In this paper the mathematical foundations of the methods for
multicriterial decision making are presented

T. Hunjak

In this paper the mathematical foundations of the methods for

multicriterial decision making are presented

Hunjak, T.

Hrčak - Portal of scientific journals of Croatia

Mathematical Communications 2(1997), 161–169 161Mathematical foundations of the methodsfor multicriterial decision making∗Tihomir Hunjak†Abstract. In this paper the mathematical foundations of the meth-ods for multicriterial decision making are presented.Key words: multicriterial decision making, Analytic Hierarchy Pro-cess, ELECTRE method, PROMETHEE method, TOPSIS methodSazˇetak.U ovom radu prikazuju se matematicˇke osnove metoda zaviˇsekriterijalno odlucˇivanje.Kljucˇne rijecˇi: viˇsekriterijalno odlucˇivanje, analiticˇki hijerarhijskiproces, metoda ELECTRE, metoda PROMETHEE, metoda TOPSIS1. IntroductionThe multicriteria decision making problem has been observed in this paper:max{f1(a), f2(a), · · · , fk(a) : a ∈ A}. (1)A is a set of n possible decision alternatives, f1, f2, ..., fk are criteria by whichthe alternatives are evaluated. If all the criteria are not equally important, theirweights can be marked by w1, w2, ..., wk. The basic information for this kind ofdecision problem can be shown in the form of an evaluation table:f1(.) f2(.) ... fj(.) ... fk(.)w1 w2 ... wj ... wka1 f1(a1) f2(a1) · · · fj(a1) · · · fk(a1)a2 f1(a2) f2(a2) · · · fj(a2) · · · fk(a2).....................ai f1(ai) f2(ai) · · · fj(ai) · · · fk(ai).....................an f1(an) f2(an) · · · fj(an) · · · fk(an)Table 1. Evaluation table∗The lecture presented at the Mathematical Colloquium in Osijek organized by CroatianMathematical Society – Division Osijek, April 11, 1997.†Department of Mathematics, Faculty of Organisation and Informatics, Pavlinska 2, HR-42 000Varazˇdin, Croatia, e-mail: Tihomir.Hunjak@foi.hr162 T. HunjakTo solve the problem (1) means to choose the best alternative or to rank allthe alternatives. This kind of decision problem solving, including a discrete setof explicitly described alternatives, is different from the multicriteria optimizationproblem solving where the set of alternatives has been determined implicitly byconstraints. To solve the second problem the vector optimization was used. Theefficient solutions or Pareto optimal solutions of the vector optimization problemcan be characterized in a few ways which enable their identification, but the conceptis not usable to solve the problem (1). To solve the problem (1) it is necessary touse the procedure which meets the following criteria: (i) aberrations in evaluationof alternatives according to the individual criteria should be considered, (ii) thescaling effect, which occurs as a result of the use of different measuring scales usedfor evaluation of alternatives according to different criteria, must be eliminated and(iii) there must exist a possibility for a clear interpretation of the weight of thecriteria.It is necessary to stress that the method must enable that for each pair ofalternatives a, b ∈ A during their comparing , one of the following statements canbe chosen:• aPb or bPa - a is preferred to b or vice versa• aIb - a and b are indifferent• aRb - a and b are incomparable.The relation P ∪I∪R is a partial ranking relation, and one can say that a preferencestructure has been introduced into the set A of alternatives together with it.The main concepts for problem solving (1) satisfying the mentioned conditionsare:(a) multiattribute utility function (value function) for deterministic case,(b) scalarizing of the problem (1),(c) introduction of an outranking relation into the set A.(a) It is necessary to assess a value function v(·) of the formv(a) = w1v1(f1(a)) + w2v2(f2(a)) + . . .+ wkvk(fk(a))where vj(fj(a)) are assumed to be strictly positive, and maximise it over the setA. The Saaty method known as the Analytic Hierarchy Process (AHP) has beenshown as an example of the method explicable by this theoretical approach(b) A few methods have been developed which characterize the efficient solutionsof the multicriteria programming problem by its transformation into a scalar form.The compromise programming i.e. the search of the solution closest to the idealsolution, is the method which can also be used for discrete problem solving. The ex-ample of this concept for problem solving can be recognised in the TOPSIS method.