Project management software selection using analytical hierarchy process.

Sutterfield, J. S. ; Swirsky, Steven ; Ngassam, Christopher 等

INTRODUCTION

Decision Analysis involving multiple variables and objectives that can be quantified is rather commonplace. The methods for solving such problems are rather well established. One simply quantifies with some measure the variables and objectives involved in the problem, then chooses the appropriate solution methodology, obtains the necessary data and calculates an answer. However, with problems involving variables and objectives that cannot be measured, or at best can be only partly measured, the solution approach is not always so clear. This is particularly true when the variables and objectives involve personal preferences. The approach frequently taken with such problems is to simply prioritize the decision considerations and try to choose a solution that maximizes the desired decision quantities at minimum cost. Although this may not be too difficult with a limited number of decision quantities, it can become very difficult when the number of such quantities is large. In addition, the problem becomes vastly more complex when some of the decision quantities are in mutual conflict. Thus, making rational decisions under such circumstances may become extraordinarily difficult.

A number of techniques are available for arriving at decisions having multi-attributes. Unfortunately, most of them require that the attributes be measurable. When the attributes are of a more qualitative nature, the multi-attribute problem becomes much more difficult to handle. Herein lies the value and power of Analytical Hierarchy Process (AHP). With AHP it is possible to give a qualitative type problem a quasi-quantitative structure, and to arrive at decisions by expressing preferences for one attribute over another, and testing whether the preferences are rational consistent.

In this paper, we use AHP to analyze the selection process for project management software (PMS). We approach this problem by treating the PMS features offered by various companies as attributes. We then have a group of PM professionals evaluate the various features as to their importance and desirability. From these evaluations, paired comparison matrices are developed. Next, a set of matrices is developed for each software provider that evaluate just how well each provider's software satisfies each attribute. A consistency ratio is then computed to determine how rigorously rational consistency has been maintained in the analysis. Ordinarily, an AHP analysis would end here, but we extend our analysis by using the Student's "t" test for small sample sizes to determine a confidence interval within which the responses should lie. This analysis fills a void in the literature by demonstrating a perfectly generalized process for solving a practical multi-attribute decision problem, and for arriving at a high level of confidence that a rational decision will have been made.

LITERATURE REVIEW

AHP was originally conceived by Thomas L. Saaty as a structured method for solving problems involving decision variables or decision attributes, at least some of which, are qualitative, and cannot be directly measured (Saaty, 1980). It met with almost immediate acceptance and was applied to a wide range of problems. Very soon it began to be applied to executive decisions involving conflicts in stakeholder requirements and strategic planning (Saaty, 1982; Arbel & Orgler, 1990; Uzoka, 2005). The real power of AHP consists in its use of fairly elementary mathematics to structure complex problems in which decisions involve numerous decision makers, and multiple decision variables. Another facet of the power of the AHP approach consists in its ability to impose a quasi-quantitative character on decision problems in which the decision variables are not necessarily quantitative. The power and versatility of AHP are demonstrated by the wide range of problems to which the approach has been applied. It was used early for such problems as the justification of flexible manufacturing systems (Canada & Sullivan, 1989), and continues to be used in such applications (Chan & Abhary, 1996; Chandra & Kodali, 1998; Albayrakoglu, 1996). It has been used in such widely different applications as business crisis management (Lee & Harrald, (1999) and pavement maintenance (Ramadhan, Wahab & Duffuaa, 1999). Other interesting applications of AHP include the evaluation of personnel during the hiring process (Taylor, Ketcham & Hoffman, 1998), determination of investor suitability in structuring capital investment partnerships (Bolster & Janjigian & Trahan, 1995), apportioning public sector funds where numerous projects usually compete for limited resources (Barbarosoglu & Pinhas, 1995), and determination of real estate underwriting factors in the underwriting industry (Norris & Nelson, 1992). In the areas of accounting and finance, AHP has seen increasing use helping to direct the limited resources of auditors to their most effective and efficient use Baranoff, 1989; Arrington, Hillison & Jensen, 1984) and in the detection of management fraud Webber, 2001; Deshmukh & millet, 1998), and the prediction of bankruptcy (Park & Han, 2002). Thus, AHP is a very powerful, versatile and generalized approach for analyzing multi-attribute decision problems in which the decision considerations do not necessarily need to be directly measurable. Although AHP has been used in a very wide range of applications, this literature search has not disclosed any in which it has been used for project management software selection.

