1. Introduction
The assignment problem (AP) is commonly employed in production and service systems, as well as in a variety of other indirect applications. The assignment problem can be defined as a set of
n jobs that must be completed by
n workers, with costs varying according to the specific tasks. Each job must be given to one and only one worker, and each worker must complete one and only one task at a time. Then, to minimize the sum, an assignment problem is developed to choose precisely one element from each row and column. Costs, on the other hand, are not deterministic numbers in many real-world situations. Many researchers have begun to examine the assignment problem and its variants in a fuzzy environment in recent years. Ref. [
1] proposed a ranking method for two types of symmetric interval-valued fuzzy numbers and obtained the optimal assignment cost. Ref. [
2] proposed a ranking method and a zero suffix method to obtain an optimal solution for the two-stage fuzzy transportation problem. Ref. [
3] developed a strategy for making decisions in a fuzzy environment, in which the goals or constraints define classes of possibilities and the boundaries are not well defined. Ref. [
4] transformed a fuzzy assignment problem into a multi-objective linear programming problem and obtained the solution that is independent of the weights. Ref. [
5] established a modified Hungarian method-based algorithm for identifying the best fuzzy AP solution presented. For a fuzzy AP, this approach is used to determine the lowest assignment cost possible. In the beginning, fuzzy numbers were used directly. Then, with the use ofthe-cut process, the fuzzy AP wasconverted into an interval AP. Ref. [
6] developed multi-criteria decision-making tool based onthe characterization object method for solving the inter-valued triangular fuzzy number. Ref. [
7] developed a ranking method to transform fuzzy assignment to crisp, solved byLINGO9.0. Ref. [
8] developed an algorithm to solve multi-interval-valued fuzzy soft sets to analyze a decision problem. Ref. [
9] presented the application of interval-valued fuzzy numbers. Ref. [
10] used linguistic variables to convert qualitative data into quantitative data to solve the fuzzy assignment problem. Ref. [
11] used a robust ranking method for defuzziffication solved by the Hungarian method. Ref. [
12] sorted membership functions by different measures of a linear and non-linear inter-valued hexagonal fuzzy number. In Ref. [
13], crisp, fuzzy, and intuitionistic fuzzy optimization problems wereformulated. A novel theorem is suggested and proved for type-2 fuzzy/type-2 intuitionistic fuzzy optimization issues. Ref. [
14] presented an algorithm fuzzy Hungarian method without conversion into crisp numbers andobtained the optimal solution. Ref. [
15] developed a fuzzy Vogel approximation method for determining the best solution to the fuzzy transportation problem. Ref. [
16] proposed an ASM method to obtain the optimal solution thatrequires the least number of iterations. Ref. [
17] represented a unique solution to the assignment problem that was implemented using Python 3.8 as the programming language. Ref. [
18] established a new ranking method to solve the unbalanced fuzzy assignment problem. Ref. [
19] formulated fuzzy numbers to crisp numbers using Yager’s ranking indices and reached the optimal solution by the Hungarian method. Ref. [
20] established the ranking method for interval-valued fuzzy numbers and reached the optimal solution by the traditional method. Ref. [
21] used symmetric fuzzy number and then formulated this to crisp by Yager’s ranking method and obtained the optimal solution. Ref. [
22] developed an average ranking method for symmetric intuitionistic fuzzy numbers and obtained the optimal solution. Ref. [
23] developed the relative preference relation byranking triangular interval-valued fuzzy numbers. Ref. [
24] proposed the fuzzy set concept, which deals with imprecision and ambiguity in real-life circumstances.
In this paper, considering the assignment problem by the proposed method, we locate the one’s position in each row as well as each column, and then the given (n × n) matrix is reduced in each and every iteration. In every iteration, one row will be assigned to one column so that we obtain the optimal schedule and assignment cost of the given problem. We applied the algorithm tosymmetric interval-valued triangular and trapezoidal fuzzy assignment problems and achieved the optimal schedule and assignment cost. The proposed algorithm is not based on the Hungarian method. Illustrated examples are given for a better understanding of the solution procedure.
