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Software Review: Expert Choice Professional: Decision Support Software Publisher: Expert Choice, Inc., 4922 Ellsworth Avenue, Pittsburgh, PA 14213; phone: 412-682-3844; fax: 412-682-7008; WWW e-mail: sales@ ahp.net Year of publication: 1996 Version reviewed: 9.0 Requirements: PC with 80386 processor or higher, VGA display or better, Windows 3.1 or higher, 10 MB hard disk space, 8MB RAM; compatible with Windows 95 Materials: Five 3.5-inch high-density diskettes, 121-page paperback tutorial, 392-page paperback manual. Price: $595; 30% educational discount available. Team version (not reviewed) $13,295 including radio base unit and eight keypads, with each additional keypad $285, and $2,840 educational discount on the software component of the team version - G. David Garson, 1997.
Software Review: Expert Choice Professional: Decision Support Software Publisher: Expert Choice, Inc., 4922 Ellsworth Avenue, Pittsburgh, PA 14213; phone: 412-682-3844; fax: 412-682-7008; WWW e-mail: sales@ ahp.net Year of publication: 1996 Version reviewed: 9.0 Requirements: PC with 80386 processor or higher, VGA display or better, Windows 3.1 or higher, 10 MB hard disk space, 8MB RAM; compatible with Windows 95 Materials: Five 3.5-inch high-density diskettes, 121-page paperback tutorial, 392-page paperback manual. Price: $595; 30% educational discount available. Team version (not reviewed) $13,295 including radio base unit and eight keypads, with each additional keypad $285, and $2,840 educational discount on the software component of the team version. Software Review: Expert Choice Professional: Decision Support Software Publisher: Expert Choice, Inc., 4922 Ellsworth Avenue, Pittsburgh, PA 14213; phone: 412-682-3844; fax: 412-682-7008; WWW e-mail: sales@ ahp.net Year of publication: 1996 Version reviewed: 9.0 Requirements: PC with 80386 processor or higher, VGA display or better, Windows 3.1 or higher, 10 MB hard disk space, 8MB RAM; compatible with Windows 95 Materials: Five 3.5-inch high-density diskettes, 121-page paperback tutorial, 392-page paperback manual. Price: $595; 30% educational discount available. Team version (not reviewed) $13,295 including radio base unit and eight keypads, with each additional keypad $285, and $2,840 educational discount on the software component of the team version.
Original Article Analytic Hierarchy Process and Expert Choice: Benefits and limitations Alessio Ishizaka. and Ashraf Labib Portsmouth Business School, University of Portsmouth, Richmond Building, Portland Street, Portsmouth PO1 3DE, UK. E-mails:;.Corresponding author. Abstract This article describes the original Analytic Hierarchy Process (AHP) as it is implemented in the software package Expert Choice.
We demonstrate its application through a practical example. In particular, we discuss problem modelling, pairwise comparisons, judgement scales, derivation methods, consistency indices, synthesis of the weights and sensitivity analysis. Finally, the limitations of the original AHP along with the new proposed development are explained. OR Insight (2009) 22, 201–220. Doi:10.1057/ori.2009.10 Keywords: AHP; decision making; review Introduction The Analytic Hierarchy Process (AHP) is a multi-criteria decision making (MCDM) method that helps the decision-maker facing a complex problem with multiple conflicting and subjective criteria (for example location or investment selection, projects ranking and so forth). Several papers have compiled the AHP success stories in very different fields (Zahedi, 1986; Golden et al, 1989; Shim, 1989; Vargas, 1990; Saaty and Forman, 1992; Forman and Gass, 2001; Kumar and Vaidya, 2006; Omkarprasad and Sushil, 2006; Ho, 2008; Liberatore and Nydick, 2008). The oldest reference we have found dates from 1972 (Saaty, 1972).
After this, a paper in the Journal of Mathematical Psychology (Saaty, 1977) precisely described the method. The vast majority of the applications still use AHP as described in this first publication and are unaware of & 2009 Operational Reasearch Society Ltd 0953-5543 OR Insight www.palgrave-journals.com/ori/ Vol. 22, 4, 201–220 Ishizaka and Labib successive developments. Chhukar mere man ko mp3 song free download. This fact is probably owing to the leading software supporting AHP, namely, Expert Choice (which still incorporates AHP as it was described in its first publication. In this article, we describe AHP through Expert Choice and provide a sketch of the major directions in methodological developments (as opposed to a discussion of applications) and the further research in this important field. The Original AHP Method Like several other MCDM methods such as ELECTRE, MacBeth, SMART, PROMETHEE, UTA and so on (Belton and Stewart, 2002; Figueira et al, 2005), AHP is based on four steps: problem modelling, weights valuation, weights aggregation and sensitivity analysis.
