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Qualitative Group Decision Making Process by Applying an Innovative Fuzzy-Neural Approach Ki-Young Song and Janusz Kozinski Earth & Space Science & Engineering York University Toronto, ON, Canada
Gerald T.G. Seniuk
Madan M. Gupta*
College of Law University of Saskatchewan Saskatoon, SK, Canada
Intelligent Systems Research Laboratory College of Engineering University of Saskatchewan Saskatoon, SK, Canada *author in correspondence:
[email protected] Abstract— Many qualitative decisions that appear crisp and certain are grounded in fuzziness as a result of the uncertain environment within which the decisions are made. We propose an innovative approach for decision making in such uncertain conditions by using a fuzzy-neural method. The key idea of this proposed approach is to adapt the synaptic neural weight by applying a variance weighting process (VWP) which reduces the fuzziness of group opinions. If such decisions were made by an individual, the fuzzy environment underlay of the apparent crisp decision may not be as visible as it would be within a group environment where the various opinions are observable. Applying VWP, the fuzzy environment can be adjusted by assigning higher neural weights for those factors with lower variance within the group and lower neural weight for those factors with higher variance within the group. In this paper, a simulation study with the proposed fuzzy-neural approach was carried out. The result of the study indicates that this proposed approach can improve the effectiveness of qualitative decision making by providing the decision maker with a new cognitive tool to assist in the reasoning process.
information available. However, in fuzzy decision environments as those listed above, there is no opportunity to correct the decision after it has been made. Therefore, such decision making environments could be assisted by providing some feedback mechanism that allows various outcomes to be assessed and evaluated.
Keywords- fuzzy environment; variance weighting process (VWP); fuzzy-neural decision making process (FNDMP)
However, in reality, the processes of decision making are hampered by various assumptions (noise), biases and uncertainties. These three factors (noise, bias and uncertainty) can be defined as a fuzzy environment, and the fuzzy environment results in the following three serious weaknesses: i) the decision does not represent reality, ii) there is growing skepticism of the validity of using general principles in the absence of specific content of knowledge, and iii) there is an increasing awareness of the biases and other limitations that characterize the thinking of individuals [6, 7]. Therefore, the continual search to develop a superior method to enhance the quality of decisions for helping decision makers overcomes these limitations.
I.
INTRODUCTION
In our daily life, we frequently face ambiguous multiple choices in our decision making. Decision making involves perception of fuzzy information, and thereby that of our cognitive process for the evaluation of suitable courses of action among several alternative choices [1]. Such decisions are made in economics, finance, medical diagnostics, public policy-making, law and indeed in any field where human judgment is called upon to choose among competing and viable options. These are decisions that are made with incomplete or conflicting evidence or data. Under such constraints, the rational option is to postpone the decision and make further studies. But many personal and societal decisions cannot be postponed – for example, whether to operate on an emergency patient, whether or how to respond militarily to a threat, or whether to hold or sell in a dynamic market. Slight improvements in the effectiveness of these cognitive processes can make significant differences in the outcomes resulting from the subsequent decisions. In order to have the best possible decisions in such cases, it would be necessary to have an opportunity to continuously improve and adjust the decision based on feedback and new
For decades, a number of methods to improve the quality of decisions have been developed by applying some new mathematical methods. Also, today tools and machines using computers have received great attention for better quality of decision making [1]. Fuzzy logic has been one of the major candidates as a mathematical tool in decision making processes. One application of fuzzy logic in decision making was to employ fuzzy linguistic quantifiers which quantify linguistic expressions such as ‘good’, ‘fair’, and ‘bad’ [2, 3]. Some other ways of using fuzzy logic were to define uncertainties with the possibility of events for strategic decisions [4]. Some other approaches of fuzzy logic on group decisions can be found in the reference [5].
In this paper, we introduce a simpler but powerful novel decision making model by applying fuzzy-neural approach, named fuzzy-neural decision making process (FNDMP). FNDMP is developed for qualitative decision making from group opinions (evaluations). The procedure of FNDMP is partitioned mainly in two segments: handling fuzzy environment (noise, bias and uncertainty) and making a group decision. The fuzzy environment from the evaluations is managed by a variance weighting process (VWP) which is a simpler way to measure the fuzziness of group opinions. Further, VWP is applied for synaptic weights of a neural net. In
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2 the group decision segment of this study, neural methods are employed to make a final group decision applying VWP and also applying classified pre-assigned weights. We present the performance of FNDMP with a generic case study as one would encounter in a qualitative environment as opposed to the crisper quantitative environment, such as in control systems. II.
