Computers ind. Engng Vol. 20, No. 3, pp. 355-365, 1991 0360-8352/91 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1991 Pergamon Press plc

ANALYTIC HIERARCHY PROCESS BASED ON DATA FLOW DIAGRAM

LIAN WANG 1 and TzuI RAZ 2 ~Corporate Research and Development, United Airlines, Executive Offices--EXOEB, P.O. Box 66100,

Chicago, IL 60666, U.S.A. 2IBM Corporation, Westlake Software Development Laboratory, 5 West Kirkwood Blvd., Roanoke,

TX 76299, U.S.A.

(Received for publication 4 December 1990)

Abstract--A new methodology is developed to assist one in the processes of selecting the best system configuration for implementing a computer-based system. Based on the Data Flow Diagram produced in System Analysis and System Design, an Alternative Evaluation Hierarchy is generated. System design criteria and characteristics of the system configuration alternatives are associated with the subsystems and the basic components in the system. The Analytic Hierarchy Process (AHP) is applied to identify the best system configuration. The use of objective measures in deriving the weights of the subsystems and the basic components in the AHP is proposed.

INTRODUCTION

The life cycle of a computer-based system consists of four basic phases: System Analysis, System Design, System Implementation and System Operation. When there is more than one system configuration available for implementation, conscious decisions are required as to which con- figuration should be chosen. Such decisions are crucial to the success of System Implementation and System Operation. For large scale systems, it is quite difficult to make such decisions because of the large number of subsystems and basic components involved and multiple characteristics associated with each configuration alternative. Some techniques are available to provide assistance in making such decisions. Examples include benchmarking [1], queuing models [2], simulation [3], scoring [4], and the analytic hierarchy process based on tasks and functional concerns (AHPTF) [5]. With the exception of the AHPTF, none of these techniques is capable of handling both tangible and intangible factors. The AHPTF incorporates all functional, operational and tech- nological factors, tangible and intangible, into a unified evaluation model. Unfortunately, it can not be easily integrated into System Analysis and System Design, especially for large scale systems.

During System Analysis and System Design, a tremendous amount of information concerning system requirements and system structure is generated. The information constitutes valuable input to the decision process. Ideally, the decision process should be integrated into System Analysis and System Design in such a manner that all important and relevant information generated is utilized and duplication of effort is minimized. Unfortunately, the decision process is often treated as a separate activity. Therefore, extra efforts to build evaluation models are required, and useful information may be lost. This paper presents a new methodology, the analytic hierarchy process based on data flow diagram (AHPDFD), to assist the decision maker in selecting the best system configuration from several alternatives. The A H P D F D combines a decision support tool, the Analytic Hierarchy Process, with a structured system analysis and design tool, the Data Flow Diagram.

The analytic hierarchy process (AHP) was first developed by Saaty [6] as a desicion support tool to assist one in describing the general decision operation by decomposing a complex problem into a multi-level hierarchic structure of objectives, criteria, subcriteria and alternatives. Eventually decision making is reduced to a set of pairwise comparisons, and the results are mathematically synthesized to obtain a ranking of the alternatives. It also provides a mechanism to enforce consistent judgments at the various levels of the decision hierarchy. To avoid unnecessary duplication, it is assumed that the reader is familiar with the AHP. Since its introduction in 1977,

355

356 LIAr~ WANG and TZVl RAZ

there have been controversies surrounding the AHP (e.g., [7] and [8]). However, the authors of this paper believe that, when applied properly, the AHP is a useful decision support tool.

The data flow diagram (DFD) is widely used in structured systems analysis and design [9, 10]. A DFD consists of symbols representing processes, data stores, and external entities (sources and destinations of data) connected by directed arcs representing data flows between them. Figure 1 (a) shows the standard symbols used to construct a DFD. Figure 2 shows a DFD that will be used later in an illustrative example. The DFD identifies the main components of a system and the relationships between them. The DFD's may be nested. Each process in the DFD may be a subsystem that can be exploded into a subordinate DFD.

