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A fuzzy multicriteria group decision making approach to select configuration items for software development
Fuzzy Sets and Systems 134 (2003) 343 – 363
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A fuzzy multicriteria group decision making approach to select con%guration items for software development Juite Wanga; ∗ , Yung-I Linb a
Department of Industrial Engineering, Feng Chia University, P.O. Box 25-097, Taichung 407, Taiwan, Republic of China b Aeronautical System Research Division, Chung-Shang Institute of Science and Technology, P.O. Box 90008-11-21, Taichung 40722, Taiwan, Republic of China Received 6 February 2001; received in revised form 5 April 2002; accepted 7 May 2002
Abstract Selection of con%guration items in software con%guration management is important to determine the software quality and reduce the development time and cost. The objective of this research is to develop a multi-criteria group decision making model based on fuzzy set theory to improve the con%guration items selection process. Since most information available in this stage is not numerical, fuzzy set theory is used to represent the evaluation ratings of candidate items. The developed model ranks candidate items into partial or complete orders that can assist decision makers in selecting more proper set of con%guration items. The consensus measures are also developed to determine the group acceptability of the obtained ranking orders. In addition, sensitivity analysis can be performed to examine the solution robustness. An example of 7ight simulator development project is used to illustrate the concept developed. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Decision analysis; Multiple criteria evaluation; Group decision-making; Con%guration management; Software development
1. Introduction The increasing dominance of computers in industry has made software become more important in the business decision-making. More human decision-making functions are embedded into the software. However, due to the changing environment of today’s marketplace, changes of requirements or speci%cation frequently occur throughout the life cycle of the software development project. Moreover, unlike manufacturing hardware products, it is di@cult to develop a set of standards and ∗
Corresponding author. Tel.: +886-2451-7250, ext. 3636; fax: +886-2451-0240. E-mail addresses: [email protected] (J. Wang), yung [email protected] (Y.-I. Lin).
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 2 8 3 - X
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363 Flight simulator
Cockpit system
Control loading system
...
...
...
nv1
...
nv2
Host computer system
Navigation/ communication task
...
nv3
nv4
I/O control system
Sound&graphic system
...
...
...
nv5
...
Radar task
nv6
Visual display system
... ...
nv7
Instructor operation system
... ...
nv8
nv9
Fig. 1. The work-breakdown structure for a 7ight simulator project.
statistics that allows people to measure the eHectiveness or quality of software until after it is built. Therefore, a systematic method is required to develop a large-scale software product to better control changes and uncertainties occurred in the development process. Con%guration management [6,7,8,15,16,26] is a key element in developing a large-scale software product. Its purpose is to maintain the integrity of products as they evolve throughout the entire system life cycle, from concept, through design, development, production, and maintenance. Con%guration management is important for managing the modern software project, because it provides stability to the production of a software system by controlling the product evolution. In addition, the entire product life cycle cost can be reduced by proper con%guration management. There are four essential functions for con%guration management [6,8,17,26]: con%guration identi%cation, con%guration control, con%guration status accounting, and con%guration audits and reviews. The purpose of con%guration identi%cation is to select a proper set of con%guration items. A con%guration item (CI) is de%ned as an aggregation of hardware or software whose functions and performance parameters must be de%ned and controlled to achieve the overall performance of a system or product. A special attention will be given to the selected con%guration items in the entire product life cycle. All physical and functional characteristics necessary to de%ne a CI throughout its life cycle should be documented and all documentation should include relevant information on change and traceability. The work breakdown structure (WBS), which is a family tree of project elements, has been usually developed by the product development teams, before entering the stage of con%guration identi%cation. It de%nes the various hardware, software, and service systems and their subcomponents that make up the speci%ed project or program. An example of WBS for a 7ight simulator project is shown in Fig. 1. Improper selection of CIs will result in the poor product quality and increase the development time and cost [8,16,26]. Selecting too many CIs will increase the cost of management control; e.g., developing related con%guration documentation for requirements, design, manufacturing and veri%cation. On the contrary, selecting too few or wrong items as CIs runs the risk of too little control and creates logistics and maintenance di@culties, due to the lack of management visibility. Ideally, the selection of CIs should be with a full view of the product life cycle cost and management
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impacts. However, due to only the incomplete software development information available at the early stage of product development, it is very di@cult to perform the quantitative cost=bene%t analysis to predict the impact of selecting a component as a CI on the entire product life cycle cost. In practice, a con%guration management board that involves members from various life cycle perspectives is organized to determine a proper set of CIs from hundreds or thousands of candidate items. A set of criteria is determined to help the selection; for example: security criticality, human safety criticality, mission criticality, %nancial criticality, design newness, interface criticality, hardware integrity, status sensitivity, technology sensitivity, etc. Some criteria are more signi%cant than others. Each candidate item is evaluated by a group of experts in the con%guration management board and the %nal decision is based on their experiences and coordination. However, evaluating and comparing a large number of candidate items is a cumbersome work in the CI selection process. The objective of this research is to develop a methodology to assist decision makers in selecting a proper set of CIs in con%guration management. The CI selection problem is modeled as a multi-criteria group decision making problem [10]. Fuzzy set theory [12] is used to represent the evaluation results of candidate items, because most information available at this stage is subjective and imprecise and is usually expressed in a nature language by individual experts. Once these individual evaluation results are collected, fuzzy set theory would permit to perform the tradeoH analysis among various criteria. A fuzzy multi-criteria group decision model based on the concept of fuzzy preference relation [19] and fuzzy majority [9,11] is developed to rank candidates into a partial order or a complete order according to their importance. Two consensus measures are also developed to determine the mutual agreements about the chosen CIs among experts. If the consensus is reached, then a candidate item with a higher ranking order will has higher priority to be chosen as a CI. Several standards for con%guration management [6,7,8,16,17] only provide guidelines or selection criteria for selecting CIs in the con%guration identi%cation stage. We have also found that most research in con%guration management is to develop a computerized con%guration management system that is focused on version=revision control in software development [15,21,30]. Recent research developed various approaches for managing the con%guration management of the emerging development paradigm; such as the object-oriented [25], component-based [30] or distributed software development [20]. There is little research in developing a decision support system for con%guration management, especially in the con%guration identi%cation stage. This is because the information available in this stage is mostly qualitative and is di@cult to be used for developing quantitative models for decision-making. This paper is organized as follows. Section 2 provides a formal description of our research problem. Next, a multi-criteria group decision making model to select a set of CIs is proposed in Section 3. In Section 4, an example of a 7ight simulator software project is used to illustrate the developed concept. The data used in the example is presented in Appendix. Finally, Section 5 concludes this paper. 2. Problem description The impacts of selecting a CI on the software quality and life cycle cost are di@cult to predict in practice. Therefore, we consider the CI selection problem as a fuzzy multi-criteria group decisionmaking problem in this paper. The research problem is described next.
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A set of candidate items
Evaluate candidate items by a group of experts
Evaluation criteria
Ratings of candidate items Rank candidate items by the multi-criteria group decision making model Partial or complete orders of candidate items Determine consensus Degree of consensus N
Is consensus reached? Y Select configuration items
Configuration items Fig. 2. The decision process to select con%guration items.
Let A be the set of n candidate items below the subsystem level of WBS under evaluation by m experts, according to a set C of q criteria. Let G t (a) = [g1t (a); g2t (a); : : : ; gqt (a)] be the evaluation result of candidate item a∈A by expert e t ∈T , where the function gkt (a) represents its rating for criterion ck ∈C. The objective of this research is to develop an approach to assist decision makers in selecting a proper set of CIs. Fig. 2 shows the CI selection process. Due to imprecise and subjective project information available in the early stage of software development, it may be di@cult to assess candidate items with certain values for some criteria. For example, to evaluate the candidate item “nv3 ” under the “Navigation=communication” task shown in Fig. 1 with the criterion “mission criticality”, the answer may not be either “yes” or “no”, but in between. The damage of this item may not lead to the total failure of the mission, but within a certain degree depending on the situation. In this paper, the rating of each criterion is described by some linguistic terms, which provide a more natural way to express the rating for assessing candidate items. These linguistic terms can be manipulated with fuzzy set theory [12] and they can be interpreted as speci%c fuzzy numbers that allow us to perform the tradeoH analysis among various criteria. An example of a linguistic scale consisting of seven linguistic values {“de%nitely not”, “probably not”, “maybe not”, “unknown”, “maybe”, “probably”, “de%nitely”} is shown in Fig. 3. The strongest agreement for a criterion is given the highest (lowest) linguistic value “de%nitely” (“de%nitely not”) in the scale.
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probably probably unknown definitely not maybe maybe definitely not not
µ(x)
1.0
0
1
2
3
4
5
6
7
8
9
10
x
Fig. 3. The linguistic scale for rating candidate items.
3. Fuzzy group decision making approach to select con guration items This paper considers the con%guration item selection problem as a multi-criteria group decisionmaking problem. A group decision-making problem is characterized by a set of alternatives and a set of experts who provide their judgments on these alternatives. The objective is to %nd an alternative or a set of alternatives that is best acceptable by the group of experts. However, criteria for selecting alternatives are not considered explicitly in the past research of group decision making models. Fuzzy set theory has been applied in the group decision-making problems for a long time [10]. Most of fuzzy group decision-making models developed in the literature were based on the fuzzy preference relations [9,10,11,19,22] that can be obtained by pairwise comparison between various alternatives by experts. The fuzzy preference relation is de%ned next. De nition (Fuzzy preference relation [19]): A fuzzy preference relation R on a set A is a fuzzy set on the product A × A, such that R : A × A → [0; 1]. Let P(a; b)∈R be the fuzzy preference relation between a and b, where a; b∈A. Then P(a; b) and P(b; a) are reciprocal, i.e., P(a; b) + P(b; a) = 1:
(1)
Let A be a set of candidate items and a; b∈A. P(a; b) = 1 indicates the maximum degree of preference of item a over b. P(a; b)∈(0:5; 1) indicates a de%nite preference of alternative a to b with the intensity of preference corresponding to the value of P(a; b). Note that the higher value means a stronger intensity of preference. P(a; b) = 0:5 indicates the indiHerence between candidate items a and b. According to the obtained fuzzy preference relations, either direct or indirect approaches can be used to derive a solution set [9,19,22]. A direct approach determines a solution directly from the set of individual fuzzy preference relation. An indirect approach derives a solution from a social fuzzy preference relation that can be obtained by aggregating a set of individual fuzzy preference relations. Since it is not necessary to achieve a full consensus in practice for a group decisionmaking problem, Kacprzyk [9] developed the concept of soft majority (e.g., “Most”) based on the fuzzy linguistic quanti%er [29] or the OWA aggregation operator [27] for dealing with this issue.
