Computers & Operations Research 30 (2003) 1487–1497
www.elsevier.com/locate/dsw
On teaching the analytic hierarchy process
Lawrence Bodin
∗, Saul I. Gass
Robert H. Smith School of Business, College of Business and Management, University of Maryland, College Park, MD 20742, USA
Abstract
We discuss aspects of the analytic hierarchy process (AHP) that we feel are important to the successful presentation of the AHP to graduate business students. We have also created a working paper that presents some of the examples and class projects that we found to be of pedagogical value. This working paper can be obtained by email from the 5rst author ([email protected]).
Scope and purpose
Since its inception nearly 25 years ago, the AHP has been widely used to solve a broad range of multi-criteria decision problems. Successful applications can be found in business, industry, government, and the military. The community of operations research (OR) practitioners has incorporated the AHP in its methodo-logical tool-bag while a number of university programs include AHP in courses covering decision-making tech-niques or in general OR courses. Instructors new to the topic need to be concerned with how the fundamental concepts of the AHP are 5rst presented to students. Based on our teaching experiences, we discuss our view of how key elements of the AHP should be presented in the classroom in the hope it will aid instructors to understand better the important decision-aiding framework of the AHP.
? 2003 Elsevier Science Ltd. All rights reserved. Keywords:Analytic hierarchy process; Teaching material
1. Introduction
Over the past 20 years, the authors have taught core MBA management science (decision-making and modeling) courses that included a module on the analytic hierarchy process (AHP). Based on student feedback regarding how they were able to use this material in their job activities and
∗Corresponding author. Tel.: +1-301-405-2210.
E-mail address:[email protected] (L. Bodin).
0305-0548/03/$ - see front matter? 2003 Elsevier Science Ltd. All rights reserved. PII: S0305-0548(02)00188-0
in other MBA courses, we feel that our pedagogical approach is an appropriate one for teaching this (still rather recent) operations research (OR) decision-aiding methodology. Since its inception, the AHP is widely used by the analytical community. For the last decade or so, material on the AHP has appeared in introductory management science textbooks where it is presented as an OR decision-aiding paradigm for solving multi-criteria problems. In general, we 5nd that the textbook coverage of AHP is very rudimentary, especially concerning the fundamental aspects of the process. In this paper, we discuss some of our teaching experiences in order to aid others who are teaching or are planning to teach the AHP within an MBA program. In Section 2, we discuss key aspects of the AHP that we believe should be covered in the classroom. In Section 3, we present examples of the AHP. In Section 4, we brieAy describe some exercises that instructors may 5nd useful for assignments and term projects. In Section 5, we oBer concluding remarks.
We have prepared a working paper containing the six examples discussed in Section 4. These examples are labeled EX1-EX6 in the working paper and in this paper. The reader can receive the working paper that contains these exercises by sending a request to the 5rst author at email address [email protected].
We assume the reader is familiar with the concepts and theory of the AHP and with the class of multi-criteria, multi-alternative decision problems to which it can be applied. We recommend using in class the expert choice (EC) software that implements the AHP, as proposed by Saaty. EC is easy to use and a free copy, suitable for most classroom exercises can be downloaded from the web site, www.expertchoice.com. Depending upon the version of EC used by the student, there may be limitations with respect to an expiration date or number of levels in the AHP tree that can be used. HIPRE and Criterium Decision Plus are other software packages that contain an implementation of AHP (we have not used either of these packages). We see no pedagogical advantage in doing the AHP computations on a spreadsheet. (We have also taught similar AHP material to undergraduate students. Although the undergraduate students can understand the material on the AHP, we 5nd their ability to apply the AHP concepts to be rote and without much imagination.)
Over time, the core MBA management science course at the University of Maryland changed from a 15 week, full-semester core course to a seven week, half-semester (1 1/2 credits) core course to the present day non-core (elective) half-semester (2 credit) course. Although material in our MBA course changed as contact hours were varied, we always kept the AHP as a 2–3 week module.
