One’s Fixing Method for a Distinct Symmetric Fuzzy Assignment Model
Abstract
:1. Introduction
2. Some Preliminaries
3. Ranking Method for Symmetric Triangular Interval-Valued Fuzzy Number (STIFN) [1]
- (i)
- Let be a symmetric triangularinterval-valued fuzzy number, then define and ; then,. Here is the multiplicative operator.
- (ii)
- Let be a symmetric trapezoidal interval-valued fuzzy number, and then define
4. Arithmetic Operations of Symmetric Interal-Valued Triangualr Fuzzy Numbers [1]
- Addition
- Subtraction
- Scalar multiplication
5. Arithmetic Operations of Symmetric Interval-Valued Trapezoidal Fuzzy Numbers [1]
- Addition
- Subtraction
- Scalar multiplication
6. Assignment Problem
Job1 | Job2 | Jobn | ||
Machine 1 | ……… | |||
Machine 2 | ……… | |||
……… | ……… | ……… | ……… | |
……… | ……… | ……… | ……… | |
Machine n | ……… |
- Step 1:
- Intially, we consider an assignment problem either with fuzzy parameters or without fuzzy parameters.
- Step 2:
- (i) If the assignment problem is formulated without fuzzy parameters, then go to Step (4).(ii) If the assignment problem formulated has only the fuzzy parameters, then go to Step (3).
- Step 3:
- Convert the symmetric fuzzy assignment problem into crisp values using the ranking method.
- Step 4:
- Check to see if the assignment problem is balanced.
- (i)
- If it is balanced, go to Step 6.
- (ii)
- It it is not balanced, go to Step 5.
- Step 5:
- Add a dummy row (or) column with a cost value ofzero to make the fuzzy assignment problem a balanced one.
- Step 6:
- Divide each element of the ith row of the matrix by the minimum element of each row, which should be written on the right hand side of the matrix. Go to Step 8 if you get a result with ones in each row and column. If not, proceed to Step 7.
- Step 7:
- Determine the smallest element of each column and write it below the jth column of the matrix; then divide each element of the jth column of the matrix, obtaining the assignment cost as one in each column.
- Step 8:
- Identify the one’s position in each (i,j)th row and column.
- Step 9:
- Find the value of each row’s succession of ones (after 1), select the maximum value, and delete the corresponding row and column of the matrix and perform allocations.
- Step 10:
- In the reduced matrix, repeat step 8 until all of the rows are assigned and the optimal allocation for assignment cost is obtained.
Row | Column | S(1) |
2 | 2,4 | 2.67 |
3 | 2 | 2.8 |
4 | 3 | 1.93 |
Row | Column | S(1) |
2 | 4 | 2.67 (max) |
4 | 3 | 1.93 |
Row | Column | S(1) |
1 | 2 | 2 |
2 | 1 | 3 |
3 | 3 | 3.5 (max) |
4 | 1 | 1.3 |
Row | Column | S(1) |
1 | 2 | 2 |
2 | 1 | 3 (max) |
4 | 1 | 1.6 |
Row | Column | S(1) |
1 | 1 | 7.4 |
2 | 2 | 5.29 |
3 | 3,4 | 28.6 (max) |
4 | 2 | 2.68 |
Row | Column | S(1) |
1 | 1 | 7.4 (max) |
2 | 1 | 5.29 |
4 | 1 | 2.68 |
7. Results and Discussion
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
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Row | Column | S(1) |
1 | 1 | 2.5 (max) |
2 | 2,4 | 2 |
3 | 2 | 1.8 |
4 | 3 | 1.93 |
Author Name | Method | Optimal Schedule | Optimal Solution | ||
---|---|---|---|---|---|
Existing Method | Proposed Method | Existing Method | Proposed Method | ||
Senthil Kumar [10] Example 8.4 | TORA Software | x11 = 4, x24 = 8, x32 = 5, x43 =7 | x11 = 4, x24 = 8, x32 = 5, x43 =7 | 24 | 24 |
Rezaul Karim [14] Example 2 | Python 3.8 programming language | x12 = 5, x21 = 3, x33 = 2, x44 = 0 | x12 = 5, x21 = 3, x33 = 2, x44 = 0 | 10 | 10 |
Amutha, Uthra [1] Example 1 | New Ranking method | x11 = 0.275, x22 = 0.385, x34 = 0.27, x43 =5.445 | x11 = 0.275, x22 = 0.385, x34 = 0.27, x43 = 5.445 | 6.38 | 6.38 |
Uthra et al. [16] Example 1 | Using Yager’s ranking method | x11 = 1.5, x22 = 1.75, x34 = 1.5, x43 = 17.25 | x11 = 1.5,x22 = 1.75, x34 = 1.5, x43 = 17.25 | 22 | 22 |
Amutha, Uthra [1] Example 1 | New Ranking method | x12 = 0.275,x24 = 0.385, x34 = 0.27, x43 = 5.445 | x12 = 0.275,x24 = 0.385, x34 = 0.27, x43 = 5.445 | 4.375 | 4.375 |
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Gomathi, S.V.; Jayalakshmi, M. One’s Fixing Method for a Distinct Symmetric Fuzzy Assignment Model. Symmetry 2022, 14, 2056. https://doi.org/10.3390/sym14102056
Gomathi SV, Jayalakshmi M. One’s Fixing Method for a Distinct Symmetric Fuzzy Assignment Model. Symmetry. 2022; 14(10):2056. https://doi.org/10.3390/sym14102056
Chicago/Turabian StyleGomathi, S. V., and M. Jayalakshmi. 2022. "One’s Fixing Method for a Distinct Symmetric Fuzzy Assignment Model" Symmetry 14, no. 10: 2056. https://doi.org/10.3390/sym14102056
APA StyleGomathi, S. V., & Jayalakshmi, M. (2022). One’s Fixing Method for a Distinct Symmetric Fuzzy Assignment Model. Symmetry, 14(10), 2056. https://doi.org/10.3390/sym14102056