The A H P D F D consists of three major steps. First, the D F D generated in System Analysis and System Design is mapped into a hierarchy, referred to as a data flow hierarchy (DFH). Then, the decision variables and the alternatives are incorporated into the D F H to produce an alternative evaluation hierarchy (AEH). Finally, the AHP is applied to the AEH to identify the best system configuration. These steps are described in detail and are illustrated by an example later in this paper. Also, the A H P D F D is compared with the AHPTF.

HIERARCHY AND SERVICE DIGRAPH

In this section, some relevant definitions are reviewed, and the concepts of service digraph and maximum service digraph are introduced.

Definition 1

An ordered set is a set S with a binary relation " ~<" (reads less than or equal to) that is:

reflexive: For all x, x ~< x; antisymmetric: If x ~< y and y ~< x, then x = y; transitive: If x ~< y and y ~< z, then x ~< z.

Data flow F

Entity E

Process P

S I Data store S

(a) Symbols used in data flow diagrams

Hierarchy H

Service DFD SD

(b) Additional symbols for data flow hierarchies

Fig. 1. Symbols of data flow hierarchy.

AHP based on DFD 357

D1 IF2

F4 ~ FFB ~ D2

F0 7 F9

Fig. 2. DFD of the automated quality control system.

I f x < ~ y a n d x ~ y , t h e n x < y . I f x < y and there is no t such that x < t < y t h e n y i s said to cover x. A simply or totally ordered set (also called a chain) is an ordered set with the additional property that if x, y e S then either x ~< y or y ~< x, otherwise S is partially ordered. For any element x in an ordered set, the set {ylx covers y} is denoted by x - and the set {YlY covers x} is denoted by x +. The elements in x - are called the descendants of x and the elements in x + are called the ancestors of x.

Definition 2

Let H be a finite partially ordered set with the largest element b. H is a hierarchy if it satisfies the following conditions:

(I) There is a partition of H into sets Lk, k = 0 . . . . . h, where L0 = {b}; (2) X ELk implies x - ~ Lk+l, k =O . . . . . h - l ; (3) x ~ L k implies x + ~_ L k_ l, k = 2 . . . . . h.

Definition 3

A directed graph (or digraph) consists of a set of nodes Y = {v,, v2 . . . . }, a set of arcs E = {et, e2 . . . . }, and a mapping of every arc onto an ordered pair of nodes (vi, vj). A digraph g is a sub-digraph of a digraph G if all the nodes and all the arcs of g are in G.

A digraph may be represented by a incidence matrix [11]. The incidence matrix of a digraph with n nodes, e arcs and no self-loops is an n by e matrix A = (aij) such that

aij = 1 if j t h arc is coming out from ith node; aij = - 1 if j t h arc is going into ith node; a~j = 0, otherwise.

Definition 4

A walk is defined as a finite alternating sequence of nodes and arcs. No arc appears more than once in a walk. A node, however, may appear more than once. For example, Fig. 3 shows a digraph and its incidence matrix. (vt, es, v3, es, v6) is a walk between vl and v6.

A D F D may be viewed as a digraph with data flows being the arcs, and external entities (sources and destinations), processes and data stores being the nodes. Henceforth, D F D and digraph will be used interchangeably. I f there is a data flow from v~ to vj, then it is said that v~ serves vj. Each node serves or/and is served by other nodes. I f there is a walk from node v~ to vj, then it is said that v, contributes to vj.

Definition 5

A service digraph g = { Vs, Us} of node vm in digraph G, is a subgraph of G, which satisfies the following conditions.

(1) v,. e vs; (2) if v, ~ Vs then v~ appears in a walk to v,, in G; (3) if u~ ~ Us then ui appears in a walk to vm in G.

CA[E 20/3--E

358 LIAN WANG and Tzv] RAZ

e I

e 2

e 4

e 1 e 2 e 3 e 4 es e 6 e 7 e a

V 1 +1 0 0 +1 +1 0 0 0

V 2 -1 +1 0 0 0 0 0 0

V 3 0 -1 +1 0 -1 +1 0 +1

V 4 0 0 -1 -1 0 0 0 0

V s 0 0 0 0 0 - I + I 0

V 6 0 0 0 0 0 0 - I - I

Fig. 3. Digraph and incidence matrix.