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Based on this concept, Kacprzyk and Fedrizzi [1,11] also developed a model of soft consensus to measure the consensus reached by most of the experts. Moreover, Herrera et al. [2] extended the numerical representation of the fuzzy preference relation with linguistic labels and developed the linguistic OWA operator to determine the collective linguistic preference from a set of individual linguistic preferences. The linguistic non-dominance degree and several linguistic consensus degrees were also determined to solve the group decision-making problem [3]. They also extended their models to the heterogeneous group decision-making problem [4,5]. Marimin et al. [14] also represented the fuzzy preference relation with linguistic labels and further characterized them by fuzzy numbers [12]. Subsequently, the arithmetic operations of fuzzy numbers and the extension of the neat OWA operator [28] were used in the direct and indirect approaches to solve the problem and to determine the degree of consensus among experts. In this paper, the proposed methodology that is adapted from Kacprzyk [11] is divided into three stages: (1) construction of fuzzy preference relations between candidate items; (2) determination of a partial order or a complete order of con%guration items under evaluation; and (3) determination of the consensus measures among experts. 3.1. Construction of fuzzy preference relations In this paper, we apply the fuzzy preference relation to model imprecise preference relations between diHerent candidate items. To reduce the cumbersome work of assigning degrees of preference between diHerent candidate items by a decision maker (about qn(n − 1)=2 times), in this paper, the fuzzy preference relation between two candidate items a and b with respect to criterion k is obtained by pair-wise comparison of linguistic evaluation ratings gkt (a) and gkt (b), which can be characterized by fuzzy numbers. The linguistic evaluation ratings are provided by an experienced decision maker (about qn times). There are several methods in the literature to determine the fuzzy preference relation between two fuzzy numbers, such as: Nakamura [18], Kolodziejczyk [13], Tseng and Klein [23]. This paper utilizes Tseng and Klein’s approach based on Hamming distance to derive the fuzzy preference relation between two candidate items for each criterion [24], because of the computational e@ciency of their approach. De nition. Let a and b be two fuzzy numbers. The fuzzy preference relations P(a; b) and P(b; a) are de%ned as follows [23]: P(a; b) =
D(a; b) + D(a ∩ b; 0) ; D(a; 0) + D(b; 0)
(2)
P(b; a) =
D(b; a) + D(a ∩ b; 0) ; D(a; 0) + D(b; 0)
(3)
where D(a; b): D(b; a): D(a; 0): D(b; 0):
Area Area Area Area
where a dominates b (none in Fig. 4(a)), where b dominates a (areas 1 and 3 in Fig. 4(a)), of a (areas 1 and 2 in Fig. 4(a)), of b (areas 2 and 3 in Fig. 4(a)),
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
a
1.0
b
c
1.0
349
d
1
µ(x)
µ(x)
6 3
4
2 0 (a)
0.5 x
1.0
0 (b)
5 0.5 x
7
1.0
Fig. 4. Examples of the fuzzy preference relation between: (a) a and b, and (b) c and d.
D(a ∩ b; 0): Intersection area of a and b (area 2 in Fig. 4(a)). For example, the fuzzy preference relation between a and b in Fig. 4(a) is calculated as follows: P(a; b) =
area 2 = 0:18; (area 1 + area 2) + (area 2 + area 3)
P(b; a) =
(area 1 + area 3) + area 2 = 0:82: (area 1 + area 2) + (area 2 + area 3)
Similarly, in Fig. 4(b), the fuzzy preference relation between c and d can be determined P(c; d) =
area 4 + area 5 = 0:346; (area 5 + area 6) + (area 4 + area 5 + area 7)
P(d; c) =
(area 6 + area 7) + area 5 = 0:654: (area 5 + area 6) + (area 4 + area 5 + area 7)
3.2. Multi-criteria group decision-making model for con>guration item selection According to the obtained fuzzy preference relations between candidate items, a multi-criteria group decision-making model is used to prioritize candidate items. Previous research usually determines a complete order of alternatives according to the degree of group support of each candidate item, either obtained by direct or indirect approaches [9,19,22]. However, in the domain of software con%guration management, it is not necessary to distinguish two candidate items that are roughly equally important, because the information collected is still vague at the early stage of software project. Therefore, in addition to the complete ranking order of candidate items, this research also determines the partial ranking order of candidate items to assist decision makers in selecting a proper set of CIs. Let rkt (a; b) be the fuzzy preference relation between candidate items a and b with respect to criterion ck for expert e t and be calculated according to Eq. (2): rkt (a; b) = P(gkt (a); gkt (b)):
(4)
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The fuzzy preference relation between candidate items a and b for expert e t can be obtained by the weighted sum of rkt (a; b) over all criteria r t (a; b) =
q
wk × rkt (a; b)
where
k=1
q
wk = 1:
(5)
k=1
Then the dominance degree to which an expert e t supports the candidate item a can be determined by the average of r t (a; b) for b∈A and b = a: 1 ht (a) = r t (a; b): (6) n−1 b∈A; b=a
Next, the group support of a candidate item can be obtained by aggregating the individual dominance degrees. Since it is not necessary to obtain unanimous agreement (i.e., full consensus) in a real situation, a candidate item can be considered as a CI if it is agreed by “the majority of” experts. Therefore, the concept of fuzzy majority [11] that is more reasonable in group decision-making is used to aggregate experts’ opinions. The concept of fuzzy majority can be modeled by the fuzzy linguistic quanti%ers or the OWA operator [1]. In this paper, the OWA operator is used to aggregate the dominance degree of a candidate item for all experts. The de%nition of OWA operator is introduced as follows [27]. De nition (OWA operator): An OWA operator of dimension n is a mapping F : I n → I (where I = [0; 1])that has an associated weighting vector V = (v1 ; v2 ; : : : ; vn )T such as (1) vi ∈[0; 1], 16i6n, and (2) ni=1 vi = 1. Furthermore F(a1 ; a2 ; : : : ; an ) = v1 b1 + v2 b2 + · · · + vn bn ;
(7)
where bj is the jth largest element in the collection a1 ; a2 ; : : : ; ; an . The OWA operator provides a continuous transition from the “pure-and” to the “pure-or” by adjusting the weighting vector V . The weighting vector V can be determined from the quanti%erguided function Q(x) in the following way: vi = Q(i=n) − Q(i − 1=n):
(8)
Q can be represented as a fuzzy subset of [0; 1] such that for each x ∈[0; 1], Q(x) indicates the degree to which the x portion of objects satis%es the concept devoted by Q. Accordingly, the group dominance degree to which “the majority of” experts support a candidate item a can be determined as h(a) = FQ (h1 (a); h2 (a); : : : ; hm (a));
(9)
where FQ is the OWA operator. The weighting vector VQ = (v1 ; v2 ; : : : ; vm ) can be determined by the quanti%er-guided function Q, where 1x − a x ¿ b a 6 x 6 b; a; b ∈ [0; 1]: (10) Q(x) = b−a 0 x¡a
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
"At least half"
"Most" 1.0
"More than 70%" 1.0
µ (x)
µ(x)
1.0
µ (x)
351
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(a)
(b)
x
x
(c)
x
Fig. 5. Three types of the quanti%er-guided function.
Fig. 5 shows three types of quanti%er-guided function “Most”, “At least half” and “More than 70%”. According to the obtained group dominance degrees, the complete order of candidate items can be obtained. The candidate item with greater group dominance degree has the higher priority to be chosen as a CI. Example. Suppose that the dominance degrees obtained from all experts for a candidate item a are [ht ] = [0:58; 0:33; 0:92; 0:08; 0:72; 0:65] and the quanti%er-guided function Q =“Most” is used for aggregation. The group dominance degree can be obtained according to Eq. (9): h(a) = FQMost (0:58; 0:33; 0:92; 0:08; 0:72; 0:65) = 0:613; where V = (0; 0:07; 0:33; 0:60; 0; 0). From the above example, we can understand that the quanti%er-guided function “Most” takes the middle part of the ordered individual dominance degrees and calculates the weighted sum of them. Other linguistic quanti%ers [28], such as “at least half”, “More than 70%”, may be used. Moreover, Eq. (6) can be rewritten with the OWA operator to model the situation where decision makers intend to evaluate a candidate item such that “most” criteria are satis%ed. In addition to the complete order of candidate items, we also develop an approach to determine the partial order among them. Based on the obtained fuzzy preference relations between pairs of candidate items (see Eq. (5)), we %rst determine the fuzzy group preference relation by aggregating r t (a; b) for all experts using the OWA operator, for all a; b∈A: R(a; b) = FQ (r 1 (a; b); r 2 (a; b); : : : ; r m (a; b));
(11)
where FQ is the OWA operator with the quanti%er-guided function de%ned in Eq. (10). From the above equation, we can build a matrix of fuzzy group preference relations between all pairs of candidate items. The fuzzy group preference relations can be used to construct the partial order of candidate items [24]: ∀a and b∈A: a is preferred to b ⇔ R(a; b) − R(b; a) ¿ $; a is indiHerent to b ⇔ |R(a; b) − R(b; a)| 6 $; where $ is the indiHerence threshold which is used to discriminate the preference between a and b. If the diHerence between R(a; b) and R(b; a) does not exceed $, then it is not considered signi%cant
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
and the item a is therefore indiHerent with b. The value of $ can be assigned by decision makers based on their experience. The partial order among candidate items can be represented by a preference graph, where a node of the preference graph represents a candidate item and an arc represents the preference relation between two items. If candidate item a is preferred to b then there is an arc from node a to b. The partial order of candidate items can help decision makers to identify poor dominated candidate items that can be eliminated from further consideration. 3.3. Determination of consensus measures In this paper, the concept of fuzzy majority is also used to determine the degree of consensus among experts, since it is not realistic to obtain a full consensus for a group decision-making problem in practice. A decision accepted by most of the experts involved in the group decisionmaking process will be good enough. Previous research determined the soft consensus measures by applying the fuzzy linguistic quanti%er or the OWA operator to aggregate the diHerences between experts’ opinions [1,11]. This paper also develops two consensus measures based on the concept of “soft” consensus [11]. The measure of dominance consensus is de%ned as the degree of agreement that the majority of the experts accept the obtained group dominance degrees of all candidate items. The degree of strict agreement of individual expert e t who agrees on the group dominance degree for candidate item a can be de%ned 1 |h(a) − ht (a)| 6 &1 ; st (a) = (12) 0 otherwise; where &1 is the indiHerence threshold which is used to diHerentiate whether the diHerence between h(a) and ht (a) is signi%cant enough to make the diHerence. If the diHerence does not exceed &1 , then we consider that expert e t agrees with the group dominance degree; otherwise s=he does not agree on it. According to Eq. (12), the degree of strict agreement of individual expert e t who agrees on the group dominance degrees for all candidate items can be determined 1 t St = s (a): (13) n a∈ A Then, the measure of dominance consensus is de%ned as the aggregation of S t , t ∈[1; m]: CD = FQ (S 1 ; S 2 ; : : : ; S m );
(14)
where FQ is also the OWA operator de%ned by Eq. (10). In addition, Eq. (12) can be replaced with the following equation, if the degree of weak agreement intended to be used st = 1 − Max {|h(a) − ht (a)|}: a∈ A
(15)
In other words, st can be determined by one minus the maximum of the diHerences between the individual dominance degree and the group dominance degree over all candidate items.
