An Interactive Data-Driven (Dynamic) Multiple Attribute Decision Making Model via Interval Type-2 Fuzzy Functions
Abstract
:1. Introduction
- Decision makers are expected to fill out tedious questionnaires to articulate their preferences over alternatives at each period. This is especially very time consuming and demanding when the number of criteria and alternatives are high, and the decision points are frequent, i.e., performance evaluation, risk assessment, etc.
- The models do not provide any mechanism to help decision makers making use of past decision making matrices when articulating their preferences at the current period. An interactive mechanism is needed to facilitate preference elicitation in the light of historical performance of alternatives.
- A dynamic MADM model is proposed based on a new IT2F functions approach.
- An interactive procedure is provided that the current decision making matrix is predicted in forms of IT2F sets. Moreover, vocabulary matching procedure is developed so that the predicted performance scores of alternatives are recommended to the decision makers through linguistic terms such as low, medium, high, etc.
- The proposed model interacts with decision makers whose subjective judgments are combined with the notion of “let the data speak for itself”. By providing decision makers with data-driven suggestions regarding the performance of alternatives, preference elicitation effort at each period is considerably reduced.
- The proposed model does not require any technical knowledge such as fuzzy sets, t-norms, t-conorms, implication functions, etc. The proposed model can be easily integrated into the legacy systems of the firms, since the crisp values are processed when providing IT2F outputs.
- A real-life personnel promotion problem is used to demonstrate the applicability of the proposed model. Rankings of employees are calculated based on past and current performance matrices with appropriate time series weights.
2. Theoretical Background
2.1. Traditional Dynamic Multiple Attribute Decision Making
2.2. Possibilistic Fuzzy Regression
2.3. Turksen’s Fuzzy Functions Approach
- Membership functions pertaining to antecedent and consequent parts of the fuzzy rules should be identified.
- Aggregation of antecedents requires selection of suitable conjunction and disjunction operators (t-norms, t-conorms).
- Proper implication operators should be identified for representation of the rules, which can be a challenging issue.
- A suitable defuzzification method should be identified.
3. Developed IT2F Model
3.1. Interval Type-2 Fuzzy Sets
3.2. IT2F Regression Model
3.3. Dynamic MADM Model via Proposed IT2F Functions
3.3.1. Phase-I: Problem Structuring
3.3.2. Phase-II: Training of Fuzzy Functions Approach
3.3.3. Phase-III: Ranking of Alternatives
- ,
- ,
- ,where is a monotonically non-decreasing function defined in the unit interval .
4. Case Study
4.1. Structering Personnel Promotion Problem
4.2. Estimating the Current Decision Matrix
4.3. Ranking of Employees
5. Discussion
