The Analytic Hierarchy Process (AHP) is a general theory of measurement. It is used to derive ratio scales from both discrete and continuous paired comparisons. These comparisons may be taken from actual measurements or from a fundamental scale which reflects the relative strength of preferences and feelings. The AHP has a special concern with departure from consistency, its measurement and on dependence within and between the groups of elements of its structure. It has found its widest applications in multicriteria decision making, planning and resource allocation and in conflict resolution [l-6]. In its general form the AHP is a nonlinear framework for carrying out both deductive and inductive thinking without use of the syllogism by taking several factors into consideration simultaneously and allowing for dependence and for feedback, and making numerical tradeoffs to arrive at a synthesis or conclusion. T. L. Saaty developed the AHP in 1971- 1975 while at the Wharton School (University of Pennsylvania, Philadelphia, Pa).
Метод анализа иерархий (МАИ) это . Он используется чтобы получить шкалу отношений с помощью дискретных и непрерывных парных сравнений. Эти сравнения могут быть получены как из фактических измерений, так и из фундаментальной шкалы, которая отражает оценку предпочтений и чувств. МАИ . Этот метод нашел широкое использование в многокритериальном принятии решений, планировании и распределении ресурсов, а также в разрешении конфликтов. В базовой форме МАИ это нелинейная структура которая позволяет использовать и дедуктивное и индуктивное мышление без использования силлогизмов, принимает во внимание множество факторов одновременно и позволяет учесть зависимости и обратную связь, чтобы в результате, сделав несколько компромиссов, прийти к синтезу или выводу. Т.Саати разработал МАИ в 1971-1975 будучи в то время в Уортонской школе бизнеса при Пенсильванском университете (Филадельфия).
For a long time people have been concerned with the measurement of both physical and psychological events. By physical we mean the realm of what is fashionably known as the tangibles as it relates to some kind of objective reality outside the individual conducting the measurement. By contrast, the psychological is the realm of the intangibles as it relates to subjective ideas and beliefs of the individual about himself or herself and the world of experience. The question is whether there is a coherent theory that can deal with both these worlds of reality without compromising either. The AHP is a method that can be used to establish measures in both the physical and the social domains.
Долгое время людей беспокоила проблема измерение как физических так и психических событий. Под физическими мы имеем ввиду
In using the AHP to model a problem one needs a hierarchic or a network structure to represent that problem and pairwise comparisons to establish relations within the structure. In the discrete case these comparisons lead to dominance matrices and in the continuous case to kernels of Fredholm operators [7], from which ratio scales are derived in the form of principal eigenvectors, or eigenfunctions, as the case may be. These matrices, or kernels, are positive and reciprocal, e.g. aij = l/aji. In particular, special efforts have been made to characterize these matrices, and several of the more theoretical papers in this issue address questions and offer new information about them. Because of the need for a variety of judgments, there has also been considerable work done to characterize the process of synthesizing diverse judgments .
Чтобы использовать МАИ для моделирования проблемы необходима иерархия или сетевая структура чтобы выразить проблему и парные сравнения чтобы установить отношения со структурой. В дискретном случаи эти сравнения представляются матрицами с диагональным преобладанием, а в непрерывном случаи ядрами Френдгольмовых операторов, из которых шкалы соотношений получаются в виде главных собственных векторов или собственных функций, в зависимости от случая. Эти матрицы или ядра положительны и обладают .
In general a hierarchical model of some societal problem might be one that descends from a focus (an overall objective), down to criteria, down further to subcriteria which are subdivisions of the criteria and finally to the alternatives from which the choice is to be made.
Как правило иерархическая модель общественной проблемы идет сверху от фокуса ( основной цели), к критериям, и далее к субкритериям которые являются подразделами критериев и наконец к вариантам из которых необходимо сделать выбор.