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Another consensus measure is to measure the degree of agreement that the majority of experts accept the obtained preference order of candidate items. For expert e t , the degree of strict agreement to which candidate item a is preferred to b is 1 |R(a; b) − r t (a; b)| 6 &2 ; t z (a; b) = (16) 0 otherwise; where &2 is the indiHerence threshold which is used to diHerentiate whether the diHerence between R and r t is signi%cant enough to make the diHerence. If the diHerence does not exceed &2 , then we consider that expert e t agrees on the group preference; otherwise s=he does not agree on it. Similarly, Eq. (16) can be replaced with the following equation, if the degree of weak agreement is intended to use z t (a; b) = 1 − |R(a; b) − r t (a; b)|:
(17)
According to Eq. (15), the degree of which expert e t agrees on the preference order between all candidate items can be calculated 1 z t (a; b): (18) Zt = n(n − 1) a∈A b∈ A b=a
Therefore, the consensus measure of preference order is de%ned as CO(H ) = FQ (Z 1 ; Z 2 ; : : : ; Z m );
(19)
where FQ is also the OWA operator de%ned by Eq. (10). 4. Illustrated example: a #ight simulator software development An example of a 7ight simulator software development project is used to illustrate the concept developed. A 7ight simulator is a computer-based training devise that is designed for the training of pilots, especially for training hazardous emergency situation. In this project, the entire system consists of seven subsystems including a cockpit system, control loading system, host computer system, I=O control system, sound-and-graphic system, visual display system, and instructor operation system (see Fig. 1). Each subsystem can be decomposed into several tasks that can be further decomposed into a number of modules and the entire system contains hundreds of modules. In the past, the selection of CIs was all based on the experiences of decision makers. However, decision makers found that it is di@cult to make tradeoHs among a set of candidate items, when there are certain limits on the number of CIs due to the budget constraint. Therefore, it is required to have a decision-aid tool to assist decision makers to perform tradeoH analyses to determine a proper set of CIs. To evaluate a candidate item, nine selection criteria were used: security criticality, human safety criticality, mission criticality, %nancial criticality, design newness, interface criticality, hardware integrity, status sensitivity, and technology sensitivity. The linguistic scale used for evaluation is shown in Fig. 3. The relative importance of the nine criteria is evaluated with the linguistic scale listed in Table 1. The weight of each criterion can be obtained by transforming the assigned linguistic term
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363 Table 1 The linguistic scale for relative importance of evaluation criteria Linguistic scale
Quantitative scale
Very important Rather important Important Less important Unimportant
5 4 3 2 1
Table 2 Normalized weights for the nine criteria Criterion
Weight
Normalized weight
1. 2. 3. 4. 5. 6. 7. 8. 9.
3 2 5 1 4 4 3 2 5
0.1034 0.0690 0.1724 0.0345 0.1379 0.1379 0.1034 0.0690 0.1724
Security criticality Human safety criticality Mission criticality Financial criticality Design newness Interface criticality Hardware integrity Status sensitivity Technology sensitivity
to the corresponding quantitative value. The obtained normalized weights for all criteria are listed in Table 2. One of the most important subsystems of a 7ight simulator is the host computer system that consists of operating systems and application software for the dynamic and real-time simulation. Due to limited space, the navigation=communication task including nine modules (see Fig. 1) and the radar task including 18 modules in a hierarchical structure (see Fig. 6) are used in this paper for demonstration purpose. Four experts who are familiar with these two subsystems were invited and the produced rating results are listed in Appendix A. The function of the navigation=communication task is to determine the position of an aircraft and charting a course for guiding the craft safely and expeditiously from one point to another. It consists of nine candidate items numbered from “nv1 ” to “nv9 ”. The fuzzy preference relations between pairs of candidate items for all experts can be calculated from the ratings provided by the experts according to Eqs. (4) and (5). An example of the obtained fuzzy preference relations between pairs of candidate items is shown in Table 3. Then the dominance degrees of all candidate items for each expert can be determined in Table 4 by Eq. (6). According to Eq. (9), the group dominance degree of each candidate item can be calculated and a complete ranking order is obtained: nv9 nv5 nv4 nv2 nv6 nv1 nv8 nv3 nv7 . To obtain the partial order of candidate items, the fuzzy group preference relations can be determined in Table 5 according to Eq. (11). Assume that the indiHerence threshold ($) is set to 0.2, the partial ranking order of candidate items can be determined in Fig. 7.
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rd1
rd2
rd3
rd4
rd5
rd6
rd7
rd8
rd9
rd11
rd12
rd13
rd14
rd15
rd16
rd17
rd10
rd18
Fig. 6. The work-breakdown structure of the radar task.
Table 3 Fuzzy preference relations between pairs of candidate items for expert e1 Item
nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
nv1 nv2 nv3 nv4 nv5 nv6 nv7 nv8 nv9
0.000 0.671 0.327 0.576 0.711 0.450 0.450 0.450 0.711
0.327 0.000 0.086 0.396 0.737 0.207 0.121 0.121 0.617
0.671 0.912 0.000 0.712 0.889 0.646 0.560 0.560 0.840
0.422 0.602 0.287 0.000 0.723 0.310 0.224 0.224 0.674
0.287 0.261 0.109 0.275 0.000 0.138 0.052 0.052 0.379
0.548 0.792 0.352 0.689 0.861 0.000 0.413 0.413 0.797
0.548 0.878 0.438 0.775 0.947 0.585 0.000 0.499 0.883
0.548 0.878 0.438 0.775 0.947 0.585 0.499 0.000 0.883
0.287 0.381 0.158 0.324 0.620 0.201 0.115 0.115 0.000
Table 4 Group dominance degrees of all candidate items for the navigation=communication subsystem
e1 e2 e3 e4 h
nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
0.419 0.394 0.359 0.328 0.370
0.624 0.512 0.441 0.425 0.468
0.254 0.354 0.304 0.340 0.313
0.525 0.574 0.525 0.671 0.545
0.727 0.628 0.517 0.708 0.649
0.365 0.373 0.573 0.583 0.452
0.290 0.276 0.357 0.271 0.281
0.290 0.382 0.418 0.167 0.314
0.723 0.571 0.697 0.714 0.691
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
Table 5 Fuzzy group preference relations between candidate items Item
nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
nv1 nv2 nv3 nv4 nv5 nv6 nv7 nv8 nv9
0.000 0.649 0.404 0.695 0.753 0.531 0.411 0.472 0.757
0.337 0.000 0.255 0.571 0.679 0.440 0.319 0.266 0.643
0.582 0.704 0.000 0.722 0.752 0.678 0.458 0.512 0.761
0.268 0.382 0.274 0.000 0.519 0.324 0.242 0.228 0.502
0.217 0.289 0.218 0.440 0.000 0.243 0.108 0.132 0.451
0.398 0.480 0.273 0.647 0.709 0.000 0.268 0.328 0.606
0.573 0.648 0.505 0.733 0.838 0.709 0.000 0.506 0.776
0.493 0.683 0.451 0.740 0.803 0.609 0.463 0.000 0.757
0.221 0.327 0.214 0.453 0.489 0.368 0.194 0.197 0.000
Level 1
nv4
nv5
nv9
Level 2
Level 3
nv1
nv6
nv2
nv8
nv7
nv3
Fig. 7. The partial ranking order of candidate items ($ = 0:2).
Fig. 8 shows the in7uence of the quanti%er-guided function to the group dominance degree of each candidate item. It is obvious that the group dominance degree of a candidate item will be decreased as the selected quanti%er-guided function is changed from “at least half” to “more than 70%”. In addition, the ranking order of candidate items may be aHected by the choice of the quanti%er-guided function and the value of indiHerence threshold (see Tables 6 and 7). As a tighter quanti%er-guided function is selected or the smaller indiHerence threshold is chosen, more discriminated partial orders of candidate items will be produced. The candidate item with a higher-ranking order has higher priority to be considered as a CI for management control. The number of selected con%guration items depends on the budget allocated to con%guration management. In this case, the project manager chooses the non-dominated set (i.e., the %rst level of the partial ranking order in Fig. 7) including nv4 , nv5 , and nv9 as CIs. If the budget is enough, then the items nv2 and nv6 at the second level of the partial ranking order (see Fig. 7) can be also taken into consideration. Item nv2 may have higher priority over nv6 , because it has a higher group dominance degree (see Table 4). According to Eqs. (12)–(19), the degrees of consensus at the current stage can be determined in Table 8, where &1 and &2 are assumed to be 0.1 and 0.15, respectively. If the project manager considers that the consensus among team members is not enough, he or she may encourage team
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
357
0.8 At least half
Group dominance degree
0.7
Most
0.6
More than 70%
0.5 0.4 0.3 0.2 0.1 0 nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
CI candidates
Fig. 8. The in7uence of the quanti%er-guided function to the group dominance degree.
Table 6 Partial ranking orders of candidate items regarding diHerent quanti%er-guided functions for the navigation= communication task ($ = 0:2) Quanti%er guided function
Ranking order
“At least half ”
Complete Partial Complete Partial Complete Partial
“Most” “More than 70%”
nv9 nv5 nv4 nv2 nv6 nv1 nv8 nv3 nv7 {nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv3 ; nv7 ; nv8 } nv9 nv5 nv4 nv6 nv2 nv1 nv8 nv3 nv7 {nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv3 ; nv7 ; nv8 } nv9 nv5 nv4 nv2 nv6 nv1 nv7 nv3 nv8 {nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv8 } {nv3 ; nv7 }
Table 7 Partial ranking orders of candidate items regarding various threshold values for the navigation=communication task (Q = “Most”) IndiHerence threshold ($)
Partial order
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
{nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv8 } {nv3 ; nv7 } {nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv3 ; nv7 ; nv8 } {nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv3 ; nv7 ; nv8 } {nv2 ; nv4 ; nv5 ; nv9 } {nv1 ; nv6 ; nv8 } {nv3 ; nv7 } {nv2 ; nv4 ; nv5 ; nv6 ; nv9 } {nv1 ; nv3 ; nv7 ; nv8 } {nv1 ; nv2 ; nv3 ; nv4 ; nv5 ; nv6 ; nv9 } {nv7 ; nv8 } {nv1 ; nv2 ; nv3 ; nv4 ; nv5 ; nv6 ; nv8 ; nv9 } {nv7 } {nv1 ; nv2 ; nv3 ; nv4 ; nv5 ; nv6 ; nv7 ; nv8 ; nv9 } {nv1 ; nv2 ; nv3 ; nv4 ; nv5 ; nv6 ; nv7 ; nv8 ; nv9 }
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363 Table 8 Consensus measures (&1 = 0:1 and &2 = 0:15)
Strict dominance Weak dominance Strict order Weak order
e1
e2
e3
e4
Consensus
0.889 0.844 0.569 0.904
0.889 0.879 0.764 0.927
0.669 0.868 0.694 0.915
0.667 0.852 0.542 0.895
0.756 0.858 0.786 0.907
Table 9 Group dominance degrees of all candidate items for the radar task Item
rd1
rd2
rd3
rd4
rd5
rd6
rd7
rd8
rd9
h
0.119
0.128
0.124
0.130
0.500
0.628
0.272
0.521
0.581
Item
rd10
rd11
rd12
rd13
rd14
rd15
rd16
rd17
rd18
h
0.620
0.595
0.639
0.623
0.597
0.619
0.557
0.594
0.624
