Abstract In this paper, a model to estimate the weights of mutually dependent cri-teria, based on cause-effect assessments of a group of professionals, is developed forproblem of multiple criteria decision making (MCDM). Here, both DEMATEL (De-cision Making Trial and Evaluation Laboratory) and TOPSIS (Technique for OrderPerformance by Similarity to Ideal Solution) models are combined and extended tohandle fuzzy evaluations where the first is used to set the weights of the interdepen-dent criteria and the second for drawing a decision from a group of professionals whouse linguistic ratings in their evaluation.
The presented model is characterized by the capability to estimate the criteriaweights when the criteria are interrelated. The strict determination of the criteriaweights prior to the assessment process is eliminated as they are computed by theDEMATEL part. A classical case-study of optimal sore throat treatment in primarycare unit is used to demonstrate the efficiency of the proposed model.
Keywords MCDM · DEMATEL · TOPSIS · Decision making
1. Introduction and Literature Review
MCDM is a sub-discipline of operations research which refers to making decisionsin the incidence of multiple and generally conflicting criteria. Specifically, MCDMrefers to screening, prioritizing and selecting a set of alternatives (also referred toas “candidates” or “actions”) under usually independent and conflicting criteria [1,2]. In practical applications, the criteria weights cannot always be assessed preciselyparticularly, when the criteria have some correlation and interdependence.
Doraid Dalalah (�)IE Department, Jordan University of Science and Technology, Jordanemail: [email protected] Al-TahatIE Department, Jordan University, Amman, JordanKhaled BatainehME Department, Jordan University of Science and Technology, Jordan
Such scenarios may arise in some decision making problems as in clinical diagno-sis where the symptoms, the signs as well as the criteria are mutually dependent andthe evaluations are linguistic in their description. Classical MCDM methods cannoteffectively handle such imprecise information, particularly when the individuals donot have the full right to set their own weights [3].
In this study, to help solve this problem, we present a fuzzy multi-criteria deci-sion making model for interdependent criteria. Here, the criteria weights are deter-mined based on their relation-prominence strength using a modified fuzzy DEMA-TEL model. Next, a modified fuzzy TOPSIS model is used for the decision makingprocess. The DEMATEL is extended to include fuzzy ratings and its structural di-graphs are modified to estimate the weights of the criteria. The TOPSIS model willthen be used to draw a final decision by computing the distances from ideal and anti-ideal solutions.
In brief, the DEMATEL is a methodology which can confirm interdependenceamong variables and aid in the development of a visual plan to reflect interrelation-ships between variables. Originated from the Geneva Research Centre of the BattelleMemorial Institute [4], the DEMATEL aimed at the fragmented and antagonistic phe-nomena of world societies to search for integrated solutions. It is practical and usefulfor visualizing the structure of complicated causal relationships with matrices or di-graphs. The matrices or digraphs portray a contextual relation between the elementsof a system, in which a numeral represents the strength of influence. DEMATEL hasbeen successfully applied in many fields, for example the authors in [5] analyzed theobstructive factors of welfare service with the DEMATEL method, and in [6], em-ployed it to design and evaluate software of displaying-screen structure in analyzinga supervisory control system.
As for TOPSIS, it identifies the best alternative by the shortest distance from apre-determined positive ideal solution (PIS), and the farthest distance from other pre-determined negative ideal solution (NIS). Since its introduction by [7], huge literatureemerged addressing TOPSIS applications, such applications include the evaluationof quality of services [8], the applications in aggregate production planning [9], intercompany comparison [10], large scale nonlinear programming [11] and facility loca-tion selection [12].
The major foreseeable contribution of our proposed model is twofold: First, theestimation of criteria weights when the criteria are interrelated. Fundamentally, strictdetermination of the weights of the criteria may impose a drastic influence on theresults in traditional models as they have shown high sensitivity to such weights. Inour model, there would not be any need to firmly decide the criteria weights. Second,the extension to handle fuzzy ratings and linguistic evaluations with respect to fuzzyideal and anti-ideal solutions, a matter that may facilitate the judgment process by aharmonized model of both DEMATEL and TOPSIS.
1.1. MCDM Literature Review
One of the powerful tools for decision making is decision support system (DSS).While MCDM is one discipline of the decision making field, it is widely used in con-junction with DSS by a huge number of decision makers in variety of areas such as
Although MCDM problems are prevalent and common in everyday life, as a disci-pline, MCDM has a relatively short history of about 30 years [13]. The advancementin computer technologies was one of the major drives for the development of MCDMdiscipline. In one hand, the revolution of information technology and extensive use ofcomputers have made a huge amount of information available, which makes MCDMprogressively more important and helpful in sustaining business decision making. Onthe other hand, the rapid growth of computer technology in recent years has made itfeasible to carry out systematic analysis of complex MCDM approaches.
A good deal of literature exists in MCDM problem [1, 14-18]. Such studies tendto solve MCDM problems under some predetermined conditions and assumptions,though some of the methods were criticized as ad hoc and unjustified on theoreticaland/or empirical grounds [19]. Since the seventies, theories and models in MCDMhave continued to grow at a steady state. A number of surveys have been conductedshowing the vitality of the field and the multitude of methods which have been devel-oped [18, 20].