AHP METHODOLOGY

As noted above, the procedure for using Analytical Hierarchy Process is well established. It is a tribute to Dr. Saaty that his original work, done more than a quarter century ago, remains virtually unmodified. Thus, the present work will follow rather closely Dr. Saaty's original approach. This statement involves the following steps:

1) A clear concise description of the decision objective. In practice, this is probably best done by a team of 4 to 6 people who have good knowledge of the objective to be achieved, and who have a stake in arriving at the best possible decision.

2) Identification of those attributes that are to be included in arriving at the desired objective defined in step 1. These attributes are also best identified by a team of 4-6 people and preferably the same team that is used to identify the objective of the analysis.

3) Determination of any sub-attributes upon which an attribute might be based.

4) Identification of a set of alternatives that are thought to achieve, at least partially, the desired objective. We say "at least partially" because probably no alternative will completely provide all desired attributes. However, the alternatives should be selected because of providing some degree of satisfaction to all attributes.

5) Once the attributes are identified, they are entered into a preference matrix, and a preference or importance number is assigned to reflect the preference for/importance of each attribute relative to all others. The strength of preference/importance is indicated by assigning a preference/importance number according to the following rating scale:

Preference Number for Then Attribute A is .....
Attribute A over over (or to) Attribute B
Attribute B

9 Absolutely more important or preferred
7 Very strongly more important or preferred
5 Strongly more important or preferred
3 Weakly more important or preferred
1 Equally important or preferred

Intermediate degrees of preference for attribute A over attribute B are reflected by assigning even numbered ratings "8", "6", "4" and "2" for one attribute over another. For example, assigning a preference number of "6" would indicate a preference for attribute "A" over attribute "B" between "Strongly more important or preferred" and "Very strongly more important or preferred." Also, the logical principle of inverse inference is used, in that assigning attribute "A" a preference rating of, say "8", over attribute "B", would indicate that attribute "B" were only "1/8" as important as attribute "A." Either such a preference matrix is developed by each participant in the process, or the participants arrive at agreement as to the preference numbers for the attributes.

6) Mathematical operations are then used to normalize the preference matrix and to obtain for it a principal or characteristic vector.

7) Next, participants collectively rate each of the possible approaches/options for satisfying the desired objective(s). This is done along the same lines as shown above for attributes. The same rating scale is used, except here the ratings reflect how well each possible approach/option satisfies each of the attributes. For example, suppose that Alternatives "1" and "2" were being compared as to how well each satisfies Attribute "A" relative to the other. If Alternative "1" were given a preference rating of, say 5, relative to Alternative "2" it would mean that for Attribute "A", Alternative "1" was thought to satisfy it 5 times as well as Alternative "2". Or conversely, Alternative "2" was believed to satisfy Attribute "A" only "1/5" as well as did Alternative "1". Thus, a set of preference matrices is developed, one matrix for each attribute, that rates each alternative as to how well it satisfies the given attribute relative to all other alternatives under consideration.

8) Again, as with the attribute preference matrices, mathematical operations are used to normalize each preference matrix for alternatives, and to obtain a principal or eigenvector for each. When complete, this provides a set of characteristic vectors comprised of one for each attribute. Each of these vectors reflects just how well a given alternative/option satisfies each attribute relative to other alternatives.

9) Then with an eigenvector from "8" above, we cast these into yet another matrix, in which the number of rows equals the number of alternatives/options, and the number of columns equals the decision attributes. This matrix measures how well each alternative/option satisfies each decision attribute.

10) Next, the matrix from "9" above is multiplied by the characteristic vector from "6" above. The result of this multiplication is a weighted rating for each alternative/option indicating how well it satisfies each attribute. That alternative/option with the greatest weighted score is generally the best choice for satisfying the decision attributes, and thus achieving the desired objective.