Definition 6

The maximum service digraph g = { Ymax, Umax } of node Vm in digraph G is a sub-digraph of G, which satisfies the following conditions:

(1) Vm~ Vmax; (2) vi ~ Vm~x if and only if vi appears in a walk to vm in G; (3) u~ e Um~x if and only if u~ appears in a walk to vm in G.

Figure 4 shows the maximum service digraph of v6 and its incidence matrix. The incidence matrix of the maximum service digraph of v~ may be obtained from the incidence

matrix of G by repeating the following two steps until no changes result from the previous iteration.

(1) Examine the row for each v~ ~ Vm in the incidence matrix. If only - l(s) appears in the row then - l ( s ) is substituted by 0(s).

(2) Examine each column in the incidence matrix. If only + l(s) appears in the column then + l(s) is substituted by 0(s).

If G is the digraph corresponding to a DFD, then the DFD corresponding to g is called a service DFD of Vm, denoted by S(vm) If g is the maximum service digraph of vm, then its corresponding DF D is called the maximum service DFD ofv~, denoted by MS(vm). The node v~ is called the master node of S(vm) and MS(vm).

In order to apply the AHP, a DFD needs to be mapped into a hierarchy referred to as data flow hierarchy (DFH). The procedure to build the D F H is presented in the next section.

(1)

(2)

ALGORITHM FOR BUILDING THE DFH

Set k = 0. Let L0 = {b }, where b represents the system being designed;

Set k = k + 1. Find the maximum service DFDs for all the external entities. The maximum service DFD of an external entiry represents the part of the system that serves the external entiry. Such maximum service DFDs form the second level of the DFH.

AHP based on DFD 359

e8

08

e 1 e 2 e 3 e 4 e s es e 7 ee

V 1 +1 0 0 0 +1 0 0 0

V 2 -1 +1 0 0 0 0 0 0

V 3 0 -1 0 0 -1 +1 0 +1

V 4 0 0 0 0 0 0 0 0

V s 0 0 0 0 0 -1 +1 0

V e 0 0 0 0 0 0 -1 -1

Fig. 4. Maximum service digraph ofv6 and incidence matfix.

(3) Set k = k + 1. Construct level k with processes and data stores in the maximum service DFDs of level k - 1.

(4) Set k = k + 1. For each process of level k - 1 that needs to be decomposed further, explode it into a more detailed DFD. The set of these more detailed DFDs form the level k.

Repeat steps (3) and (4) until no node in the system needs to be decomposed further. The nodes with no descendants in the D F H are called basic components. Figure 1 shows the symbols that may be used in the representation of the DFH.

Figure 5 is an example of a three level DFH. At the level zero of the DFH, H represents the entire system. There are two external entities E 1 and E2. Their maximum service DFDs, MS(E 1) and MS(E2), form the level one. H - = {MS(E1), MS(E2)}. Entities E1 and E2, processes P1 and P2, and data stores D1 and D2 form the level two. MS(E1)- = MS(E2)- = {El, E2, P1, P2, D1, D2}. The performance of the system depends upon the performances of the subsystems MS(E1) and MS(E2) whose performances in turn depend on those of El , E2, P1, P2, DI and D2.

D2

Fig. 5. The data flow hierarchy of the automated quality control system.

360 LIAN WANG and Tzvl RAZ

BUILDING ALTERNATIVE EVALUATION HIERARCHY

To associate the decision variables (design criteria) and system configuration alternatives with the subsystems and the basic components, the AEH is created. The AEH is obtained by adding more levels to the DFH. Let B denote the set of basic components. Let A denote the set of the system configuration alternatives under consideration. The AEH may be built following the three steps below.

(1) For each x e B, select decision variables dl, d2 . . . . . dk(x) and let x - = {dl, d2 . . . . . dk(x)}. Let B = { B - x } U x -

(2) Repeat step (1) until no decision variables need to be further decomposed. (3) For each decision variable x e B, let x - = A.

