All methods discussed in this section are appropriate for the following general MCDM problem:
Max[C1(x), C2(x), . . ., Ck(x)] (5.21) Subject to x∈X
In the context of an MCSP,
C1(x) = thelth attribute for alternativex, θl(x) x∈X= the set of available alternatives
x= any alternative
θjl =θl(xj) =lth attribute value forjth alternative 5.5.1 Max–Min Method
This method is based on the assumption that the DM is very pessimistic in his/her outlook and wants to maximize, over the decision alternatives, the achievement in the weakest crite- rion. Alternatives are characterized by the minimum achievement among all of its criterion values. It, thus, uses only a portion of the available information by ignoring all other crite- rion values. To make intercriterion comparison possible, all criterion values are normalized.
Geometrically, this method maximizes the minimum normalized distance from the anti-ideal solution along each criterion for all available alternatives.
Mathematically, this method works as follows (assuming linear normalization):
Max
Min
Cl(x)−Ll∗
Hl∗−Ll∗
, l= 1,2, . . ., k
(5.22) Subject to x∈X
where
Hl∗= ideal solution value for thelth criterion Ll∗= anti-ideal solution value for thelth criterion 5.5.2 Min–Max (Regret) Method
In this method, it is assumed that the DM wants to minimize the maximum opportunity loss. Opportunity loss is defined as the difference between the ideal (solution) value of a criterion and the achieved value of that criterion in an alternative. Thus, the Min–Max method tries to identify a solution that is close to the ideal solution. Geometrically, this method finds a solution that minimizes the maximum normalized distance from the ideal solution along each criterion for all available alternatives.
Mathematically, this method is represented by the following problem, Min
Max
Hl∗−Cl(x) Hl∗−Ll∗
, l= 1,2, . . ., k
(5.23) Subject to x∈X
where
Hl∗= ideal solution value for thelth criterion Ll∗= anti-ideal solution value for thelth criterion
5.5.3 Compromise Programming
Compromise programming (CP) identifies the preferred solution (alternative) that is as close to the ideal solution as possible. That is, it identifies the solution whose distance from the ideal solution is minimum. Distance is measured with one of the metrics,Mp, defined below. Distance is usually normalized to make it comparable across criteria units. Note that CP is also known as the global criterion method. Mathematically, compromise programming involves solving the following problem:
MinMp(x) (5.24)
Subject to x∈X
where the metricMpis defined (using linear normalization) as follows:
Mp(x) = i=k
i=1
Hi∗−Ci(x) Hi∗−Li∗
p 1/p
, 1≤p≤ ∞ (5.25)
AsHi∗≥Ci(x) is always true for Equation 5.25, the CP problem formulation can be restated as follows:
MinMp(x) = i=k
i=1
Hi∗−Ci(x) Hi∗−Li∗
p1/p
, 1≤p≤ ∞ (5.26)
Subject to x∈X
Using Equation 5.26 is simpler and it is the form commonly used in compromise programming.
Note that geometrically, the distance measures in a CP problem have different meanings depending on the value ofpchosen. For p= 1,M1(x) measures the “city-block” or “Man- hattan block” distance (sum of distances along all axes) fromH∗; forp= 2,M2(x) measures the straight-line distance fromH∗; forp=∞,M∞(x) measures the maximum of the axial distances fromH∗.
5.5.4 TOPSIS Method
TOPSIS (technique for order preference by similarity to ideal solution) was originally pro- posed by Hwang and Yoon [2] for the MCSP. TOPSIS operates on the principle that the preferred solution (alternative) should simultaneously be closest to the ideal solution,H∗, and farthest from the negative-ideal solution,L∗. TOPSIS does not require the specifica- tion of a value (utility) function but it assumes the existence of monotonically increasing value (utility) function for each (benefit) criterion. The method uses an index that combines the closeness of an alternative to the positive-ideal solution with its remoteness from the negative-ideal solution. The alternative that maximizes this index value is the preferred alternative. In TOPSIS, the pay-off matrix is first normalized as follows:
rij = θij
iθij21/2 i= 1, . . ., m;j= 1, . . ., k (5.27) Next, the weighted pay-off matrix,Q, is computed:
qij =λjrij i= 1,2, . . ., m;j= 1,2, . . ., k (5.28) whereλj is the relative importance weight of thejth attribute; λj≥0 and
λj= 1.