In the next sections, we will review these four steps used by AHP and its evolutions based on a simple problem: the selection of a car to buy. Problem modelling As with all decision-making processes, the facilitator will sit a long time with the decision-maker(s) to structure the problem, which can be divided into three parts: goal (buy a car), criteria (initial cost, maintenance cost, prestige, quality and its sub-criteria) and alternatives (Fiat Uno, Nissan Maxima 4 Doors, Mercedes Benz 290, Volvo 840, Ford Fiesta) (Figure 1).
AHP has the advantage of permitting a hierarchical structure of the criteria, which provides users with a better focus on specific criteria and sub-criteria when allocating the weights. Pairwise comparisons At each node of the hierarchy, a matrix will collect the pairwise comparisons of the decision-maker (for example Figure 2). Psychologists argue that it is easier and more accurate to express one’s opinion on only two alternatives than simultaneously on all the alternatives. It also allows consistency and cross checking between the different pairwise comparisons (see Consistency part under the section The original AHP method). AHP uses a ratio scale, which, contrary to methods using interval scales (Kainulainen et al, 2009), requires no units in the comparison.
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The judgement is a relative value or a quotient a/b of two quantities a and b having the same units (intensity, meters, utility and so on). The decision-maker does not need to provide a numerical judgement; instead a relative verbal appreciation, more familiar in our daily lives, is sufficient. 202 & 2009 Operational Reasearch Society Ltd 0953-5543 OR Insight Vol. 22, 4, 201–220 Analytic Hierarchy Process and Expert Choice Figure 1: Example of hierarchy. Figure 2: Comparison matrix of the first node. If the matrix is perfectly consistent, then the transitivity rule (1) holds for all comparisons aij: aij ¼ aik akj ð1Þ For example, if team A beats team B two-zero and team B beats team C three-zero, then it is expected with the transitivity rule (1) that team A beats team C six-zero (3 2 ¼ 6).
However, this is seldom the case because our world is inconsistent by nature. As a minimal consistency is required to derive meaningful priorities, a test must be done (see Consistency part under the section The original AHP method). Webber et al (1996) state that the order in & 2009 Operational Reasearch Society Ltd 0953-5543 OR Insight Vol. 22, 4, 201–220 203 Ishizaka and Labib Figure 3: Numerical scale. Figure 4: Verbal scale. Which the comparisons are entered in the matrix may affect the successive judgments.
Judgement scales One of AHP’s strengths is the possibility to evaluate quantitative as well as qualitative criteria and alternatives on the same preference scale of nine levels. These can be numerical (Figure 3), verbal (Figure 4) or graphical (Figure 5).
Priorities derivation Once the comparisons matrices are filled, priorities can be calculated. The traditional AHP uses the eigenvalue method.
For some users this method seems quite obscure. In order to explain it, we start from the case of a consistent matrix with known priorities pi. In this case, the comparison of the p alternatives i and j is given by pi/pj, which multiplied by the priority vector results in: = 204 & 2009 Operational Reasearch Society Ltd 0953-5543 OR Insight Vol.
22, 4, 201–220 Analytic Hierarchy Process and Expert Choice Figure 5: Graphical scale. Or grouped: A p ¼ n p ð2Þ where p: vector of the priorities; n: dimension of the matrix; A: comparison matrix Equation (2) is the formulation of an eigenvector problem.
The calculated priorities are exact for a consistent matrix. When slight inconsistencies are introduced, priorities should vary only slightly according to the perturbation theory (Saaty, 2003). Consistency As priorities make sense only if derived from consistent or near consistent matrices, a consistency check must be applied. Saaty (1977) has proposed a consistency index (CI), which is related to the eigenvalue method: CI ¼ lmax n n1 ð3Þ where lmax ¼ maximal eigenvalue The consistency ratio, the ratio of CI and RI, is given by: CR ¼ CI RI ð4Þ where RI is the random index (the average CI of 500 randomly filled matrices).
If CR is less than 10 per cent, then the matrix can be considered as having an acceptable consistency. Saaty (1977) calculated the RIs shown in Table 1. Other researchers have run simulations with different numbers of matrices (Lane and Verdini, 1989; Tummala and Wan, 1994; Alonso and Lamata, 2006) or incomplete matrices (Forman, 1990).
Their RIs are different but close to Saaty’s. & 2009 Operational Reasearch Society Ltd 0953-5543 OR Insight Vol. 22, 4, 201–220 205 Ishizaka and Labib Table 1: Random indices from Saaty (1977) n 3 4 5 6 7 8 9 10 RI 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49 Figure 6: Priorities with the distributive mode. Aggregation The last step is to synthesize the local priorities across all criteria in order to determine the global priority.
The historical AHP approach (called later distributive mode) adopts an additive aggregation (5) with normalization of the sum of the local priorities to unity (Figure 6): pi ¼ X wj lij ð5Þ j where pi: global priority of the alternative i; lij: local priority; wj: weight of the criterion j. The ideal mode uses a normalization by dividing the score of each alternative only by the score of the best alternative under each criterion (Figure 7).