Once the AEH is obtained, it is treated as a standard decision hierarchy in the AHP. The local weights of the nodes at each level of the hierarchy are obtained from the results of a series of pairwise comparisons between the nodes. The results of the pairwise comparisons are organized in so called comparison matrices. The global weights are synthesized from the local weights to rank the alternatives.

In the standard AHP, the entries in the comparison matrices are based on the judgements of the decision maker. A quantity referred to as a Consistency Ratio is used to measure the consistency of the evaluation process. The authors believe that the efficiency and the consistency of the A H P D F D may be improved if the weights are derived from objective measures instead of subjective judgements. The next section presents an approach to deriving the entries of the comparison matrices based on the amount of information contributed by the various nodes being compared.

OBJECTIVE EVALUATION OF NODE WEIGHTS

The weight of a node in the AEH should reflect the contribution of the node to the entire system. The amount of information that a node provides to the rest of the system is a measure of the node's contribution. The more a node contributes to the system, the larger its weight should be. Shannon's measure of information amount is probably the most widely used, mainly because of its attractive mathematical properties. In this section a method for deriving node weights based on Shannon's measure of information amount [12] is proposed. It should be noted that it is possible to develop other objective measures to quantify node contributions. The objective evaluation of the node weights not only improves the consistency of the decision-making process but also reduces the judging workload on the part of the decision maker.

A data flow may be considered as a random variable whose sample space is the set of data passing through it. The sample space and the distribution of the random variable are determined by the characteristics of the system and should be known or can be approximated at the end of System Analysis and System Design. The average number of data units (records) passing through data flow F per time unit is referred to as the intensity of data flow F, denoted by N(F). If P~ denotes the probability that value i is observed, i = 1,2 . . . . . m, then the entropy of data flow F, H(F), is defined as

H(F) = - ~ Pi log P~. (1) i=1

Definition 7 F~, i = 1 . . . . . m, is a data flow leaving node u in a service digraph S(v), with intensity N(F~).

The sample space of Ft consists of data units 1, 2 . . . . . and n(i) appearing Nt~, N,2 . . . . . and N~n~0 times, respectively. The information contribution of node u to S(v) is defined as:

m n(i)

H(u) = - 2 2 Pq log P,j (2) i = l j = l

where Pij = Nifl[N(F~) + N(F2) + " " + N(Fm)].

A H P b a s e d o n D F D 361

Definition 8 Consider a service digraph S(u). Fi, i = 1 . . . . . m, is a data flow going into node u with intensity

N(Fi). The sample space of F~ consists of data units l, 2 . . . . . n(i), appearing N~I, N,~, . . . , N~n(0 times respectively. The information contribution of service digraph S(u) is defined as:

m n(i)

n[S(u)] = - ~ ~ P0 log P~j (3) i = l j = l

where P0 = No/[N(Ft) 4- N(F2) 4- " " 4- N(Fm)] With the objective measures introduced here, each entry in the comparison matrix may be

calculated as the ratio of the information contributions of the two nodes being compared. One may have noticed that the above objective measures are not sensitive to the relative importance of the data flows. To remedy this, weighted information contribution may be introduced similarly to the weighted entropy [13].

A N E X A M P L E

In this example, the task is to select the most suitable microcomputer system for an automated quality control system. The automated quality control system is designed to inspect items produced, collect data on a lot basis, calculate lot fraction nonconforming, compare to control limits, generate messages to notify the production department of the inspection results, and record the inspection history.

Table 1 shows the characteristics of the four candidate microcomputer systems, A, B, C and D. The data are adopted from Arbel and Seidmann [5]. The decision variables appear in the first column of Table 1 followed by the four alternatives in the columns two through five. The DFD of the automated quality control system appears in Fig. 2. Each entity, process, data store, and data flow is described in Table 2. For simplicity, it is assumed that, for each data flow, record occurrences are uniformly distributed. It is further assumed that the size of the sample space of each data flow is equal to the intensity of the data flow.