Using the weighted pay-off matrix, ideal and negative-ideal solutions (H∗ and L∗) are identified as follows:
H∗={q∗j, j= 1,2, . . ., k}={Maxqij,for alli;j= 1,2, . . ., k}
(5.29) L∗={q∗j, j= 1,2, . . ., k}={Minqij,for alli;j= 1,2, . . ., k}
Based on these solutions, separation measures for each solution (alternative) are calculated:
Pi∗=
j(qij−q∗j)2 1/2
, i= 1,2, . . ., m
(5.30) P∗i=
j(qij−q∗j)2 1/2
, i= 1,2, . . ., m
where Pi∗ is the distance of the ith solution (alternative) from the ideal solution andP∗i
is the distance of the same solution from the negative-ideal solution. TOPSIS identifies the preferred solution by minimizing the similarity index, D, defined below. Note that all the solutions can be ranked by their index values; a solution with a higher index value is preferred over that with index values smaller than its value.
Di=P∗i/(Pi∗+P∗i), i= 1,2, . . ., m (5.31) Note that 0≤Di≤1; Di= 0 when the ith alternative is the negative-ideal solution and Di= 1 when theith alternative is the ideal solution.
5.5.5 ELECTRE Method
ELECTRE method, developed by Roy [6], falls under the category called outranking methods. It compares two alternatives at a time (i.e., uses pairwise comparison) and attempts to build an outranking relationship to eliminate alternatives that are dominated using the outranking relationship. Six successive models of this method have been developed over time. They are: ELECTRE I, II, III, IV, Tri, and IS. Excellent overviews of the history and foundations of ELECTRE methods are given by Roy [6,7] and Rogers et al. [8]. We will explain only ELECTRE I in this section. The outcome of ELECTRE I is a (smaller than original) set of alternatives (called the kernel) that can be presented to the DM for the selection of “best solution.” Complete rank ordering of the original set of alternatives is possible with ELECTRE II.
An alternativeAi outranks another alternativeAj (i.e.,Ai→Aj) when it is realistic to accept the risk of regardingAi as at least as good as (or not worse than)Aj, even whenAi
does not dominateAj mathematically. This outranking relationship is not transitive. That is, it is possible to haveAp→Aq,Aq→ArbutAr→Ap. Each pair of alternatives (Ai,Aj) is compared with respect to two indices: a concordance index,c(i,j), and a discordance index, d(i,j). The concordance indexc(i,j) is a weighted sum of the number of criteria in which Ai is better thanAj. The discordance indexd(i,j) is the maximum weighted difference in criterion levels among criteria for whichAi is worse than Aj.
Let θip=pth criterion achievement level for alternativeAi
λp = relative importance weight of criterionp;k
p=1λp= 1 and λp>0 rip=pth criterion achievement level (normalized) forAi; use any appropriate
normalization scheme to deriverip from θip (see previous section) qip=λp rip
gis the criterion index for whichqip > qjp
lis the criterion index for whichqip < qjp
eis the criterion index for whichqip=qjp
sis the index of all criteria (s=g+l+e) Then,
c(i, j) =
p∈g
λp+
p∈e
ϕpλp (5.32)
d(i, j) = Max|qip−qjp|,∀l
Max|qip−qjp|,∀s whereϕp is usually set equal to 0.5.
ELECTRE assumes that criterion levels are measurable on an interval scale for the dis- cordance index. Ordinal measures are acceptable for the concordance index. Weights should be ratio scaled and represent relative importance to unit changes in criterion values. Two threshold values,αandβ, are used and these are set by the DM. Sensitivity analysis with respect toαandβ is needed to test the stability of the outranking relationship.
AlternativeAioutranks alternativeAj iffc(i,j)≥αandd(i,j)≤β. Based on the outrank- ing relation developed, the preferred set of alternatives, that is, a kernel (K), is defined by the following conditions:
1. Each alternative inK is not outranked by any other alternative inK.
2. Every alternative not inK is outranked by at least one alternative inK.
5.5.6 Analytic Hierarchy Process
The analytic hierarchy process (AHP) method was first proposed by Saaty [4,9]. AHP is applicable only for MCSP. With AHP, value (utility) function does not need to be evaluated, nor does it depend on the existence of such a function. To use this method, the decision problem is first structured in levels of a hierarchy. At the top level is the goal or overall purpose of the problem. The subsequent levels represent criteria, subcriteria, and so on.
The last level represents the decision alternatives.
After the problem has been structured in the form of a hierarchy, the next step is to seek value judgments concerning the alternatives with respect to the next higher level subcriteria.
These value judgments may be obtained from available measurements or, if measurements are not available, from pairwise comparison or preference judgments. The pairwise compar- ison or preference judgments can be provided using any appropriate ratio scale. Saaty has proposed the following scale for providing preference judgment.