members exchanging ideas between them and reevaluate again until the group consensus is reached. From this study, we found that the strict consensus measures are more meaningful than the weak consensus measures, if the values of &1 and &2 are set properly. In practice, decision makers can tolerate certain diHerences between his=her ideal solution and the proposed solution. The values of indiHerence thresholds are chosen based on the experience of decision makers. The function of the radar task is to locate objects beyond the range of vision, and to determine their distance. It includes 18 candidate items numbered from “rd1 ” to “rd18 ” in a hierarchical structure. Using the developed multi-criteria group decision-making model, the group dominance degree of each candidate item can be calculated (see Table 9) and a complete ranking order is obtained: rd12 rd6 rd18 rd13 rd10 rd15 rd14 rd11 rd17 rd9 rd16 rd8 rd5 rd7 rd4 rd2 rd3 rd1 . The partial ranking orders of candidate items with respect to various values of indiHerence threshold are listed in Table 10. In this case, the con%guration management board selected the set {rd5 ; rd6 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } as CIs, according to their experience. This result is the same as the set identi%ed in the %rst level of the obtained partial order, as $ = 0:5. If there is a budget constraint, we may decrease the value of $ and check which items can be removed from further consideration. We can observed from Table 10 that item rd8 is removed from the non-dominated set as $ = 0:4 and rd5 is also removed from the non-dominated set in addition to rd8 as $ = 0:3. Therefore, we may consider removing rd5 and rd8 from the set of CIs, if the budget is not enough. If a candidate item (e.g., rd5 ) is not determined as a CI, but it contains other sub-modules that have been determined as CIs, then some attention should be given for the item during the software development. We call the item a “pseudo” CI. This is because we still need to ensure that the interfaces among submodules of a pseudo CI will be properly designed and built to achieve its expected performance. In practice, the management cost of a “pseudo” CI can be neglected. In
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
359
Table 10 Partial ranking orders of candidate items regarding various threshold values for radar task (Q = “Most”) IndiHerence threshold ($)
Partial order
0.1
{rd12 ; rd13 ; rd15 } {rd6 ; rd14 ; rd17 } {rd10 ; rd18 } {rd9 ; rd11 } {rd8 ; rd16 } {rd5 } {rd7 } {rd2 ; rd3 ; rd4 } {rd1 } {rd6 ; rd10 ; rd12 ; rd13 ; rd15 ; rd17 } {rd9 ; rd11 ; rd14 ; rd16 ; rd18 } {rd5 ; rd8 } {rd7 } {rd2 ; rd3 ; rd4 } {rd1 } {rd6 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd5 ; rd8 } {rd7 } {rd2 ; rd3 ; rd4 } {rd1 } {rd5 ; rd6 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd8 } {rd7 } {rd1 ; rd2 ; rd3 ; rd4 } {rd5 ; rd6 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd1 ; rd2 ; rd3 ; rd7 } {rd4 } {rd5 ; rd6 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd1 ; rd2 ; rd3 ; rd4 ; rd7 } {rd5 ; rd6 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd1 ; rd2 ; rd3 ; rd4 ; rd7 } {rd5 ; rd6 ; rd7 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd1 ; rd2 ; rd3 ; rd4 } {rd5 ; rd6 ; rd7 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd1 ; rd2 ; rd3 ; rd4 }
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Table 11 The evaluation results and the selected CIs for four expertsa
e1 e2 e3 e4 a
nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
Selected CIs
1 −1 0 −3
−1 −1 2 −2
−6 −4 −3 −3
1 −2 4 8
7 1 4 5
−6 −3 5 2
−6 −4 −1 −8
−6 −5 1 −7
5 −1 8 6
nv1 ; nv4 ; nv5 ; nv9 nv5 nv1 ; nv2 ; nv4 ; nv5 ; nv6 ; nv8 ; nv9 nv4 ; nv5 ; nv6 ; nv9
The number in a cell indicates the diHerence between the numbers of ‘+’ and ‘−’
addition, according to the experienced project manager, the partial order of candidate items provides more information than the complete order in this research problem. It is preferred to classify the set of candidate items into diHerent subsets from the “best” to “worst”, rather than ranking candidate items into a complete order. A simple CI selection procedure [16] that is often used in the real world was applied to verify the result of the proposed approach. The four experts were asked to evaluate each candidate item with “yes” or “no” for the nine criteria. If the number of “yes” is greater than or equal to the number of “no”, then the candidate item will be considered as a CI. The navigation=communication task is used in this experiment. According to the above procedure, Table 11 lists the evaluation results for all candidate items and the selected sets of CIs for four experts. From the result, we can know that diHerent experts may have diHerent opinions about the proper set of CIs. For example, the attitude of expert e3 is
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
quite conservative; on the contrast, expert e2 is very optimistic. However, we can %nd that the items nv4 , nv5 , and nv9 are commonly chosen by three experts. These three items are also the top-three items in the identi%ed complete order and the non-dominated set in the partial order of the candidate items (see Fig. 7). This %nding is consistent with the result obtained by our approach. In addition, the obtained ranking order of the candidate item can help con%guration management board to select more proper set of CIs. Decision makers can perform sensitivity analysis by changing indiHerence threshold or the quanti%er-guided function to make more robust selection decisions. 5. Conclusion This research developed a multi-criteria group decision-making model based on fuzzy set theory to improve the con%guration items selection process. The model constructs imprecise preference relations among candidate items based on the concept of fuzzy majority and determines ranking orders among them. The consensus measures were also developed to determine the group acceptability of the obtained ranking orders. Decision makers can perform sensitivity analysis to examine the solution robustness. We concluded that the developed methodology can assist decision makers in selecting more proper set of CIs that can reduce the software life cycle cost. Acknowledgements This research is partially supported by grant No. NSC 91-2213-E-035-032 from the National Science Council of the Republic of China. Appendix A. Evaluation ratings by four experts for the #ight simulator software development The evaluation ratings provided by four experts for the nine candidate items in the navigation= communication subsystem are listed in the Tables 12–15, respectively. The set of linguistic terms Table 12 Linguistic ratings of nine candidate items regarding nine criteria by expert e1 Criteria
1 2 3 4 5 6 7 8 9
Candidate items nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
pn my py pn pn py py my pn
mn my py mn mn py mn my pn
pn mn uk pn dn my pn mn pn
my mn my my mn my pn mn my
my py py my mn py my my my
mn mn uk pn pn my pn mn pn
mn mn uk pn pn my pn mn pn
mn mn uk pn pn my pn mn pn
my py py my mn py mn my my
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
361
Table 13 Linguistic ratings of nine candidate items regarding nine criteria by expert e2 Criteria
1 2 3 4 5 6 7 8 9
Candidate items nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
dn dn py pn dn dy mn dy dy
dn dn dy dn dn my uk py uk
dn dn my dn dn uk mn py mn
dn dn dy mn dn py dy uk mn
dn dn dy dy pn dy dy dy mn
dn dn my dn dn dy pn dy pn
dn dn uk dn dn py dn my pn
dn dn dy dn dn py mn dn pn
dn dn dy mn dn py dy py mn
Table 14 Linguistic ratings of nine candidate items regarding nine criteria by expert e3 Criteria
1 2 3 4 5 6 7 8 9
Candidate items nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
dy uk py dn dn dy dy dn dn
dy uk dy dn dn dy dy dy dn
dn dn dy dn dn dy dn dy dn
dn uk dy uk dy dy dy uk dy
dn uk dy uk dy dy dy uk dy
pn uk dy uk dy dy dy dy dy
dn uk uk dn dn dy dy dy dn
dn dy dy dn dn dy dy dy dn
dy dy dy uk dy dy dy dy dy
Table 15 Linguistic ratings of nine candidate items regarding nine criteria by expert e4 Criteria
1 2 3 4 5 6 7 8 9
Candidate items nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
my dn dy pn dn dy pn dn dn
py mn dy pn mn py pn uk pn
pn dn my dn dn my dy pn dn
dy py dy uk my py dy dy my
dy uk dy pn my dy py py uk
py my dy uk mn my mn py mn
pn dn mn dn dn pn uk mn dn
pn dn my dn dn dn pn dn dn
dy uk dy mn dy dy dy dy my
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used in the evaluation is {“de%nitely not (dn)”, “probably not (pn)”, “maybe not (mn)”, “unknown (uk)”, “maybe (my)”, “probably (py)”, “de%nitely (dy)”}. References [1] M. Fedrizzi, J. Kacprzyk, Consensus degrees under fuzzy majorities and preferences using ordered weighted average operators, Proceedings of the 5th IFSA World Congress, Seoul, Korea, 1993, pp. 624 – 626. [2] F. Herrera, E. Herrera-Viedma, J.L. Verdegra, A sequential selection process in group decision making with a linguistic assessment approach, Inform. Sci. 85 (1995) 223–239. [3] F. Herrera, E. Herrera-Viedma, J.L. Verdegra, A model of consensus in group decision making under linguistic assessments, Fuzzy Sets and Systems 78 (1996) 73–87. [4] F. Herrera, E. Herrera-Viedma, J.L. Verdegra, A rational consensus model in group decision making using linguistic assessments, Fuzzy Sets and Systems 88 (3) (1997) 31–49. [5] F. Herrera, E. Herrera-Viedma, J.L. Verdegra, Choice processes for non-homogeneous group decision making in linguistic setting, Fuzzy Sets and Systems 94 (3) (1998) 287–308. [6] IEEE Guide to Software Con%guration Management, The Institute of Electrical and Electronics Engineers, 1988. [7] IEEE Standard for Software Con%guration Management Plans, The Institute of Electrical and Electronics Engineers, 1998. [8] ISO-10007, Quality Management—Guidelines for Con%guration Management, 1995. [9] J. Kacprzyk, Group decision making with a fuzzy linguistic majority, Fuzzy Sets and Systems 11 (1986) 105–118. [10] J. Kacprzyk, M. Fedrizzi (Eds.), Multiperson decision making models using fuzzy sets and possibility theory, Kluwer, Dordrecht, 1990. [11] J. Kacprzyk, M. Fedrizzi, H. Nurmi, Group decision making and consensus under fuzzy preferences and fuzzy majority, Fuzzy Sets and Systems 49 (1992) 21–31. [12] G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, New Jersey, 1995. [13] W. Kolodziejczyk, Orlovsky’s concept of decision-making with fuzzy preference relation—further results, Fuzzy Sets and Systems 19 (1) (1986) 11–20. [14] M.U. Marimin, I. Hatono, H. Tamura, Linguistic labels for expressing fuzzy preference relations in fuzzy group decision making, IEEE Trans. Systems Man Cybernet.—Part B: Cybernet. 28 (2) (1998) 205–218. [15] A.K. Midha, Software con%guration management for the 21st century, Bell Labs Tech. J. 2 (1) (1997) 154–165. [16] MIL-STD-483A, Con%guration Management Practices for Systems, Equipment, Munitions, and Computer Programs. [17] MIL-STD-973, Con%guration Management. [18] K. Nakamura, Preference relations on a set of fuzzy utilities as a basis for decision making, Fuzzy Sets and Systems 20 (2) (1986) 147–162. [19] H. Nurmi, Approaches to collective decision making with fuzzy preference relations, Fuzzy Sets and Systems 6 (1981) 249–259. [20] P. O’Connell, R. McCrindle, Using SOAP to clean up con%guration management, Proc. IEEE Computer Society’s International Computer Software and Applications Conference, Chicago, IL, 2001, pp. 555 –560. [21] S. Srinath, R. Ramakrishnan, D.J. Ram, Generic model for semantics-based versioning in projects, IEEE Trans. Systems Man Cybernet. Part A: Systems Humans 30 (2) (2000) 108–123. [22] T. Tanino, Fuzzy preference orderings in group decision making, Fuzzy Sets and Systems 12 (1984) 117–131. [23] T.Y. Tseng, C.M. Klein, New algorithm for the ranking procedure in fuzzy decision making, IEEE Trans. Systems Man Cybernet. 19 (5) (1989) 1289–1296. [24] J. Wang, A fuzzy outranking method for conceptual design evaluation, Internat. J. Production Res. 35 (4) (1997) 995–1010. [25] B. Westfechtel, B.P. Munch, R. Conradi, A layered architecture for uniform version management, IEEE Trans. Software Eng. 27 (12) (2001) 1111–1133. [26] D. Whitgift, Software Con%guration Management: Methods and Tools, Wiley, England, 1991. [27] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Systems Man Cybernet. 18 (1988) 183–190.