We believe that the AHPs ability to resolve a wide class of important decision problems must be in the common-knowledge base of an educated MBA. At the same time, we realize that we are not try-ing to turn the MBA students into OR analysts. The objectives of our introductory MBA course were: (a) To enable the students to become conversant with ideas and techniques for resolving multiple
criteria decision problems, and
(b) To understand how these ideas can be used in the business world.
We wanted to ensure that our MBA graduates had the background and experience to ask the right questions of their staBs and/or fellow workers when faced with such problems. Throughout the course, we stressed that the OR methodologies, especially the AHP, are decision-aids. They do not make decisions. Instead, these methods furnish information to the ultimate decision maker (DM) who must evaluate and integrate all relevant information when choosing a “best” alternative solution or when ranking a set of alternatives.
2. Key elements of the AHP to cover in a class
The following elements are essential for understanding the theory and application of AHP and should be covered in any class on the AHP.
• The AHP Fundamental Pairwise Comparison Scale. • Inconsistency and Sensitivity Analysis.
• Ratio Scales. • The Ratings Model.
• The Team Approach for Solving the AHP. • The AHP and Resource Allocation.
Further, the notion of rank reversal could be covered in the class if time permits but is not essential to a basic understanding of the AHP. These elements are now described.
2.1. AHP fundamental pairwise comparison scale
Central to the resolution of a multi-criteria problem by the AHP is the process of determining the weights of the criteria and the 5nal solution weights of the alternatives with respect to the criteria. As the true weights are unknown, they must be approximated. To do so, the AHP requires answers (either numerical or verbal) to a sequence of questions that compare two criteria or two alternatives. The numerical answers are given using a fundamental 1–9 scale, while the verbal answers are converted to their equivalent numeric values on that 1–9 scale. When he was developing the AHP, Saaty investigated a wide range of possible numerical comparison scales for comparing two items (27 scales to be exact, p. 56, Saaty [1]) before deciding to use the fundamental 1–9 scale. Saaty showed that the 1–9 scale works exceptionally well in its ability to encapsulate a problem’s quantitative and qualitative information as required by the pairwise comparison mode of the AHP. Other researchers have proposed diBerent scales involving logarithms, geometric powers, and negative numbers, etc. Proponents of these scales claim that the weights produced by these scales are more acceptable than those found using the fundamental 1–9 scale. We do not 5nd these claims persuasive and we see no reason to deviate from the 1–9 scale. The 1–9 scale has proven to be a most adequate measurement scale that enables a DM to approximate the unknown weights for a wide and important class of multi-criteria problems.
It is important to emphasize that in making a pairwise comparison, the DM is really trying to estimate unknown weights. Let anElementof the decision problem be either a criterion, subcriterion, or alternative. The pairwise comparison question asked by the DM, “Is Element A more important than (or preferred to) Element B and by how much?”, should be interpreted as taking the ratio of the DMs estimate of A’s unknown weight to B’s unknown weight and normalizing the fraction so that the denominator is always 1. In essence, when answering a pairwise comparison question, the DM estimates the true but unknown weights based on insight and experience relative to the multi-criteria decision problem. Saaty [1,2] has developed a number of examples such as estimating areas, distances, etc. that show that the AHP is able to generate very accurate weights in situations where the underlying basis for making these comparisons is unknown to the person answering the pairwise comparison questions. The ratio estimates should be looked at as coming from an implied
ratio scale. If, using the verbal scale associated with the 1–9 scale, Element A is “strongly more” important than Element B, the resultant ratio is 5/1. This is equivalent to saying, for example, that the user may have estimated Element A’s unknown weight to be about 40 and Element B’s unknown weight to be about 8, so that the relative ratio is 5/1.
