 10 criteria. The aggregate weights of the 11 criteria arepresented inTable 6.
Then, the aggregate fuzzy weights of the alternatives are computed using Eq. (4).For example, the aggregate rating foralternative A1 for criterion C1 given by the three decision makers is computed as follows:
aij =mink
{7, 5, 7}, bij =1
3
3k=1
(9+7+9), cij =maxk
{9, 9, 9} =(5, 8.33, 9).
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Table 4
Linguistic assessments for the 11 criteria.
Criteria Decision makers
D1 D2 D3
Accessibility (C1) VH VH VH
Security (C2) VH H VH
Connectivity to multimodal transport (C3) VH H VH
Costs (C4) H VH VH
Environmental impact (C5) M H MProximity to customers (C6) VH H VH
Proximity to suppliers (C7) H VH H
Resource availability (C8) M H H
Conformance to sustainable freight regulations (C9) H H VH
Possibility of expansion (C10) VH H VH
Quality of service (C11) VH VH VH
Table 5
Linguistic assessments for the three alternatives.
Criteria Alternatives Decision makers
D1 D2 D3
C1
A1 VG G VG
A2 G VG VG
A3 VG VG VG
C2
A1 G VG G
A2 M G M
A3 P P M
C3
A1 M G M
A2 M M M
A3 M G G
C4
A1 G G M
A2 M G M
A3 M G G
C5
A1 G M G
A2 G M M
A3 G P G
C6
A1 G VG VG
A2 VG VG G
A3 VG VG G
C7
A1 M VG M
A2 G G M
A3 G G G
C8
A1 M M G
A2 G M M
A3 M M M
C9
A1 M M M
A2 M P P
A3 P VP VP
C10
A1 G G VG
A2 VG G VG
A3 G G G
C11
A1 VG G VG
A2 VG G G
A3 G VG VG
Likewise, the aggregate ratings for the three alternatives (A1, A2, A3) with respect to the 11 criteria are computed. Theaggregate fuzzy decision matrix for the alternatives is presented inTable 7.
In the next step, we perform normalization of the fuzzy decision matrix of alternatives using Eqs. (8)(10).For example,the normalized rating for alternative A1 for criterion C1 (Accessibility) is given by
cj =maxi
{9, 9, 9} =9
rij =
59
, 8.339
, 99
=(0.56, 0.926, 1).
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Table 6
Aggregate fuzzy weights for criteria.
Criteria Decision makers Aggregated fuzzy weights
D1 D2 D3
C1 (7, 9, 9) (7, 9, 9) (7, 9, 9) (7, 9, 9)
C2 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)
C3 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)
C4 (5, 7, 9) (7, 9, 9) (7, 9, 9) (5, 8.33, 9)
C5 (3, 5, 7) (5, 7, 9) (3, 5, 7) (3, 5.66, 9)C6 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)
C7 (5, 7, 9) (7, 9, 9) (5, 7, 9) (5, 7.66, 9)
C8 (3, 5, 7) (5, 7, 9) (5, 7, 9) (3, 6.33, 9)
C9 (5, 7, 9) (5, 7, 9) (7, 9, 9) (5, 7.66, 9)
C10 (7, 9, 9) (5, 7, 9) (7, 9, 9) (7, 8.33, 9)
C11 (7, 9, 9) (7, 9, 9) (7, 9, 9) (7, 9, 9)
Table 7
Aggregate fuzzy weights for alternatives.