There has been extensive work on how to structure hierarchies for practical problems. Two general types of hierarchies are the forward and the backward process hierarchies. All problems have been found to fall into one or the other of these two categories. Planning combines them in an iterative fashion [4]. The elements of a hierarchy are grouped in clusters according to homogeneity (see Axiom 2 in Section 3) and a level may consist of one or several homogeneous clusters. The elements in each level may be regarded as constraints, refinements or decompositions of the elements above. In a “complete” hierarchy, as in the example of “choosing the best college” given below, all the elements in one level have all the elements in the succeeding level as descendants. In this case the levels are single homogeneous clusters. Otherwise a hierarchy is “incomplete”.
Была проделана обширная работа о структуризации иерархий для политических проблем. Два основных вида иерархий — это иерархии прямого и обратного процесса. Все проблемы попадают в одну из этих двух категорий. Планирование соединяет их
Pairwise comparisons are fundamental in the use of the AHP. The members of parliament must first establish priorities for their main criteria by judging them in pairs for their relative importance, thus generating a pairwise comparison matrix. Judgments which are represented by numbers from the fundamental scale below are used to make the comparisons. The number of judgments needed for a particular matrix of order n, the number of elements being compared, is n(n - 1)/2 because it is reciprocal and the diagonal elements are equal to unity. The paper by Harker [lo, this issue, pp. 353-3601 gives conditions under which it is possible to use fewer judgments and still obtain accurate results.
парные сравнения являются неотъемлемой частью МАИ. Членам парламентом сначала нужно установить приоритеты к основным критериям сравнивая их в парах чтобы выяснить их относительную важность, тем самым формируя матрицу попарных сравнений. Количество суждений для матрицы размерности n, количество сравниваемых элементов, равняется n(n-1)/2 поскольку матрица … и диагональные элементы равны единице. Статья
On examining the matrices below, we note that a pair of elements (i,j) in a level of the hierarchy are compared with respect to a parent element in the level immediately above as a common property or criterion used to judge as to which one has it more and by how much. The typical way to phrase a question to fill an entry in the matrix of comparisons is: when considering two elements, i on the left side of the matrix and j on the top, which has the property more, or which one satisfies the criterion more, i.e. which one is considered more important under that criterion and how much more (using the fundamental scale values from Table l)? This gives us aij (oraji). The reciprocal value is then automatically entered for the transpose.
Осмотрев матрицы ниже, можно заметить что пара элементов (I,j) на уровне иерархии сравнивается
The question asked in making a pairwise comparison can influence the judgments provided and hence also the priorities. It must be made clear from the start what the focus of the hierarchy is and how the elements in the second level either serve to fulfill that focus or are its consequence, and so on down the hierarchy for each parent element and its descendants.
Вопрос, который мы задаем проводя попарное сравнение может повлиять на наше решении и следственно приоритеты. С самого начала должно быть ясно в чем заключается фокус иерархии и как элементы на втором уровня служат либо удовлетворению фокуса либо являются его последствиями, и это же должно выполнятся ниже в иерархии для каждого элемента предка и его потомков.
The next step is to derive the scale of priorities (or weights). It has been shown that this scale is obtained by solving for the principal eigenvector of the matrix and then normalizing the result. This is called the local derived scale before weighting by the priority of its parent criterion (which for the second-level elements is always equal to unity, the weight of the focus). After weighting, it is called the global derived scale. It has also been shown that the principal eigenvector is the only way to obtain the derived scale that makes use of all the dominance information given in the matrix when the latter is inconsistent. In computing the principal eigenvector all possible intransitivities and chains of intransitivity enter into the calculations as the matrix is raised to powers.
Следующий шаг это получить шкалу приоритетов (или весов). Было показано что эта шкала получается
The decomposition principle is applied by structuring a simple problem with the elements in a level being independent from those in succeeding levels, working downward from the focus in the top level, to criteria bearing on the focus in the second level, followed by subcriteria in the third level, and so on, from the more general (and sometimes uncertain) to the more particular and concrete. Saaty [ 121 makes a distinction between two types of dependence which he calls functional and structural. The former is the familiar contextual dependence of elements on other elements in performing their function, whereas the latter is the dependence of the priority of elements on the priority and number of other elements. Absolute measurement, sometimes called scoring, is used when it is desired to ignore such structural dependence among elements, while relative measurement is used otherwise.