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Period | Past Decision Matrices | Corresponding Time Series | Input Matrix | Output Matrix |
---|---|---|---|---|
t1 | ||||
t2 | ||||
t3 | ||||
t4 |
Period | C1 | C2 | C3 | C4 | C5 |
---|---|---|---|---|---|
1 | 6 | 7 | 3 | 3 | 7 |
2 | 6 | 7 | 2 | 2 | 6 |
3 | 7 | 8 | 3 | 3 | 5 |
4 | 6 | 9 | 4 | 5 | 5 |
5 | 6 | 8 | 3 | 6 | 5 |
6 | 6 | 9 | 4 | 8 | 4 |
7 | 7 | 8 | 4 | 8 | 4 |
8 | 6 | 7 | 5 | 7 | 4 |
9 | 6 | 6 | 6 | 8 | 5 |
10 | 6 | 5 | 6 | 8 | 5 |
11 | 7 | 4 | 7 | 9 | 6 |
12 | 6 | 3 | 6 | 10 | 6 |
13 | 5 | 4 | 6 | 10 | 6 |
14 | 5 | 4 | 8 | 8 | 6 |
15 | 4 | 4 | 6 | 7 | 8 |
16 | 4 | 3 | 6 | 7 | 7 |
17 | 4 | 3 | 6 | 7 | 8 |
18 | 4 | 2 | 6 | 9 | 8 |
Alternative | Criteria | Parameters | ||
---|---|---|---|---|
c | m | Lagged_Periods | ||
1 | 1 | 2 | 1.6 | 5 |
2 | 5 | 1.6 | 5 | |
3 | 4 | 2.1 | 4 | |
4 | 5 | 2.1 | 5 | |
5 | 4 | 1.6 | 4 | |
2 | 1 | 4 | 1.6 | 5 |
2 | 5 | 2.1 | 5 | |
3 | 2 | 1.1 | 5 | |
4 | 5 | 2.1 | 5 | |
5 | 5 | 1.6 | 5 | |
3 | 1 | 2 | 2.1 | 5 |
2 | 2 | 1.6 | 5 | |
3 | 3 | 1.6 | 5 | |
4 | 5 | 2.1 | 5 | |
5 | 5 | 1.6 | 5 |
Variable | IT2F Regression Coefficients | IT2F Coefficients | ||||
---|---|---|---|---|---|---|
b | f | g | p | q | ||
1 | 10.572 | 8.65 × 10−9 | 0.031 | 9.30 × 10−11 | 9.13 × 10−11 | ((10.572, 10.572, 10.603;1), (10.572, 10.572, 10.603;1)) |
−8.524 | 3.39 × 10−9 | 0.070 | 6.45 × 10−11 | 6.44 × 10−11 | ((−8.524, −8.524, −8.454;1), (−8.524, −8.524, −8.454;1)) | |
−4.550 | 2.04 × 10−9 | 0.016 | 2.79 × 10−11 | 2.78 × 10−11 | ((−4.55, −4.55, −4.534;1), (−4.55, −4.55, −4.534;1)) | |
13.466 | 7.27 × 10−9 | 0.085 | 7.34 × 10−11 | 7.34 × 10−11 | ((13.466, 13.466, 13.551;1), (13.466, 13.466, 13.551;1)) | |
0.017 | 0.146043 | 0.010 | 2.67 × 10−11 | 2.59 × 10−11 | ((−0.129, 0.017, 0.027;1), (−0.129, 0.017, 0.027;1)) | |
−0.613 | 1.24 × 10−9 | 0.005 | 0.1 | 0.185714 | ((−0.713, −0.613, −0.423;1), (−0.613, −0.613, −0.609;1)) | |
−0.211 | 1.44 × 10−9 | 0.004 | 1.67 × 10−11 | 1.63 × 10−11 | ((−0.211, −0.211, −0.207;1), (−0.211, −0.211, −0.207;1)) | |
0.822 | 0.078111 | 0.126 | 1.48 × 10−11 | 1.42 × 10−11 | ((0.744, 0.822, 0.948;1), (0.744, 0.822, 0.948;1)) | |
0.038 | 1.44 × 10−9 | 0.006 | 1.36 × 10−11 | 1.34 × 10−11 | ((0.038, 0.038, 0.044;1), (0.038, 0.038, 0.044;1)) |
Variable | IT2F Regression Coefficients | IT2F Coefficients | ||||
---|---|---|---|---|---|---|
b | f | g | p | q | ||
1 | −10.907 | 1.28 | 1.018 | 4.25 × 10−9 | 4.59 × 10−9 | ((−12.184, −10.907, −9.889;1), (−12.184, −10.907, −9.889;1)) |
−18.964 | 3.03 × 10−9 | 0.000 | 1.44 × 10−8 | 1.44 × 10−8 | ((−18.964, −18.964, −18.964;1), (−18.964, −18.964, −18.964;1)) | |
15.077 | 1.79 × 10−9 | 0.000 | 3.39 × 10−9 | 3.61 × 10−9 | ((15.077, 15.077, 15.077;1), (15.077, 15.077, 15.077;1)) | |
−3.447 | 3.12 × 10−9 | 0.000 | 7.20 × 10−1 | 1.16 | ((−4.167, −3.447, −2.289;1), (−3.447, −3.447, −3.447;1)) | |
−0.107 | 1.50 × 10−9 | 0.000 | 8.08 × 10−10 | 8.56 × 10−10 | ((−0.107, −0.107, −0.107;1), (−0.107, −0.107, −0.107;1)) | |
−0.725 | 9.06 × 10−10 | 0.000 | 3.64 × 10−7 | 0.024974 | ((−0.725, −0.725, −0.7;1), (−0.725, −0.725, −0.725;1)) | |