Scale value Explanation
1 Equally preferred (or important) 3 Slightly more preferred (or important) 5 Strongly more preferred (or important) 7 Very strongly more preferred (or important) 9 Extremely more preferred (or important) 2, 4, 6, 8 Used to reflect compromise between scale values
After the value judgments of alternatives with respect to subcriteria and relative impor- tances (or priorities) of the sub-criteria and criteria have been received (or computed), composite values indicating overall relative priorities of the alternative are then determined by finding weighted average values across all levels of the hierarchy.
AHP is based on the following set of four axioms. The description of the axioms is based on Harker [5].
Axiom 1: Given two alternatives (or subcriteria)AiandAj, the DM can stateθij, the pairwise comparison (or preference judgment), with respect to a given criterion from a set of criteria such that θji= 1/θij for alliandj. Note that θij indicates how strongly alternativeAi is preferred to (or better than) alternativeAj. Axiom 2: When judging alternativesAiandAj, the DM never judges one alternative
to be infinitely better than another, that is,θij=∞with respect to any criterion.
Axiom 3: One can formulate the decision problem as a hierarchy.
Axiom 4: All criteria and alternatives that impact the decision problem are repre- sented in the hierarchy (i.e., it is complete).
When relative evaluations of subcriteria or alternatives are obtained through pairwise comparison, Saaty [9] has proposed a methodology (the eigenvector method) for computing the relative values of alternatives (and relative weights of subcriteria). With this method, the principal eigenvector is computed as follows:
θv=λmaxv (5.33)
wherev= vector of relative values (weights) andλmax= maximum eigenvalue.
According to Harker [5], the principal eigenvector can be determined by raising the matrix θto increasing powerskand then normalizing the resulting system:
v= lim
k→∞
(θke)
(eTθke) (5.34)
where eT= (1,1, . . .,1,1). The v vector is then normalized to the w vector, such that i=n
i=1 wi= 1. See Equation 5.20 for an easy heuristic proposed by Harker [5] for estimatingv.
Once thew vector has been determined,λmax can be determined as follows:
λmax=
j=n j=1θ1jwj
w1
(5.35) As there is scope for inconsistency in judgments, the AHP method provides for a measure of such inconsistency. If all the judgments are perfectly consistent, then λmax=n(wherenis the number of subcriteria or alternatives under consideration in the current computations);
otherwise,λmax> n. Saaty defines consistence index (CI) as follows:
CI = (λmax−n)
(n−1) (5.36)
For different sizes of comparison matrix, Saaty conducted experiments with randomly gen- erated judgment values (using the 1–9 ratio scale discussed before). Against the means of
the CI values of these random experiments, called random index (RI), the computed CI values are compared, by means of the consistency ratio (CR):
CR =CI
RI (5.37)
As a rule of thumb, CR≤0.10 indicates acceptable level of inconsistency.
The experimentally derived RI values are:
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
RI 0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 1.51 1.48 1.56 1.57 1.59
After the relative value vectors, w, for different sub-elements of the hierarchy have been computed, the next step is to compute the overall (or composite) relative values of the alternatives. A linear additive function is used to represent the composite relative evaluation of an alternative. The procedure for determining the composite evaluation of alternatives is based on maximizing the “overall goal” at the top of the hierarchy. When multiple DMs are involved, one may take geometric mean of the individual evaluations at each level. For more on AHP, see Saaty [9]. An excellent tutorial on AHP is available in Forman and Gass [10].
5.5.7 PROMETHEE Method
The preference ranking organization method of enrichment evaluations (PROMETHEE) methods have been developed by Brans and Mareschal [11] for solving MCSP.
PROMETHEE I generates a partial ordering on the set of possible alternatives, while PROMETHEE II generates a complete ordering of the alternatives. The PROMETHEE methods seek to enrich the usual dominance relation to generate better solutions for the general selection type problem. Only PROMETHEE I will be summarized in this section.
We assume that there exists a set of n possible alternatives, A= [A1, A2, . . ., An], and k criteria,C= [C1, C2, . . ., Ck], each of which is to be maximized. In addition, we assume that the relative importance weights,λ= [λ1, λ2, . . ., λk], associated with thekcriteria, are known in advance. We further assume that the criteria achievement matrixθis normalized, using any appropriate method, toRso as to eliminate all scaling effects.