There are many variations on the theme of MCDM depending on the theoretical ba-sis used for the modeling. In fact, multi-criteria may include both multiple attributesand multiple objectives which fall under the platform of multiple attribute utility the-ory (MAUT) and multiple objective linear programming (MOLP), respectively. Withthe introduction of MCDM, the first contributions to truly scientific approach to deci-sion making were made, but the problem was to deal with human preferences whichcan hardly reach a momentous degree of consistency or coherence.
Following [18, 20], MCDM methods may be divided into the following major cat-egories: the outranking category, the category/categories of value and utility theory,the interactive multiple objective programming category and the category of groupdecision and negotiation theories.
The first category depends on the ranking of alternatives. Ranking may be basedon the degree of optimality as in [21], Hamming distance [22], comparison function[23], fuzzy mean and spread [24], proportion to the ideal and anti-ideal as in TOPSIS,left and right scores [25], centroid index [26], area measurement [27] and linguisticranking methods [28].
A good example of the second family is the analytical hierarchy process [29].In this family, the relative importance is being assessed for multiple attributes. Othermethods such as fuzzy simple additive weighting [21, 30], fuzzy conjunctive/disjunctivemethods [31], fuzzy outranking methods [32] and maximin methods [33].
The third category includes fuzzy mathematical programming which involves ex-tensive methods such as linear programming [34], flexible programming [35], possi-bilistic programming [36], robust programming [37], possibilistic programming withfuzzy preference relations [38], possibilistic linear programming with fuzzy goals[39]. A good survey of the developments in fuzzy programming was presented in[16, 40, 41].
The fourth category emerges from the growing field of group decision making andnegotiation, especially from the viewpoints of management science and operations
research. In this family, the decision effectiveness, errors and biases, group think,evaluation of group processes are all addressed for improving group decision making[42]. The study in [20] presented a good survey of group decision making methods.
The majority of multi-criteria approaches are based on crisp comparisons of pref-erences and strict weight values [2]. However, in practical applications, the com-parisons may be held using linguistic scales, such as “High”, “Medium” and “Low”with linguistic weights. Such evaluations cannot be handled using crisp scales, rather,fuzzy sets may be more powerful in such evaluations. A rich literature exists on fuzzyMCDM such as the research and surveys was presented in [7, 43-45].
Contrary to conventional MCDM literature studies where the criteria weights aredetermined in a strict manner, our model incorporates the strength of DEMATELto pull the criteria weights out of the interdependent relations between the criteriathemselves. This way, the weights are computed by utilizing the structural DEMTELdiagraphs. The calculated weights will be plugged into a modified TOPSIS model torank the alternatives using an optimized closeness coefficient. Up to our knowledge,such hybrid model has not been addressed in previous related literature, a certaintywhich demonstrates the major contribution in this study.
This paper is organized as follows: The new model is described in Section 2,presenting the modified DEMATEL, TOPSIS, the membership degree and the imple-mentation procedure. A numerical example of a classical decision making problemis demonstrated in Section 3 followed by the conclusions in Section 4.
2. The Model
Generally, an MCDM problem is characterized by a) the ratings of each alternativewith respect to each criterion and b) the weights given to each criterion. In thispaper, a hybrid model is proposed that implements a modified DEMATEL-TOPSISapproach to calculate the weights and process the fuzzy ratings, respectively. Specifi-cally, a modified DEMATEL model will be used to estimate the weight of each crite-rion based on a causal assessment performed by a group of experts. Later, a modifiedfuzzy TOPSIS model will be used to process the weights with the correspondingassessment of each alternative against the different criteria. In such an approach,the model can capture the experts’ knowledge and preferences precisely, where theweights do not have to be strictly predetermined prior to the rating process.
Our approach mainly depends on two initial matrices, namely, the direct-relationfuzzy matrix and the fuzzy decision matrix. Both matrices are evaluated through dif-ferent assessment processes. The first assessment process begins with a survey ofthe cause-effect relations between the criteria themselves. This method quantifies thetotal effect of a criterion on the other remaining criteria. In this modified approach,a group of experts is consulted to provide such assessment. The assessment pro-cess is performed by using a fuzzy domain. For instance, one could say criterion ahas a high influence on criterion b. Similar statements can be drawn using differentlinguistic variables, particularly, no influence (No), low influence (L), moderate in-fluence (M), high influence (H) and very high influence (VH). After the cause-effectrelations among the criteria have been assessed, we can establish the average direct-relation fuzzy matrix and proceed to calculate the ultimate criteria weights that will
Fuzzy Inf. Eng. (2012) 2: 195-216 199
be plugged into the modified TOPSIS model by which the contrast among the alterna-tives is calculated. The ratings of alternatives with respect to criteria are measured bysimilar linguistic variables, particularly, very poor (VP), poor (P), fair (F), good (G),and very good (VG). Those ratings symbolize the entries of the initial fuzzy decisionmatrix.
The most widely used fuzzy membership function is a triangular fuzzy numberwhich is based on three values, the lower (�), the most possible (m) and the upperbound (u). In our analysis, a triangular membership function is defined by these threevalues, where the most possible value is tied to unity. For instance, a fuzzy set a canbe represented as a = (a�, am, au).
An MCDM problem can be concisely expressed in matrix format as:
C1 C2 · · · Cn
Y =
A1
A2...
Am
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
y11 y12 · · · y1n
y21 y22 · · · y2n....... . ....
ym1 yn2 · · · ymn
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
, (1)
W = [w1 w2 · · ·wn],
where A1, A2, · · · , Am are possible alternatives, C1,C2, · · · ,Cn are the criteria withwhich the alternative performance are measured, yi j is the rating of alternative Ai
with respect to criterion C j, and wj is the weight of criterion C j. In our approach,a hybrid fuzzy model is established using the DEMATEL and TOPSIS techniques.A modified fuzzy DEMATEL model will be used to get the criteria weights throughfinding the causal diagram of the criteria.