Принцип декомпозиции применяется структурируя простую проблему так чтобы элементы на уровне были независимы от
The principle of comparative judgments is applied to construct pairwise comparisons of the relative importance of elements in some given level with respect to a shared criterion or property in the level above, giving rise to the kind of matrix encountered above and its corresponding principal eigenvector.
Принцип сравнительных суждений используется чтобы создать попарные сравнения относительной важности элементов данного уровня с учетом общего с уровнем выше критерия или свойства, в результате получая матрицу как выше и ей соответствующий главный собственный вектор.
The third principle is that of synthesizing the priorities. In the AHP priorities are synthesized from the second level down by multiplying local priorities by the priority of their corresponding criterion in the level above and adding, for each element in a level according to the criteria it affects. (The second-level elements are multiplied by unity, the weight of the single top-level goal). This gives the composite or global priority of that element, which in turn is used to weight the local priorities of the elements in the level below compared to each other with it as the criterion, and so on to the bottom level.
Третий принцип говорит о синтезировании приоритетов. В МАИ приоритеты синтезируются
Let U be a finite set of n elements called alternatives. Let K be a set of properties or attributes with respect to which elements in U are compared. We will refer to the elements of a as criteria. A criterion is a primitive.
Пусть конечный набор n элементов называемых вариантами. Пусть K набор качеств или характеристик по которым элементы в сравниваются. Критерий это
We perform binary comparisons on the elements in U according to a criterion in c. Let >c be a binary relation on 9t representing “more preferred than” with respect to a criterion C in 6. Let -c be the binary relation “indifferent to” with respect to a criterion C in B. Hence, given two elements Ai, AjcA, either Ai >.Aj or Aj >=Ai or Ai -c Aj, VC 6. We use Ai kc Aj to indicate more preferred or indifferent. A given family of binary relations >c with respect to a criterion C in 6 is a primitive.
Проведем бинарные сравнения элементов из в соответствии с критерием из c.
Let Bbe the set of mappings from U x U to R+ (the set ofpositive reals). Let f: K -> B. Let Pc gf(C) for CE K. P, assigns a positive real number to every pair (Ai, Aj)c‘91 x 2l. Let Pc(Ai, Aj) = UijE R ‘, Ai, Aj E 91. For each C E K;, the triple (%!I x VI, R +, PC) is a fundumentul or primitiw scale. A fundamental scale is a mapping of objects to a numerical system.
if Ai > c Aj, we say that Ai dominates Aj with respect to C E Cs. Thus, PC represents the intensity or strength of preference for one alternative over another.
Axiom 1 (the reciprocal axiom) For all Ai, Aj‘U and CE, Pc(Ai, Aj) = l/Pc(A,, Ai). This axiom says that the comparison matrices we construct are formed of paired reciprocal comparisons, for if one stone is judged to be five times heavier than another, then the other must perforce be one-fift
Аксиома 1
Let A = (aij) = (P&A,, Aj)) be the set of paired comparisons of the alternatives with respect to a criterion C E 6. By the definition of PC and Axiom 1, A is a positive reciprocal matrix. The object is to obtain a scale of relative dominance (or rank order) of the alternatives from the paired comparisons given in A
Пусть
We will now show how to derive the relative dominance of a set of alternatives from a pairwise comparison matrix A. Let RMM(“) be the set of (n x n) positive reciprocal matrices A = (aij) = (PJA,, Ai)), VCEE. Let [0, 11” be the n-fold Cartesian product of [0, l] and let W: &n, > [0, 11” for AE RMM(“), W(A) is an n-dimensional vector whose components lie in the interval [O, 11. The triple (RMM(“), [0, l]“, W) is a derived scale. A derived scale is a mapping between two numerical relational systems.