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
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[28] R.R. Yager, Families of OWA operators, Fuzzy Sets and Systems 59 (1993) 125–148. [29] L.A. Zadeh, A computational approach to fuzzy quanti%ers in natural languages, Comput. Math. Appl. 9 (1) (1983) 149–183. [30] L. Zhang, H. Mei, H. Zhu, A con%guration management system supporting component-based software development, International Computer Software and Applications Conference, Chicago, IL, 2001, pp. 25 –30.
www.elsevier.com/locate/fss
A fuzzy multicriteria group decision making approach to select con%guration items for software development Juite Wanga; ∗ , Yung-I Linb a
Department of Industrial Engineering, Feng Chia University, P.O. Box 25-097, Taichung 407, Taiwan, Republic of China b Aeronautical System Research Division, Chung-Shang Institute of Science and Technology, P.O. Box 90008-11-21, Taichung 40722, Taiwan, Republic of China Received 6 February 2001; received in revised form 5 April 2002; accepted 7 May 2002
Abstract Selection of con%guration items in software con%guration management is important to determine the software quality and reduce the development time and cost. The objective of this research is to develop a multi-criteria group decision making model based on fuzzy set theory to improve the con%guration items selection process. Since most information available in this stage is not numerical, fuzzy set theory is used to represent the evaluation ratings of candidate items. The developed model ranks candidate items into partial or complete orders that can assist decision makers in selecting more proper set of con%guration items. The consensus measures are also developed to determine the group acceptability of the obtained ranking orders. In addition, sensitivity analysis can be performed to examine the solution robustness. An example of 7ight simulator development project is used to illustrate the concept developed. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Decision analysis; Multiple criteria evaluation; Group decision-making; Con%guration management; Software development
1. Introduction The increasing dominance of computers in industry has made software become more important in the business decision-making. More human decision-making functions are embedded into the software. However, due to the changing environment of today’s marketplace, changes of requirements or speci%cation frequently occur throughout the life cycle of the software development project. Moreover, unlike manufacturing hardware products, it is di@cult to develop a set of standards and ∗
Corresponding author. Tel.: +886-2451-7250, ext. 3636; fax: +886-2451-0240. E-mail addresses: [email protected] (J. Wang), yung [email protected] (Y.-I. Lin).
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 2 8 3 - X
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363 Flight simulator
Cockpit system
Control loading system
...
...
...
nv1
...
nv2
Host computer system
Navigation/ communication task
...
nv3
nv4
I/O control system
Sound&graphic system
...
...
...
nv5
...
Radar task
nv6
Visual display system
... ...
nv7
Instructor operation system
... ...
nv8
nv9
Fig. 1. The work-breakdown structure for a 7ight simulator project.
statistics that allows people to measure the eHectiveness or quality of software until after it is built. Therefore, a systematic method is required to develop a large-scale software product to better control changes and uncertainties occurred in the development process. Con%guration management [6,7,8,15,16,26] is a key element in developing a large-scale software product. Its purpose is to maintain the integrity of products as they evolve throughout the entire system life cycle, from concept, through design, development, production, and maintenance. Con%guration management is important for managing the modern software project, because it provides stability to the production of a software system by controlling the product evolution. In addition, the entire product life cycle cost can be reduced by proper con%guration management. There are four essential functions for con%guration management [6,8,17,26]: con%guration identi%cation, con%guration control, con%guration status accounting, and con%guration audits and reviews. The purpose of con%guration identi%cation is to select a proper set of con%guration items. A con%guration item (CI) is de%ned as an aggregation of hardware or software whose functions and performance parameters must be de%ned and controlled to achieve the overall performance of a system or product. A special attention will be given to the selected con%guration items in the entire product life cycle. All physical and functional characteristics necessary to de%ne a CI throughout its life cycle should be documented and all documentation should include relevant information on change and traceability. The work breakdown structure (WBS), which is a family tree of project elements, has been usually developed by the product development teams, before entering the stage of con%guration identi%cation. It de%nes the various hardware, software, and service systems and their subcomponents that make up the speci%ed project or program. An example of WBS for a 7ight simulator project is shown in Fig. 1. Improper selection of CIs will result in the poor product quality and increase the development time and cost [8,16,26]. Selecting too many CIs will increase the cost of management control; e.g., developing related con%guration documentation for requirements, design, manufacturing and veri%cation. On the contrary, selecting too few or wrong items as CIs runs the risk of too little control and creates logistics and maintenance di@culties, due to the lack of management visibility. Ideally, the selection of CIs should be with a full view of the product life cycle cost and management
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345
impacts. However, due to only the incomplete software development information available at the early stage of product development, it is very di@cult to perform the quantitative cost=bene%t analysis to predict the impact of selecting a component as a CI on the entire product life cycle cost. In practice, a con%guration management board that involves members from various life cycle perspectives is organized to determine a proper set of CIs from hundreds or thousands of candidate items. A set of criteria is determined to help the selection; for example: security criticality, human safety criticality, mission criticality, %nancial criticality, design newness, interface criticality, hardware integrity, status sensitivity, technology sensitivity, etc. Some criteria are more signi%cant than others. Each candidate item is evaluated by a group of experts in the con%guration management board and the %nal decision is based on their experiences and coordination. However, evaluating and comparing a large number of candidate items is a cumbersome work in the CI selection process. The objective of this research is to develop a methodology to assist decision makers in selecting a proper set of CIs in con%guration management. The CI selection problem is modeled as a multi-criteria group decision making problem [10]. Fuzzy set theory [12] is used to represent the evaluation results of candidate items, because most information available at this stage is subjective and imprecise and is usually expressed in a nature language by individual experts. Once these individual evaluation results are collected, fuzzy set theory would permit to perform the tradeoH analysis among various criteria. A fuzzy multi-criteria group decision model based on the concept of fuzzy preference relation [19] and fuzzy majority [9,11] is developed to rank candidates into a partial order or a complete order according to their importance. Two consensus measures are also developed to determine the mutual agreements about the chosen CIs among experts. If the consensus is reached, then a candidate item with a higher ranking order will has higher priority to be chosen as a CI. Several standards for con%guration management [6,7,8,16,17] only provide guidelines or selection criteria for selecting CIs in the con%guration identi%cation stage. We have also found that most research in con%guration management is to develop a computerized con%guration management system that is focused on version=revision control in software development [15,21,30]. Recent research developed various approaches for managing the con%guration management of the emerging development paradigm; such as the object-oriented [25], component-based [30] or distributed software development [20]. There is little research in developing a decision support system for con%guration management, especially in the con%guration identi%cation stage. This is because the information available in this stage is mostly qualitative and is di@cult to be used for developing quantitative models for decision-making. This paper is organized as follows. Section 2 provides a formal description of our research problem. Next, a multi-criteria group decision making model to select a set of CIs is proposed in Section 3. In Section 4, an example of a 7ight simulator software project is used to illustrate the developed concept. The data used in the example is presented in Appendix. Finally, Section 5 concludes this paper. 2. Problem description The impacts of selecting a CI on the software quality and life cycle cost are di@cult to predict in practice. Therefore, we consider the CI selection problem as a fuzzy multi-criteria group decisionmaking problem in this paper. The research problem is described next.
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A set of candidate items
Evaluate candidate items by a group of experts
Evaluation criteria
Ratings of candidate items Rank candidate items by the multi-criteria group decision making model Partial or complete orders of candidate items Determine consensus Degree of consensus N
Is consensus reached? Y Select configuration items
Configuration items Fig. 2. The decision process to select con%guration items.
Let A be the set of n candidate items below the subsystem level of WBS under evaluation by m experts, according to a set C of q criteria. Let G t (a) = [g1t (a); g2t (a); : : : ; gqt (a)] be the evaluation result of candidate item a∈A by expert e t ∈T , where the function gkt (a) represents its rating for criterion ck ∈C. The objective of this research is to develop an approach to assist decision makers in selecting a proper set of CIs. Fig. 2 shows the CI selection process. Due to imprecise and subjective project information available in the early stage of software development, it may be di@cult to assess candidate items with certain values for some criteria. For example, to evaluate the candidate item “nv3 ” under the “Navigation=communication” task shown in Fig. 1 with the criterion “mission criticality”, the answer may not be either “yes” or “no”, but in between. The damage of this item may not lead to the total failure of the mission, but within a certain degree depending on the situation. In this paper, the rating of each criterion is described by some linguistic terms, which provide a more natural way to express the rating for assessing candidate items. These linguistic terms can be manipulated with fuzzy set theory [12] and they can be interpreted as speci%c fuzzy numbers that allow us to perform the tradeoH analysis among various criteria. An example of a linguistic scale consisting of seven linguistic values {“de%nitely not”, “probably not”, “maybe not”, “unknown”, “maybe”, “probably”, “de%nitely”} is shown in Fig. 3. The strongest agreement for a criterion is given the highest (lowest) linguistic value “de%nitely” (“de%nitely not”) in the scale.
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probably probably unknown definitely not maybe maybe definitely not not
µ(x)
1.0
0
1
2
3
4
5
6
7
8
9
10
x
Fig. 3. The linguistic scale for rating candidate items.
3. Fuzzy group decision making approach to select con guration items This paper considers the con%guration item selection problem as a multi-criteria group decisionmaking problem. A group decision-making problem is characterized by a set of alternatives and a set of experts who provide their judgments on these alternatives. The objective is to %nd an alternative or a set of alternatives that is best acceptable by the group of experts. However, criteria for selecting alternatives are not considered explicitly in the past research of group decision making models. Fuzzy set theory has been applied in the group decision-making problems for a long time [10]. Most of fuzzy group decision-making models developed in the literature were based on the fuzzy preference relations [9,10,11,19,22] that can be obtained by pairwise comparison between various alternatives by experts. The fuzzy preference relation is de%ned next. De nition (Fuzzy preference relation [19]): A fuzzy preference relation R on a set A is a fuzzy set on the product A × A, such that R : A × A → [0; 1]. Let P(a; b)∈R be the fuzzy preference relation between a and b, where a; b∈A. Then P(a; b) and P(b; a) are reciprocal, i.e., P(a; b) + P(b; a) = 1:
(1)
Let A be a set of candidate items and a; b∈A. P(a; b) = 1 indicates the maximum degree of preference of item a over b. P(a; b)∈(0:5; 1) indicates a de%nite preference of alternative a to b with the intensity of preference corresponding to the value of P(a; b). Note that the higher value means a stronger intensity of preference. P(a; b) = 0:5 indicates the indiHerence between candidate items a and b. According to the obtained fuzzy preference relations, either direct or indirect approaches can be used to derive a solution set [9,19,22]. A direct approach determines a solution directly from the set of individual fuzzy preference relation. An indirect approach derives a solution from a social fuzzy preference relation that can be obtained by aggregating a set of individual fuzzy preference relations. Since it is not necessary to achieve a full consensus in practice for a group decisionmaking problem, Kacprzyk [9] developed the concept of soft majority (e.g., “Most”) based on the fuzzy linguistic quanti%er [29] or the OWA aggregation operator [27] for dealing with this issue.