Suppose that a system consists of a set of criteria C = {C1,C2, · · · ,Cn}, the re-lations between the criteria factors can be determined through an influence assess-ment. The influence pair-wise comparison scale is partitioned into five distinct levelsas mentioned previously, that is “No”, “L”, “M”, “H” and “VH”. Fig.1 presents themembership relations between these scales. The initial fuzzy direct-relation matrix Zis n×n matrix that symbolizes the pair-wise influence between criteria. In this matrix,zi j denotes the extent to which the criterion Ci influences criterion C j. Consequently,the principal diagonal elements zi j of matrix Z are put to be zero [46].
C1 C2 · · · Cn
Z(k) =
C1
C2...
Cn
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0 z(k)12 · · · z(k)
1n
z(k)21 0 · · · z(k)
2n....... . ....
z(k)n1 z(k)
n2 · · · 0
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, k = 1, 2, · · · , p, (2)
where zi j = (zi j,l, zi j,m, zi j,u), k refers to a specific expert among the group and p isthe number of surveyed professionals. To transform the criteria scales into compa-rable ones, the linear scale transformation is implemented as a normalization. Thenormalized direct-relation fuzzy matrix of expert k, denoted as X(k) is thus given by:
Similar to the crisp DEMATEL method, it is assumed here that at least one i such
thatn∑
j=1z(k)
i j,u < r(k) ∀k = 1, · · · , p, which is usually achieved in practical cases. The
average matrix, denoted by X of X(1), X(2), · · · , X(p), thusly can be given as:
X =X(1) ⊕ X(2) ⊕ · · · ⊕ X(p)
p(4)
and hence
X =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
x11 x12 · · · x1n
x21 x22 . . . x2n...... · · · ...
xn1 xn2 · · · xnn
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(5)
and
xi j =
p∑k=1
x(k)i j
p, (6)
where xi j consists of (xi j,�, xi j,m, xi j,u) and the operator⊕ represents element-by-elementaddition. The fuzzy matrix X is called the average direct-relation fuzzy matrix. Here,we implement the arithmetic mean to group the collected data of the experts. This
Fuzzy Inf. Eng. (2012) 2: 195-216 201
approach can emphasize the differences within individuals, which is better than ag-gregating all the data of the experts right after the initial direct-relation fuzzy matrixZ(k) is obtained.
After computing the average direct-relation fuzzy matrix, the fuzzy numbers in-side this matrix can be split into detached sub-matrices X�, Xm and Xu. Hence, any ofthe sub-matrices X�, Xm or Xu represents the sub-stochastic matrix that results from anabsorbing type Markov chain which is obtained by removing all columns and rows ofthe absorbing states [46]. Note that lim
w→∞(Xs)w = O and limk→∞
(I + Xs + X2s + · · ·+ Xk
s ) =
(I − Xs)−1, ∀s = �,m, u, where O is the null matrix and I is the identity matrix. Thetotal-relation fuzzy matrix T can be acquired by calculating the following term:
T = limw→∞(X + X2 + · · · + Xw) = X(I − X)−1, (7)
that is, T� = X�(I − X�)−1, Tm = Xm(I − Xm)−1 and Tu = Xu(I − Xu)−1. The setof equations in (11) includes both inverse and matrix multiplication (i.e., it is notelement by element computation). Accordingly, T will then look like:
T =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
t11 t12 · · · t1n
t21 t22 . . . t2n....... . ....
tn1 tn2 · · · tnn
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (8)
where ti j = (ti j,�, ti j,m, ti j,u). The row and column sums of the sub-matrices T�,Tm,Tu
denoted by Di and Ri respectively, can be obtained through the following two formu-las:
Di =
n∑
j=1
ti j (i = 1, 2, · · · , n), (9)
Ri =
n∑
i=1
ti j ( j = 1, 2, · · · , n), (10)
where the summation above is held via element-by-element addition. After the de-fuzzification of Di and Ri, we get Dde f
i and Rde fi for the list of criteria. Here, the point
that splits the triangular fuzzy number into two even areas is considered as the de-fuzzification point. The causal diagram represents the mapping of the pairs of (Dde f
i
+ Rde fi ) and (Dde f
i - Rde fi ), where the horizontal axis (Dde f
i + Rde fi ) is called “Promi-
nence” and the vertical axis (Dde fi - Rde f
i ) is called “Relation”. The prominence axis inthe causal diagram demonstrates how significant a criterion with regard to the groupof criteria, whereas the relation axis may divide the criteria into the cause and effectgroups. When the value (Dde f
i - Rde fi ) is positive, the criterion belongs to the cause
group. If the value (Dde fi - Rde f
i ) is negative, the criterion belongs to the effect group.Hence, causal diagrams can visualize the complicated causal relationships betweencriteria into a visible structural model and provide valuable insights for problem solv-ing. In this context, we suggest the following expression to measure the relative im-portance of a criterion which accounts for the interdependence, the cause-effect mapas well as the interrelations between the criteria:
The weights above have to be normalized so that they sum to unity, which simplycan be held by:
wi =Wi
n∑i=1
Wi
. (11)
Now that the weight values are found, they can be used in the modified fuzzy TOP-SIS model. The main concept in the traditional TOPSIS is that the chosen alternativeshould have the shortest distance from the PIS and the farthest from the NIS.