An important aspect of the AHP is the idea of consistency. If one has a scale for a property possessed by some objects and measures that property in them, then their relative weights with respect to that property are fixed. In this case there is no judgmental inconsistency (although if one has a physical scale and applies it to objects in pairs and then derives the relative standing of the objects on the scale from the pairwise comparison matrix, it is likely that inaccuracies will have occurred in the act of applying the physical scale and again there would be inconsistency). But when comparing with respect to a property for which there is no established scale or measure, we are trying to derive a scale through comparing the objects two at a time. Since the objects may be involved in more than one comparison and we have no standard scale, but are assigning relative values as a matter of judgment, inconsistencies may well occur. In the AHP consistency is defined in the following way, and we are able to measure inconsistency.
Важный аспект МАИ это идея
Definition. The mapping PC is said to be consistent iff PcAAit Aj)Pc(Aj, AA = PAA,, Ad, Vi, j, k. Similarly, the matrix A is consistent iff aijajr = air, Vi, j, k.
Дефиниция. Отображение
In a partially ordered set, we define x < y to mean that x < y and x # y; y is said to cover (dominate) x. If x < y, then x < t < y is possible for no t. We use the notation x- = {yl x covers y} and x’ = {y ) y covers x}, for any element x in an ordered set.
Let H be a finite partially ordered set. Then H is a hierarchy if it satisfies the following conditions: (a) there is a partition of H into sets L,, k = 1,. , h, for some h where L1 = {b}, b is a single element; (b) XEL, implies x-EL,,+, k = l,...,h - 1; (c) XEL, implies x+EL-, k = 2 ,..., h.
Definition. Given a positive real number p > 1, a nonempty set x- c Lk+ 1 is said to be p-homogeneous with respect to x E Lk if for every pair of elements y1 , yz E .x -, l/p d P&r, y2) < p. In particular the reciprocal axiom implies that Pc(yi,yi) = 1.
Дефиниция.
Axiom 2 (the homogeneity axiom) Given a hierarchy H, XEH and XEL, x- c L,,, is p-homogeneous for k = 1,. , h - 1. Homogeneity is essential for meaningful comparisons, as the mind cannot compare widely disparate elements. For example, we cannot compare a grain of sand with an orange according to size. When the disparity is great, elements should be placed in separate clusters of comparable size, or in different levels altogether. Given L,, L k+ 1 5 H, let us denote the local derived scale for YE x- and x E L, by $k+ ,(y/x), k = 2,3,. , h - 1. Without loss of generality we may assume that tik+ r(y/x) = 1. Consider the matrix IC/,JLk/Lk- ,) whose columns are local derived scales of elements in L, with respect to elements in Lk _ 1.
Аксиома 2
Definition. A set A is said to be outer dependent on a set C if a fundamental scale can be defined on A iyith respect to every C E 6. The process of relating elements (e.g. alternatives) in one level of the hierarchy according to the elements of the next higher level (e.g. criteria) expresses the dependence (what is called outer dependence) of the lower elements on the higher so that comparisons can be made between them. The steps are repeated upward in the hierarchy through each pair of adjacent levels to the top element, the focus or goal. The elements in a level may also depend on one another with respect to a property in another level. Inputtoutput of industries is an example of the idea of inner dependence, formalized as follows.
Дефиниция. Набор A называется внешне зависимым от набора C если фундаментальная шкала
Definition. Let U be outer dependent on K. The elements in 91 are said to be inner dependent with respect to C E Q if for some A E ‘?I, PI is outer dependent on A.
Дефиниция
Axiom 3 Let H be a hierarchy with levels L,, LX ,..., L,. For each L,, k = 1, 2 ,_.., h - I: (1) L!i+, is outer dependent on Lk; (2) L!i+1 is not inner dependent with respect to all XE L,; (3) L, is not outer dependent on Lk+,
Аксиома 3
Axiom 4 (the axiom of expectations) C c H - L,, A=Lh. This axiom is merely the statement that thoughtful individuals who have reasons for their beliefs should make sure that their ideas are adequately represented in the model. All alternatives, criteria and expectations (explicit and implicit) can be and should be represented in the hierarchy. This axiom does not assume rationality. People are known at times to harbor irrational expectations and such expectations can be accommodated.