METHODOLOGY
The basic model we build upon is the confluence of fuzzy logic and neural methods [8]. Using this approach, however, we propose one significant modification to the synaptic weights, namely a variance weighting process (VWP) which is the process of modifying the weights with the variance in rankings of different evaluators. When we examined the qualitative language of decision making in a group, it became apparent that the opaqueness of the process hides the impact of this dimension of variance in the decision making process. Without allowing for the incorporation of VWP, the decision making will be plagued by the variance resulting from the undue influence of the unscreened subjective ranking tacitly present in any qualitative decision making process. A. Variance weighting process (VWP) In this process, qualitative language is translated to quantitative language. Zadeh [9] described this process as ‘defuzzification’. In a group decision making process, the defuzzifier will be the various decision-makers (evaluators) who will each provide a numerical score to reflect their qualitative evaluation of a particular set that they are each scoring. For example, if someone were assessing the quality of various tea brands, the assessor may consider a quality of “tasty” as being “very high” whereas another may score it as “good, but tart”. To defuzzify their assessments, the first tester might score the taste as “9” out of 10, while the second might score it as “6”. These are the subjective values assigned, and this is the most common method used for the defuzzification of the subjective evaluation. Such rankings are the normal starting position, from which we also begin our study. In the nature of decision making, some of the more general language of evaluation research [10] is used to capture the decision making process. Let us imagine a generic decision-making case considering m evaluators and n factors as shown in Table 1.
minimum and maximum range. The averages of each factor are calculated as 1
,
1,2, … ,
(1)
As found in the literature, current decision-making processes consider only the average of the individual scores, which does not quite signify the nature of the data (evaluations). In our approach presented in this paper, we use both the averages and variances of the individual scores on each factor, and this process in a decision-making represents the nature of for each factor is the data more precisely. The variance determined by the spread of the evaluators’ subjective evaluations of a particular factor, from which the development variance weighting process (VWP) was motivated to handle the fuzzy environment (noise, bias and uncertainty) of evaluations in each factor. During group decision making processes, it is commonly observed that some evaluations from different individual evaluators for certain factors are expressed by scores in some close interval. It is also to be noted that some diversity of scores in the evaluation process for certain factors arises due to the different experiences, subjectivity and cognitions of individual evaluators. The diversity indicates that the scoring in each factor is subjective to individual experience and his perception and cognition. In order to assess the agreement of evaluations for each factor, it is necessary that some weight be assigned to each factor in the evaluation process. This basic fuzzy logic principle of variance weighting process (VWP) is expressed as follows: •
If the evaluations for a factor are in general agreement, that is the variance is relatively low, then the weight of the factor becomes higher.
•
If the evaluations for a factor are not in general agreement, that is the variance is relatively high, then the weight of the factor becomes lower.
The variance and an inverse variance function are defined respectively in Eqns.(2) and (3). 1
,
1,2, … ,
(2)
1,2, … ,
(3)
TABLE 1: EVALUATION MATRIX (m × n) FOR FNDMP
exp
FACTORS
F1
F2
…
Fj
…
Fn
s11 : si1 : sm1
s12 : si2 : sm2
… : … : …
s1j : sij : smj
… : … : …
s1n : sin : smn
μ1 μ2 … μj … Average (µ) v1 v2 … vj … Variance (v) w1 w2 … wj … Factor Weight (w) (m = number of evaluators and n = number of factors)
μn vn wn
EVALUATORS
E1 : Ei : Em
Factors are the characteristics of a case to be evaluated, and the evaluators evaluate the factors with scores within some
,
where exp • is a exponential function, and α is the gain that affects the weighting function as shown in Figure 1.
wj α
vj Figure 1. An inverse weight function to express the principle of VWP. α is representing the importance of variances, Eqn.(3). As α increases, the slope of the weight function becomes steeper.