We have found that trying to 5ne-tune or micro-manage the ratios by using more than one decimal place does not aid in determining the 5nal weights. Splitting the diBerence between a 5/1 and a 6/1 by using a single decimal point may be appropriate. Unless there is a rationale for doing otherwise, we suggest, based on work by Standard [3], that any indecision between two ratios, say 5/1 and 6/1, be resolved by using 5.5/1. Of course, all AHP studies should include the appropriate sensitivity analyses to determine break points for which the 5rst and second ranked alternatives (as well as others) change positions. These ideas are discussed further in Section 2.2.
We suggest that after the instructor discusses and uses the 1–9 scale and works through a couple of examples, the instructor has the class solve the geometric validation example developed by Saaty [2]. This example, called EX6 in the working paper, calls for the visual qualitative comparison of 5ve geometric 5gures in terms of their areas. The AHP solution yields weights wi that represent
(approximate) the ratio of 5gure i’s area to the total area encompassed by all 5ve 5gures. To accomplish this exercise in class, the instructor acts as the facilitator and the students act as DMs. The instructor asks the students to vote on the pairwise comparison value for each pair of 5gures and sets the value of the pairwise comparison based on the consensus of the class. The value of the pairwise comparisons can have one decimal place such as 5.5 in the case of a vote that is almost tied. The 5nal results are generally very accurate, assuming “oB-the-wall” values are rejected. (See discussion below on the team approach for a more accurate way of “averaging” the values.)
The geometric validation example also demonstrates that the DM must exhibit care in maintaining consistency in the comparisons by remembering previous comparison values, that is, which geometric 5gure has a greater area with respect to the others. This example illustrates that exact consistency is neither required nor desired. The AHP, by requiring the analyst to make all pairwise comparisons, captures and measures any deviation from consistency.
For a reasonably large problem, some students complain about all the work and data input required. They should be told that no matter what decision method is used, much eBort and time must be expended in data gathering and analysis, and so it is with the AHP.
2.2. Inconsistency and sensitivity analysis
In making a sequence of pairwise comparisons, especially for systems that have 5ve or more criteria and/or alternatives, we would expect that the estimates of the unknown weights, as reAected by the weight estimates (ratios) given in answer to the pairwise comparison questions, need not be exact or consistent. The AHP measures inconsistency by comparing the DMs data to a set of random results that assumes, for the same size matrix, that the estimates were random. Saaty [1] developed a measure of inconsistency, called the inconsistency ratio (IR) that is based on fundamental theoretical results on the size of the largest eigenvalue for the matrices in question. The IR can go from zero (true consistency of the input) to a very large positive number. The implications of a large IR are discussed below.
Based on the underlying theory of the AHP, Saaty stated that an IR of 0.10 or less should be accepted, while any larger number should cause the DM to rethink the input and attempt to 5nd
any anomalies in the comparisons. In our experience, two of the most common mistakes that cause a large IR are as follows:
(a) The DM states an intransitive relationship in the pairwise comparisons. For example, when making the set of pairwise comparisons, the DM states that A is preferred to B, B is preferred to C, and C is preferred to A. This situation can easily occur, especially when comparing 5 or more items. Finding intransitivities may take some eBort. Gass [4] describes procedures for 5nding intransitive relationships in pairwise comparison matrices.
(b) A comparison is made that has A preferred to B when in reality B is preferred to A. This can sometimes happen (inadvertently) by not “inverting” properly in the EC system. Incorrect inversions can usually be found by applying the ECs inconsistency check (available when the comparison matrix is viewed).
The IR60:10 (10% rule) is not hard and fast. If an IR = 0:15 and the decision-maker feels con5dent that the input should not be changed, then the results should be accepted. The chances are that a sensitivity study will show that the 5nal weights for the alternatives will maintain the same rank order and the individual weights will not change by much. If the user wants to accept the results of a set of pairwise comparisons when the IR¿0:10, then the user has to be careful that the pairwise comparisons really reAect the user’s beliefs rather than contain an error. This situation occurred in the problem of ranking the San Diego Padres baseball players for the Major League expansion draft held in November, 1997 (see Bodin and Epstein [5]). Eddie Epstein, the Director of Baseball Operations for the San Diego Padres, wanted to give higher weights to players who excelled in certain categories (especially on-5eld performance). To accomplish this, he accepted some IRs larger than 0.10.