−0.262 | 3.18 × 10−9 | 0.000 | 6.04 × 10−10 | 6.37 × 10−10 | ((−0.262, −0.262, −0.262;1), (−0.262, −0.262, −0.262;1)) | |
0.735 | 3.18 × 10−9 | 0.000 | 5.26 × 10−10 | 5.27 × 10−10 | ((0.735, 0.735, 0.735;1), (0.735, 0.735, 0.735;1)) | |
0.150 | 1.16 × 10−5 | 0.001 | 5.74 × 10−10 | 6.10 × 10−10 | ((0.15, 0.15, 0.151;1), (0.15, 0.15, 0.151;1)) |
Alternative | Criteria | Proposed IT2F Functions | IT2F Regression | ||
---|---|---|---|---|---|
RMSE | MAPE | RMSE | MAPE | ||
1 | 1 | 0.2307 | 2.3883 | 0.6974 | 11.4445 |
2 | 0.1948 | 3.3149 | 0.7844 | 15.1122 | |
3 | 0.2519 | 4.0679 | 0.9828 | 14.0217 | |
4 | 0.375 | 3.4459 | 0.6952 | 7.2826 | |
5 | 0.3606 | 5.6463 | 0.8976 | 16.7376 | |
2 | 1 | 0.2894 | 6.0687 | 0.4449 | 13.6456 |
2 | 0.2392 | 2.4306 | 0.6793 | 7.5079 | |
3 | 0.3733 | 4.2811 | 0.9343 | 11.083 | |
4 | 0.4283 | 5.1677 | 0.5934 | 7.8132 | |
5 | 0.3035 | 4.0027 | 0.5657 | 7.9788 | |
3 | 1 | 0.1457 | 1.722 | 0.7285 | 8.0463 |
2 | 0.1686 | 3.9912 | 0.5833 | 17.4631 | |
3 | 0.2417 | 2.6302 | 0.6196 | 6.6876 | |
4 | 0.2142 | 2.135 | 0.3221 | 3.4379 | |
5 | 0.2205 | 2.7494 | 0.5567 | 7.9517 |
Criteria | Alternatives | ||
---|---|---|---|
A1 | A2 | A3 | |
C1 | ((1.694, 2.99, 4.515;1), | ((2.163, 3.73, 5.516;1), | ((5.223, 7.902, 11.27;1), |
(2.064, 2.99, 3.822;1)) | (2.864, 3.73, 4.204;1)) | (5.601, 7.902, 10.396;1)) | |
C2 | ((1.354, 2.255, 3.33;1), | ((4.992, 8.419, 12.522;1), | ((2.662, 3.919, 5.449;1), |
(1.891, 2.255, 2.623;1)) | (5.964, 8.419, 10.802;1)) | (3.204, 3.919, 4.596;1)) | |
C3 | ((2.986, 5.358, 8.46;1), | ((2.194, 4.143, 5.991;1), | ((6.016, 9.229, 12.788;1), |
(4.13, 5.358, 6.362;1)) | (3.144, 4.143, 5.238;1)) | (6.68, 9.229, 11.589;1)) | |
C4 | ((7.187, 10.353, 14.08;1), | ((0.469, 1.617, 3.242;1), | ((5.678, 8.415, 11.417;1), |
(8.084, 10.353, 12.123;1)) | (0.933, 1.617, 2.178;1)) | (6.071, 8.415, 10.508;1)) | |
C5 | ((2.409, 5.252, 9.04;1), | ((4.427, 6.446, 9.052;1), | ((3.715, 6.246, 8.94;1), |
(3.372, 5.252, 7.208;1)) | (5.168, 6.446, 7.487;1)) | (4.566, 6.246, 7.472;1)) |
Linguistic Terms | IT2F Number | |
---|---|---|
Symbol | Explanation | |
VL | Very Low | ((0, 0, 1;1), (0, 0, 0.5;1)) |
L | Low | ((0, 1, 3;1), (0.5, 1, 2;1)) |
ML | Medium Low | ((1, 3, 5;1), (2, 3, 4;1)) |
M | Medium | ((3, 5, 7;1), (4, 5, 6;1)) |
MH | Medium High | ((5, 7, 9;1), (6, 7, 8;1)) |
H | High | ((7, 9, 10;1), (8, 9, 9.5;1)) |
VH | Very High | ((9, 10, 10;1), (9.5, 10, 10;1)) |
Criteria | Alternatives | ||
---|---|---|---|
A1 | A2 | A3 | |
C1 | ML | ML | MH |
C2 | ML | H | ML |
C3 | M | M | H |
C4 | H | L | H |
C5 | M | MH | MH |
Criteria | Alternatives | ||
---|---|---|---|
A1 | A2 | A3 | |
C1 | ML | MH | MH |
C2 | ML | H | ML |
C3 | M | M | H |
C4 | H | L | MH |
C5 | M | H | MH |
Periods | Distance to Positive Ideal Solution | ||
---|---|---|---|
Employee 1 | Employee 2 | Employee 3 | |
t1 | 0.1458 | 0.1101 | 0.1111 |
t2 | 0.2259 | 0.1569 | 0.1084 |
t3 | 0.2155 | 0.1546 | 0.1161 |
t4 | 0.1604 | 0.1203 | 0.0850 |
t5 | 0.1730 | 0.1313 | 0.0938 |
t6 | 0.1700 | 0.0919 | 0.0833 |
t7 | 0.1700 | 0.0977 | 0.0750 |
t8 | 0.1714 | 0.0709 | 0.0643 |
t9 | 0.1256 | 0.0872 | 0.0750 |