3
B. Neural model for decision making [11, 12] After applying VWP for individual evaluations, the next process of decision making is to analyze the consensus by a classification method. Neural networks which are composed of many neural units are one of the most powerful tools for classification, and neural networks provide a superiority of identification and classification and have therefore been widely used in this type of research [13]. A neural unit was inspired by the study of biological neurons such as the synaptic operation and the somatic operation. In the synaptic operation, new input information is perceived through the memory cell (w). In the somatic operation, the perceived information is processed by a linear or nonlinear mapping function ( Ф • ). A sigmoidal function is commonly applied for a mapping function due to its special characteristics exhibiting a progression from small beginning to an accelerated end as natural processes [14]. Considering the features of conventional neural networks, a new neural model and neural structures were developed as shown in Figure 2 and Figure 3. The neural model consists of three layers: one input layer, one hidden layer and one output layer. In this study, we do not incorporate a dynamic feedback (such as back-propagation algorithm) in the neural model; therefore, we apply a static model. The input layer of the neural model corresponds to the input vectors which are the averages of evaluations of each factor. Since the average cannot handle the fuzzy environment (noise, bias and uncertainty) of group evaluations, variance weighting process (VWP) is applied as the factor weight (W) (see Figure 3(a)) as discussed in the previous section. The average and VWP are defined in Eqns. (1)~(3). The synaptic operation ( ) and the somatic operation ( ) of neural units in the input layer are defined as (4)
where is the input vector (outputs from the input layer), k is the number of factor categories 1,2, … , , Ф • is a is a gain. linear function, and
Figure 2. A new neural net model for group decision making process. This net is composed of an input layer, a hidden layer and an output layer.
(a)
Neural structure of the input layer
∑
(b)
Neural structure of the hidden layer
(5)
Ф where is average of a factor is a gain. linear function, and
(7)
Ф
Input vector (outputs of input layer)
As shown in Figure 1, it should be noted that as variance increases, the significance of the weight decreases. Also, the relationship between and can be tuned by α which implies the significance of the relationship.
1,2, … ,
, Ф • is a
In the next step, the outputs in the input layer are transmitted to the hidden layer. In this study, we define the hidden layer of the neural model as categorization of factors. In most cases of outlining factors for evaluation, it is common that some factors are grouped in the same category where these factors have some common characteristics. Some categories are more important for a decision, but some are not. The importance of a category is pre-assigned by unanimous group (value of category) (see Figure 3(b)). The assessments as number of neural units in the hidden layer is determined by the ) number of categories of factors. The synaptic operation ( ) of neural units in the hidden and the somatic operation ( layer are defined as (6)
∑
(c)
Neural structure of the output layer
Figure 3. New neural structures of the input layer, the hidden layer and the output layer for group decision assistant process.
The output layer in the neural model completes the group decision by accumulating the outputs from the hidden layer. The neural weights in the output layer are given a value of “1” representing equal importance of inputs (see Figure 3(c)). The number of neural units in this layer will be taken based on the ) number of decisions to make. The synaptic operation (
4 and the somatic operation (Y) of neural units in the output layer are defined as (8) Ф
(9)
where is the output from the hidden layer, k is the number is threshold, Ф • is of factor categories 1,2, … , , is a gain which represents the a sigmoidal function, and crispness of a decision. The model that is being proposed in this paper is an analytical tool to aid in the decision process. This is a generic mathematical decision analysis model that can be applied in a variety of real life decisions made in a qualitative language environment. The advantage of applying a defuzzifier in the model to the qualitative (fuzzy) environment is that it allows for more precisiation [15, 16] without at the same time artificially losing the fuzzy richness of the reality under consideration. III.
RESULTS AND DISCUSSION
IV.