Criteria Alternatives Decision makers Aggregate ratings
D1 D2 D3
C1
A1 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)
A2 (5, 7, 9) (7, 9, 9) (7, 9, 9) (5, 8.33, 9)A3 (7, 9, 9) (7, 9, 9) (7, 9, 9) (7, 9, 9)
C2
A1 (5, 7, 9) (7, 9, 9) (5, 7, 9) (5, 7.66, 9)
A2 (3, 5, 7) (5, 7, 9) (3, 5, 7) (3, 5.66, 9)
A3 (1, 3, 5) (1, 3, 5) (3, 5, 7) (1, 3.66, 7)
C3
A1 (3, 5, 7) (5, 7, 9) (3, 5, 7) (3, 5.66, 9)
A2 (3, 5, 7) (3, 5, 7) (3, 5, 7) (3, 5, 7)
A3 (3, 5, 7) (5, 7, 9) (5, 7, 9) (3, 6.33, 9)
C4
A1 (5, 7, 9) (5, 7, 9) (3, 5, 7) (3, 6.33, 9)
A2 (3, 5, 7) (5, 7, 9) (3, 5, 7) (3, 5.66, 9)
A3 (3, 5, 7) (5, 7, 9) (5, 7, 9) (3, 6.33, 9)
C5
A1 (5, 7, 9) (3, 5, 7) (5, 7, 9) (3, 6.33, 9)
A2 (5, 7, 9) (3, 5, 7) (3, 5, 7) (3, 5.66, 9)
A3 (5, 7, 9) (1, 3, 5) (5, 7, 9) (1, 5.66, 9)
C6 A1 (5, 7, 9) (7, 9, 9) (7, 9, 9) (5, 8.33, 9)A2 (7, 9, 9) (7, 9, 9) (5, 7, 9) (5, 8.33, 9)
A3 (7, 9, 9) (7, 9, 9) (5, 7, 9) (5, 8.33, 9)
C7
A1 (3, 5, 7) (7, 9, 9) (3, 5, 7) (3, 6.33, 9)
A2 (5, 7, 9) (5, 7, 9) (3, 5, 7) (3, 6.33, 9)
A3 (5, 7, 9) (5, 7, 9) (5, 7, 9) (5, 7, 9)
C8
A1 (3, 5, 7) (3, 5, 7) (5, 7, 9) (3, 5.66, 9)
A2 (5, 7, 9) (3, 5, 7) (3, 5, 7) (3, 5.66, 9)
A3 (3, 5, 7) (3, 5, 7) (3, 5, 7) (3, 5, 7)
C9
A1 (3, 5, 7) (3, 5, 7) (3, 5, 7) (3, 5, 7)
A2 (3, 5, 7) (1, 3, 5) (1, 3, 5) (1, 3.66, 7)
A3 (1, 3, 5) (1, 1, 3) (1, 1, 3) (1, 1.66, 5)
C10
A1 (5, 7, 9) (5, 7, 9) (7, 9, 9) (5, 7.66, 9)
A2 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)
A3 (5, 7, 9) (5, 7, 9) (5, 7, 9) (5, 7, 9)
C11
A1 (7, 9, 9) (5, 7, 9) (7, 9, 9) (5, 8.33, 9)
A2 (7, 9, 9) (5, 7, 9) (5, 7, 9) (5, 7.66, 9)
A3 (5, 7, 9) (7, 9, 9) (7, 9, 9) (5, 8.33, 9)
The normalized value of alternative A1 for criterion C4 (Costs) is given by
aj =mini
(3, 3, 3)= 3
rij =
3
9,
3
6.33,
3
3
=(0.33, 0.473, 1).
Likewise, we compute the normalized values of the alternatives for the remaining criteria. The normalized fuzzy decisionmatrix for the three alternatives is presented inTable 8.
Then, thefuzzy weighted decision matrix forthe three alternatives is constructed using Eq.(11). The rijvalues from Table 8and wj values fromTable 6are used to compute the fuzzy weighted decision matrix for the alternatives. For example, for
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Table 8
Normalized fuzzy decision matrix for alternatives.
Criteria aj c
j Normalized ratings
A1 A2 A3
C1 5 9 (0.56, 0.926, 1) (0.56, 0.926, 1) (0.78, 1, 1)
C2 1 9 (0.56, 0.852, 1) (0.33, 0.629, 1) (0.11, 0.407, 0.78)
C3 3 9 (0.33, 0.629, 1) (0.33, 0.56, 0.78) (0.33, 0.703, 1)
C4 3 9 (0.33, 0.473, 1) (0.33, 0.53, 1) (0.33, 0.473, 1)
C5 1 9 (0.11, 0.157, 0.333) (0.11, 0.177, 0.33) (0.11, 0.177, 1)
C6 5 9 (0.56, 0.926, 1) (0.56, 0.926, 1) (0.56, 0.96, 1)
C7 3 9 (0.33, 0.703, 1) (0.33, 0.703, 1) (0.56, 0.78, 1)
C8 3 9 (0.33, 0.629, 1) (0.33, 0.629, 1) (0.33, 0.56, 0.78)
C9 1 7 (0.428, 0.714, 1) (0.14, 0.528, 1) (0.14, 0.24,0.71)
C10 5 9 (0.56, 0.85, 1) (0.56, 0.926, 1) (0.56, 0.78, 1)
C11 5 9 (0.56, 0.93, 1) (0.56, 0.85, 1) (0.56, 0.926, 1)
Table 9
Weighted normalized alternatives, FPIS and FNIS.