6. Assignment Problem
The cost of assigning the ith machine to the jth job can be expressed as anxn matrix termed a cost matrix, where cij is the cost of assigning the ith machine to the jth job. Consider a case in which you must assign n machines to n jobs. (one machine to one job). Let cijrepresent the unit cost of assigning the ith machine to the jth job.
Let
.
| Job1 | Job2 | | Jobn |
Machine 1 | | | ……… | |
Machine 2 | | | ……… | |
| ……… | ……… | ……… | ……… |
| ……… | ……… | ……… | ……… |
Machine n | | | ……… | |
Formulation of the fuzzy assignment problem:
The objective function is Subject to
where
and
is the symmetric interval-valued fuzzy number.
Now, we introduce a new algorithm, namely, one’s orientation algorithm for finding an optimal solution for the assignment problem with and without fuzziness
One’s Orientation algorithm:
- 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.
We exposed the architecture of the proposed algorithm in
Figure 3Now, the proposed algorithm can be illustrated with the help of the following numerical examples.We computed four examples; first, we assumed two problems (Example 1 and Example 2) based on non-symmetric data, and we took another two problems (Example 3 and Example 4) based on symmetric data from [
1].
Example 1. [10]Find the minimum assignment cost of the matrix. Solution:
Step 1: Consider an assignment problem without fuzzy parameters.
Step 4: The numbers of rows and columns are equal. Therefore, it is a balanced assignment problem.
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.
Step 7: Here, there are no ones in fourth column, so determine the smallest element from column 4 and write it below the jth column of the matrix; then divide each element of the jth column of the matrix.
Therefore, we obtained the assignment cost as one in each row and each column.
Step 8: Identify the one’s position in each (i,j)th row and column.
Step 9: Determine the value of each row’s succession of ones, pick the highest one, (see
Table 1) and delete the associated row and column of the matrix and perform allocations.
Assign row 1 column 1 and delete the corresponding rows and columns.
In the reduced matrix, repeat step 8 until all of the rows are assigned and the optimal allocation for the assignment cost is obtained.
Now, the reduced matrix is
Locating the one’s position
Row | Column | S(1) |
2 | 2,4 | 2.67 |
3 | 2 | 2.8 |
4 | 3 | 1.93 |
Assign row 3 column 2 and delete the corresponding rows and columns.
Now, the reduced matrix is
Locating the one’s position
Row | Column | S(1) |
2 | 4 | 2.67 (max) |
4 | 3 | 1.93 |
Assign row 2 column 4 and row 4 column 3.
Step 10: the assignment schedule is
The minimum assignment cost is 24.
Example 2. [14]Find the minimum assignment cost of the matrix Solution:
Step 1: Consider the problem not in a fuzzy environment. Step 4: the assignment problem is not balanced.
Step 5, add a dummy column to make a balanced assignment problem.
Step 6: Divide each element of the ithrow of the matrix by the minimum element of each row, which should be written on the right hand side of the matrix.
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.
Therefore, we obtained the assignment cost as a one in each row and each column.
Step 8: Identify the one’s position ineach (i, j)th row and column.
Step 9: Determine the value of each row’s succession of ones, pick the highest one, and delete the associated row and column of the matrix and perform allocations.
Locating the One’s Position
Row | Column | S(1) |
1 | 2 | 2 |
2 | 1 | 3 |
3 | 3 | 3.5 (max) |
4 | 1 | 1.3 |
Assign row 3 column 3 and delete the corresponding rows and columns.
In the reduced matrix, repeat step 8 until all of the rows are assigned and the optimal allocation for assignment cost is obtained.