A. Validation of FNDMP with Group Decision First, we calculated averages for neural inputs of the neural model and processed the variance weighting process (VWP) for factor weights applying the inverse function with gain α 1. After that, the obtained data were processed in the neural model with pre-assigned weights as described in Figure 2 and Figure 3. Since ‘not relevant (X)’ evaluations were not accounted in the decision, ‘X’ was excluded from the calculation, and the modified equations for averages, variances and VWP are defined as 1
A CASE STUDY: NEW PRODUCT DEVELOPMENT
In industries, it is fundamental to research on a new product if the product can be competitive to competitors’ and/or be successful in the market before deciding a new product line. Accurate decision making on the new product development becomes more important, and a committee of experts evaluates the idea of the new product with various factors. Recently, many applications of fuzzy set theory have been found in industrial engineering researches by employing evaluations and survey analysis [17-19]. We apply FNDMP to a group decision of a new product development with 50 decision-makers (Evaluators) considering 14 factors for this new product. A survey (benchmark) was carried out using these 50 evaluators, and their evaluations were expressed by scores between -10 and 10 as given in Table 2. Table 2 shows evaluation for each factor by individual evaluators as well as averages, variances and variance weights for each factor. ‘X’ in the table indicates that the evaluator could not evaluate the factor because of the following reasons: i) the evaluator considered that the factor was not important to this decision, or ii) the evaluator did not consider himself familiar with the factor. This allowed the application of VWP to the process with the value determined by the degree of variance. For this particular case study having 14 factors, we divided these factors into the following four categories with weights ( ) as follows: • • • •
In bivalent “YES” or “NO” decisions, FNDMP determines the degree of confidence in the decision ( 100%) between -1 (representing ‘100% NO’) and 1 (representing ‘100% YES’). We define that over 30% confident of a decision will authenticate the decision.
Category 1: F2, F5, F6, F7, F8, F9 and F11, Category 2: F2, F3, F5, F6, F7, F9, F11 and F12, Category 3: F2, F5, F6, F7, F9, F11 and F13, 1 Category 4: F1, F4, F10 and F14,
15 5
10
It should be noted that some of the factors are common to various categories such as F2 is included in Categories 1, 2 and 0 and the gains 3. We also assigned threshold 1, 0.25 for the somatic operations by postulating that this case study did not require high crispness on a decision but only some degree of confident in the decision.
(10)
1
(11) exp
(12)
where is the number of ‘X’s in the evaluation of -th factor 1,2, … , , 50 , and 14 . The values of the neural inputs and factor weights are listed in Table 2. After completing the FNDMP, the decision value became 0.1 which meant that FNDMP considered the group agreed that the answer was “NO”. However, the group might not be confident with the decision since the degree of confidence was 10% (0.16 out of 1). The benchmark group from which we preassigned means and standard deviations resulted in 44% of evaluators determining the answer was “NO” with 21% 100% ). The group decisions from confident ( FNDMP and the benchmark group are illustrated in Figure 4. As seen, both group decisions lie in the same judgment and in the low confidence range where the group decision may be easily changed from positive to negative or negative to positive when some of evaluators change their opinions. The lower the confidence range is, the easier the decision is changed. Considering potential errors in the randomly generated evaluation values from the assigned benchmark values, it can be concluded that the proposed FNDMP with pre-assigned parameters is valid for use in such decision making environments. B. Fact-Finding Process of Individual Evaluator by FNDMP After validating FNDMP with the group decision, all the factors could be ranked by the values of weights. The process of making such rankings in itself would aid the decision maker in his or her thinking. “Did the evaluator place too much emphasis on that factor?” is a question that would then be open to meaningful review, either by others or the individual alone. The exploration of an evaluation is called “fact-finding process”. Thus, we applied FNDMP to review the decision of
5 an individual evaluator. In this review process, the decisions of individual evaluators were examined with feedbacks by varying their evaluations on certain factors. From the results shown in Table 2, it was found that F4 and F10 gave the higher values of factor weight indicating that these two factors were most dominating factors in decision making. In this case study, we explored E1’s evaluation. Before changing the evaluations, we found that the decision confidence of E1 was -0.07 applying FNDMP with the evaluations of E1. The results of the fact-finding process of E1 are shown in Figure 5. The lower values of factor weights had little effect on the individual decision process. In the figure, the decision changes of E1 with different evaluation of F1 are shown (red squares in Figure 5). The red squares are
accumulated at certain point, which represents that the decisions of E1 are mostly same regardless of E1’s evaluations of F1. The changes of F2, F3, F5, F6, F7, F8, F11, F12, F13 and F14 resulted in the similar outcomes. However, the other factors (F4, F9 and F10) played significant roles in a decision making. In the figure, the decision changes of E1 with different evaluation of F10 are described (yellow circles in Figure 5). The yellow circles are widely distributed along the decision confidence, which indicates that the decisions of E1 are actually changing with E1’s different evaluation of F10. Some of the decisions are located in confident zone as well as in different judgment. The changes of F4 and F9 resulted in the similar outcomes. Regarding the remaining Evaluators’ evaluations, the group decision might be dramatically varied and altered.