(c) According to the information from the evaluation table, the relation which ranksalternatives partially or totally has been introduced into the set A. The methodsELECTRE (I and II) and PROMETHEE (I and II) are described in this paper asillustratory examples of this approach.Multicriterial decision making 1632. Analytic hierarchy processThe Analytic Hierarchy Process (AHP) developed by T. Saaty [1] is useful as adecision making methodology when multiple costs and multiple benefits are relevantfor determining the priorities of the alternatives.The basic steps in constructing and examining an AHP model are: (1) decom-pose the problem into a hierarhical structure, (2) perform judgments to establishpriorities for the elements of the hierarchy, (3) synthesis of the model, (4) performa sensitivity analysis.There exist different types of AHP hierarchies, but for this paper it is enough tounderstand the basic AHP model which includes the goal, criteria and alternatives(Figure 1).Criterion 1 Criterion 2 Criterion kAlt. 1 Alt. 2 Alt. nGoalFigure 1. Basic AHP model with goal, criteria and alternativesAfter constructing AHP hierarchy, it is necessary to perform judgments to es-tablish priorities for the elements of the hierachy. The decision maker’s judgmentsabout the relative importance between two elements in each pair of all elements onthe same level of the hierarchy are expressed by the help of the following nine-pointintensity scale (The Fundamental Scale):Definition Intensity of preferenceequally important 1moderately more important 3strongly more important 5very strongly more important 7extremely more important 9Table 2. Fundamental Scale164 T. HunjakOnce all judgments have been performed, they are all synthesized by the helpof a related mathematical model (2) which is briefly described here.Let n be the number of criterion (or alternatives) for which we want to find theweights wi. Let aij = wi/wj , where wi is the weight of i-th criterion (or priorityof i-th alternative) be the element of matrix A. The pairwise comparison matrix Aand vector w satisfy the equationAw = nw (2)Because of a special form of the matrix A (each row is a constant multiple of thefirst row, all elements are positive and aij = 1/aji), rank of A is one, all eigenvaluesare zero, except for one, and nonzero eigenvalue has a value of n.If the matrix A contains inconsistencies, the vector w of the weights can beobtained using the equations(A− λmaxI)w = 0∑wi = 1where λmax is the largest eigenvalue of matrix A. Because of the characteristics ofthe matrix A, λmax ≥ n and the difference λmax − n can be used for measuringinconsistencies. A consistency index CI has been constructingCI =λmax − nn− 1and consistency ratio is defined as CR = CIRI , where RI is a random index (randomindex is the consistency index of many randomly generated pairwise comparisonmatrices of size n). If the value of CR is less or equal to 0, 10, the pairwise compar-isons are considered to be acceptable. Otherwise, the comparisons must be repeatedin order to resolve the inconsistencies.3. TOPSIS methodLet us denote f∗i = maxa∈Afi(a). Criteria functions can be benefit functions or costfunctions. In the latter the function with the opposite sign is presumed to bemaximized. The vector F ∗ = (f∗1 , f∗2 , . . . , f∗k ) is an ideal solution (the ideal point,the ideal vector) of the problem (1). If there is the alternative a∗ ∈ A so thatfi(a∗) = f∗i , ∀i ∈ {1, 2, . . . , k}, then a∗ is a perfect solution of the problem (1).Most often such a solution does not exist. In such a case it is possible to find thealternative from the set A for which the vector F (a) = (f1(a), f2(a), . . . , fk(a)) isclosest to the ideal solution. This solution is called a compromise solution. Differentmetrics can be chosen for measuring distances from the ideal solution and theresult of the choice are the possibilities for different interpretations of compromisesolutions for problem (1).For a ∈ A to measure the distance of the vector F (a) = (f1(a), f2(a), . . . , fk(a))from the ideal solution Minkowski’s Lp metrics is most frequently useddp(a) =( k∑i=1(f∗i − fi(a))p) 1p.Multicriterial decision making 165The main characteristics of this metrics are d1 ≥ dp ≥ d∞, d∞ = maxi{f∗i − fi(a)},at which the choice of parameters p depends on the decision maker’s attitude to theinterpretations tied to its extreme values (p = 1 gives a maximal total gain, andp = ∞ is a minimal total deviation). One of the very important characteristics ofthe compromise solution is connected to a value choice of the parameter p wherefor 1 ≤ p < ∞ solutions are Pareto optimal or efficient, while it does not have tobe the same for p =∞, but in this case at least one compromise solution is at thesame time an efficient one.If functions fi, i ∈ {1, 2, . . . , k}, have different weights wi, i ∈ {1, 2, . . . , k}, thefollowing functiondp(a) =( k∑i=1wpi (f∗i − fi(a))p) 1p.is used for the decision maker who uses them as the solving criteria for the distancemeasuring of the vector to the ideal solution. Different from the described com-promise ranking where the alternative from the set A closest to the ideal solutionis being