Traditionally, alternative A1 is said to dominate alternative A2 iff θ1j≥θ2j, ∀j and θ1j> θ2j, for at least one j. However, this definition does not work very well in situa- tions whereA1 is better than A2 with respect to the first criteria by a very wide margin whileA2 is better thanA1 with respect to the second criteria by a very narrow margin or A1 is better thanA2with respect to criterion 1 by a very narrow margin and A2 is better than A1 with respect to criterion 2, again by a very narrow margin, or A1 is marginally better thanA2 with respect to both criteria.
To overcome such difficulties associated with the traditional definition of dominance, the PROMETHEE methods take into consideration the amplitudes of the deviations between the criteria. For each of thekcriteria, consider all pairwise comparisons between alternatives.
Let us define the amplitude of deviation, di, between alternative aand alternative b with respect to criterionias
di =Ci(a)−Ci(b) =θai−θbi
The following preference structure summarizes the traditional approach:
Ifdi>0 thenais preferred toband we writeaPb. Ifdi= 0 thenais indifferent tob and we writeaIb. Ifdi<0 thenbis preferred toaand we writebPa.
In PROMETHEE a preference function for criteria i,Pi(a,b), is introduced to indicate the intensity of preference of alternative a over alternative b with respect to criterion i.
(Note:Pi(b,a) gives the intensity of preference of alternativebover alternativea.)Pi(a,b) is defined such that 0≤Pi(a,b)≤1 and
Pi(a,b) = 0 ifdi≤0 (equal to or less than 0), indicating “no preference” betweena andb
Pi(a,b)≈0 ifdi>0 (slightly greater than 0), indicating “weak preference” ofaoverb Pi(a,b)≈1 ifdi>>0 (much greater than 0), indicating “strong preference” ofaoverb Pi(a,b) = 1 ifdi>>>0 (extremely greater than 0), indicating “strict preference” of a
overb
Next functionEi(a,b) is defined, which can be of six forms. The most commonly used form is the Gaussian (or normal distribution); we will use this in this section. For each criterion, the decision maker and the analyst must cooperate to determine the parameter s where 0< s <1. This parameter is a threshold delineating the weak preference area from the strong preference area. Once this parameter is established the values are calculated as follows:
Ei(a, b) =
⎧⎨
⎩
1−e−(di2s)22 , di≥0 0, di<0
(5.38) IfEi(a,b)>0, thenais preferred tob with respect to criterioni.π(a,b), preference index function, is defined next as follows:
π(a, b) =
j=k
j=1
λjEj(a, b) (5.39)
where λj is the weight of criterion j. The preference index function expresses the global preference of alternativeaover alternativeb. Note:
π(a, a) = 0 0≤π(a,b)≤1
π(a,b)≈0 implies a weak global preference ofaoverb π(a,b)≈1 implies a strong global preference ofaoverb π(a,b) expresses intensity of dominance of aoverb π(b, a) expresses intensity of dominance of bovera
Usingπ(a, b), positive-outranking flow,ϕ+(a), and negative-outranking flow,ϕ−(a), are calculated as follows. Positive-outranking expresses how alternative “a” outranks all other alternatives and, therefore, the higher the value the better the alternative is. The negative- outranking expresses how “a” is outranked by all other alternatives and, therefore, the lower the value the better the alternative is.
ϕ+(a) =
i=n
i=1
π(a, Ai) (5.40)
ϕ−(a) =
i=n
i=1
π(Ai, a) (5.41)
The following function is defined to assist in ordering the preference of the alternatives:
⎧⎨
⎩
a S+b, ϕ+(a)> ϕ+(b)
a I+b, ϕ+(a) =ϕ+(b) (5.42)
⎧⎨
⎩
a S−b, ϕ−(a)< ϕ−(b)
a I−b, ϕ−(a) =ϕ−(b) (5.43)
The PROMETHEE I partial relation is the intersection of these two pre-orders. There are three possible conclusions when making pairwise comparisons between alternatives.
Conclusion I:aoutranksb
a PI b if
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
a S+ banda S− b a S+ banda I− b a I+ banda S− b
(5.44)
In this case the positive flow exceeds or is equal to the positive flow ofband the negative flow ofbexceeds or is equal to the negative flow ofa. The flows agree and the information is sure.
Conclusion II:aindifferent tob
a II b ifa I+ b and a I− b (5.45)
In this case the positive and negative flows of the two alternatives are equal. The alternatives are concluded to be roughly equivalent.
Conclusion III:aincomparable tob
This will generally occur if alternativeaperforms well with respect to a subset of the criteria for whichb is weak whilebperforms well on the criteria for which ais weak.