Now, the reduced matrix is
Locating the one’s position
Row | Column | S(1) |
1 | 2 | 2 |
2 | 1 | 3 (max) |
4 | 1 | 1.6 |
Assign row 2 to column 1 and delete the corresponding rows and columns.
Now, the reduced matrix is . Here, we got the 2 × 2 reduced matrix;the possible assignments are row1 column 2 and row 4 column 4(nothing is assigned to P4)
Step 10: the assignment schedule is
The minimum assignment cost is 10.
Note: As a result of the proposed algorithm, the assignment cost was derived under crisp parameters. Now, we apply the technique to fuzzy parameters such as triangular and trapezoidal fuzzy numbers with symmetric interval-valued fuzzy assignments.
Example 3. P1, P2, P3, and P4 are four professional programmers who work at a computer center. The center requires the development of four application program: A1, A2, A3, and A4. After carefully reviewing the program to be built, the director of the computer center estimates the computing time in minutes required by the experts forthe application program using symmetric interval-valued triangular fuzzy numbers. Assume that . Assign programmers to the program in a method that uses the least amount of computing time. The fuzzy assignment problem’s cost matrix is
Solution:
Step 1: Consider the symmetric triangular fuzzy assignment problem
Step 3: Using the new ranking method [
1],
the above fuzzy assignment problem becomes
Step 4: number of rows = number of columns.
Step 5: Therefore, it is a balanced assignment problem.
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.
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.
Therefore, we obtained the assignment cost as one in each row and each column.
Step 8: Identify the one’s position in each (i,j)th row and column.
Step 9: Determine the value of each row’s succession of ones, pick the highest one, and delete the associated row and column of the matrix and perform allocations.
Locating the One’s Position
Row | Column | S(1) |
1 | 1 | 7.4 |
2 | 2 | 5.29 |
3 | 3,4 | 28.6 (max) |
4 | 2 | 2.68 |
Assign row 3 column 4 and delete the corresponding rows and columns.
In the reduced matrix, repeat step 8 until all of the rows are assigned and the optimal allocation for asignment cost is obtained.
Now, the reduced matrix is
Locating the one’s position
Row | Column | S(1) |
1 | 1 | 7.4 (max) |
2 | 1 | 5.29 |
4 | 1 | 2.68 |
Assign row 1 column 1 and delete the corresponding rows and columns.
Now, the reduced matrix is , again following step 6, so that matrix has ones in each and column. Then proceed to step 7 and step 8.
Assign row 2 column 2 & row 4 column 3.
Step 10: the assignment schedule is and the total minimum assignment cost is 6.380.
Hence, the minimum symmetric fuzzy triangular assignment cost is [(16,24,24),(12,20,20)].
Example 4. The Navy’s top officer has four subordinates and four responsibilities to complete. The efficiency of the subordinates varies, as do the duties’ inherent difficulty. His estimates of how long each subordinate would take to complete the each task (in hours) are expressed as symmetric interval-valued trapezoidal fuzzy numbers. Assume that . Determine the best assignment for reducing the total number of workinghours. The fuzzy assignment problem’s cost matrix is
Solution:
Step 1: Consider the symmetric trapezoidal fuzzy assignment problem
Step 3: Using the new ranking method [
1],
the above fuzzy assignment problem becomes
Based on step 4 and step 5, the number of rows = the number of columns.
Therefore, it is a balanced assignment problem.
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.
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.
Therefore, we obtained the assignment cost as one in each row and each column.
Step 8: Identify the one’s position ineach (i,j)th row and column.
Step 9: Determine the value of each row’s succession of ones, pick the highest one, and delete the associated row and column of the matrix and perform allocations.
In the reduced matrix, repeat step 8 until all of the rows are assigned and the optimal allocation for the assignment cost is obtained.
Step 10: the assignment schedule is , and the total minimum assignment cost is 4.375.
Hence, the minimum symmetric fuzzy trapezoidal assignment cost is [(5,12,16,16),(1,8,12,12)].