11) Last, it is necessary to calculate a consistency ratio (C. R.) to ensure that the preference choices in the problem have been made with logical coherence. The procedure for calculating the C. R. may be found in a number of works on AHP (Canada & Sullivan, 1989) and will not be repeated here. According to Saaty, if choices have been made consistently the C. R. should not exceed 0.10. If it does, it then becomes necessary to refine the analysis by having the participants revise their preferences and re-computing all of the above. Thus, the entire process is repeated until the C. R. is equal to or less than 0.10.

12 Having completed steps "1" thru "11", that alternative/option with the greatest weighted score is selected as best satisfying the decision alternatives, and thus achieving the desired objective.

APPLICATION OF AHP METHODOLOGY

The usual practice for developing an AHP analysis is to develop a separate set of matrices for each of the participants, or to have participants brainstorm until they arrive at a group decision for the required matrices. However, in the instant case, time and distance made the usual approach impossible. Consequently, the authors have done the next best thing and have averaged participants' ratings. The average ratings are used in the subsequent analysis, and the sensitivity analysis done on the averaged ratings would be analogous to reconvening the participants in order to refine the ratings and thus arrive at a satisfactory C. R. The results of this are shown in Figures 1 thru 7 below. The codes for the Software Features and Companies (alternatives/options) used in the figures immediately follow:

Software Feature Code Company

Multiple Networks Supported A 1
Simultaneous Multiple User Access B 2
Full Critical Path C 3
Time/Cost D
Multi-project PERT Chart E
Resources Used F

ANALYSIS OF RESULTS

As indicated above, the attribute preference matrices for participants were averaged for each attribute, so that the final ratings shown in Fig. 1 are the result of an attribute by attribute average, rather than a matrix by matrix average.

[FIGURE 1 OMITTED]

Fig. 2 below shows the normalized group preference matrix resulting from normalizing the averaged values in Fig. 1. The characteristic vector for this matrix is shown in the rightmost column.

As can be seen in the last row of Fig. 2, the C. R. for these averaged attributes is 0.05, a value well within that prescribed by Saaty for logical coherence. This result is very interesting because only one of the participants achieved a C. R. of less than 0.10, this being a C. R. of 0.09. The other six had C. R.s ranging from 0.140 to 0.378. This means that the group as a whole was more consistent in rating the attributes than was each individual. Since the averaged ratings resulted in a C. R. of less than 0.10, it was unnecessary at this point to perform sensitivity analysis on the averaged attribute ratings. However, had sensitivity analysis been necessary, each individual rating would have been adjusted slightly upward or downward until a C. R. of equal to or less than 0.10 were obtained. Since only one of the participants had a C. R. less than 0.10, the averaging process was analogous to calling the participants into caucus and having them arrive at mutually agreeable ratings for all attributes. The conclusion to be drawn from this is that a group of individuals will frequently make more rational choices than will a single individual. As a practical matter, it would seem to be more expeditious to first obtain the individual attribute preference matrices, average them and then call participants together to negotiate more consistent ratings. Alternatively, each individual would have to revise his individual results and have a new C. R. computed. This process would then have to be continued until all participants had individually arrived at C. R.s of equal to or less than 0.10. The former approach of averaging and caucusing employs a Delphi approach to decision making, which has been shown usually to lead to better decision outcomes than decisions made in isolation.

Ordinarily, selection of the options/alternatives (in our case software providers) for satisfying the above attributes would also be made by the primary participants. Each participant would rate each option/alternative as to how well it satisfied each attribute. However, in the present case this part was done by the authors based on an analysis of several project management software products and their common and unique features. Those ratings, as with the attribute preference ratings, were then cast into matrices and C. R.s were calculated for each software provider. The final results for this are shown in Figs. 3 thru 8.