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Based on this concept, Kacprzyk and Fedrizzi [1,11] also developed a model of soft consensus to measure the consensus reached by most of the experts. Moreover, Herrera et al. [2] extended the numerical representation of the fuzzy preference relation with linguistic labels and developed the linguistic OWA operator to determine the collective linguistic preference from a set of individual linguistic preferences. The linguistic non-dominance degree and several linguistic consensus degrees were also determined to solve the group decision-making problem [3]. They also extended their models to the heterogeneous group decision-making problem [4,5]. Marimin et al. [14] also represented the fuzzy preference relation with linguistic labels and further characterized them by fuzzy numbers [12]. Subsequently, the arithmetic operations of fuzzy numbers and the extension of the neat OWA operator [28] were used in the direct and indirect approaches to solve the problem and to determine the degree of consensus among experts. In this paper, the proposed methodology that is adapted from Kacprzyk [11] is divided into three stages: (1) construction of fuzzy preference relations between candidate items; (2) determination of a partial order or a complete order of con%guration items under evaluation; and (3) determination of the consensus measures among experts. 3.1. Construction of fuzzy preference relations In this paper, we apply the fuzzy preference relation to model imprecise preference relations between diHerent candidate items. To reduce the cumbersome work of assigning degrees of preference between diHerent candidate items by a decision maker (about qn(n − 1)=2 times), in this paper, the fuzzy preference relation between two candidate items a and b with respect to criterion k is obtained by pair-wise comparison of linguistic evaluation ratings gkt (a) and gkt (b), which can be characterized by fuzzy numbers. The linguistic evaluation ratings are provided by an experienced decision maker (about qn times). There are several methods in the literature to determine the fuzzy preference relation between two fuzzy numbers, such as: Nakamura [18], Kolodziejczyk [13], Tseng and Klein [23]. This paper utilizes Tseng and Klein’s approach based on Hamming distance to derive the fuzzy preference relation between two candidate items for each criterion [24], because of the computational e@ciency of their approach. De nition. Let a and b be two fuzzy numbers. The fuzzy preference relations P(a; b) and P(b; a) are de%ned as follows [23]: P(a; b) =
D(a; b) + D(a ∩ b; 0) ; D(a; 0) + D(b; 0)
(2)
P(b; a) =
D(b; a) + D(a ∩ b; 0) ; D(a; 0) + D(b; 0)
(3)
where D(a; b): D(b; a): D(a; 0): D(b; 0):
Area Area Area Area
where a dominates b (none in Fig. 4(a)), where b dominates a (areas 1 and 3 in Fig. 4(a)), of a (areas 1 and 2 in Fig. 4(a)), of b (areas 2 and 3 in Fig. 4(a)),
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
a
1.0
b
c
1.0
349
d
1
µ(x)
µ(x)
6 3
4
2 0 (a)
0.5 x
1.0
0 (b)
5 0.5 x
7
1.0
Fig. 4. Examples of the fuzzy preference relation between: (a) a and b, and (b) c and d.
D(a ∩ b; 0): Intersection area of a and b (area 2 in Fig. 4(a)). For example, the fuzzy preference relation between a and b in Fig. 4(a) is calculated as follows: P(a; b) =
area 2 = 0:18; (area 1 + area 2) + (area 2 + area 3)
P(b; a) =
(area 1 + area 3) + area 2 = 0:82: (area 1 + area 2) + (area 2 + area 3)
Similarly, in Fig. 4(b), the fuzzy preference relation between c and d can be determined P(c; d) =
area 4 + area 5 = 0:346; (area 5 + area 6) + (area 4 + area 5 + area 7)
P(d; c) =
(area 6 + area 7) + area 5 = 0:654: (area 5 + area 6) + (area 4 + area 5 + area 7)
3.2. Multi-criteria group decision-making model for con>guration item selection According to the obtained fuzzy preference relations between candidate items, a multi-criteria group decision-making model is used to prioritize candidate items. Previous research usually determines a complete order of alternatives according to the degree of group support of each candidate item, either obtained by direct or indirect approaches [9,19,22]. However, in the domain of software con%guration management, it is not necessary to distinguish two candidate items that are roughly equally important, because the information collected is still vague at the early stage of software project. Therefore, in addition to the complete ranking order of candidate items, this research also determines the partial ranking order of candidate items to assist decision makers in selecting a proper set of CIs. Let rkt (a; b) be the fuzzy preference relation between candidate items a and b with respect to criterion ck for expert e t and be calculated according to Eq. (2): rkt (a; b) = P(gkt (a); gkt (b)):
(4)
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The fuzzy preference relation between candidate items a and b for expert e t can be obtained by the weighted sum of rkt (a; b) over all criteria r t (a; b) =
q
wk × rkt (a; b)
where
k=1
q
wk = 1:
(5)
k=1
Then the dominance degree to which an expert e t supports the candidate item a can be determined by the average of r t (a; b) for b∈A and b = a: 1 ht (a) = r t (a; b): (6) n−1 b∈A; b=a
Next, the group support of a candidate item can be obtained by aggregating the individual dominance degrees. Since it is not necessary to obtain unanimous agreement (i.e., full consensus) in a real situation, a candidate item can be considered as a CI if it is agreed by “the majority of” experts. Therefore, the concept of fuzzy majority [11] that is more reasonable in group decision-making is used to aggregate experts’ opinions. The concept of fuzzy majority can be modeled by the fuzzy linguistic quanti%ers or the OWA operator [1]. In this paper, the OWA operator is used to aggregate the dominance degree of a candidate item for all experts. The de%nition of OWA operator is introduced as follows [27]. De nition (OWA operator): An OWA operator of dimension n is a mapping F : I n → I (where I = [0; 1])that has an associated weighting vector V = (v1 ; v2 ; : : : ; vn )T such as (1) vi ∈[0; 1], 16i6n, and (2) ni=1 vi = 1. Furthermore F(a1 ; a2 ; : : : ; an ) = v1 b1 + v2 b2 + · · · + vn bn ;
(7)
where bj is the jth largest element in the collection a1 ; a2 ; : : : ; ; an . The OWA operator provides a continuous transition from the “pure-and” to the “pure-or” by adjusting the weighting vector V . The weighting vector V can be determined from the quanti%erguided function Q(x) in the following way: vi = Q(i=n) − Q(i − 1=n):
(8)
Q can be represented as a fuzzy subset of [0; 1] such that for each x ∈[0; 1], Q(x) indicates the degree to which the x portion of objects satis%es the concept devoted by Q. Accordingly, the group dominance degree to which “the majority of” experts support a candidate item a can be determined as h(a) = FQ (h1 (a); h2 (a); : : : ; hm (a));
(9)
where FQ is the OWA operator. The weighting vector VQ = (v1 ; v2 ; : : : ; vm ) can be determined by the quanti%er-guided function Q, where 1x − a x ¿ b a 6 x 6 b; a; b ∈ [0; 1]: (10) Q(x) = b−a 0 x¡a
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
"At least half"
"Most" 1.0
"More than 70%" 1.0
µ (x)
µ(x)
1.0
µ (x)
351
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(a)
(b)
x
x
(c)
x
Fig. 5. Three types of the quanti%er-guided function.
Fig. 5 shows three types of quanti%er-guided function “Most”, “At least half” and “More than 70%”. According to the obtained group dominance degrees, the complete order of candidate items can be obtained. The candidate item with greater group dominance degree has the higher priority to be chosen as a CI. Example. Suppose that the dominance degrees obtained from all experts for a candidate item a are [ht ] = [0:58; 0:33; 0:92; 0:08; 0:72; 0:65] and the quanti%er-guided function Q =“Most” is used for aggregation. The group dominance degree can be obtained according to Eq. (9): h(a) = FQMost (0:58; 0:33; 0:92; 0:08; 0:72; 0:65) = 0:613; where V = (0; 0:07; 0:33; 0:60; 0; 0). From the above example, we can understand that the quanti%er-guided function “Most” takes the middle part of the ordered individual dominance degrees and calculates the weighted sum of them. Other linguistic quanti%ers [28], such as “at least half”, “More than 70%”, may be used. Moreover, Eq. (6) can be rewritten with the OWA operator to model the situation where decision makers intend to evaluate a candidate item such that “most” criteria are satis%ed. In addition to the complete order of candidate items, we also develop an approach to determine the partial order among them. Based on the obtained fuzzy preference relations between pairs of candidate items (see Eq. (5)), we %rst determine the fuzzy group preference relation by aggregating r t (a; b) for all experts using the OWA operator, for all a; b∈A: R(a; b) = FQ (r 1 (a; b); r 2 (a; b); : : : ; r m (a; b));
(11)
where FQ is the OWA operator with the quanti%er-guided function de%ned in Eq. (10). From the above equation, we can build a matrix of fuzzy group preference relations between all pairs of candidate items. The fuzzy group preference relations can be used to construct the partial order of candidate items [24]: ∀a and b∈A: a is preferred to b ⇔ R(a; b) − R(b; a) ¿ $; a is indiHerent to b ⇔ |R(a; b) − R(b; a)| 6 $; where $ is the indiHerence threshold which is used to discriminate the preference between a and b. If the diHerence between R(a; b) and R(b; a) does not exceed $, then it is not considered signi%cant
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
and the item a is therefore indiHerent with b. The value of $ can be assigned by decision makers based on their experience. The partial order among candidate items can be represented by a preference graph, where a node of the preference graph represents a candidate item and an arc represents the preference relation between two items. If candidate item a is preferred to b then there is an arc from node a to b. The partial order of candidate items can help decision makers to identify poor dominated candidate items that can be eliminated from further consideration. 3.3. Determination of consensus measures In this paper, the concept of fuzzy majority is also used to determine the degree of consensus among experts, since it is not realistic to obtain a full consensus for a group decision-making problem in practice. A decision accepted by most of the experts involved in the group decisionmaking process will be good enough. Previous research determined the soft consensus measures by applying the fuzzy linguistic quanti%er or the OWA operator to aggregate the diHerences between experts’ opinions [1,11]. This paper also develops two consensus measures based on the concept of “soft” consensus [11]. The measure of dominance consensus is de%ned as the degree of agreement that the majority of the experts accept the obtained group dominance degrees of all candidate items. The degree of strict agreement of individual expert e t who agrees on the group dominance degree for candidate item a can be de%ned 1 |h(a) − ht (a)| 6 &1 ; st (a) = (12) 0 otherwise; where &1 is the indiHerence threshold which is used to diHerentiate whether the diHerence between h(a) and ht (a) is signi%cant enough to make the diHerence. If the diHerence does not exceed &1 , then we consider that expert e t agrees with the group dominance degree; otherwise s=he does not agree on it. According to Eq. (12), the degree of strict agreement of individual expert e t who agrees on the group dominance degrees for all candidate items can be determined 1 t St = s (a): (13) n a∈ A Then, the measure of dominance consensus is de%ned as the aggregation of S t , t ∈[1; m]: CD = FQ (S 1 ; S 2 ; : : : ; S m );
(14)
where FQ is also the OWA operator de%ned by Eq. (10). In addition, Eq. (12) can be replaced with the following equation, if the degree of weak agreement intended to be used st = 1 − Max {|h(a) − ht (a)|}: a∈ A
(15)
In other words, st can be determined by one minus the maximum of the diHerences between the individual dominance degree and the group dominance degree over all candidate items.
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Another consensus measure is to measure the degree of agreement that the majority of experts accept the obtained preference order of candidate items. For expert e t , the degree of strict agreement to which candidate item a is preferred to b is 1 |R(a; b) − r t (a; b)| 6 &2 ; t z (a; b) = (16) 0 otherwise; where &2 is the indiHerence threshold which is used to diHerentiate whether the diHerence between R and r t is signi%cant enough to make the diHerence. If the diHerence does not exceed &2 , then we consider that expert e t agrees on the group preference; otherwise s=he does not agree on it. Similarly, Eq. (16) can be replaced with the following equation, if the degree of weak agreement is intended to use z t (a; b) = 1 − |R(a; b) − r t (a; b)|:
(17)
According to Eq. (15), the degree of which expert e t agrees on the preference order between all candidate items can be calculated 1 z t (a; b): (18) Zt = n(n − 1) a∈A b∈ A b=a
Therefore, the consensus measure of preference order is de%ned as CO(H ) = FQ (Z 1 ; Z 2 ; : : : ; Z m );
(19)
where FQ is also the OWA operator de%ned by Eq. (10). 4. Illustrated example: a #ight simulator software development An example of a 7ight simulator software development project is used to illustrate the concept developed. A 7ight simulator is a computer-based training devise that is designed for the training of pilots, especially for training hazardous emergency situation. In this project, the entire system consists of seven subsystems including a cockpit system, control loading system, host computer system, I=O control system, sound-and-graphic system, visual display system, and instructor operation system (see Fig. 1). Each subsystem can be decomposed into several tasks that can be further decomposed into a number of modules and the entire system contains hundreds of modules. In the past, the selection of CIs was all based on the experiences of decision makers. However, decision makers found that it is di@cult to make tradeoHs among a set of candidate items, when there are certain limits on the number of CIs due to the budget constraint. Therefore, it is required to have a decision-aid tool to assist decision makers to perform tradeoH analyses to determine a proper set of CIs. To evaluate a candidate item, nine selection criteria were used: security criticality, human safety criticality, mission criticality, %nancial criticality, design newness, interface criticality, hardware integrity, status sensitivity, and technology sensitivity. The linguistic scale used for evaluation is shown in Fig. 3. The relative importance of the nine criteria is evaluated with the linguistic scale listed in Table 1. The weight of each criterion can be obtained by transforming the assigned linguistic term
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363 Table 1 The linguistic scale for relative importance of evaluation criteria Linguistic scale
Quantitative scale
Very important Rather important Important Less important Unimportant
5 4 3 2 1
Table 2 Normalized weights for the nine criteria Criterion
Weight
Normalized weight
1. 2. 3. 4. 5. 6. 7. 8. 9.