The selection problem is solved by both the AHPDFD and the AHPTF, and comparisons between the two methods are made. The software Expert Choice [14] is used to build the decision hierarchies and to perform the computations on an IBM PC microcomputer.

Following the steps of the AHPDFD described in previous sections, the DFH shown in Fig. 5 is obtained. The root node of the DFH represents the whole system. There are two external entities, E1 and E2. Their respective maximum service digraphs form the level one. Each of them is further decomposed into the component nodes at the next level. In this example, all the processes and data stores of the system appear in both of the maximum service DFSs. Thus, MS(E1)- = MS(E2)- = (El, E2, P1, P2, D1, D2}. Since the system is quite simple, there is no need to carry on the decomposition further.

The AEH shown in Fig. 6 is obtained by adding three levels to the DFH. The first level added consists of seven decision variable groups--CPU parameters, main memory, auxiliary storage, interface, software, development tools and special features. Each decision variable group is decomposed into more detailed decision variables at the next level. Finally, the four alternatives under consideration form the last level.

The third step involves entering the values of the pairwise comparison matrices at all the levels of the AEH, and applying the standard AHP procedure to calculate local and global weights. The pairwise comparisons at the three lowest levels of the AEH are based on the subjective judgements of the decision maker. At levels one and two, the entries of the comparison matrices are derived based on the information contribution defined earlier. For example, the comparison matrix at level two of the AEH in Fig. 6 with respect to MS(E1) and H is the following:

El E2 D1 D2 P1 P2

El 1.0 0.6 1.0 0.7 0.6 0.6 E2 1.0 1.7 1.2 | .0 1.2 D1 1.0 0.7 0.6 0.6 D2 1.0 0.8 1.1 P1 1.0 1.1 P2 1.0

362 LIAN WANG and Tzvl RAZ

Table 1. Decision variables and the characteristics of the alternatives

Alternative systems

Decision variables A B C D

CPU parameters Word size 8 8 8 16 Repertoire size 38 42 28 72 Clock cycle time ~ sec) 0.12 0.15 0.29 0.08 General purpose registers 8 4 28 [6 Interrupt modes Vectored Vectored & Fast & Vectored

nonvectored normal Direct addressing (K Bytes) 64 64 48 800 Addressing modes 6 10 8 14

Main memory Present capacity (K Bytes) 32 48 32 256 Maximum capacity (K Bytes) 64 48 48 500 Storage allocation Good Fair Very good Very good

Auxiliary storage Capability (K Bytes) 280 320 720 320 Data transfer rate (KBits/sec) 25 30 54 30 Average access time (msec/record) 175 175 310 250 DOS Vendor's CPM & Vendor's & CPM &

vendor's partial CPM vendor's Interface

Analog conversion Very good Very good Good Fair DMA A~ ailable Available Available Available Serial/parallel Moderate Slow Moderate Fast Synchronous/asynchronous Available Not available Available Available Number of ports 16 12 24 10

Software/Language System utilities Few Many Many Moderate Applications packages Need to develop Good Very good Need to develop Type* A,PL,F,B, A,PL,CB,B A,F,B A,PL,PS,C Relative capability Moderate Fair Very good Good

Development tools Cross assemblers Not available Available Available Not available Software simulators Very good Good Good Fair Hardware emulators Good Fair Very good Excellent Development systems Excellent Very good Good Fair

Special features Many/important Few/important Moderate/important Few/important

*A, Assembler; B, Basic; C, C; CB, Cobol; F, Fortran; PS, Pascal; PL, PL/M.

Table 2. Description of entities, processes, data stores and data flows

Intensity Symbols Description (recs/hr)

El Quality control department, receives records from quality control process P2 via D2 and sends analysis results to D2.

E2 Production department, sends items produced to the testing process PI and generates BOL (Beginning Of Lot), EOL (End Of Lot), BOB (Beginning Of Batch), and EOB (End of Batch) signals by interrupts.