TABLE 2: EVALUATION OF FACTORS AND AVERAGES, VARIANCES AND WEIGHTS OF EACH FACTOR.
Evaluators
E1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 E 10 E 11 E 12 E 13 E 14 E 15 E 16 E 17 E 18 E 19 E 20 E 21 E 22 E 23 E 24 E 25 E 26 E 27 E 28 E 29 E 30 E 31 E 32 E 33 E 34 E 35 E 36 E 37 E 38 E 39 E 40 E 41 E 42 E 43 E 44 E 45 E 46 E 47 E 48 E 49 E 50 (×10‐3)
F1 5 X ‐3 5 1 2 ‐2 3 4 4 2 5 X ‐1 3 X 1 5 ‐2 3 6 ‐1 3 X 7 4 5 2 X 1 2 1 3 ‐1 ‐1 X ‐3 X 3 X ‐2 5 X 4 ‐1 ‐4 6 1 3 1
F2 ‐1 X ‐4 ‐2 ‐1 ‐1 ‐2 ‐1 ‐3 ‐2 1 ‐5 ‐3 2 ‐5 ‐2 3 1 ‐5 ‐4 ‐3 ‐2 X X ‐7 2 ‐2 ‐7 ‐6 ‐3 ‐3 ‐1 4 ‐2 ‐5 ‐5 ‐4 1 ‐1 X ‐1 ‐6 ‐2 ‐1 ‐1 ‐2 ‐1 ‐2 ‐2 ‐8
F3 ‐1 5 9 X 4 5 3 2 2 ‐2 5 3 ‐2 X ‐1 8 4 ‐2 5 7 5 ‐1 1 6 1 2 ‐2 2 1 1 2 1 3 3 3 7 X 6 2 1 ‐1 ‐3 4 2 4 8 9 ‐2 X 1
1.93
‐2.26
2.61
7.97 0.3
6.82 1.1
9.98 0
F4 X 1 2 1 X ‐3 1 X 1 ‐1 1 1 1 2 ‐1 X ‐1 1 1 3 ‐1 X 1 1 1 X 2 ‐1 ‐1 ‐1 X ‐1 ‐2 X X X 1 1 ‐2 X X 2 ‐1 ‐3 ‐1 ‐2 1 ‐1 ‐1 X
F5 ‐6 ‐3 ‐2 ‐6 X ‐1 ‐6 ‐4 X ‐7 ‐4 ‐3 X ‐5 ‐7 ‐6 4 ‐4 ‐6 ‐5 X X ‐3 X 3 ‐6 ‐6 X X ‐4 X ‐3 ‐3 ‐1 1 ‐5 ‐2 ‐3 ‐3 ‐2 1 ‐4 ‐2 2 ‐3 2 ‐2 ‐2 1 ‐4
F6 ‐5 ‐2 ‐1 ‐4 ‐5 ‐6 1 2 ‐1 1 6 X X 1 2 ‐1 ‐4 ‐2 4 1 3 ‐2 ‐1 3 ‐2 ‐6 ‐2 ‐1 X 2 ‐1 1 ‐2 ‐1 2 3 ‐1 ‐4 X ‐1 ‐4 ‐3 ‐3 X X X X ‐5 1 1
0.05
‐2.90
‐0.86
Factors F7 1 ‐2 ‐7 1 ‐2 X ‐2 ‐4 X ‐1 ‐4 ‐4 ‐3 2 ‐2 X 5 1 ‐1 ‐6 ‐4 ‐2 4 ‐3 ‐6 X ‐7 ‐8 3 3 ‐10 ‐3 ‐4 ‐1 ‐5 3 ‐2 ‐7 3 ‐2 ‐6 ‐2 ‐8 ‐2 ‐2 1 ‐8 ‐3 1 ‐4 ‐2.37
F8 3 5 9 10 ‐1 2 2 5 8 6 5 3 4 6 10 7 5 ‐1 3 10 10 3 5 X 1 7 2 9 7 4 10 7 8 8 8 3 3 9 ‐2 3 10 7 1 5 6 4 6 5 ‐1 3
F9 X ‐5 X X ‐2 X ‐2 1 ‐4 ‐1 ‐5 ‐2 ‐1 ‐1 ‐2 1 ‐2 3 ‐3 ‐2 ‐1 X X X ‐5 ‐1 ‐4 1 ‐3 4 ‐1 1 ‐4 ‐2 X ‐2 X 2 ‐1 ‐3 ‐2 ‐2 4 ‐1 X 1 ‐1 1 2 ‐2
F10 X ‐1 1 ‐1 X X 2 X 1 X X X 2 1 1 ‐1 1 1 1 X 1 1 X 1 X 2 X 1 2 X 1 ‐1 X 1 X X X X X ‐1 X 1 X X X X 2 X ‐1 ‐2
F11 ‐2 4 ‐2 ‐3 X ‐1 ‐4 1 1 3 ‐6 X ‐3 7 1 X ‐4 ‐1 ‐1 X 2 X 1 1 X 4 X 5 ‐1 ‐6 ‐1 ‐3 ‐4 ‐2 3 4 6 2 ‐4 ‐1 1 ‐2 ‐1 1 1 1 ‐3 ‐3 ‐3 ‐4
F12 1 5 6 2 5 6 3 5 ‐7 5 4 1 2 5 X X 1 3 8 ‐2 ‐6 ‐4 3 ‐4 ‐1 3 ‐2 5 3 7 2 X 2 ‐1 1 ‐4 X 9 4 4 5 4 ‐2 8 4 3 ‐3 ‐4 ‐1 3
F13 4 2 ‐1 6 X 1 6 ‐1 8 4 ‐2 6 6 7 2 7 6 2 8 ‐2 8 9 10 5 7 5 1 X 2 1 1 8 6 ‐1 1 4 ‐1 4 9 2 3 6 ‐1 3 X 1 1 ‐2 2 X
F14 3 7 4 5 ‐1 5 4 X 2 7 X 5 3 5 9 7 7 9 4 3 ‐1 3 4 2 3 1 6 6 5 9 2 5 ‐1 2 7 ‐3 7 X ‐1 X X ‐3 1 X 6 3 4 4 7 2
5.14
‐1.15
0.62
‐0.37