Criteria Alternatives FPIS (A) FPNS (A)
A1 A2 A3
C1 (3.89, 8.33, 9) (3.89, 8.33, 9) (5.44, 9, 9) (9, 9, 9) (3.89, 3.89, 3.89)
C2 (2.78, 7.09, 9) (1.67, 5.24, 9) (0.56, 3.38, 7) (9, 9, 9) (0.56, 0.56, 0.56)C3 (1.67, 5.238, 9) (1.67, 4.63, 7) (1.67, 5.86, 9) (9, 9, 9) (1.67, 1.67, 1.67)
C4 (1.67, 3.947, 9) (1.67, 4.41, 9) (1.67, 3.95, 9) (9, 9, 9) (1.67, 1.67, 1.67)
C5 (0.33, 0.894, 3) (0.33, 1, 3) (0.33, 1, 9) (9, 9, 9) (0.33, 0.33, 0.33)
C6 (2.78, 7.709, 9) (2.77, 7.7, 9) (2.78, 7.71, 9) (9, 9, 9) (2.78, 2.78, 2.78)
C7 (1.67, 5.38, 9) (1.67, 5.38, 9) (2.78, 5.96, 9) (9, 9, 9) (1.67, 1.67, 1.67)
C8 (1, 3.98, 9) (1, 3.987, 9) (1, 3.52, 7) (9, 9, 9) (1, 1, 1)
C9 (2.14, 5.47, 9) (0.714, 4, 9) (0.7, 1.8, 6.4) (9, 9, 9) (0.71, 0.714, 0.714)
C10 (2.778, 7.09, 9) (2.778, 7.7, 9) (2.78, 6.48, 9) (9, 9, 9) (2.78, 2.78, 2.78)
C11 (3.89, 8.33, 9) (3.89, 7.6, 9) (3.89, 8.33, 9) (9, 9, 9) (3.89, 3.89, 3.89)
Table 10
Distancesdv (Ai,A)anddv (Ai.,A
)for alternatives.
Criteria dv (A1,A) dv (A2,A
) dv (A3,A) dv (A1,A
) dv (A2,A) dv (A3,A
)
C1 3.908 3.908 4.266 2.974 2.974 2.051C2 6.296 5.611 4.063 3.754 4.753 5.961
C3 4.709 3.520 4.875 4.75 5.0596 4.602
C4 4.431 4.518 4.431 5.138 4.991 5.138
C5 1.572 1.586 5.015 7.673 7.636 6.806
C6 4.584 4.584 4.584 3.666 3.666 3.666
C7 4.747 4.747 4.946 4.715 4.715 3.995
C8 4.927 4.927 3.755 5.447 5.447 5.713
C9 5.576 5.146 3.359 4.448 5.579 6.496
C10 4.371 4.584 4.178 3.754 3.666 3.873
C11 3.908 3.667 3.908 2.974 3.048 2.974
alternative A1, the fuzzy weight for criterion C1 (Accessibility) is given by
vij =(0.56, 0.926, 1)(.)(7, 9, 9) = (3.89, 8.33, 9).
Likewise, we compute the fuzzy weights of the three alternatives for the remaining criteria (Table 9). Then, we compute thefuzzy positive ideal solution (A) and the fuzzy negative ideal solutions (A) usingEqs. (12)(13)for the three alternatives.For example, for criterion C1 (Accessibility),A =(9, 9, 9) andA =(3.89, 3.89, 3.89). Similar computations are performedfor the remaining criteria. The results are presented in the last two columns ofTable 9.
Then, we compute the distance dv (.) for each alternative from the fuzzy positive ideal matrix (A) and fuzzy negative
ideal matrix (A) using Eqs.(2),(12)and(13).For example, for alternative A1 and criterion C1, the distances dv (A1,A)and
dv (A1,A)are computed as follows:
dv (A1,A) =
1
3[(3.899)2 +(8.339)2 +(99)2] =2.974
dv (A1,A)=
1
3[(3.893.89)2 +(8.333.89)2 +(93.89)2] =3.907.
Likewise, we compute the distances for the remaining criteria for the three alternatives. The results are shown in Table 10.
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Table 11
Closeness coefficients (CCi) of the three alternatives.
A1 A2 A3 Ranking order
di 49.033 46.803 47.385
A1> A3> A2d+i 49.302 51.539 51.281
CCi 0.498 0.475 0.4802
Then, we compute the distances di anddi usingEqs. (14)(15).For example, for alternative A1 and criterion C1, thedistancesdi andd
i are given by
di =
1
3[(3.899)2 +(8.339)2 +(99)2] +
1
3[(2.789)2 +(7.099)2 +(99)2] +
+
1
3[(3.899)2 +(8.339)2 +(99)2] =49.3
di =
1
3[(3.893.89)2 +(8.333.89)2 +(93.89)2] +
1
3[(2.780.56)2 +(7.090.56)2 +(90.56)2]
+ +
1
3[(3.893.89)2 +(8.333.89)2 +(93.89)2] =49.03.
Using the distancesdi anddi , we compute the closeness coefficient for the three alternatives using Eq.(16).For example,for alternative A1, the closeness coefficient is given by
CCi =di /(d
i +d
+i ) = 49.03/(49.3+49.03) = 0.499.
Likewise,CCiis computed for the other two alternatives. The final results are shown in Table 11.