Аксиома 4
Based on the concepts in Axiom 3 we can now develop a weighting function. For each XE H, there is a suitable weighting function (whose nature depends on the phenomenon being hierarchically structured): w,: x + [0, l] such that c W,(Y) = I. Note that h = 1 is the last level for which x- is not empty. The sets Li are the levels of the hierarchy, and the function w, is the priority function of the elements in one level with respect to the objective x. We observe that even if x- # Lk (for some level Lk), w, may be defined for all of L, by setting it equal to zero for all elements in L, not in x. The weighting function is one of the more significant contributions towards the application of hierarchy theory.
Основываясь на
Definition. A hierarchy is complete if, Vx c L,, x+ c Lt_, We can state the central question: Basic Problem. Given any element x E L,, and subset S c L,, (E < p), how do we define a function w,,: S + [0, l] which reflects the properties of the priority functions on the levels L,, k = cc,. . . ,p - 1. Specifically, what is the function wb.Lh. .Lh 4 [O, l]? In less technical terms, this can be paraphrased thus: Given a social (or economic) system with a major objective b, and the set L, of basic activities, such that the system can be modeled as a hierarchy with largest element b and lowest level L,. What are the priorities of the elements of any level and in particular those of L,, with respect to b?
Дефиниция. Иерархия называется полной если,
We now present a method to solve the Basic Problem. Assume that Y = {yi,. ..,y,,} c L, and that X = {xi,. ..,x,,_+,} c LL+l. Without loss of generality we may assume that X = L,, 1, and that there is an element z E Lk such that yi E z-. Then consider the priority functions w,: Y + [O, l] and wyj: X + [0, l]j = 1,. . . ,mk. Construct the priority function of the elements in X with respect to z, denoted w, w: X + [0, 11, by W(Xi) = F wy,(xi)wz(Yj)t i = l,...,mk+r
It is obvious that this is no more than the process of weighting the influence of the element yj on the priority of xi by multiplying it with the importance of xi with respect to z. The algorithms involved will be simplified if one combines the w,j(xi) into a matrix B by setting bij = w,j(xi). If one further sets wi = w(xi) and WI = w,(yj), then the above formula becomes wi = z bijwJ, i = l,...,nk+,. j=l Thus, one may speak of the priority vector w and, indeed, of the priority matrix B of the (k + 1)th level; this gives the final formulation, w = Bw’. The following theorem is easy to prove.
Очевидно что
Theorem. Let H be a complete hierarchy with largest element b and h levels. Let BI, be the priority matrix of the kth level, k = 1,. . . , h. If w’ is the priority vector of the pth level with respect to some element z in the (p - 1)th level, then the priority vector w of the qth level (p < 4) with respect to z is given by w =BqB,_,...B,+Iw’. Thus, the priority vector of the lowest level with respect to the element b is given by w = BhB,_ 1 . . . Bzb, if L, has a single element, b, = 1. Otherwise, b, is a prescribed vector. We note that the pairwise comparison process takes into consideration nonlinearities. Such nonlinearities are captured by the composition weighting process.
Теорема. Положим H полная иерархия с наибольшим элементом b и h уровнями.
Often alternatives depend on criteria and criteria on alternatives and there should be a cycle connecting the two which is more accurately studied with the network feedback approach. The AHP has been generalized to deal with feedback as shown below, although people generally prefer to simplify and arrange their thinking in terms of a linear hierarchy even if the answers are only approximate.
Часто альтернативы зависят от критериев и критерии от альтернатив и должен существовать цикл, соединяющий их что более подробно изучается в подходе с сетевой обратной связью. МАИ был обобщен чтобы обходится с обратной связью как показано ниже, хотя люди обычно предпочитают упрощать и думать в рамках линейной иерархии даже если результаты только приближены.