In our modified TOPSIS model, we use fuzzy assessments instead of crisp val-ues. Linguistic variables are used to assess the ratings of each alternative with re-spect to each criterion. According to the notion of TOPSIS, a fuzzy negative idealsolution (FNIS) and a fuzzy positive ideal solution (FPIS) are defined. Next, the dis-tances from FPIS and FNIS are calculated to all alternatives. Finally, to better portraythe contrast between the ideal alternative and the remaining alternatives, an optimalmembership degree function is established which will eventually determine the rank-ing order of all alternatives. Higher values of this membership degree indicate that analternative is closer to FPIS and farther from FNIS.
Following (1), a fuzzy multi-criteria group decision making problem can be con-cisely expressed in a fuzzy matrix format as:
C1 C2 · · · Cn
y(k) =
A1
A2...
Am
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
y(k)11 y(k)
12 · · · y(k)1n
y(k)21 y(k)
22 · · · y(k)2n
....... . ....
y(k)m1 y(k)
n2 · · · y(k)mn
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
,(12)
W = [w1 w2 · · ·wn],
where yi j is a linguistic variable represented by triangular fuzzy numbers, yi j =
(yi j,�, yi j,m, yi j,u). Suppose that a decision group of p experts is to conduct the assess-ment, the average rating of alternatives with respect to each criterion can be calculatedas:
yi j =1p
[y(1)i j ⊕ y(2)
i j ⊕ · · · ⊕ y(p)i j ], (13)
where y(k)i j is the rating of the kth decision maker. The collection of all yi j constitutes
the aggregated matrix Y . Taking into consideration the weights of each criterion, wecan construct the weighted-normalized fuzzy decision matrix as:
V = [vi j]m×n, i = 1, 2, · · · ,m, j = 1, 2, · · · , n, (14)
where vi j = yi j(•)wj, ∀ j =1, · · · , n. The elements vi j of the weighted normalizedfuzzy decision matrix are normalized positive triangular fuzzy numbers within theclosed interval [0, 1]. Accordingly, we may characterize the FPIS A∗ and the FNISA− as:
Fuzzy Inf. Eng. (2012) 2: 195-216 203
A∗ = (v∗1, v∗2, · · · , v∗n),
A− = (v−1 , v−2 , · · · , v−n ),
(15)
v∗j ={maxi(vi j) if C j ∈ C; maxi(vi j) if C j ∈ C’},v−j ={maxi(vi j) if C j ∈ C; maxi(vi j) if C j ∈ C’},
where C is the set of benefit criteria, i.e., more of each criterion is better, and C’ isthe set of detriment criteria, i.e., less is better. The ideal distance of each alternativefrom A∗ and A− can be calculated as:
d∗i =n∑
j=1d(vi j, v∗j), ∀i = 1, 2, · · · ,m,
d−i =n∑
j=1d(vi j, v−j ), ∀i = 1, 2, · · · ,m,
(16)
where d(•, •) is the distance measure between any two fuzzy numbers. The distancecan be calculated as follows: let a and b be two triangular fuzzy numbers. Using thevertex method, we can calculate the distance between the two fuzzy numbers as:
d(a, b) = [13
((a� − b�)2 + (am − bm)2 + (au − bu)2))]0.5. (17)
Let variable ui denote the global evaluation for alternative Ai in regard to all criteria,which in turn will provide the ranking order of all alternatives. Now, from the of fuzzysets theory point of view, ui interprets the membership degree of alternative Ai withrespect to the notion of “optimum to all criteria”, i.e., the ideal alternative, while 1-ui
denotes the membership degree to the anti-ideal. In order to better explain the differ-ence between the ideal alternative with respect to other alternatives, the membershipdegree ui can be related as a weight for the fuzzy ideal weight distance. Hence, theweighted fuzzy ideal weight distance is given by:
Dgi = uid∗i , (18)
and its counter part:Db
i = (1 − ui)d−i . (19)
Following the optimality principle presented in [47], the optimal evaluation of themembership degree ui can be solved by minimizing the square sum of the weightedfuzzy ideal weight distance Dg
i and its counter part Dbi , i.e., the objective function will
The expression in (21) represents the overall evaluation of each alternative withrespect to all criteria. The best alternative can be easily identified and the rankingorder of all alternatives can be determined through the membership degree of eachalternative. Equation (21) is comparable in its current form to TOPSIS closenesscoefficient. The closeness coefficient in TOPSIS is calculated through the distanceto anti-ideal solution divided by the summation of the ideal and anti-ideal solutions[48].
Noticeably, according to (21), an alternative Ai would be closer to FPIS (A∗) andfarther from FNIS (A−) as the membership degree ui approaches 1. In other words,the membership degree can determine the ranking order of all alternatives and hencewill indicate the best one among a set of given feasible alternatives.
The described hybrid approach can be implemented by following given steps be-low:
Step 1: Form a committee of p decision-makers, identify the interdependent evalu-ation criteria and alternatives and acquire the assessments of the group of p expertsto measure the relationship between the criteria C = {C1,C2, · · · ,Cn} in terms of lin-guistic evaluations. Hence, p direct-relation fuzzy matrices Z(1), Z(2), · · · , Z(p) willresult, each of which corresponds to a different expert.