By comparing theCCi values of the three alternatives (Table 11), we find that A1 > A3 > A2. Therefore, location A1 isselected as the best location for implementing the new urban distribution center for the logistics company.
4.1. Sensitivity analysis
To investigate the impact of criteria weights on the location selection of an urban distribution center, we conducted asensitivity analysis. 16 experiments were conducted. The details of the 16 experiments are presented inTable 12.
It can be seen inTable 12that in the first five experiments, the weights of all criteria are set equal to (1, 1, 3), (1, 3, 5),(3, 5, 7), (5, 7, 9) and (7, 9, 9). In experiments 615, the weight of one criterion is set as the highest and the remaining are setto the lowest value. For example, in experiment 6, the criterion C1 has the highest weight = (7, 9, 9) whereas the remainingcriteria have weight=(1, 1, 3). In experiment 17, the weights of all the cost category criteria are set as the highest, that is,criteria C3 and C4 at the highest weight = (7,9, 9), and the weights of benefit category criteria (C1C2, C5C11) are set as thelowest weight=(1, 1, 3). In experiment 18, the weight of the cost category criteria (C3, C4) is set as the lowest weight =(1,1, 3) and the weights of the benefit category criteria (C1C2, C5C11) are set as the highest weight =(7, 9, 9).
The results of the sensitivity analysis are presented in Fig. 4. It can be seen that for all 18 experiments, location A1has emerged as the best location in 14 experiments. In experiments 6, 10, 12 and 17, the location A3 has emerged as thewinner. Therefore, we can say that the location decision is relatively insensitive to benefit criteria weights; however whenthe weights of cost criteria (C3, C4) are set as the highest, then the best solution is changed from A1 to A3.
5. Conclusion
In this paper, we present a multi-criteria decision making framework for location planning for urban distributioncenters under a fuzzy environment. The proposed approach comprises four steps. In step 1, we identify the criteria forevaluating potential locations for distribution centers. These criteria are: Accessibility, Security, Connectivity to multimodaltransport, Costs, Environmental impact, Proximity to customers, Proximity to suppliers, Resource availability, Conformanceto sustainable freight regulations, Possibility of expansion, and Quality of service. In step 2, the potential locations forimplementing urban distribution centers are identified. In step 3, the decision makers provide ratings for the criteria andthe potential locations. Fuzzy TOPSIS is used to determine aggregate scores for all potential locations and the one with thehighest score is finally chosenfor implementation. Sensitivity analysis is performed to assessthe influenceof criteria weightson the decision making process.
The strength of our work is the ability to deal with multiple criteria and model uncertainty in location planning for urbandistribution centers through the use of linguistic parameters and fuzzy theory. The proposed approach can be practically
applied by logistics operators in implementing new distribution centers, considering the sustainable freight regulationsproposed by municipal administrations.
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Table 12
Experiments for sensitivity analysis.
Experiment number CCi Description
A1 A2 A3
1 0.416 0.405 0.410 All criteria weights= (1, 1, 3)
2 0.461 0.448 0.451 All criteria weights= (1, 3, 5)
3 0.470 0.453 0.456 All criteria weights= (3, 5, 7)
4 0.477 0.456 0.459 All criteria weights= (5, 7, 9)
5 0.513 0.483 0.486 All criteria weights= (7, 9, 9)
6 0.440 0.429 0.447 Weight of criteria 1=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
7 0.468 0.437 0.413 Weight of criteria 2=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
8 0.432 0.408 0.430 Weight of criteria 3=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
9 0.424 0.417 0.419 Weight of criteria 4=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
10 0.382 0.373 0.414 Weight of criteria 5=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
11 0.440 0.429 0.434 Weight of criteria 6=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
12 0.435 0.425 0.442 Weight of criteria 7=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
13 0.432 0.422 0.413 Weight of criteria 8=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
14 0.453 0.423 0.401 Weight of criteria 9=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
15 0.436 0.429 0.427 Weight of criteria 10=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
16 0.440 0.425 0.434 Weight of criteria 11=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
17 0.393 0.387 0.421 Weight of criteria 3 & 4=(7, 9, 9)
Weight of remaining criteria=(1, 1, 3)
18 0.540 0.504 0.491 Weight of criteria 3 & 4=(1, 1, 3)Weight of remaining criteria=(7, 9, 9)
0
0.1
0.2
0.3
0.4
0.5
0.6
Experiment 1
Experiment 2
Experiment 3
Experiment 4
Experiment 5
Experiment 6
Experiment 7
Experiment 8
Experiment 9
Experiment 10
Experiment 11
Experiment 12
Experiment 13
Experiment 14
Experiment 15
Experiment 16
Experiment 17
Experiment 18
Sensitivity Analysis
A1
A2
A3
Fig. 4. Results of sensitivity analysis.
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