DI The data store in which item specifications and quality standards are stored. D2 Quality history of items and analysis results obtained by quality control department. Pl Measures an item automatically and sends the digital testing results to P2. P2 Calculates fraction defective for a lot based on the testing results from Pl, compares it

with control limits, generates appropriate messages to production department, and writes quality history.

FI Specifications and quality standards of items to be inspected. 1000 F2 Acceptable fraction defective for a lot of a specific kind of items. 10,000 F3 Analog signal carrying the raw data of items with a unique item code. 10,000,000 F4 Testing results. 10,000,000 F5 Records from quality control process P2. 1,000,000 F6 Same as F5. 1,000,000 F7 Analysis results obtained by quality control department. 10,000 F8 Messages being sent to production department. 1,000,000 F9 Same as F7. 10,000

Entry (3, 4) corresponds to the relative weight of D1 with respect to D2, and is calculated as H(D1)/H(D2) -- log[N/(F1) + N(F2)]/Iog[N(F9)] = log(1000 + 10,000)/log(l,000,000) = 0.7. The rest of the entries in the matrix are calculated in the similar manner. It is obvious that entry (i,j) is the reciprocal of entry ( j , i). Since the comparison matrices are quite voluminous, they are not reproduced here.

AHP based on DFD 363

For the purpose of comparison, the AHPTF model is built as shown in Fig. 7. It was obtained by simplifying the AHPTF model developed by Arbel and Seidmann in [5]. In this example, the lowest three levels of the AHPTF model are as same as the lowest three levels of the A H P D F D model.

The final results obtained by both the A H P D F D and the A H P TF are summarized in Table 3. The overall ranks of the candidate systems obtained by both approaches are the same. The consistency ratio of the A H P D F D hierarchy is slightly lower than the consistency ratio of the A HP TF hierarchy. The number of pairwise comparisons required by the A H P D F D is about 50% less than the AHPTF.

From this example, it can be seen that the A H P TF treats a system as a collection of tasks and functions while the A H P D F D treats a system as a collection of subsystems, and basic components. It is possible to combine the A H P D F D with the AHPTF. One can decompose a complex system into simpler subsystems, and then decompose the subsystems into tasks, functions, and so on.

Generally speaking, as the scale of the problem increases the number of nodes in an AHPTF model does not increase while the number of nodes in an A H P D F D model does. For different problems, the AHPTF models are almost identical with major system tasks at level one, functional concerns at level two, decision variables at level three, and alternatives at level four. Each node at each level is defined by Arbel adn Seidmann. The A H P D F D models (or AEHs) are problem specific. Every A H P D F D model is specially tailored for a specific system. The more nodes in a DFD, the more nodes in the corresponding A H P D F D model.

In order to obtain the comparison matrices, the AHPTF requires subjective pairwise com- parisons at every level. Subjective pairwise comparisons at higher levels can be more difficult to make than at lower levels because of the lack of knowledge about the system. The A H P D F D only

Ioara e 'ersll Main I memory

I Word size ] Present cap. ] Reper. size I Maximum cap.

ICycle time I Storage a oc. I IG-P registers I

I Ilnter. modes I IDirect addres. I IAddres. modes I

I Auxiliary storage

J Capability Data tran.

Rate Ave. access

Time DOS

I I n*e'a°e I J

Analog conv. DMA Seral/paral Synch./asnch. # of ports

+ P Im

r

I Sys. utilities Applications Type Relative capability

L

1o2

Develop- Special ment tools features

J Cross assem. I Software sim. I

IHardware emu. I I°eve'°0 s's i

Fig. 6. The AHPDFD alternative evaluation hierarchy for the microcomputer selection problem.

364 LIAN WANG and TzvI RAZ

Select the most suitable

micro-computer

__i__ J J

mPrn°tCoe~Sg ] [processing ] [ C°rcrtec~i:e ] [ - - Data

] [ - - I _ _ I Data

conversion

I

] l - - Main

memory

Reliability

I I _

Computational] ability J

f

Programm- ability

CPU parameters

flex.