The results from this part of the process, as can be seen, turned out quite well. Only the C. R. for the third attribute, "Full Critical Path," turned out to have a C. R. greater than 0.10. This C. R. was originally 0.54, a value well above the allowable. Sensitivity analysis was performed on this matrix until the present C. R. of 0.05 was obtained. The remainder of the C.R.s were well below 0.10, as can be seen. Remarkably, the C. R. for "Multi-project PERT Chart" turned out to be 0.000, indicating perfect consistency among the choices. An interesting problem arose for the "Full Critical Path" attribute in that neither Company 2 nor 3 offered this feature. It would be expected that a company would receive a rating of "0" for failure to provide a feature. However, this would have resulted in Company 1 being infinitely preferred over Companies 2 and 3. This would have been inconsistent with the rating scale, which only ranges from "1" to "9". To address this problem, a preference rating of "9" was assigned for Company 1 over Companies 2 and 3. This resulted in ratings of 1/9 or 0.11 being assigned as to the preference for Companies 2 and 3 over Company 1. Thus, the final values for these ratings turned out near "0", though not exactly "0". Once all matrices were assured to have a C. R. of equal to or less than 0.10, their values were cast into a final matrix as shown in Fig. 9 below.

In this matrix, the top row of six numbers will be seen to be the principal or characteristic vector from the attribute preference matrix. Each of the six columns of three numbers under this top row will be seen to be the principal vectors for Companies 1, 2 and 3 as obtained from the analysis described in the immediately preceding paragraph. The rightmost column titled "Weighted Alternative Evaluations" contains the results of this analysis. It will be seen that Company 1 best satisfies the original attributes in that it has a rating of 0.43. Companies 2 and 3 follow in order, with ratings of 0.36 and 0.21, respectively. These ratings are quite stable. Sensitivity analysis was done on these results by varying the ratings in the original attribute preference matrix. However, these results remained virtually unchanged. Company 1 would then be selected as best satisfying the desired attributes.

In order to test for the consistency of the participant attribute preferences, the C. R.s were subjected to a Student's "t" test to obtain a confidence interval. The "t" test is used for applications in which the sample size is smaller than about twenty, the samples reasonably can be assumed to be normally distributed and the distribution of the sample standard deviation is independent of the distribution of the sample values. Having made these assumptions, we obtained the following results:

Next a "t" test is performed with six degrees of freedom and a confidence interval of 95%. This yields theoretical lower and upper confidence limits of -0.07 and 0.47. Since a C.R. of zero means perfectly consistent choices, a C.R. of less than zero is impossible. This means that the practical value of the lower confidence limit for this application is zero. Then for a 95% confidence level, all of the C.R. values for our PMs should lie within a range from 0.00 to 0.47. As can be seen from the above C.R.s, all do indeed lie within this confidence interval as obtained from the "t" test. Thus, the responses of all PMs were found to be highly consistent both from the standpoint of the C.R. obtained from the averaged attribute responses, and the confidence interval obtained from the "t" distribution. A decision maker having the foregoing analysis could thus conclude that Software Provider #1 would provide the best software package for the application envisioned.

CONCLUSIONS

The thesis of this paper has been that Analytical Hierarchy Process offers a very powerful, flexible and general approach to decision situations in which the decision variables are not necessarily quantifiable. Although it was not possible to quantify any of the attributes in the above example, it was, nonetheless possible to rationally express preferences for one above another. We say rationally because a decision maker can always express a preference for one attribute over another once a definite objective is established. In this case one preference is more desirable than another in so far as it better satisfies or achieves the objective(s), or vice versa. We have shown in the foregoing analysis that even though none of the C.R.s for a group of respondents may satisfy the 0.1 criterion specified by Saaty, that this obstacle may be overcome by averaging the responses for each attribute. We have demonstrated how to handle the problem in which a given provider does not offer one or more of the desired attributes/features. We have also shown through sensitivity analysis that once the C.R.s for all matrices have been brought to values equal to or less than 0.10 using sensitivity analysis, that the results are remarkably stable. They are relatively insensitive to fairly large changes in the original attribute preferences. We tested the consistency of the responses using the Student's "t" distribution and found that they can be placed within a 95% confidence interval, which indicates a very good degree of consistency among PM respondents. Finally, in using AHP in the problem of software selection, we have demonstrated the great power and flexibility of this little known process in solving a very wide range of practical decision problems which are otherwise virtually intractable.