The EC inconsistency check enables one to 5nd the most, 2nd most, etc. inconsistent pairwise comparison ratio. If pursued further, EC will indicate how much the ratio in question should be changed. We have found that in some cases the suggested magnitude of a change is excessive, but the direction of change is correct and of value.
Some students try to adjust their pairwise comparisons to achieve an IR =0. The instructor should discourage this practice. A little inconsistency is acceptable. It allows Aexibility in capturing the student’s understanding of the decision problem, as reAected by the pairwise comparison data. It is better to represent the user’s view, even if it is a little inconsistent, than to be consistent (IR=0) and wrong! One would not expect to 5nd a 7×7 pairwise comparison matrix to be perfectly consistent. The diBerent types of sensitivity analyses embedded in EC can be of great value in looking for inconsistencies, as well as answering “What if ?” questions. The instructor should demonstrate the sensitivity options as part of the choosing the best automobile exercise given as EX1 in the working paper. For example, by using the dynamic sensitivity option in EC, the student can readily observe how the solution changes as the weights on price are varied.
2.3. Ratio scales
Most of us do not give much thought to the properties of numbers. We know when they can be added, multiplied, or divided in terms of what are legal operations. The power of the AHP is that the resultant weights are ratio-scale numbers. The only way we can justify taking the ratio of
two numbers, say 4/2, and concluding that one is twice as big as the other, is for the numbers to be from a ratio scale that has the same units or from a units free ratio scale. The components of a pairwise comparison matrix’s maximum eigenvector are ratio-scale numbers (all positive and normalized to one). The AHP 5nal weights for the alternatives are formed by summed products of these components, with the resultant weights normalized ratio-scale numbers. These weights are units free. They allow us to compare two alternatives, say X with weight 0.340 and Y with weight 0.170, and conclude that X is twice as preferred as Y. Other number systems do not enable us to make such statements. Other multi-criteria decision methods do not produce ratio scales; they may involve ordinal or interval numbers.
The importance of ratio-scale weights has to be emphasized to the students. Even though they may be familiar with ratio, ordinal, and interval scale numbers, they tend not to pay attention to their properties, e.g., the ratio of two ordinal scale numbers or two interval scale numbers have no meaning, or such numbers cannot be added together. (An interval scale is usually used to evaluate instructors, the eponymous Likert scale; see Gass [6] for further discussion on this issue.)
2.4. The ratings mode
The EC software has two modes for analyzing a multi-criteria problem using AHP methodology. They are (1) direct (relative) comparison and (2) indirect (absolute) ratings.
The direct comparison mode is the usual way the AHP structures a problem. As implemented by the EC software, a direct comparison model can contain no more than nine elements (criteria or alternatives) at any level of an AHP hierarchy. The solution (synthesis) of a direct comparison model assigns each alternative a positive ratio-scale weight, with the sum of all the weights equal to one. The alternative with the largest weight is best.
Many AHP-based problems, however, have a large number of alternatives to compare. For example, ranking employees or MBA programs. Saaty [1] notes that when the number of items gets to be between 7±2 it becomes diRcult to maintain some semblance of consistency in the pairwise comparisons. Thus, for more than nine alternatives, one should use the EC ratings mode.
Here the alternatives are evaluated individually against the criteria using intensity (achievement) levels that measure how well an alternative accomplishes a criterion. For example (excellent, very good, good, fair, poor) for an employee under the criterion of “quality of work.” The ratings mode solution assigns a positive ratio-scale weight to each alternative. This weight is determined indepen-dently from the other alternatives, with an alternative’s weight being less than or equal to 1.0. A weight of 1.0 indicates that the alternative is a “top” performer with respect to the other alternatives. Alternatives can have the same weight value. Using these weights, the alternatives can be ranked from high to low.