t10 | 0.1303 | 0.0874 | 0.0750 |
t11 | 0.0960 | 0.0797 | 0.1086 |
t12 | 0.1146 | 0.0901 | 0.1397 |
t13 | 0.1204 | 0.0961 | 0.1393 |
t14 | 0.1004 | 0.0914 | 0.1050 |
t15 | 0.1226 | 0.1401 | 0.0797 |
t16 | 0.1264 | 0.1330 | 0.0972 |
t17 | 0.1422 | 0.1873 | 0.1001 |
t18 | 0.1404 | 0.2335 | 0.0981 |
tC | 0.1725 | 0.1955 | 0.1143 |
Periods | Distance to Negative Ideal Solution | ||
---|---|---|---|
Employee 1 | Employee 2 | Employee 3 | |
t1 | 0.1409 | 0.1556 | 0.1541 |
t2 | 0.1169 | 0.1524 | 0.2462 |
t3 | 0.1161 | 0.1334 | 0.2155 |
t4 | 0.0898 | 0.0698 | 0.1613 |
t5 | 0.0995 | 0.0805 | 0.1762 |
t6 | 0.1011 | 0.1636 | 0.1836 |
t7 | 0.0983 | 0.1278 | 0.1857 |
t8 | 0.0744 | 0.1697 | 0.1807 |
t9 | 0.0684 | 0.1259 | 0.1385 |
t10 | 0.0563 | 0.0950 | 0.1336 |
t11 | 0.0811 | 0.1102 | 0.0717 |
t12 | 0.0997 | 0.1397 | 0.0750 |
t13 | 0.0817 | 0.1311 | 0.0750 |
t14 | 0.0680 | 0.1050 | 0.0914 |
t15 | 0.0927 | 0.0850 | 0.1247 |
t16 | 0.0810 | 0.1000 | 0.1244 |
t17 | 0.1218 | 0.1050 | 0.1719 |
t18 | 0.2143 | 0.1173 | 0.1966 |
tC | 0.1873 | 0.1631 | 0.1751 |
Periods | Closeness Coefficients | ||
---|---|---|---|
Employee 1 | Employee 2 | Employee 3 | |
t1 | 0.4915 | 0.5856 | 0.5811 |
t2 | 0.3411 | 0.4928 | 0.6942 |
t3 | 0.3501 | 0.4633 | 0.6499 |
t4 | 0.3588 | 0.3671 | 0.6549 |
t5 | 0.3652 | 0.3801 | 0.6527 |
t6 | 0.3729 | 0.6405 | 0.6878 |
t7 | 0.3664 | 0.5666 | 0.7123 |
t8 | 0.3028 | 0.7053 | 0.7376 |
t9 | 0.3525 | 0.5907 | 0.6487 |
t10 | 0.3016 | 0.5208 | 0.6404 |
t11 | 0.4580 | 0.5802 | 0.3978 |
t12 | 0.4652 | 0.6078 | 0.3493 |
t13 | 0.4043 | 0.5769 | 0.3500 |
t14 | 0.4039 | 0.5347 | 0.4653 |
t15 | 0.4304 | 0.3776 | 0.6099 |
t16 | 0.3906 | 0.4291 | 0.5614 |
t17 | 0.4613 | 0.3593 | 0.6321 |
t18 | 0.6041 | 0.3345 | 0.6671 |
tC | 0.5206 | 0.4547 | 0.6051 |
Model Components | Models | |||
---|---|---|---|---|
Proposed Model | Model 1 | Model 2 | Model 3 | |
IT2F Functions | Yes | Yes | Yes | Yes |
Vocabulary Matching | Yes | Yes | No | Yes |
Modified preferences (interactive) | Yes | No | No | Yes |
DWA Operator | Yes | Yes | Yes | No |
Model | Overall Assessment | |||||
---|---|---|---|---|---|---|
Employee 1 | Employee 2 | Employee 3 | ||||
CC | Rank | CC | Rank | CC | Rank | |
Proposed Model | 0.4138 | 3 | 0.4984 | 2 | 0.5895 | 1 |
Model 1 | 0.4176 | 3 | 0.4926 | 2 | 0.5967 | 1 |
Model 2 | 0.4185 | 3 | 0.4905 | 2 | 0.5951 | 1 |
Model 3 | 0.5206 | 2 | 0.4547 | 3 | 0.6051 | 1 |
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Baykasoğlu, A.; Gölcük, İ. An Interactive Data-Driven (Dynamic) Multiple Attribute Decision Making Model via Interval Type-2 Fuzzy Functions. Mathematics 2019, 7, 584. https://doi.org/10.3390/math7070584
Baykasoğlu A, Gölcük İ. An Interactive Data-Driven (Dynamic) Multiple Attribute Decision Making Model via Interval Type-2 Fuzzy Functions. Mathematics. 2019; 7(7):584. https://doi.org/10.3390/math7070584
Chicago/Turabian StyleBaykasoğlu, Adil, and İlker Gölcük. 2019. "An Interactive Data-Driven (Dynamic) Multiple Attribute Decision Making Model via Interval Type-2 Fuzzy Functions" Mathematics 7, no. 7: 584. https://doi.org/10.3390/math7070584