An Intelligent Tutoring System for AHP Alessio Ishizaka and Markus Lusti WWZ / Information Systems, University of Basel, 4003 Basel, Switzerland E-mail:
[email protected] Keywords. Intelligent Tutoring Systems, explanation, multi-criteria decisions, AHP, comparison matrix, consistency Abstract. AHP is a multi-criteria decision method widely used by non-experts. We have been implementing an Intelligent Tutoring System teaching the underlying theory, in particular the construction of consistent or near consistent AHP matrices, and the calculation of priorities and consistency indexes. AHP Tutor’s architecture supports adaptable exercises and dynamic explanations of solution paths. 1 Introduction The Analytic Hierarchy Process (AHP) is a multi-criteria decision making method, which proves its worth in many domains. End users frequently do not know its theoretical basis. This makes it difficult to exploit and critically assess its potential. To facilitate the understanding of the underlying basics, we have developed AHP TUTOR, an Intelligent Tutoring System implemented in Prolog. The following sections introduce Intelligent Tutoring Systems (ITS), describe the teaching objectives of AHP TUTOR and discuss selected implementation aspects. 2 Intelligent Tutoring Systems Intelligent (or knowledge based) Tutoring Systems try to cover aspects of the domain and pedagogic competence of human tutors. A complete ITS can be described by its expert, student, tutorial, and communication modules (Lusti,1992). Since non-trivial student and tutorial modules for complex domains are difficult to realise, we have been developing a series of tutors concentrating on the expert and communication modules, in particular on generaliseable explanation and authoring components (see for example Trapp 1998, and Lusti 1995). The tutors cover algorithmic domains from Operations Research (project planning, linear programming, and AHP), database design, and financial analysis. Table 1 summarises their domain independent features: Authoring component
Tutoring component
customizeable case knowledge (open number of text problems) customizeable domain knowledge customizeable explanation and help components authoring language to define traceable (explainable) procedures articulate expert modules protocol driven hypertext explanations contradiction learning extended hypertext
Table 1: Domain independent features of our problem solving monitors 1
AHP TUTOR concentrates on the expert module. The student module is simple and evaluates the student’s work during the problem-solving process by a straightforward performance model. The student has access to a next step only if the level of his/her knowledge is judged as being satisfactory. 3 Authoring Component The authoring subsystem allows the teacher to compose new case problems. Decision alternatives can be freely added, deleted and modified. The author can also select a comparison criterion, the maximum of the comparison scale, the tolerated error in the transitivity rule and the number of errors allowed to succeed in a particular exercise. On the basis of the author’s selections the wording of the exercise is composed. 4 Curriculum The curriculum describes the choice and the sequence of the teaching objectives. A student is enabled to identify comparisons, to build a consistent or near consistent comparison matrix, and to calculate priorities and consistency indexes. 4.1 Consistency a) Description Exercises allow the student to practice the following steps to reach a consistent or near consistent comparison matrix (Ishizaka 2002): Step 1 Comparisons on the principal diagonal set all decision alternatives against themselves. The resulting comparison values are trivial. Step 2 Independent comparisons are not linked to other comparisons by the transitivity rule (1) or the reciprocity rule (2). For pedagogical reasons it is best to choose the first diagonal above the principal diagonal for independent comparisons. Step 3 Transitive comparisons are deduced by the transitivity rule (1) from the first diagonal entered in step 2. Step 4 The reciprocal comparisons can be deduced by the reciprocity rule (2). (1)
ai,j = ai,i+1 · ai+1,i+2 · … · aj-1,j ±
e⋅h 100
(2)
1 a = ij a ji
where aij = comparison between alternative i and j h = highest value of the comparison scale e = tolerated error
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b) Implementation Each comparison has to be submitted to tests before being accepted (fig. 2). The input must be within the comparison scale and no steps must be skipped. Furthermore, the consistency rules (1) and (2) have to be respected. The comparisons of step 2 must not lead to step 3 values exceeding the comparison scale (see example below). Example: Suppose that the maximum height of the comparison scale is h = 9, the tolerated error is e = 20 % and the dimension of the matrix is n = 5. In the second step, the user enters successively a12 = 3, a23 = 2, a34 = 2 (fig. 1). According to the transitivity rule, the comparison a14 will be outside the 20 ⋅ 9 scale: a14 = 3 · 2 · 2 ± = 12 ± 1.8. 100 1
3
12±1.8
1
2 1
2 1 1
Fig. 1: The first diagonal leads to a step 3 comparison outside the scale. Input
outside of the comparison scale
Yes
Error (1)
Yes
Yes
comparison =1 Yes
main diagonal
No
No
valid diagonal and upper part
No
Yes
lower part
first diagonal
No
Yes matrix is complete
Error (2) Yes
comparison is induced outside the comparison scale
No No Yes
transitivity respected
Error (3)
No
Error (4) Write comparison
Error 1: Error 2:
No
reciprocity is respected Yes
No
Error (5) Yes
Error (6) No
Error (7)
The entered comparison is outside the scale. The comparisons on the main diagonal must be 1.