3 2 5 1 4 4 3 2 5
0.1034 0.0690 0.1724 0.0345 0.1379 0.1379 0.1034 0.0690 0.1724
Security criticality Human safety criticality Mission criticality Financial criticality Design newness Interface criticality Hardware integrity Status sensitivity Technology sensitivity
to the corresponding quantitative value. The obtained normalized weights for all criteria are listed in Table 2. One of the most important subsystems of a 7ight simulator is the host computer system that consists of operating systems and application software for the dynamic and real-time simulation. Due to limited space, the navigation=communication task including nine modules (see Fig. 1) and the radar task including 18 modules in a hierarchical structure (see Fig. 6) are used in this paper for demonstration purpose. Four experts who are familiar with these two subsystems were invited and the produced rating results are listed in Appendix A. The function of the navigation=communication task is to determine the position of an aircraft and charting a course for guiding the craft safely and expeditiously from one point to another. It consists of nine candidate items numbered from “nv1 ” to “nv9 ”. The fuzzy preference relations between pairs of candidate items for all experts can be calculated from the ratings provided by the experts according to Eqs. (4) and (5). An example of the obtained fuzzy preference relations between pairs of candidate items is shown in Table 3. Then the dominance degrees of all candidate items for each expert can be determined in Table 4 by Eq. (6). According to Eq. (9), the group dominance degree of each candidate item can be calculated and a complete ranking order is obtained: nv9 nv5 nv4 nv2 nv6 nv1 nv8 nv3 nv7 . To obtain the partial order of candidate items, the fuzzy group preference relations can be determined in Table 5 according to Eq. (11). Assume that the indiHerence threshold ($) is set to 0.2, the partial ranking order of candidate items can be determined in Fig. 7.
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
355
rd1
rd2
rd3
rd4
rd5
rd6
rd7
rd8
rd9
rd11
rd12
rd13
rd14
rd15
rd16
rd17
rd10
rd18
Fig. 6. The work-breakdown structure of the radar task.
Table 3 Fuzzy preference relations between pairs of candidate items for expert e1 Item
nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
nv1 nv2 nv3 nv4 nv5 nv6 nv7 nv8 nv9
0.000 0.671 0.327 0.576 0.711 0.450 0.450 0.450 0.711
0.327 0.000 0.086 0.396 0.737 0.207 0.121 0.121 0.617
0.671 0.912 0.000 0.712 0.889 0.646 0.560 0.560 0.840
0.422 0.602 0.287 0.000 0.723 0.310 0.224 0.224 0.674
0.287 0.261 0.109 0.275 0.000 0.138 0.052 0.052 0.379
0.548 0.792 0.352 0.689 0.861 0.000 0.413 0.413 0.797
0.548 0.878 0.438 0.775 0.947 0.585 0.000 0.499 0.883
0.548 0.878 0.438 0.775 0.947 0.585 0.499 0.000 0.883
0.287 0.381 0.158 0.324 0.620 0.201 0.115 0.115 0.000
Table 4 Group dominance degrees of all candidate items for the navigation=communication subsystem
e1 e2 e3 e4 h
nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
0.419 0.394 0.359 0.328 0.370
0.624 0.512 0.441 0.425 0.468
0.254 0.354 0.304 0.340 0.313
0.525 0.574 0.525 0.671 0.545
0.727 0.628 0.517 0.708 0.649
0.365 0.373 0.573 0.583 0.452
0.290 0.276 0.357 0.271 0.281
0.290 0.382 0.418 0.167 0.314
0.723 0.571 0.697 0.714 0.691
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
Table 5 Fuzzy group preference relations between candidate items Item
nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
nv1 nv2 nv3 nv4 nv5 nv6 nv7 nv8 nv9
0.000 0.649 0.404 0.695 0.753 0.531 0.411 0.472 0.757
0.337 0.000 0.255 0.571 0.679 0.440 0.319 0.266 0.643
0.582 0.704 0.000 0.722 0.752 0.678 0.458 0.512 0.761
0.268 0.382 0.274 0.000 0.519 0.324 0.242 0.228 0.502
0.217 0.289 0.218 0.440 0.000 0.243 0.108 0.132 0.451
0.398 0.480 0.273 0.647 0.709 0.000 0.268 0.328 0.606
0.573 0.648 0.505 0.733 0.838 0.709 0.000 0.506 0.776
0.493 0.683 0.451 0.740 0.803 0.609 0.463 0.000 0.757
0.221 0.327 0.214 0.453 0.489 0.368 0.194 0.197 0.000
Level 1
nv4
nv5
nv9
Level 2
Level 3
nv1
nv6
nv2
nv8
nv7
nv3
Fig. 7. The partial ranking order of candidate items ($ = 0:2).
Fig. 8 shows the in7uence of the quanti%er-guided function to the group dominance degree of each candidate item. It is obvious that the group dominance degree of a candidate item will be decreased as the selected quanti%er-guided function is changed from “at least half” to “more than 70%”. In addition, the ranking order of candidate items may be aHected by the choice of the quanti%er-guided function and the value of indiHerence threshold (see Tables 6 and 7). As a tighter quanti%er-guided function is selected or the smaller indiHerence threshold is chosen, more discriminated partial orders of candidate items will be produced. The candidate item with a higher-ranking order has higher priority to be considered as a CI for management control. The number of selected con%guration items depends on the budget allocated to con%guration management. In this case, the project manager chooses the non-dominated set (i.e., the %rst level of the partial ranking order in Fig. 7) including nv4 , nv5 , and nv9 as CIs. If the budget is enough, then the items nv2 and nv6 at the second level of the partial ranking order (see Fig. 7) can be also taken into consideration. Item nv2 may have higher priority over nv6 , because it has a higher group dominance degree (see Table 4). According to Eqs. (12)–(19), the degrees of consensus at the current stage can be determined in Table 8, where &1 and &2 are assumed to be 0.1 and 0.15, respectively. If the project manager considers that the consensus among team members is not enough, he or she may encourage team
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0.8 At least half
Group dominance degree
0.7
Most
0.6
More than 70%
0.5 0.4 0.3 0.2 0.1 0 nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
CI candidates
Fig. 8. The in7uence of the quanti%er-guided function to the group dominance degree.
Table 6 Partial ranking orders of candidate items regarding diHerent quanti%er-guided functions for the navigation= communication task ($ = 0:2) Quanti%er guided function
Ranking order
“At least half ”
Complete Partial Complete Partial Complete Partial
“Most” “More than 70%”
nv9 nv5 nv4 nv2 nv6 nv1 nv8 nv3 nv7 {nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv3 ; nv7 ; nv8 } nv9 nv5 nv4 nv6 nv2 nv1 nv8 nv3 nv7 {nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv3 ; nv7 ; nv8 } nv9 nv5 nv4 nv2 nv6 nv1 nv7 nv3 nv8 {nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv8 } {nv3 ; nv7 }
Table 7 Partial ranking orders of candidate items regarding various threshold values for the navigation=communication task (Q = “Most”) IndiHerence threshold ($)
Partial order
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
{nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv8 } {nv3 ; nv7 } {nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv3 ; nv7 ; nv8 } {nv4 ; nv5 ; nv9 } {nv2 ; nv6 } {nv1 ; nv3 ; nv7 ; nv8 } {nv2 ; nv4 ; nv5 ; nv9 } {nv1 ; nv6 ; nv8 } {nv3 ; nv7 } {nv2 ; nv4 ; nv5 ; nv6 ; nv9 } {nv1 ; nv3 ; nv7 ; nv8 } {nv1 ; nv2 ; nv3 ; nv4 ; nv5 ; nv6 ; nv9 } {nv7 ; nv8 } {nv1 ; nv2 ; nv3 ; nv4 ; nv5 ; nv6 ; nv8 ; nv9 } {nv7 } {nv1 ; nv2 ; nv3 ; nv4 ; nv5 ; nv6 ; nv7 ; nv8 ; nv9 } {nv1 ; nv2 ; nv3 ; nv4 ; nv5 ; nv6 ; nv7 ; nv8 ; nv9 }
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363 Table 8 Consensus measures (&1 = 0:1 and &2 = 0:15)
Strict dominance Weak dominance Strict order Weak order
e1
e2
e3
e4
Consensus
0.889 0.844 0.569 0.904
0.889 0.879 0.764 0.927
0.669 0.868 0.694 0.915
0.667 0.852 0.542 0.895
0.756 0.858 0.786 0.907
Table 9 Group dominance degrees of all candidate items for the radar task Item
rd1
rd2
rd3
rd4
rd5
rd6
rd7
rd8
rd9
h
0.119
0.128
0.124
0.130
0.500
0.628
0.272
0.521
0.581
Item
rd10
rd11
rd12
rd13
rd14
rd15
rd16
rd17
rd18
h
0.620
0.595
0.639
0.623
0.597
0.619
0.557
0.594
0.624
members exchanging ideas between them and reevaluate again until the group consensus is reached. From this study, we found that the strict consensus measures are more meaningful than the weak consensus measures, if the values of &1 and &2 are set properly. In practice, decision makers can tolerate certain diHerences between his=her ideal solution and the proposed solution. The values of indiHerence thresholds are chosen based on the experience of decision makers. The function of the radar task is to locate objects beyond the range of vision, and to determine their distance. It includes 18 candidate items numbered from “rd1 ” to “rd18 ” in a hierarchical structure. Using the developed multi-criteria group decision-making model, the group dominance degree of each candidate item can be calculated (see Table 9) and a complete ranking order is obtained: rd12 rd6 rd18 rd13 rd10 rd15 rd14 rd11 rd17 rd9 rd16 rd8 rd5 rd7 rd4 rd2 rd3 rd1 . The partial ranking orders of candidate items with respect to various values of indiHerence threshold are listed in Table 10. In this case, the con%guration management board selected the set {rd5 ; rd6 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } as CIs, according to their experience. This result is the same as the set identi%ed in the %rst level of the obtained partial order, as $ = 0:5. If there is a budget constraint, we may decrease the value of $ and check which items can be removed from further consideration. We can observed from Table 10 that item rd8 is removed from the non-dominated set as $ = 0:4 and rd5 is also removed from the non-dominated set in addition to rd8 as $ = 0:3. Therefore, we may consider removing rd5 and rd8 from the set of CIs, if the budget is not enough. If a candidate item (e.g., rd5 ) is not determined as a CI, but it contains other sub-modules that have been determined as CIs, then some attention should be given for the item during the software development. We call the item a “pseudo” CI. This is because we still need to ensure that the interfaces among submodules of a pseudo CI will be properly designed and built to achieve its expected performance. In practice, the management cost of a “pseudo” CI can be neglected. In
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
359
Table 10 Partial ranking orders of candidate items regarding various threshold values for radar task (Q = “Most”) IndiHerence threshold ($)
Partial order
0.1