A network is a set of nodes (each of which consists of a set of elements) and a set of arcs which indicate the order of interaction among the components. The priorities of the elements in each node are components of the principal eigenvector of the matrix of pairwise comparisons of the relative impact of these elements with respect to an element or node with which they interact. The interaction is indicated by an arc of the network. All such eigenvectors define what is known as a supermatrix of impact priorities. By weighting the eigenvectors corresponding to each component by the priority of that component in the system, the supermatrix is transformed into a stochastic matrix. The limiting impact priorities are obtained by computing large powers of this matrix.
Сеть — это набор вершин (каждая из которых состоит из набора элементов) и набора дуг, которые показывают порядок взаимодействий между элементами. Приоритеты элементов в каждой вершине — это части главного собственного вектора матрицы попарных сравнений сравнительного влияния этих элементов в отношении элементов вершины, с которой они взаимодействуют. Это взаимодействие демонстрируется дугой сети. Все такие собственные векторы задают так называемую суперматрицу приоритетов воздействия. Задав веса собственных векторов
Definition. A partially ordered set S is a network system if (a) there is a partition of S into sets C,, k = 1,. , s; (b) there is an ordering on C,, k = 1,. . , s such that x c CI, implies either x- or x’ is in CE, for some kj or both x- E Ck,, xc E Ckj for one or more kj; (c) For each XES, there is a suitable weighting function w,: x- -+ [0, l] such that 1 w,(y) = 1 YEX - and for Ck c S, k = 1,. , s, there is a weighting function where C; = { Ch 1 Ck covers C,}
Дефиниция. Частично упорядоченный набор S является
There is an infinite number of ways to derive the vector of priorities from the matrix (aij). But emphasis on consistency leads to an eigenvalue formulation. If aij represents the importance of alternative i over alternativej and ajk represents the importance of alternative j over alternative k then aik, the importance of alternative i over alternative k, must equal aijajk for the judgments to be consistent. If we do not have a scale at all, or do not have it conveniently, as in the case of some measuring devices, we cannot give the precise values of \vi/wj but only an estimate. Our problem becomes A’w’ = imaxw’, where I.,,, is the largest or principal eigenvalue of A’ = (a,!j) the perturbed value of A = (aij) with aji = l/aij forced. To simplify the notation we shall continue to write Aw = k maxi, where A is the matrix of pairwise comparisons. The solution is obtained by raising the matrix to a sufficiently large power then summing over the rows and normalizing to obtain the priority vector w = (wl,. , w,). The process is stopped when the difference between components of the priority vector obtained at the kth power and at the (k + 1)th power is less than some predetermined small value. An easy way to get an approximation to the priorities is to normalize the geometric means of the rows. This result coincides with the eigenvector for n < 3. A second way to obtain an approximation is by normalizing the elements in each column of the judgment matrix and then averaging over each row.
Существует конечное число способов
We would like to caution the reader that for important applications one should only use the eigenvector derivation procedure because approximations can lead to rank reversal in spite of the The AHP---what it is and how it is used 171 closeness of the result to the eigenvector [13]. It is easy to prove that for an arbitrary estimate x of the priority vector, ,lirn, & Akx = cw ““ax where c is a positive constant and w is the principal eigenvector of A. This may be interpreted roughly to say that if we begin with an estimate and operate on it successively by A/&,,,, to get new estimates, the result converges to a constant multiple of the principal eigenvector.