3
Error 3: Error 4: Error 5: Error 6: Error 7:
Since the entries of step 3 are linked to those of step 2 by the transitivity rule (1), each comparison of step 2 must not take a value which would lead to a step 3 value outside the comparison scale. The transitivity rule (1) is not obeyed. You skipped a step. The matrix is completed. The comparisons are frozen. The reciprocity rule (2) is not obeyed.
Fig. 2: Step by step consistency and associated error types Each rule transgression is followed by an explanation of one of the following types: •
A Stereotype-explanation is a static text, linked to a determinate piece of knowledge. It is used for simple, “flat” explanations, which cannot be hierarchically detailed.
•
A How-explanation justifies an answer by a top down tree from the requirements to the solution. The student can navigate freely through the entire tree, gradually looking for deeper explanations.
•
A What if-explanation or hypothetical reasoning gives a hypothetical solution under modified premises.
The explanation type depends on the type of error (tab. 2).
Error 1 Error 2 Error 3 Error 4 Error 5 Error 6 Error 7
Stereotype – explanation x x
How explanation
What explanation
x x
x
x x x
Table 2: Explanation and error types 4.2 Priorities a) Description Determining the priorities of decision alternatives in AHP is an eigenvalue problem. As soon as the comparison matrix is successfully filled the priorities can be calculated iteratively. The Power Method is composed by the following iterative steps: Step 1 Square the comparison matrix. Step 2 Calculate and normalise the row sums. Step 3 Using the resulting matrix, repeat steps 1 and 2. Step 4 Repeat step 3 until the difference between the the sums of two consecutive calculations is smaller than a (student-defined) stop criterion.
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b) Implementation The calculation of priorities follows the algorithm in fig. 3: A1=A02 FOR EACH row i pi1 = normalised sum of the row END FOR
(where A0 is the comparison matrix) (priority pi after the first iteration)
REPEAT Aj=Aj-12 FOR EACH row i pij = normalised sum of the row END FOR difference = 0 FOR EACH pij differencei = pij - pij-1 IF difference > differencei THEN difference = differencei END IF END FOR UNTIL difference > stop criterion
(initialisation: iteration j = 2) (priority pi after the j iteration) (error calculation)
Fig. 3: Algorithm of the priorities calculation For each iteration the student can ask the current values of the priorities and the difference between it and the value of the past iteration (in bold in fig. 3). How-explanations show how the priorities are calculated (fig. 4). Priorities after the 2. iteration are:
0,1 = 40,522/ 405.103
...
0,3 = 121,527/ 405,103
0,6 = 243.054/ 405,103
... 121,527 is the sum of the 2. line of the matix
405,103 is the sum of the vectorial elements
... 13,51 is the element (2,1) of the matrix
26,997 is the element (2,2) of the matrix
81,018 is the element (2,3) of the matrix
... Fig. 4: Extract of an explanation tree
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Fig. 5: Explanation in a dialogue window Because of the size of a typical explanation tree, navigation facilities have been implemented. A dialogue window can also be used for the navigation through the tree (fig. 5). A What if-explanation can be obtained by changing the stop criterion or the values of the comparison matrix. 4.3 Consistency Indexes a) Description The student has to calculate the Consistency Index (3) and the Consistency Ratio (4). (3)
C.I . =
λ−n n
(4)
C.R. =
C.I . R.I .
where C.I. = Consistency Index C.R. = Consistency Ratio R.I. = Randomize Index λ = highest eigenvalue n = dimension of the matrix b) Implementation We extract λ of the eingenvalue equation A·p = λ·p (5).
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n
(5)
λ=
∑a i =1
1i
⋅ pi
p1
where λ = eigenvalue a1i = comparison of element 1 and i pi = priority of element i A How-explanation and a What if-explanation are implemented for the calculation of the Consistency Index and the Consistency Ratio. 5 Help
The goal of our system is to propose exercise problems. The theory itself has to be acquired in another way, for example in class, from textbooks or by conventional computer assisted learning. AHP TUTOR offers a hypertext based help component. It contains the theory necessary to solve the exercises and has been implemented in DHTML. 6 Conclusion
This paper describes the implementation of an Intelligent Tutoring System for AHP. We have been using it to accompany conventionally taught AHP courses. Its explanation component has been proved very helpful. We will extend AHP TUTOR to handle additional problem types demonstrating the validity of the eigenvalue approach. 7 References
Ishizaka A., Lusti M., 2002. An Expert Module to Improve the Consistency of an AHP Matrix, The sixteenth triennial conference of the International Federation of Operational Research Societies, Edinburgh, (Downloadable from website http://www.wwz.unibas.ch/wi/members/Ishizaka/IFORS2002.pdf) Lusti, M.,1992.Intelligente tutorielle Systeme, R.Oldenbourg Verlag, München Lusti, M., 1995. An Authoring Component for Protocol Driven Hypertext Explanations, Proceedings of the 7th World Conference on AI in Education, Aug. 16-19, Washington, pp. 290-298 Trapp, U., Lusti, M. 1998. A Dynamic Link Library of Explanation Functions, In: Proceedings of the 4th International Conference on Computer Aided Learning and Instruction in Science and Engineering (CALISCE’98), Göteborg, pp.372-380
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