Step 2: Acquire the decision makers’ opinions to get the fuzzy ratings y(1), y(2), · · · ,y(p), where in each matrix an expert will evaluate alternative Aiunder criterion C j forall criteria and alternatives.
Step 3: Acquire the normalized direct-relation fuzzy matrix X(k), ∀k = 1, · · · , p andcalculate the average matrix X using (4).
Step 4: Compute the total-relation fuzzy matrix T using (7).
Step 5: Compute Di and Ri then defuzzify to find Dde fi and Rde f
i .
Step 6: Find the criteria weights using the formula in (11).
Step 7: Compute the aggregated matrix Y from the results of Step 2.
Step 8: Use the results in Step 6 and 7 to construct the weighted normalized fuzzydecision matrix as in (14).
Step 9: Determine FPIS and FNIS as in (15).
Step 10: Calculate the distance of each alternative from the FPIS and FNIS respec-tively using (16).
Step 11: Calculate the membership degree ui of each alternative using (21).
Step 12: According to the optimal membership degrees, determine the ranking orderof all alternatives.
In this numerical example, we will apply the proposed approach to a classical sorethroat optimal treatment problem, a frequent illness for which primary care physi-cians are consulted [2, 49, 50]. Known as pharyngitis or tonsillitis, a sore throat is aninfection that can be either viral or bacterial which affects the pharynx. Contagiousviral infection (such as cold, flu or mononucleosis) are the most common causes forsore throat, although some bacterial infection may cause serious throat infections(such as strep, mycoplasma or hemophilus). It is well known in the medical societythat bacterial sore throats react rapidly to antibiotics as compared to viral infections.Researchers have long attempted to decide how best to treat strep throat, yet it re-mains a leading cause for physician visits.
The major concern is the extent to which the clinical likelihood of a group. Astreptococcal infection should influence the decision of clinicians. In this case study,we will conduct a multi-criteria decision analysis using the proposed model in orderto determine the best treatment of pharyngitis.
The objective of our case study is to recognize the optimal office management ofpatients aged 18 years or older who are suffering of a chief complaint of strep throat(sore throat). There will be four major interdependent criteria which are commonlyimplemented for the treatment, particularly: shortening the illness duration, reducingthe antibiotic side effects, optimal antibiotics use and preventing infectious complica-tions. Two sub-criteria are included within the infectious complications, namely, localcomplications (peri-tonsillar abscess) and systemic complications (acute rheumaticfever). The antibiotics side effects are usually divided into minor and major: gas-trointestinal distress/anaphylaxis and rash. The best (optimal) antibiotics use is alsodivided into avoiding over-treatment to reduce unnecessary antibiotic use as well asthe development of bacterial antibiotic resistance and avoiding under-treatment withantibiotics to decrease the chance of preventable spread of disease to family and closecontacts. Five diagnostic management strategies are examined in this study, the pro-vided strategies are known to be common in treating sore throat in the medical com-munity, specifically:
A1: No test no rx, meaning no testing and no treatment.
A2: Rapid strep, here, we obtain a rapid streptococcal antigen test, then wetreat patients whose test results are positive, do not treat patients whose testresults are negative.
A3: Culture, we conduct a throat culture and treat patients whose test resultsare positive, don’t treat if negative.
A4: Rapid strep and culture, we obtain a rapid streptococcal antigen test andtreat patients whose test results are positive, obtain a throat culture on patientswhose test results are found to be negative and treat if the culture result ispositive.
A5: Empiric rx, with no further diagnostic testing, do treat everyone.
The list of interdependent criteria will thusly be:
Fig.2 shows the interdependence of the criteria. For instance, we all know thatthe use of antibiotics may affect the illness duration, the illness duration has an effecton the infectious complications (systemic or local). Medication side effects may berelated to over-treatments and so on. Such complicated interdependence may lead toconfusion in the decision making process. It is not clear which criterion should havethe highest weight from the perspectives of the decision maker. Our presented modelwill address such decision making problem.
Fig. 2 Criteria sample interdependence
The criteria assessment against the alternatives was held using the linguistic vari-ables mentioned before, where Table 2 shows the surveyed ratings.
Using the cause-effect relationship matrix given in Table 1, we can calculate theaverage fuzzy matrix X and the total relation fuzzy matrix T . The resulting matricesare given in Table 3 and 4.
Correspondingly, from the above table we can calculate Di and Ri as seen in Table5 and 6.
Now using Table 5 and 6, we can simply calculate the prominence and the relationterms for each alternative as illustrated in Table 7. The causal diagram which relatesthe prominence to the relation is depicted in Fig.3.
Each alternative can now be weighted using (11), the corresponding weights areshown in Table 7.
Fuzzy Inf. Eng. (2012) 2: 195-216 207
Table 1: Initial direct-relation fuzzy matrices, where e1, e2 and e3 constitute the setof three experts.
C5 H H H VH VH VH H H H H H H 0 0 0 NO NO NO H H H
C6 M M M M M M M M M H H H H H H 0 0 0 M M M
C7 NO NO NO M M M H H H M M M M M M M M M 0 0 0
Table 2: Criteria versus alternatives assessment.