Хотим предупредить читателя
A simple way to obtain the exact value (or an estimate) of &,,% when the exact value (or an estimate) of w is available in normalized form is to add the columns of A and multiply the resulting vector with the vector w. The resulting number is A,,,,, (or an estimate). This follows from and The problem is now, how good is the principal eigenvector estimate w? Note that if we obtain w = (wi,. . , w,) by solving this problem, the matrix whose entries are wi/wj is a consistent matrix which is our consistent estimate of the matrix A. The original matrix A itself need not be consistent. In fact, the entries of A need not even be transitive; i.e. A, may be preferred to A2 and A2 to A, but A, may be preferred to A,. What we would like is a measure of the error due to inconsistency. It turns out that A is consistent iff A,,,,, = n and that we always have II,,, 2 n. This suggests using i “ax - n as an index of departure from consistency. But where Li, i = 1,. . . , n are the eigenvalues of A. We adopt the average value (A,,,,, - n)/(n - l), which is the (negative) average of E.i, i = 2,. . . , n (some of which may be complex conjugates). It is interesting to note that 2(1.,,, - n)/(n - 1) is the variance of the error incurred in estimating aij. This can be shown by writing aij = (Wi/Wj)Eij,Eij > 0 and eij = 1 + dij,dij > - 1, and substituting in the expression for &,,ax. It is dij that concerns us as the error component and its value 16ij( < 1 for an unbiased estimator. Normalizing the principal eigenvector yields a unique estimate of a ratio scale underlying the judgments.
Простой способ получить точное значение
The C.I. of a matrix of comparisons is given by C.I. = (3,,,, - n)/(n - 1). The consistency ratio (C.R.) is obtained by comparing the C.I. with the appropriate one of the following set of numbers, each of which is an average random consistency index (R.I.) derived from a sample of size 500, of a randomly generated reciprocal matrix using the scale l/9, l/8, . . . , 1, . .8, 9 to see if it is about 0.10 or less. If it is not less than 0.10, study the problem and revise the judgments:
Why tolerate 10% inconsistency? The priority of consistency to obtain a coherent explanation of a set of facts must differ by an order of magnitude from the priority of inconsistency which is an error in the measurement of consistency. Thus, on a scale from O-1, inconsistency should not exceed 0.10 by very much. Note that the requirement of 10% should not be made much smaller such as 1% or 0.1%. The reason is that inconsistency itself is important, for without it new knowledge which changes preference order cannot be admitted. Assuming all knowledge to be consistent contradicts experience which requires continued adjustment in understanding. Thus the objective of developing a wide-ranging consistent framework depends on admitting some inconsistency. This also accounts for why the number of elements compared should be small. If the number of elements is large, their relative priorities would be small and error can distort these priorities considerably. If the number of items is small and the priorities are comparable a small error does not affect the order of magnitude of the answers and hence the relative priorities would be about the same. For this to happen, the items must be < 10 so their values on the whole would be > 10% each and hence remain relatively unaffected by 1% error for example.
Зачем допускать 10% несоответствия?
The consistency index (C.I.) for an entire hierarchy is defined by where wij = 1 for j = 1, and nij+, is the number of elements of the (j + 1)th level with respect to the ith criterion of the jth level.
the ith criterion of the jth level. Let ICJ be the number of elements of C; , and let w (kJ(h) be the priority of the impact of the hth component on the kth component, i.e. wck)(,,) = w,,(C,,) or wcki: C,, -+ w()(,,,. If we label the components of a system along lines similar to those we followed for a hierarchy, and denote by wjk the limiting priority of the jth element in the kth component, we have ICC1 cs = i g wjk c WCk)(h)k(j,h), k=l j=l /I=1 where pk(j,h) is the C.I. of the pairwise comparison matrix of the elements in the kth component with respect to the jth element in the hth component.
Cognitive psychologists have recognized for some time that there are two kinds of comparisons, absolute and relative. In absolute comparisons alternatives are compared with a standard in one’s memory that has been developed through experience; in relative comparisons alternatives are compared in pairs according to a common attribute. The AHP has been used with both types of comparisons to derive ratio scales of measurement. We call such scales absolute and relative measurement scales, respectively. Relative measurement in the AHP is well-developed and its use has already been illustrated in the school selection example in this paper. Here is a brief description of absolute measurement. Incidentally the software package Expert Choice [14] also includes this method of measurement under the name of “ratings”.