REFERENCES

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J. S. Sutterfield, Florida A&M University

Steven Swirsky, Florida A&M University

Christopher Ngassam, Florida A&M University

PM C.R.
1 0.14
2 0.15
3 0.14
4 0.20
5 0.09
6 0.38
7 0.32

y-bar = 0.20
sample standard deviation = 0.11

Fig 2: Normalized group attribute performance
preference matrix

 A B C D E F
A. 1.000 1.000 0.200 0.500 1.000 0.200
B. 1.000 1.000 0.250 1.000 1.000 1.000
C. 5.000 4.000 1.000 4.000 5.000 1.000
D. 2.000 1.000 0.250 1.000 3.000 0.500
E. 1.000 1.000 0.200 0.333 1.000 0.250
F. 5.000 1.000 1.000 2.000 4.000 1.000

 Decimal Equivalents

 A B C D E F
A. 0.067 0.111 0.069 0.057 0.067 0.051
B. 0.067 0.111 0.086 0.113 0.067 0.253
C. 0.333 0.444 0.345 0.453 0.333 0.253
D. 0.133 0.111 0.086 0.113 0.200 0.127
E. 0.067 0.111 0.069 0.038 0.067 0.063
F. 0.333 0.111 0.345 0.226 0.267 0.253
 1.000 1.000 1.000 1.000 1.000 1.000

 Row Row
 Sums Averages
A. 0.421 0.070 1.000 1.000 0.200 0.500
B. 0.697 0.116 1.000 1.000 0.250 1.000
C. 2.162 0.360 5.000 4.000 1.000 4.000
D. 0.770 0.128 2.000 1.000 0.250 1.000
E. 0.414 0.069 1.000 1.000 0.200 0.333
F. 1.536 0.256 5.000 1.000 1.000 2.000
 1.000

A. 1.000 0.200 x 0.070 = 0.442
B. 1.000 1.000 x 0.116 = 0.729
C. 5.000 1.000 x 0.360 = 2.287
D. 3.000 0.500 x 0.128 = 0.809
E. 1.000 0.250 x 0.069 = 0.434
F. 4.000 1.000 x 0.256 = 1.614

 D = 6.3171 6.2846.3528 6.320 6.285 6.305

Lambda Max = 6.310C.I. = 0.0621 C.R. = 0.050

Figure 3: Multiple networks supported

 P Q R
Company 1 1.000 0.500 4.000
Company 2 2.000 1.000 7.000
Company 3 0.250 0.143 1.000
Column totals =

 1.000 0.500 4.000
 2.000 1.000 7.000
 0.250 0.143 1.000

 D = 3.002 3.004 3.0008

 Lambda max = 3.0023 C.I. = 0.0012 C.R. = 0.002

 Decimal equivalents Row Row
 P Q R sums averages
Company 1 0.308 0.304 0.333 0.945 0.315
Company 2 0.615 0.609 0.583 1.807 0.602
Company 3 0.077 0.087 0.083 0.247 0.082
Column totals = 1.000 1.000 1.000 1.000

 0.315 0.946
 x 0.602 = 1.810
 0.082 0.247

 D = 3.002 3.004 3.0008

 Lambda max = 3.0023 C.I. = 0.0012 C.R. = 0.002

Figure 4: Simultaneous multiple user access

 P Q R

Company 1 1.000 1.000 4.000
Company 2 1.000 1.000 6.000
Company 3 0.250 0.167 1.000

 Decimal equivalents
 Row Row
 P Q R sums averages

Company 1 0.444 0.461 0.364 1.270 0.423
Company 2 0.444 0.461 0.545 1.451 0.484
Company 3 0.111 0.077 0.091 0.279 0.093
Column totals = 1.000 1.000 1.000 1.000

Company 1 1.000 1.000 4.000 x 0.423 = 1.279
Company 2 1.000 1.000 6.000 x 0.484 = 1.465
Company 3 0.250 0.167 1.000 x 0.093 = 0.280