1 ] 1 I

Configuration ]

l _ _ Dale

logging

L 1 Capacity

1[ --are/ [ Oeve o0 [ S0eca Auxiliary Interface storage language ment tools features

[ I I ] ] I wor~size Present cap. 1 Capab,,y Anaiog~nv. Sys. ut,ities ICrossassem. 1 Reper. size Maximum cap. I Data tran. DMA Applications [Software sim. [ Cycle time Storage alloc. ] Rate Seral/paral Type [Hardware emu. / G-P registers I Ave. access Synch./asnch. Relative IDevelop. sys. Inter. modes / c " ~ Direct addres. DOS Time # of ports ddres.~.~ modes

Fig. 7. The AHPTF decision hierarchy for the microcomputer selection problem.

requires subjective pairwise comparisons at the lower levels where more detailed information concerning the system is available. In the example, the AHPTF requires subjective pairwise comparisons at every level while the AHPDFD requires subjective pairwise comparisons only at levels three, four and five. The comparison matrices at the higher levels of the AHPDFD model are obtained based on the information contributions of different subsystems and basic components in the system.

With the AHPTF, to make decisions with respect to subsystems, separate models have to be built, one for each subsystem. Also, it is difficult to synthesize the subsystem results to make decisions with respect to the whole system. The AHPDFD eliminates the need to build a separate model for each subsystem. All one has to do is to "cut off" the sub-hierarchies corresponding to the subsystems. To synthesize the subsystem results, one can simply put the nodes that represent the subsystems back into the original hierarchy and proceed with the standard AHP. For the example in this paper, if the task is to select one computer for P1 and one for P2, the AHPTF requires two models identical to the one in Fig. 7 except that the symbols for the node at level zero are P1 and

Table 3. Summary of the results

AHPTF AHPDFD

Global weight of: Alternative A 0.336 0.337 Alternative B 0.288 0.299 Alternative C 0.195 0.186 Alternative D 0.181 0.178

Computing time (sec ) 465 236 Consistency ratio of the hierarchy 0.03 0.02 Number of pairwise comparisons 5850 2923

AHP based on DFD 365

P2, respectively, instead o f H. With the A H P D F D , one can simply delete all the ancestors o f P1 and P2 to obtain P1 and P2 sub-hierarchies.

CONCLUSIONS

The A H P D F D combines the A H P with the D F D to provide a new approach to decision making in computer-based system design. The A H P D F D has the following advantages.

First, it allows the decision maker to decompose a large and complex system into smaller and simpler subsystems and basic components . The decision making process is integrated into System Analysis and System Design rather than treated as a separate activity. Useful informat ion is incorporated into the evaluation model and the efforts to build the evaluation model are reduced.

Second, it introduces objective measures for deriving the weights o f the nodes in the higher levels o f the evaluat ion model. Subjective pairwise compar isons are only required at the lower levels where experts with better informat ion are often available. Therefore, better decision making may be achieved.

Finally, there is no need to build separate models for subsystem decision making. When necessary, it is easy to synthesize the subsystem results to make decisions with respect to the whole system.

REFERENCES

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(1984). 6. T. L. Saaty. The Analytic Hierarchy Process, McGraw-Hill, New York (1980). 7. J. S. Dyer. Remarks on the analytic hierarchy process. Mgmt Sci. 36, No. 3 (1990). 8. T. L. Saaty. An exposition of the AHP in reply to the paper "Remarks on the analytic hierarchy process", Mgmt Sci.

36, No. 3 (1990). 9. C. P. Gane and T. Sarson. Structured Systems Analysis. Prentice-Hall, Englewood Cliffs, NJ (1979).

10. T. DeMarco. Structured Analysis and System Specification, Prentice-Hall, Englewood Cliffs, NJ (1979). 11. N. Deo. Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall, Englewood Cliffs, NJ

(1974). 12. C. E. Shannon. The Mathematical Theory of Communication, University of Illinois Press, Urbana (1959). 13. I. Csiszar and J. Korner. Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic Press, New

York (1981). 14. Decision Support Software, Inc. Expert Choice, Decision Support Software Inc., McLean, VA (1985).