Когнитивные психологи уже некоторое время разделяют два вида сравнения: абсолютное и относительное. При абсолютном сравнениях варианты сравниваются с стандартом в памяти человека, который был приобретен с опытом; при относительном сравнении варианты сравниваются в парах по общему признаку. МАИ использовался с обоими видами сравнений для получения шкал соотношений измерений. Такие шкалы называют абсолютными и относительными шкалами измерений соответственно. Относительное измерение в МАИ хорошо развито и его использование уже было про иллюстрировано в примере про школы в этой статье.
Absolute measurement (sometimes called scoring) is applied to rank the alternatives in terms of the criteria or else in terms of ratings (or intensities) of the criteria; e.g. excellent (A), very good (B), good (C), average (D), below average (E), poor and very poor (F). After setting priorities on the criteria (or subcriteria, if there are any) pairwise comparisons are also performed on the ratings themselves to set priorities for them under each criterion. Finally, alternatives are scored by checking off their rating under each criterion and summing these ratings for all the criteria. This produces a ratio scale score for the alternative. The scores thus obtained of the alternatives can be normalized.
Абсолютное сравнение (которые иногда называют счетом)
Absolute measurement needs standards, often set by society for its convenience, and sometimes having little to do with the values and objectives of the judge making the comparisons. In completely new decision problems or in old problems where no standards have been established, we must continue to use relative measurement comparing alternatives in pairs to identify the best. The question now is: “What happens to rank when using relative measurement and alternatives are added or deleted?”
Абсолютному измерению требуются стандарты, часто заданные обществом для удобства, и часто имеют мало общего с ценностями и целями человека производящего сравнения.
When relative measurement is used to, for example, buy a car, even when the priorities of the criteria are set in advance independently of the alternatives, the car that qualifies in the end depends on the number of cars examined. Adding a new car to the collection being examined may cause reversal in the rank of the original cars. This phenomenon can be accounted for by considering the normalization operation as a structural criterion which has to do with information generated in the measurement process. With relative measurement the priority of such a criterion changes when new alternatives are added or old ones deleted and hence the priorities of the old alternatives, which depend on all the criteria, including this structural criterion, would change and a different rank order may occur among the old alternatives. The analogy can be made with any mathematical model, e.g. linear programming, when a new variable or a new constraint is added. There is no necessary relation between the solution of the new problem and the old one
Когда относительное измерение используется к примеру чтобы купить машину даже если приоритеты критериев задана заранее в не зависимости от вариантов, машина прошедшая квалификацию будет зависеть от количества сравниваемых машин. Добавление новой машин в набор сравниваемых машин может вызвать
As it should, rank is unaffected when only one criterion is involved and the judgments are consistent. More generally, one can show that with consistency, the rank order of two alternatives is unaffected when the judgment values of one dominate those of the other in every pairwise comparison matrix under the criteria. However, the final rank can change even when the judgments are consistent when an alternative dominates another under one criterion but is dominated by it under another.
Как и должно быть,
When dealing with group judgments, Saaty has proposed that any rule to combine the judgments of several individuals should also satisfy the reciprocal property. A proof that the geometric mean, which makes no requirement on who should vote first, satisfies this condition was later generalized in a paper by Aczel and Saaty [S] and by Aczel and Alsina [15, this issue, pp. 31 l-3201. Group judgment differences can be resolved through a consistency check. When several people propose radically different judgments in certain positions of the matrix these can be tested with other judgments on which there is wide agreement by solving the problem separately for each controversial judgment and measuring the consistency. The judgment yielding the highest consistency The AHP-what it is and how it is used 175 in the overall problem is retained. The following consistency comparison for each individual’s judgments with those of the scale vector w derived from group judgments has been proposed: t bijwj/wi - n2 - 0.1. i,j= 1 Probabilistic judgments have been studied extensively by Vargas [16]. In particular he showed that when the judgments are given by a y-distribution the derived vector belongs to a Dirichlet distribution with a /?-distribution of each component.
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