{rd12 ; rd13 ; rd15 } {rd6 ; rd14 ; rd17 } {rd10 ; rd18 } {rd9 ; rd11 } {rd8 ; rd16 } {rd5 } {rd7 } {rd2 ; rd3 ; rd4 } {rd1 } {rd6 ; rd10 ; rd12 ; rd13 ; rd15 ; rd17 } {rd9 ; rd11 ; rd14 ; rd16 ; rd18 } {rd5 ; rd8 } {rd7 } {rd2 ; rd3 ; rd4 } {rd1 } {rd6 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd5 ; rd8 } {rd7 } {rd2 ; rd3 ; rd4 } {rd1 } {rd5 ; rd6 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd8 } {rd7 } {rd1 ; rd2 ; rd3 ; rd4 } {rd5 ; rd6 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd1 ; rd2 ; rd3 ; rd7 } {rd4 } {rd5 ; rd6 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd1 ; rd2 ; rd3 ; rd4 ; rd7 } {rd5 ; rd6 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd1 ; rd2 ; rd3 ; rd4 ; rd7 } {rd5 ; rd6 ; rd7 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd1 ; rd2 ; rd3 ; rd4 } {rd5 ; rd6 ; rd7 ; rd8 ; rd9 ; rd10 ; rd11 ; rd12 ; rd13 ; rd14 ; rd15 ; rd16 ; rd17 ; rd18 } {rd1 ; rd2 ; rd3 ; rd4 }
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Table 11 The evaluation results and the selected CIs for four expertsa
e1 e2 e3 e4 a
nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
Selected CIs
1 −1 0 −3
−1 −1 2 −2
−6 −4 −3 −3
1 −2 4 8
7 1 4 5
−6 −3 5 2
−6 −4 −1 −8
−6 −5 1 −7
5 −1 8 6
nv1 ; nv4 ; nv5 ; nv9 nv5 nv1 ; nv2 ; nv4 ; nv5 ; nv6 ; nv8 ; nv9 nv4 ; nv5 ; nv6 ; nv9
The number in a cell indicates the diHerence between the numbers of ‘+’ and ‘−’
addition, according to the experienced project manager, the partial order of candidate items provides more information than the complete order in this research problem. It is preferred to classify the set of candidate items into diHerent subsets from the “best” to “worst”, rather than ranking candidate items into a complete order. A simple CI selection procedure [16] that is often used in the real world was applied to verify the result of the proposed approach. The four experts were asked to evaluate each candidate item with “yes” or “no” for the nine criteria. If the number of “yes” is greater than or equal to the number of “no”, then the candidate item will be considered as a CI. The navigation=communication task is used in this experiment. According to the above procedure, Table 11 lists the evaluation results for all candidate items and the selected sets of CIs for four experts. From the result, we can know that diHerent experts may have diHerent opinions about the proper set of CIs. For example, the attitude of expert e3 is
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J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
quite conservative; on the contrast, expert e2 is very optimistic. However, we can %nd that the items nv4 , nv5 , and nv9 are commonly chosen by three experts. These three items are also the top-three items in the identi%ed complete order and the non-dominated set in the partial order of the candidate items (see Fig. 7). This %nding is consistent with the result obtained by our approach. In addition, the obtained ranking order of the candidate item can help con%guration management board to select more proper set of CIs. Decision makers can perform sensitivity analysis by changing indiHerence threshold or the quanti%er-guided function to make more robust selection decisions. 5. Conclusion This research developed a multi-criteria group decision-making model based on fuzzy set theory to improve the con%guration items selection process. The model constructs imprecise preference relations among candidate items based on the concept of fuzzy majority and determines ranking orders among them. The consensus measures were also developed to determine the group acceptability of the obtained ranking orders. Decision makers can perform sensitivity analysis to examine the solution robustness. We concluded that the developed methodology can assist decision makers in selecting more proper set of CIs that can reduce the software life cycle cost. Acknowledgements This research is partially supported by grant No. NSC 91-2213-E-035-032 from the National Science Council of the Republic of China. Appendix A. Evaluation ratings by four experts for the #ight simulator software development The evaluation ratings provided by four experts for the nine candidate items in the navigation= communication subsystem are listed in the Tables 12–15, respectively. The set of linguistic terms Table 12 Linguistic ratings of nine candidate items regarding nine criteria by expert e1 Criteria
1 2 3 4 5 6 7 8 9
Candidate items nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
pn my py pn pn py py my pn
mn my py mn mn py mn my pn
pn mn uk pn dn my pn mn pn
my mn my my mn my pn mn my
my py py my mn py my my my
mn mn uk pn pn my pn mn pn
mn mn uk pn pn my pn mn pn
mn mn uk pn pn my pn mn pn
my py py my mn py mn my my
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
361
Table 13 Linguistic ratings of nine candidate items regarding nine criteria by expert e2 Criteria
1 2 3 4 5 6 7 8 9
Candidate items nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
dn dn py pn dn dy mn dy dy
dn dn dy dn dn my uk py uk
dn dn my dn dn uk mn py mn
dn dn dy mn dn py dy uk mn
dn dn dy dy pn dy dy dy mn
dn dn my dn dn dy pn dy pn
dn dn uk dn dn py dn my pn
dn dn dy dn dn py mn dn pn
dn dn dy mn dn py dy py mn
Table 14 Linguistic ratings of nine candidate items regarding nine criteria by expert e3 Criteria
1 2 3 4 5 6 7 8 9
Candidate items nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
dy uk py dn dn dy dy dn dn
dy uk dy dn dn dy dy dy dn
dn dn dy dn dn dy dn dy dn
dn uk dy uk dy dy dy uk dy
dn uk dy uk dy dy dy uk dy
pn uk dy uk dy dy dy dy dy
dn uk uk dn dn dy dy dy dn
dn dy dy dn dn dy dy dy dn
dy dy dy uk dy dy dy dy dy
Table 15 Linguistic ratings of nine candidate items regarding nine criteria by expert e4 Criteria
1 2 3 4 5 6 7 8 9
Candidate items nv1
nv2
nv3
nv4
nv5
nv6
nv7
nv8
nv9
my dn dy pn dn dy pn dn dn
py mn dy pn mn py pn uk pn
pn dn my dn dn my dy pn dn
dy py dy uk my py dy dy my
dy uk dy pn my dy py py uk
py my dy uk mn my mn py mn
pn dn mn dn dn pn uk mn dn
pn dn my dn dn dn pn dn dn
dy uk dy mn dy dy dy dy my
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used in the evaluation is {“de%nitely not (dn)”, “probably not (pn)”, “maybe not (mn)”, “unknown (uk)”, “maybe (my)”, “probably (py)”, “de%nitely (dy)”}. References [1] M. Fedrizzi, J. Kacprzyk, Consensus degrees under fuzzy majorities and preferences using ordered weighted average operators, Proceedings of the 5th IFSA World Congress, Seoul, Korea, 1993, pp. 624 – 626. [2] F. Herrera, E. Herrera-Viedma, J.L. Verdegra, A sequential selection process in group decision making with a linguistic assessment approach, Inform. Sci. 85 (1995) 223–239. [3] F. Herrera, E. Herrera-Viedma, J.L. Verdegra, A model of consensus in group decision making under linguistic assessments, Fuzzy Sets and Systems 78 (1996) 73–87. [4] F. Herrera, E. Herrera-Viedma, J.L. Verdegra, A rational consensus model in group decision making using linguistic assessments, Fuzzy Sets and Systems 88 (3) (1997) 31–49. [5] F. Herrera, E. Herrera-Viedma, J.L. Verdegra, Choice processes for non-homogeneous group decision making in linguistic setting, Fuzzy Sets and Systems 94 (3) (1998) 287–308. [6] IEEE Guide to Software Con%guration Management, The Institute of Electrical and Electronics Engineers, 1988. [7] IEEE Standard for Software Con%guration Management Plans, The Institute of Electrical and Electronics Engineers, 1998. [8] ISO-10007, Quality Management—Guidelines for Con%guration Management, 1995. [9] J. Kacprzyk, Group decision making with a fuzzy linguistic majority, Fuzzy Sets and Systems 11 (1986) 105–118. [10] J. Kacprzyk, M. Fedrizzi (Eds.), Multiperson decision making models using fuzzy sets and possibility theory, Kluwer, Dordrecht, 1990. [11] J. Kacprzyk, M. Fedrizzi, H. Nurmi, Group decision making and consensus under fuzzy preferences and fuzzy majority, Fuzzy Sets and Systems 49 (1992) 21–31. [12] G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, New Jersey, 1995. [13] W. Kolodziejczyk, Orlovsky’s concept of decision-making with fuzzy preference relation—further results, Fuzzy Sets and Systems 19 (1) (1986) 11–20. [14] M.U. Marimin, I. Hatono, H. Tamura, Linguistic labels for expressing fuzzy preference relations in fuzzy group decision making, IEEE Trans. Systems Man Cybernet.—Part B: Cybernet. 28 (2) (1998) 205–218. [15] A.K. Midha, Software con%guration management for the 21st century, Bell Labs Tech. J. 2 (1) (1997) 154–165. [16] MIL-STD-483A, Con%guration Management Practices for Systems, Equipment, Munitions, and Computer Programs. [17] MIL-STD-973, Con%guration Management. [18] K. Nakamura, Preference relations on a set of fuzzy utilities as a basis for decision making, Fuzzy Sets and Systems 20 (2) (1986) 147–162. [19] H. Nurmi, Approaches to collective decision making with fuzzy preference relations, Fuzzy Sets and Systems 6 (1981) 249–259. [20] P. O’Connell, R. McCrindle, Using SOAP to clean up con%guration management, Proc. IEEE Computer Society’s International Computer Software and Applications Conference, Chicago, IL, 2001, pp. 555 –560. [21] S. Srinath, R. Ramakrishnan, D.J. Ram, Generic model for semantics-based versioning in projects, IEEE Trans. Systems Man Cybernet. Part A: Systems Humans 30 (2) (2000) 108–123. [22] T. Tanino, Fuzzy preference orderings in group decision making, Fuzzy Sets and Systems 12 (1984) 117–131. [23] T.Y. Tseng, C.M. Klein, New algorithm for the ranking procedure in fuzzy decision making, IEEE Trans. Systems Man Cybernet. 19 (5) (1989) 1289–1296. [24] J. Wang, A fuzzy outranking method for conceptual design evaluation, Internat. J. Production Res. 35 (4) (1997) 995–1010. [25] B. Westfechtel, B.P. Munch, R. Conradi, A layered architecture for uniform version management, IEEE Trans. Software Eng. 27 (12) (2001) 1111–1133. [26] D. Whitgift, Software Con%guration Management: Methods and Tools, Wiley, England, 1991. [27] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Systems Man Cybernet. 18 (1988) 183–190.
J. Wang, Y.-I. Lin / Fuzzy Sets and Systems 134 (2003) 343 – 363
363
[28] R.R. Yager, Families of OWA operators, Fuzzy Sets and Systems 59 (1993) 125–148. [29] L.A. Zadeh, A computational approach to fuzzy quanti%ers in natural languages, Comput. Math. Appl. 9 (1) (1983) 149–183. [30] L. Zhang, H. Mei, H. Zhu, A con%guration management system supporting component-based software development, International Computer Software and Applications Conference, Chicago, IL, 2001, pp. 25 –30.