2.5. The AHP and resource allocation
From a general OR perspective, it is important for the instructor to show how an AHP model can be combined synergistically with other OR decision methods. For example, AHP determines weights for other models (see examples EX2 and EX3 in the working paper), or results from other methods are used as input to the AHP. Students are thus able to broaden their view of the world of
OR modeling, to wit: a particular model’s eBectiveness can be increased by combining it with other methods, and the results of combined models can be more meaningful than using one approach.
A powerful example of combined methods is the AHP and resource allocation (capital budgeting). Here, a subset of proposed projects is to be selected so that the total budget of the subset is less than or equal to a given available budget. The AHP is used to determine the weights for each of the projects, where the weight of a project measures the DMs preference for the project. (The DM can be an individual or a team, as described in Section 2.6). The weights are determined by the AHP from the associated multi-criteria problem. This process is illustrated in EX2 and EX3 in our examples.
These weights are then used as objective function coeRcients in a maximizing (0–1) programming (knapsack) problem. The resultant solution determines the subset of projects that satis5es the budget constraint and gives the largest possible value to the objective function. As required, additional constraints such as labor hours and space requirements can be added to the optimization problem. When the number of projects is no more than nine, the direct comparison mode should be used to determine the weights. Otherwise, the ratings mode should be used.
2.6. The team approach for using the AHP
Generally, the AHP is used by a single DM. In many circumstances, however, a team approach for solving the AHP can be carried out. An example of the team approach for solving the AHP is the evaluation of proposed research projects by representatives from Research and Development, Manufacturing, Marketing, Logistics, Finance, and Human Resources of a pharmaceutical manufac-turer. In this situation, the team must agree upon the AHP structure and derive a consensus on the pairwise comparisons.
The team expert choice (TEC) software carries out an AHP analysis by integrating the team’s individual inputs, TEC [9]. Using a handheld radio transponder, each member of the team submits a pairwise comparison to TEC. A pairwise comparison for the team is the geometric mean of the individual comparisons. The geometric mean is the correct way of “averaging” such team data. (Note that the reciprocal of the geometric mean is the geometric mean of the reciprocals of the individual comparisons; the standard average does not produce the proper reciprocals. For more details, see AczSel and Saaty [10], Gass and RapcsSak [11]). TEC does the geometric mean “averaging” auto-matically, with the resultant AHP analysis being that of the team. Further, each member’s pairwise comparisons can be reviewed and discussed by the team. “What if’s?” like changing one’s mind or arguing that such a comparison is right or wrong, along with the usual range of sensitivity analyses, can be evaluated readily. Such discussions help to facilitate the forging of a consensual decision. The instructor, combined with the TEC, can be regarded as the facilitator of the process.
As we did not have the transponders in the classroom, we emulated the TEC by taking the class’s most popular value of each pairwise comparison. This approach usually led to an enthusiastic discussion and gave the students insight into how collaborative types of decisions can be made. To quickly illustrate the team approach, the instructor should carry out the geometric validation example given as example EX6 in the working paper. (See Condon et al. [12] for additional discussion on the AHP team approach).
2.7. Rank reversal
Rank reversal of alternatives has been cited as a weakness of the AHP. Rank reversal can occur when a problem is changed to include an additional alternative that was not considered to be a candi-date for selection when the problem was 5rst stated. For example, assume that a fourth automobile is added to the problem of choosing the best automobile to buy (example EX1 in the working paper). The weights that add up to 1.000 over the three alternatives in the original problem now must be distributed among four alternatives. It should be clear that we have a new problem and there is a possibility that the automobile that was best for the three automobile problem may not be best for the four automobile problem. The added automobile or one of the other two automobiles may turn out to be the best. To our mind, this is understandable and appropriate, as we are now analyzing a new problem.