A1 A2 A3 A4 A5
e1 e2 e3 e1 e2 e3 e1 e2 e3 e1 e2 e3 e1 e2 e3
C1 P VP P G G G VP P VP G F F G G VG
C2 F F F F F F G F F F G G G F VG
C3 P P P F G G G G F VG G F VG G VG
C4 G VG F P F F P F P P VP F VP VP VP
C5 VG G F P P P P P F VP VP P P VP VP
C6 G G G P F P P G F P F F VP VP F
C7 VP VP P F F F F F P F G P F VG VG
The causal diagram in Fig.3 shows that criterion 1 through 3 fall in the category ofeffect, i.e., they can be influenced by the remaining criteria factors. As a mater of fact,shortening duration of illness, preventing local and systemic infectious complications(Peri-tonsillar abscess and Rheumatic fever) definitely fall in the category of symp-toms rather than causes. Clearly, the assessment model shows that medication sideeffects are one of the most important factors in treating sore throat followed by the lo-
cal infectious complications (peri-tonsillar abscess). This ranking of weights is basedon the assessment process that has been conducted through the group of experts.
The over all rating of all expert (Y) is shown in Table 8, while the weighted nor-malized decision fuzzy matrix V is shown in Table 9.
According to the table above, we can find ideal and anti-ideal solutions using (15)as shown in Table 10. In our case study, all the criteria fall in the detriment (negative)group.
The distance of each alternative Ai (i =1,2,· · · ,m) from A∗ and A− can be calcu-lated using (16) as shown in Table 11 and 12.
Using the last row entry in Table 11 and 12, we can now employ (21) to find theranking of all treatment options for this particular case study. Table 13 presents thefinal score (ui) of each alternative.
The scores of the five alternatives show that: to culture every one, who visitsthe primary care unit, will be the most reasonable option taking in consideration
Fuzzy Inf. Eng. (2012) 2: 195-216 209
Table 5: The fuzzy number Di and its defuzzified term Dde fi .
i Di Dde fi
1 0.2402 0.8834 4.162 1.63
2 0.3811 1.1392 4.8314 1.97
3 0.2788 0.9767 4.4055 1.75
4 0.5345 1.4627 5.6575 2.38
5 0.7183 1.6264 5.8586 2.56
6 0.563 1.5088 5.7754 2.44
7 0.4169 1.2097 5.0073 2.05
Table 6: The fuzzy number Ri and its defuzzified term Rde fi .
i Ri Rde fi
1 0.49 1.32 5.2911 2.20671
2 0.57 1.5 5.4811 2.35409
3 0.5 1.41 5.5448 2.317
4 0.45 1.26 5.1552 2.12984
5 0.51 1.43 5.589 2.3404
6 0.26 0.84 4.048 1.5813
7 0.35 1.04 4.5885 1.84395
the relative interdependence among the criteria and experts’ judgment. Our resultshighlight the outcomes of [2, 49] where the culture option is the best among theothers. The next preferred option is to obtain a rapid streptococcal antigen test andtreat patients who test positive, obtain a throat culture on patients who test negativeand treat if the culture is positive, (rapid strep and culture). No-test No-treatment aswell as empiric treatment, both achieve almost similar scores, though, they are theleast preferred options. Although the literature studies in [49, 50] state that A2 and A5
Table 7: Alternatives prominence, relation and weights.
i Dde fi + Rde f
i Dde fi − Rde f
i Wi wi
1 3.83 -0.58 3.88 0.130
2 4.32 -0.39 4.34 0.146
3 4.06 -0.57 4.10 0.138
4 4.51 0.25 4.52 0.152
5 4.90 0.22 4.91 0.165
6 4.02 0.86 4.11 0.138
7 3.90 0.21 3.90 0.131
are almost the same in rank, it was not the case in our model. Our model conforms tothe literature results in the rank of A3, A4 and A5 and presents a better contrast for theremaining alternatives.
Although it is pointed by the baseline analysis in [49, 50] that there is no supe-rior patient management course of action as it would be difficult to mark a definitivetreatment, however, the presented model tends to spot the best management amongthe available ones.
Remarks: From the presented case study, we notice that not only our model couldpresent more insights to the alternatives and conform to the found literature results,but it also presented higher rank contrasts in which equal alternatives in [49, 50] are
Fuzzy Inf. Eng. (2012) 2: 195-216 211
Table 8: The matrix of the aggregated opinions Y .
found to have different ranks.Moreover, although the group of experts don’t have to provide any criteria weights,
indirectly, the model can compute the weights out of the interrelations between thecriteria themselves. However, the interrelations cannot just be resolved by any person,they have to be determined by the experts as they will result in the relative significanceof the criteria. Such approach will eliminate the bias towards certain criteria due toindividual preferences. Our presented model is most useful for the cases where anindividual is not completely free to impose his own preferences, instead, the criteriainterrelations are the most responsible for stating their relative importance. Such adecision model can suite medical diagnosis systems where the physicians have todecide which action plan to take for the best treatment or in the situations whenthe criteria importance is not necessarily related personal aspects as much as to the
interrelation structure.Not only the weights-which now can be imputed out of the interrelations-come
into sight of this study contributions, the combination of DEMATEL and TOPSIS al-together as well as extending them to manage fuzzy information is another part whichcould improve the contrast in the final ratings. In addition, data pooling across theexperts is accomplished via fuzzy evaluations where the calculated averages remainin the fuzzy domain. With the definition of fuzzy ideal and fuzzy anti-ideal alter-natives, the fuzzy weight distances could be calculated using a geometrical distance
Fuzzy Inf. Eng. (2012) 2: 195-216 213
Table 12: Distance from FNIS.
d(vi j, v−j )
A1 A2 A3 A4 A5
C1 0.080 0.009 0.087 0.029 0.000
C2 0.029 0.029 0.019 0.009 0.000
C3 0.080 0.027 0.027 0.019 0.000
C4 0.000 0.040 0.051 0.058 0.084
C5 0.000 0.062 0.051 0.077 0.077
C6 0.000 0.054 0.033 0.043 0.070
C7 0.084 0.038 0.048 0.038 0.000
d−i 0.273 0.259 0.316 0.273 0.230
Table 13: Final rank of each alternative treatment option.