1.98
3.54
3.82
2.27 7.74 8.13 12.77 10.58 5.26 1.37 9.91 14.69 11.54 9.22 102.8 0.4 0.3 0 0 5.2 255.1 0 0 0 0 ‘X’ represents ‘not relevant’. The two highest weights corresponding to the two lowest variances are in bold and italic.
6 the evaluators can achieve more crisp decisions. FNDMP is a useful tool to be applied for decision making in qualitative language environments such as in business or in the court of law. REFERENCES [1]
[2] [3] Figure 4. Group decisions by FNDMP and evaluators. Both group decision lie in the lower confident zone.
[4] [5]
[6] [7] [8] [9] Figure 5. Fact-finding of Evaluator 1 varying the factor evaluation from -10 to 10. Red square shows decision change varying Factor 1. Yellow circle represents decision change varying Factor 10. The dash lines indicate the boundary of confident zones.
V.
CONCLUSIONS
In this paper, we introduced a novel approach for decision making process, the fuzzy-neural decision making process (FNDMP), employing a new fuzzy-neural approach for quantitative group decision making. The key idea of this proposed approach is to change the synaptic neural weight by applying a variance weighting process (VWP) which provides the importance of each factor, thereby reduces the fuzziness of group decision making. A case study of a group decision making of a “YES” or “NO” variety on a new product development was carried out to demonstrate the application of FNDMP. The proposed decision analysis algorithm outlined in this paper provided more information and a more secure outcome. The results from the study showed that FNDMP can assist such decision making processes. The result of FNDMP reveals and identifies which variances of which factors can impact the group. Further, FNDMP was applied for individual fact-finding process using the outcomes from group decision. FNDMP showed how the individual decision could be changed by altering the evaluation of certain factors. After changing individual decisions, the group decision may also be changed. In summary, FNDMP can assist evaluators to make a decision by weighing and ranking factors to be evaluated, and
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