The concern with respect to rank reversal occurs when we add (or remove) what is termed an
irrelevant alternative. An alternative is irrelevant, if, when it is compared to an alternative from the original set of alternatives, it is never preferred (for example, adding a fourth automobile that is either the same as one of the original three automobiles or is not preferred over any of the original three automobiles). This concern over rank reversal comes from the independence of irrelevant alternative axiom of utility theory: If an alternative is non-optimal, it cannot be made optimal by adding new alternatives to the problem, Luce and RaiBa [7]. This axiom is necessary under the axiomatic structure of utility theory, but is not necessary or included in the axioms of the AHP, Saaty [1]. In fact, Luce and RaiBa, as well as others, cite decision examples in which rank reversal does occur, but they rule it out to maintain the consistency of the axiomatic approach of utility theory.
To address the rank reversal problem, the EC software allows the user two modes of analysis: (1) the distributive (closed) pairwise comparative mode if the user is not concerned about rank reversal (the basic AHP approach), or (2) the ideal (open) pairwise comparative mode if the user is con-cerned about rank reversal (an extension of the basic AHP approach). We only used the distributive mode in class and did not even discuss rank reversal. Studies using Monte Carlo simulations and real-world applications of the AHP show that rank reversal can occur but it is not likely, and, in most instances, the top ranked alternative does not change. A further discussion of rank reversal can be found in Gass and Forman [8] and at the web sites: http://mdm.gwu.edu/FormanGass.pdf. and http://mdm.gwu.edu/FormanGass.doc.
3. AHP examples
The EC software [13] has a number of examples that can be used to illustrate the basic features of the AHP and the building of an AHP model. Instructors are encouraged to review them and to select the models that are appropriate for their classes.
To illustrate the pairwise comparison mode, we suggest using the ice cream store site location problem (located in EC models 5le) and the buying of a new car problem (example EX1 in the working paper). For the ratings mode, we suggest using the employee evaluation problem (located in EC samples 5le).
The following lists some basic and advanced models that we have found to be of pedagogical value.
(a) Ranking the San Diego baseball players for the 1997 expansion draft in major league baseball using the AHP ratings model, Bodin and Epstein [5].
(b) The use of conjoint analysis and the direct comparison approach of AHP for making marketing research decisions, Bodin and Krapfel [14]. This paper shows how AHP can be used as a surrogate for conjoint analysis and how these approaches can be used together in analyzing a problem. Conjoint analysis is a popular approach for making marketing research decisions. It can use pairwise comparisons to determine how consumers trade oB diBerent criteria (called attributes in conjoint analysis) in preferring one product to another. A copy of this paper can be obtained from the 5rst author at [email protected].
(c) The use of the AHP ratings model for evaluating employees for a raise (see employee evaluation model in EC [13]).
(d) A question that comes up when modeling a problem using the AHP is whether to include the cost and bene5t criteria in a single AHP hierarchy or tree or perform a bene5t/cost analysis using two hierarchies. In the 5rst case, the analysis is performed where cost and bene5t criteria are both included in the same AHP hierarchy. In the bene5t/cost analysis, two AHP problems are solved. One AHP problem determines the weights for the alternatives with regard to cost. The second AHP problem determines the weights for the alternatives with regard to bene5t. Then, the ratios of the bene5ts to cost for each AHP alternative are determined and sorted from high to low. Alternatives with a very low bene5t weight can be omitted from the ratio analysis since their bene5t is so low that they did not pass the organization’s minimum threshold for bene5t. This issue is discussed further in Saaty [1] and Clayton et al. [15].
Other sources for AHP papers and examples include the EC, Inc. web site, www.expertchoice.com, Gass and Forman [8], ISAHP [16], ISAHP [17], Bodin [18], Liberatore and Nydick [19], Golden et al. [20], and this special issue of Computers & Operations Research.
4. Examples in the working paper
For a multi-criteria problem, unless an alternative is the best along all criteria, it should be clear that there is no one “optimal” solution. All multi-criteria solution methods yield a compromise
solution. It is a compromise in that the DM has traded-oB the weights of the criteria (and alternatives) based on the DMs understanding of the problem environment, experience, and even biases. These are captured by the AHP through the answers to the pairwise comparison questions. Other DMs with their own weights might select a diBerent solution.