A1 A2 A3 A4 A5
ui 0.585 0.665 0.706 0.682 0.586
measure. Such notation helped in the robustness of the defuzzification as the decisiondeployment is held in the fuzzy domain.
4. Conclusion
In this paper, a fuzzy DEMATEL-TOPSIS model has been proposed. The modelaids in estimating the weights of mutually dependent criteria based on cause-effectassessments of a group of professionals. In this model, both DEMATEL and TOPSIShave been extended to handle fuzzy evaluations where the first is used to set theweights of the interdependent criteria and the second is used to draw a decision froma group of professionals who use linguistic rating in their evaluations.
With the proposed methodology, the complex interactions between the criteria canbe transformed into a visible structural model and then translated into criteria weights.The calculated weights of the modified DEMATEL model are plugged into a modifiedfuzzy TOPSIS model to help draw a decision. We modified the procedures in bothDEMATEL and TOPSIS techniques in which a decision maker can use linguistic
variables to handle the vagueness and uncertainty in the judgments and the criteriainterdependence. In the modified TOPSIS, an optimality function is introduced tocalculate the optimal membership degree of an alternative to both FPIS and FNIS.The farther the degree of membership from the FNIS and the closer to the FPIS themore favored the alternative. A classical numerical example was given to find theoptimal treatment of sore throat in a typical primary care unit. The demonstratedexample has shown better results that are almost identical to the literature with aimproved insights and rank contrasts.
The major contributions of the presented model are mainly as follows: first, theestimation of criteria weights when the criteria are interrelated in which traditionalMCDM may fail or mislead the decision maker. Second, the prevention of having tostrictly determine the criteria weights in case of mutual interdependence. Third, theextension of both DEMATEL and TOPSIS to handle the fuzzy ratings and linguisticevaluations of a group of experts. Finally, the introduction of miscellaneous aggre-gating procedures and distance measures to harmonize the two approaches to fit ourhybrid model.
Acknowlegements
I would like to thank the editor and the anonymous reviewers of the InternationalJournal of Fuzzy Information and Engineering for their valuable comments to helpimprove the quality of this paper.
References
1. Hwang C L, Yoon K (1981) Multiple attributes decision making methods and application. Springer-Verlag, Berlin Heidelberg
2. Doraid Dalalah, Sami Magableh (2008) Diagnosis and management of strep throat: remote multi-criteria fuzzy reasoning approach. Telemedicine and Ehealth Journal 14(7): 656-665
3. Chen S J Hwang C L (1992) Fuzzy multiple attribute decision making methods and applications.Springer-Verlag, Berlin Heidelberg
4. Fontela E, Gabus A (1976) The DEMATEL observer, DEMATEL 1976 report. Switzerland, Geneva,Battelle Geneva Research Center
5. Yamazaki M, Ishibe K, Yamashita S (1997) An analysis of obstructive factors to welfare service usingDEMATEL method. Reports of the Faculty of Engineering, Yamanashi University 48: 25-30
6. Hori S, Shimizu Y (1999) Designing methods of human interface for supervisory control systems.Control Engineering Practice 7(11): 1413-1419
7. Hwang C L, Yoon K (1981) Multiple attributes decision making methods and applications. Springer,Berlin Heidelberg
8. Tsuar S H, Chang T Y, Yen C H (2002) The evaluation of airline service quality by fuzzy MCDM.Tourism Management 23: 107-115
9. Wang R C, Liang T F (2004) Application of fuzzy multi-objective linear programming to aggregateproduction planning. Computers and Industrial Engineering 46: 17-41
10. Deng H, Yeh C H, Willis R J (2000) Inter-company comparison using modified TOPSIS with objec-tive weights. Computers and Operations Research 27: 963-973
11. Abo-Sinha M A, Amer A H (2005) Extensions of TOPSIS for multi-objective large-scale nonlinearprogramming problems. Appl. Math. Comput. 162: 243-256
12. Chu T C (2002) Facility location selection using fuzzy TOPSIS under group decisions. InternationalJournal of Uncertainty, Fuzziness and Knowledge-Based Systems 10(6): 687-701
13. Xu D L, Yang J B (2001) Introduction to multi-criteria decision making and the evidential reason-ing approach. Working Paper Series No. 0106, Manchester School of Management, University of
Fuzzy Inf. Eng. (2012) 2: 195-216 215
Manchester Institute and Technology ISBN: 1-86115-111-X14. Robert Fuller, Peter Majlender (2003) On weighted possibilistic mean and variance of fuzzy numbers.
Fuzzy Sets and Systems 136: 363-37415. Carlsson C, Robert Fuller, Markku Heikkila, Peter Majlender (2007) A fuzzy approach to R&D
project portfolio selection. International Journal of Approximate Reasoning 44: 93-10516. Carlsson C, Fuller R (1994) Interdependence in fuzzy multiple objective programming. Fuzzy Sets
and Systems 65: 19-2917. Carlsson C, Fuller R (1995) Multiple criteria decision making. The case for interdependence. Com-
puters & Operations Research 22(3): 251-26018. Carlsson C, Robert Fuller (1996) Fuzzy multiple criteria decision making: Recent developments.