 D = 3.023 3.028 3.0057

Lambda max = 3.0189 C.I. = 0.0095 C.R. = 0.019

Figure 5: Full critical path

 Decimal equivalents

 P Q R P Q R

Company 1 1.000 9.000 9.000 0.818 0.750 0.857
Company 2 0.111 1.000 0.500 0.091 0.083 0.048
Company 3 0.111 2.000 1.000 0.091 0.167 0.095
 Column totals = 1.000 1.000 1.000

 Row Row
 sums averages

Company 1 2.425 0.808 1.000
Company 2 0.222 0.074 0.111
Company 3 0.353 0.118 0.111
 1.000

Company 1 9.000 9.000 x 0.808 = 2.532
Company 2 1.000 0.500 x 0.074 = 0.222
Company 3 2.000 1.000 x 0.118 = 0.355

 D = 3.132 3.009 3.0208

Lambdamax = 3.0539 C.I. = 0.027 C.R. = 0.054

Figure 6: Time/cost trade-off

 Decimal equivalents

 P Q R P Q R

Company 1 1.000 0.500 0.333 0.167 0.143 0.182
Company 2 2.000 1.000 0.500 0.333 0.286 0.273
Company 3 3.000 2.000 1.000 0.500 0.571 0.546
 Column totals = 1.000 1.000 1.000

 Row Row
 sums averages

Company 1 0.491 0.164 1.000 0.500
Company 2 0.892 0.297 2.000 1.000
Company 3 1.617 0.539 3.000 2.000
 1.000

Company 1 0.333 x 0.164 = 0.492
Company 2 0.500 x 0.297 = 0.894
Company 3 1.000 x 0.539 = 1.625

 D = 3.004 3.008 3.0144

Lambdamax = 3.0088 C.I = 0.004 C.R. = 0.009

Figure 7: Multi-project PERT chart

 Decimal equivalents

 P Q R P Q R

Company 1 1.000 1.000 2.000 0.400 0.400 0.400
Company 2 1.000 1.000 2.000 0.400 0.400 0.400
Company 3 0.500 0.500 1.000 0.200 0.200 0.200
 Column totals = 1.000 1.000 1.000

 Row Row
 sums averages

Company 1 1.200 0.400 1.000 1.000
Company 2 1.200 0.400 1.000 1.000
Company 3 0.600 0.200 0.500 0.500
 1.000

Company 1 2.000 x 0.400 = 1.200
Company 2 2.000 x 0.400 = 1.200
Company 3 1.000 x 0.200 = 0.600

 D = 3.000 = 3.000 3

Lambda max = 3 C.I. = 0 C.R. = 0.000

Figure 8: Resources used

 Decimal equivalents

 P Q R P Q R

Company 1 1.000 0.143 0.200 0.077 0.097 0.048
Company 2 7.000 1.000 3.000 0.538 0.678 0.714
Company 3 5.000 0.333 1.000 0.385 0.226 0.238
 Column totals = 1.000 1.000 1.000

 Row Row
 sums averages

Company 1 0.221 0.074 1.000 0.143
Company 2 1.930 0.643 7.000 1.000
Company 3 0.848 0.283 5.000 0.333
 1.000 D = 3.013

Company 1 0.200 x 0.074 = 0.222
Company 2 3.000 x 0.643 = 2.008
Company 3 1.000 x 0.283 = 0.866
 D = 3.121 3.063

Lambdamax = 3.0657 C.I. = 0.0328 C.R. = 0.066

Figure 9: Weighted alternative evaluations

 A B C

 0.070 0.116 0.360
Company 1 0.315 0.423 0.808
Company 2 0.602 0.484 0.074
Company 3 0.082 0.093 0.118

 D E F Weighted
 Alternative
 0.128 0.069 0.256
Company 1 0.164 0.400 0.074 0.430
Company 2 0.297 0.400 0.643 0.355
Company 3 0.539 0.200 0.283 0.215
 Total = 1.000