searched upon, the TOPSIS Method (Technique for Order Preference bySimilarity to Ideal Solution) introduces into the problem analysis so called negativeideal solution (the ideal solution is the most wished possibility; the negative idealsolution would thus have the least desirable characteristics).Let f−i = mina∈Afi(a) be a mark for the weakest criterial value for the criteriumfi, i ∈ {1, 2, . . . , k}. Vector F− = (f−1 , f−2 , . . . , f−k ) is called the negative idealsolution of the problem (1). For the alternative a ∈ A the distance of the vectorF (a) = (f1(a), f2(a), . . . , fk(a)) from the negative ideal solution isd−p (a) =(k∑i=1wpi (f−i − fi(a))p) 1p.To be able to identify in the set A the alternative closest to the ideal solution andthe same time farthest from the negative ideal solution and in accordance with theTOPSIS Method it is necessary to form the following functionsDp(a) =d−p (a)dp(a) + d−p (a), p = 1, 2,∞.TOPSIS solution of the problem (1) is the alternative for which the maximal valueof this function has been achieved.4. ELECTRE methodsThere are four types of the ELECTRE method and all have the same base.Concordance indexThe concordance index clk is defined for each pair of alternatives al, ak ∈ A as asum of criteria weight according to which al is not weaker than ak:clk =∑i/fi(al)≥fi(ak)wi.166 T. HunjakThe values of all the concordance indexes are being written into the concordancematrix C. The concordance index is a measure of intensity of the domination ofthe alternative al over the alternative ak.Discordance indexThis index measures the resistance of one alternative against the domination of theother. Because of different measuring scales belonging to different criteria, first itis necessary to transform all criteria values to comparable scales. It can be done ina few ways, but the authors suggest the procedure of vector normalization for theELECTRE method.In order to simplify the picture of the procedure for the values from the evalu-ation table (table 1) the following mark is introduced xij = fj(ai).The elements of the normalized evaluation table can be calculated asxij√∑kx2kj.All criteria do not have to be equally important. Because of that, every columnin the normalized evaluation table is multiplied by a weight of the associated cri-terium and by this one gets a ponderized normalized table of values which elementsarex∗ij = wjxij√∑kx2kj.The discordance indexes dkl can be calculated using these valuesdkl =maxj/fj(al)>fj(ak)|x∗kj − x∗lj |maxj∈J|x∗kj − x∗lj |(the maximum difference among valuing of alternatives according to the criteriawhere the alternative al is better than ak is divided by the maximum difference ofvaluing according to all the criteria). The discordance matrix D is formed from thediscordance indexes.Let c i d be average values of the concordance indexes i.e. of the discordanceindexes. According to this, the MI matrix is formed from the concordance matrixand discordance matrix. Its elements aremij ={1, iff cij > c and dij < d0, otherwise.If mij = 1, it is presumed that the alternative ai dominates over the alternative aj(the intensity of the domination of aj over aj is higher than the average and theresistance of the alternative aj to that domination is weaker than the average.) Thematrix can be considered to be a matrix of graph indexes where the alternatives areknots and only those with dominance are connected by arches. The exit knot of thearch belongs to the alternative which dominates over the one with the correspondingentrance knot.Multicriterial decision making 167The nondominated alternative make the kernel of the graph. The final decisionis reached according to the stability analysis of the kernel taking into the accountthe value changes of the indexes c i d and the weight criteria.If one wants completely to rank the alternatives in the set A, the procedure ofthe ELECTRE II method is used. In that case it is necessary to calculate the purevalue of the concordance with the dominance ck =n∑i=1, i6=kcki−n∑i=1, i 6=kcik and purevalue of the discordance with the domination dk =n∑i=1,i 6=kdki −n∑i=1,i6=kdik. Thealternatives are ranked then according to the highest average rank.5. PROMETHEE methodsThe PROMETHEE Method (Preference Ranking Organization METHod for En-richment Evaluations) can be described in three steps:Step 1. Enrichment of the preference structureThe generalized criteria are introduced to enable considering the span of de-viation in valuing alternatives to individual criteria.Step 2. Enrichment of the dominance relationThe outranking relation is built according to the estimate of the alternativesto all criteria. The total level of the preference with which one alternativedominates over the other is calculated for each pair of alternatives.Step 3. Decision analysisThe