It can be seen that the two approaches do not necessarily recommend the same ranking (see the rectangles in Figures 6 and 7). If we already know the priorities, the distributive mode is the only approach that will retrieve these priorities. However, if a copy (Belton and Gear, 1983) or a near copy (Dyer, 1990b) of an alternative is introduce or removed (Troutt, 1988), a rank reversal of the alternatives may appear. This phenomenon has been criticized from one side (Dyer, 1990a, b; Holder, 1990, 1991; Stam and Duarte Silva, 2003) and legitimized by another (Saaty, 1986, 1990, 1991, 1994, 1995, 2006; Harker and Vargas, 1987, 1990; Pe ´rez, 1995). This rank 206 & 2009 Operational Reasearch Society Ltd 0953-5543 OR Insight Vol.
22, 4, 201–220 Analytic Hierarchy Process and Expert Choice Figure 7: Priorities with the ideal mode. Reversal phenomenon is not unique to AHP but to all additive models (Triantaphyllou, 2001; Wang and Luo, 2009). Millet and Saaty (2000) gave some guidance on which normalization to use. If we are in a closed system (that is no alternative will be added or removed), then the distributive mode should be used. If we are in an open system (that is alternatives can be added or removed) and we allow our preferences for alternatives to be dependent on other alternatives (in other words, we accept the rank reversal phenomenon), then the distributive mode is indicated.
If we are in an open system and you do not want that other alternatives to affect the outcome, then the ideal mode is recommended. Based on these observations, Expert Choice has been modified to support the two modes. Sensitivity analysis The last step of the decision process is the sensitivity analysis, where the input data are slightly modified in order to observe the impact on the results. If the ranking does not change, the results are said to be robust. The sensitivity analysis is best performed with an interactive graphical interface. Expert Choice allows different sensitivity analyses, where the main difference is the various graphical representations (Figure 8).
Limitations of the Original AHP and Expert Choice Problem structuring This step is important, because a different structure may lead to a different final ranking. Several authors (Stillwell et al, 1987; Weber et al, 1988; Po ¨yho ¨nen et al, 1997) have observed that criteria with a large number of & 2009 Operational Reasearch Society Ltd 0953-5543 OR Insight Vol.
22, 4, 201–220 207 Figure 8: An example of four possible graphical sensitivity analyses in Expert Choice. Ishizaka and Labib 208 & 2009 Operational Reasearch Society Ltd 0953-5543 OR Insight Vol.
22, 4, 201–220 Analytic Hierarchy Process and Expert Choice sub-criteria tend to receive more weight than when they are less detailed. Brugha (2004) has provided a complete guideline to structure a problem hierarchically.
A book (Saaty and Forman, 1992) compiling hierarchies in different applications has been written. When setting up the AHP hierarchy with a large number of elements, the decision-maker should attempt to arrange these elements in clusters so they do not differ in extreme ways (Saaty, 1991; Ishizaka, 2004a, b). Pairwise comparisons Comparisons are recorded in a positive reciprocal matrix, as in Figure 2. In special cases, such as in currencies exchanges, not reciprocal matrices can be used (Hovanov et al, 2008). Even if not reciprocal matrices are then treated similarly to traditional matrices, Expert Choice does not offer the possibility not to be reciprocal. Judgement scales The use of verbal comparisons is intuitively appealing, user-friendly and more common in our everyday lives than numbers. It may also allow some ambiguity in non-trivial comparisons.
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This ambiguity in the English language has also been criticized (Donegan et al, 1992). Owing to its pairwise comparisons AHP needs ratio scales. Barzilai (2005) claims that preferences cannot be represented with ratio scales, because in his opinion an absolute zero does not exists, as with temperature or electrical tension.
Saaty (1994) states that ratio scales are the only possible measurement if we want to be able to aggregate measurement, as in a weighted sum. Dodd and Donegan (1995) have criticized the absence of a zero in the preference scale. To derive priorities, the verbal comparisons must be converted into numerical ones.
In Saaty’s AHP the verbal statements are converted into integers from one to nine. Theoretically there is no reason to be restricted to these numbers and verbal gradation. If the verbal gradation has been little investigated, several other numerical scales have been proposed (Table 2; Figures 9 and 10). Harker and Vargas (1987) have evaluated a quadratic and a root square scale in only one simple example and argued in favour of Saaty’s 1–9 scale. However, one example seems not enough to conclude the superiority of the 1–9 linear scale. Lootsma (1989) argued that the geometric scale is preferable to the 1–9 linear scale.
Salo and Ha ¨ma ¨la ¨inen (1997) point out that the integers from one to nine yield local weights, which are unevenly dispersed, so that there is lack of sensitivity when comparing elements, which are preferentially close to each other. Based on this observation, they propose a balanced scale where the local weights are evenly dispersed over the weight range 0.1, 0.9. Earlier Ma and Zheng (1991) have calculated a scale where & 2009 Operational Reasearch Society Ltd 0953-5543 OR Insight Vol.