The power of the AHP is that it provides a consistent, easily understood, analytically proven, and well-tested framework by which multi-criteria problems can be analyzed. Thus, in the examples brieAy described below and given in detail in the working paper, we give no answers. We have found these examples to be useful for classroom presentations, homework assignments and term projects. As noted earlier, the examples contained in the working paper can be obtained by email from the 5rst author at [email protected].
These examples are as follows:
• EX1 contains a simple direct comparison model for the purchase of a new automobile. The criteria and the alternatives are speci5ed.
• EX2 and EX3 are problems involving the integration of the ratings model version of the AHP with a resource allocation problem.
• EX4 contains an analysis of alternative income tax structures. The criteria to be used are not explicitly speci5ed. The student must determine a set of criteria and alternative tax strategies (over and above the tax strategies speci5ed in the example). This problem works well for teams of 3–5 students.
• EX5 is a problem of determining the best long distance telephone service. The student or team must collect data from the Internet, determine a set of criteria, and develop a set of alternatives for the associated ratings model.
• EX6 contains the analysis of the relative size of 5ve geometric 5gures. It is designed to validate the use of the fundamental 1–9 scale, i.e., to show that the 1–9 scale works very well in capturing a DMs intuitive pairwise comparison evaluations, in this case, for qualitative information. This validation example should be presented soon after AHP fundamentals and examples of the AHP are discussed.
5. Discussion
The teaching of the AHP is relatively straightforward and can be covered in about 2–3 weeks of class time in an MBA class. Because of these time restrictions, the instructor must be comprehensive, well organized, and have the EC examples worked out in advance.
Our experience has shown that the AHP is a winning topic for MBA students. The MBA students like the AHP, they easily learn how to use the AHP and, in many cases, they get very enthusiastic about the AHP. We often have to “rein-in” the students because they get so excited about the material. AHP should be a required topic for any introductory MBA course in decision-making.
Acknowledgements
The authors wish to thank Professors Bruce Golden and Ed Wasil for their insightful comments on this paper and for their interest and encouragement.
References
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Lawrence D. Bodin received his Ph.D. and M.S. at the University of California, Berkeley, and his A.B. degree in mathematics from Northeastern University. He is currently Professor Emeritus in the Robert H. Smith School of Business at the University of Maryland in College Park. Professor Bodin has consulted for numerous organizations including Federal Express, United Parcel Service and RouteSmart Technologies. Prior to joining the University of Maryland, Professor Bodin worked for IBM and Network Analysis Corporation and was an Associate Professor at the State University of New York at Stony Brook. Professor Bodin’s research areas include network analysis, vehicle routing and large-scale mathematical modeling.
Saul I. Gass received his B.S. in Education and M.A. in Mathematics from Boston University, and his Ph.D. in En-gineering Science/Operations Research from the University of California, Berkeley. He is currently Professor Emeritus at the Robert H. Smith School of Business, University of Maryland, College Park. He is a past president of the Operations Research Society of America (ORSA) and Omega Rho, the international operations research honor society. He has served as vice-president for international activities of the Institute of Operations Research and the Management Sciences (IN-FORMS). He was a 1995–1996 Fulbright Research Scholar. He is a recipient of ORSAs Kimball Medal for service to the society and the profession, INFORMSs Expository Writing Award, and the Military Operations Research Society’s Jacinto Steinhardt Memorial Award for outstanding contributions to military operations research. In 1998, he was a Distinguished Scholar-Teacher at University of Maryland. In January 2000, he was appointed Dean’s Lifetime Achievement Profes-sor at the Robert H. Smith School of Business. His research interests include linear programming, large-scale systems, model validation and evaluation, game theory, multi-objective decision analysis, and the application of operations research methodologies.