Fuzzy Sets and Systems 78: 139-15319. Stewart TJ (1992) A critical survey on the status of multiple criteria decision making theory and
practice. OMEGA International Journal of Management Science 20(5-6): 569-58620. Umm-e-Habiba (2009) A survey on multicriteria decision making approaches. International Confer-
ence on Emerging Technologies (ICET 2009) Islamabad 321-32521. Kwakernaak H (1979) An algorithm for rating multiple-aspect alternatives using fuzzy sets. Auto-
matica 15: 615-61622. Buckley J J, Chanas S (1989) A fast method of ranking alternatives using fuzzy numbers. Fuzzy Sets
and Systems 30: 337-33923. Dubois D, Prade H (1983) Ranking fuzzy numbers in the setting of possibility theory. Information
Sciences 30(3): 183-22424. Lee E S, Li, R J (1988) Comparison of fuzzy numbers based on the probability measure of fuzzy
events. Computers and Mathematics with Applications 15(10): 887-89625. Chen S J, Hwang C L (1992) Fuzzy multiple attribute decision making: Methods and applications,
Springer-Verlag, New York26. Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval. Information Sciences
24(2): 141-16127. Yager R (1978) Fuzzy decision making including unequal objectives. Fuzzy Sets and Systems 1:
87-9528. Bonissone P (1982) A fuzzy sets based linguistic approach: theory and applications, in approximate
reasoning in decision analysis. North-Holland Publishing Company 329-33929. Saaty T L (1988) The analytic hierarchy process. University of Pittsburgh: Pittsburgh30. Chang Y M, McInnis B (1980) An algorithm for multiple attribute, multiple alternative decision
problem based on fuzzy sets with application to medical diagnosis. IEEE Transactions on System,Man, and Cybernetics SMC-10(10): 645-650
31. Dubois D, Prade H (1988) Possibility theory: an approach to the computerized processing of uncer-tainty. Plenum Press, New York
32. Takeda (1982) Interactive identification of fuzzy outranking relations in a multi-criteria decision prob-lem. Fuzzy Information and Decision Processes 301-307
33. Bellman R, Zadeh L A (1970) Decision making in a fuzzy environment. Manag. Sci. 17: 141-16434. Tomas Gal, Thomas Hanne (2006) Nonessential objectives within network approaches for MCDM.
European Journal of Operational Research 168(2): 584-59235. Fuller R, Zimmermann H J (1993) Fuzzy reasoning for solving fuzzy mathematical programming
problems. Fuzzy Sets and Systems 60: 121-13336. Tanaka H, Uejima S, Asai K (1982) Fuzzy linear model fuzzy linear regression model. IEEE trans.
System, Man and Cybernetics 12: 903-90737. Negoita C V (1981) The current interest in fuzzy optimization. Fuzzy Sets and Systems 6: 261-26938. Orlovsky S A (1978) Decision making with a fuzzy preference relation. Fuzzy Sets and Systems
1(3): 155-16739. Inuiguchi M , Ichihashi H (1990) Relative modalities and their use in possibilistic linear program-
ming. Fuzzy Sets and Systems 35: 303-32340. Inuiguchi M, Ichihashi H, Tanaka H (1990) Fuzzy programming: a survey of recent developments.
Stochastic Versus Fuzzy Approaches to Multi objective Mathematical Programming under Uncer-
tainty. Kluwer Academic Publishers, Dordrecht 45-6841. Ricardo C Silva, Carlos Cruz, Jose L Verdegay, Akebo Yamakami (2010) A survey of fuzzy convex
programming models. Studies in Fuzziness and Soft Computing 254: 127-14342. Wang J G, Stanley Zoints (2008) Negotiating wisely: considerations based on MCDM/MAUT. Euro-
pean Journal of Operational Research 191-20543. Doraid Dalalah, Omar Bataineh (2008) A fuzzy logic approach to the selection of the best silicon
crystal slicing technology. Expert Systems with Applications 36(2): 3712-371944. Doraid Dalalah, Mohammed Hayajneh, Farhan Batieha (2011) A fuzzy multi-criteria decision mak-
ing model for supplier selection. Expert Systems Appl. 38(7): 8384-839145. Saghafian S, Reza Hejazi S (2005) Multi-criteria group decision making using a modified fuzzy
TOPSIS procedure. IEEE Processing of the 2005 Int. conf. on comp. intelligence for modeling,control and automation CIMCA’05: 215-221
46. Lin C J, Wu W W (2008) A causal analytical method for group decision-making under fuzzy envi-ronment. Expert Syst. Appl. 34(1): 205-213
47. Fu G (2008) A fuzzy optimization method for multicriteria decision-making, An application to reser-voir flood control operation. Expert Systems Appl. 34(1): 145-149
48. Chen C T (2000) Extension of the TOPSIS for group decision-making under fuzzy environment.Fuzzy Sets and Systems 114: 1-9
49. Singh S, James G D, Centor R M (2006) Optimal management of adults with pharyngitis-a multi-criteria decision analysis. BMC Medical Informatics & Decision Making 6: 14
50. McIsaac W J, Kellner J D, Aufricht P, Vanjaka A, Low D E (2004) Empirical validation of guidelinesfor the management of pharyngitis in children and adults. JAMA 291(13): 1587-1595