PROMETHEE I method gives a partial ranking of the set A. The infor-mation on the incomparable alternatives is also given. The PROMETHEE IImethod gives a complete ranking of the set A.Step 1. Generalized criteriaLet fi be a criterion and as such it needs to be maximalized. The comparing ofthe alternatives from the set A in pairs and with the help of this criterion gives asimple preference structure which defines the dominance relation in the set Afi(a) > fi(b) ⇔ aPbfi(a) = fi(b) ⇔ aIb.To be able to take into account different measuring scales belonging to individ-ual criteria and different deviations within those scales the generalized criterion isassociated to each criterion. The preference relation Pi(a, b) is defined for eachcriterion. It also defines the level of preference intensity from a in relation to b ac-cording to that criterion. It is supposed that Pi(a, b) is a function of the differenced = fi(a) − fi(b) and 0 ≤ Pi(a, b) ≤ 1. The Pi(a, b) value interpretations in termsof preference between a and b are the following:Pi(a, b) = 0 if d ≤ 0, no preference (or indifference)Pi(a, b) ≈ 0 if d > 0, weak preferencePi(a, b) ≈ 1 if d >> 0, strong preferencePi(a, b) = 1 if d >>> 0, strict preference.168 T. HunjakIt is obvious that the generalized criterion is a non-decreasing function of d.Six types of functions of the generalized criterion are used in the PROMETHEEMethods. It is necessary to establish the values of parameters (not more that two)for individual criteria. The clear economic interpretation must be applicable to thevalues of parameters. The decision maker decides upon the choice of the generalizedcriterium type and the values of the corresponding parameters.Step 2: Outranking relationWhen the generalized criterium is chosen (fi(·), Pi(·, ·)) the multicriteria preferenceindex of a over b is defined for every criterium fi(·) taking into account all thecriteriapi(a, b) =k∑i=1wiPi(a, b),k∑i=1wi = 1wi > 0 (i = 1, . . . , k) are weights associated to each criterion.The properties of pi(a, b) are:pi(a, a) = 0 and 0 ≤ pi(a, b) ≤ 1, ∀a, b ∈ Api(a, b) ≈ 0 implies a weak global preference of a over b,pi(a, b) ≈ 1 implies a strong global preference of a over b,pi(a, b) expresses with which degree a is preferred to over all the criteria. The valuespi(a, b) and pi(b, a) are calculated for each pair of alternatives a, b ∈ A. In this waya complete valued outranking relation is constructed on A.Step 3: Decision analysisIn this step two outranking flows are defined:φ+(a) =1n− 1∑x∈Api(a, x) - the positive outranking flowφ−(a) =1n− 1∑x∈Api(x, a) - the negative outranking flowThe positive outranking flow φ+(a) represents the outranking character of a - howmuch this alternative is outranking all the others. The negative outranking flowφ−(a) represents the outranked character of a - how much this alternative is out-ranked by all the others.The PROMETHEE I partial ranking.With positive and negative outranking flows two rankings (S+, I+) and (S−, I−) ofthe alternatives are defined:aS+b iff φ+(a) > φ+(b), aI+b iff φ+(a) = φ+(b)aS−b iff φ−(a) < φ−(b), aI−b iff φ−(a) = φ−(b).The PROMETHEE I partial ranking is the intersection of these two rankings:aP Ib if and only if ((aS+b and aS−b) or (aS+b and aI−b) or (aI+b and aS−b))aIIb if and only if aI+b and aI−bMulticriterial decision making 169aRb otherwise.P , I and R denote preference, indifference and incomparability, respectively.It is important to point out that PROMETHEE I Method does not rank the setA completely. If a decision maker wants a complete ranking of the alternatives, hecan use the PROMETHEE II Method.The PROMETHEE II complete ranking.The set A can be ranked completely in the way that the net outranking flow φ(a) =φ+(a)− φ−(a) is introduced for each alternative a ∈ A. It is obvious that if φ(a) isstronger, the alternative a is better.The PROMETHEE II complete ranking is then defined:aP IIb iff φ(a) > φ(b)aIIIb iff φ(a) = φ(b).In this way it is possible to compare all alternatives but it does not have to be anadvantage in relation to the PROMETHEE I Method.References[1] T. L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, 1980.[2] C. L. Hwang, K. Yoon, Multiple Attribute Decision Making, Lecture Notesin Economics and Mathematical Systems, Vol.164, Springer, Berlin, 1979.[3] J. P. Brans, B. Mareschal, Ph. Vincke, How to select and how to rankprojects: The PROMETHEE method, Europ. J.Operat. Res. 24(1986), 228-238.[4] M. Roubens, P. Vincke, Preference modelling, Lecture Notes in Economicsand Mathematical Systems, Vol.250, Springer, Berlin, 1985.[5] K. Yoon, A Reconcilation Among Discrete Compromise Solutions, J.Opl. Res.Soc. 38(1987), 277-286.

Mathematical foundations of the methods for multicriterial decision making

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