An algorithm for solving unbalanced intuitionistic fuzzy assignment problem using triangular intuitionistic fuzzy number
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The document presents an algorithm for solving unbalanced intuitionistic fuzzy assignment problems using triangular intuitionistic fuzzy numbers. It addresses uncertainty in real-life assignments where costs are not deterministic, proposing a method to transform fuzzy costs into crisp values for solution application. The method is deemed computationally simple and applicable to various decision-making problems in real-life scenarios.
![An Algorithm for Solving Unbalanced Intuitionistic Fuzzy Assignment 301
Problem Using Triangular Intuitionistic Fuzzy Number
The degree of acceptance of the production of toys for the DM increases if the
production of toys increases from 148 to 182; while it decreases if the production of toys
increases from 182 to 262. Beyond (182, 342), the level of acceptance or the level of
satisfaction for the DM is zero. The DM is totally satisfied or the production of toys
is totally acceptable if the production is 262. The degree of non-acceptance of the
production for the DM decreases if the production increases from 148 to 262 while it
increases if the production increases from 262 to 376. Beyoned (148, 376), the
production of toys is totally un-acceptable.
Assuming that ( )I
Z
pm is membership value (degree of acceptance) and ( )I
Z
pJ is non-
membership value (degree of non-acceptance) of production p . Then the degree of
acceptance of the production p is ( )100 %I
Z
pm for the DM and the degree of non
acceptance is ( )100 %Z
pn for the DM . DM is not sure by ( ) ( )( )100 1 %I I
Z Z
p pm J- -
that he/she should accept the production p or not. Values of ( )I
Z
pm and ( )I
Z
pJ at
different values of p can be determined using equations given below.
( )
0 for 182
182
for 182 262
80
1 for 262
342
for 262 342
80
0 for 342
Z
p
p
p
p p
p
p
p
m
<Ï
Ô -
Ô £ £
Ô
Ô
= =Ì
Ô -
Ô £ £
Ô
Ô >Ó
( )
1 for 148
262
for 148 262
114
0 for 262
262
for 262 376
114
1 for 376
Z
p
p
p
p p
p
p
p
J
<Ï
Ô -
Ô £ £
Ô
Ô
= =Ì
Ô -
Ô £ £
Ô
Ô >Ó
6. Conclusion
In this paper we have proposed a simple computational method for solving unbalanced
intuitionistic fuzzy assignment problem. This proposed method gives the opportunity to
find the optimum schedule and optimum objective value of UIFAP directly. On the
basis of the present study, it can be concluded that the UIFAP , BIFAP and MIFAP
which can be solved by the existing methods (Kumar and Hussain (2014)) can also be
solved by the proposed method. Hence the proposed method is computationally very
simple and easy to understand and it can be easily applied by decision maker to solve real
life UIFAP . This technique can also be used in solving other types of problems like,
project schedules, transportation problems and network flow problems.
References
[1] Amit Kumar and Anila Gupta, Assignment and Travelling Salesman Problems with Coefficients as
LR Fuzzy Parameters, International Journal of Applied Science and Engineering, (2012) 10, 3:155-
170.
[2] Amit Kumar, Amarpreet Kaur and Anila Gupta, Fuzzy Linear Programming Approach for Solving
Fuzzy Transportation probles with Transshipment, J Math Model Algor, (2011), 10:163-180.
[3] Amit Kumar, Anila Gupta and Amarpreet Kumar, Method for Solving Fully Fuzzy Assignment
Problems Using Triangular Fuzzy Numbers, International Journal of Computer and Information
Engineering, 3: 4 (2009).
[4] Annie Varghese and Sunny Kuriakose, Centroid of an intuitionistic fuzzy number, Notes on
Intuitionistic Fuzzy Sets, Vol.18, (2012), No.1, 19-24.](https://image.slidesharecdn.com/analgorithmforsolvingunbalancedintuitionisticfuzzyassignmentproblemusingtriangularintuitionisticfuzz-180706171435/85/An-algorithm-for-solving-unbalanced-intuitionistic-fuzzy-assignment-problem-using-triangular-intuitionistic-fuzzy-number-4-320.jpg)
![302 P. Senthil Kumar and R. Jahir Hussain
[5] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems, Vol.20, no.1 (1986), pp.87-96.
[6] D. Avis, L. Devroye, An analysis of a decomposition heuristic for the assignment problem, Oper. Res.
Lett., 3 (6) (1985), 279-283.
[7] R. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, management sci.,17 (B) (1970),
141-164.
[8] M. L. Balinski, A Competitive (dual) simplex method for the assignment problem, Math. Program, 34
(2) (1986), 125-141.
[9] R. S. Barr, F. Glover and D. Klingman, The alternating basis algorithm for assignment problems,
Math. Program, 13 (1) (1977), 1-13.
[10] M. S. Chen, On a Fuzzy Assignment Problem, Tamkang Journal, 22 (1985), 407-411.
[11] P. K. De and Bharti Yadav, A General Approach for Solving Assignment Problems Involving with
Fuzzy Costs Coefficients, Modern Applied Science, Vol.6, No.3; March (2012).
[12] S. P. Eberhardt, T. Duad, A. Kerns, T. X. Brown and A. S. Thakoor, Competitive neural architecture
for hardware solution to the assignment problem, Neural Networks, 4 (4) (1991), 431-442.
[13] M. S. Hung and W. O. Rom, Solving the assignment problem by relaxation, Oper. Res., 24 (4) (1980),
969-982.
[14] R. Jahir Hussain and P. Senthil Kumar, The Transportation Problem in an Intuitionistic Fuzzy
Environment, International Journal of Mathematics Research, Vol.4 Number 4 (2012), pp.411-420.
[15] R. Jahir Hussain and P. Senthil Kumar, An Optimal More-for-Less Solution of Mixed Constrains
Intuitionistic Fuzzy Transportation Problems, Int. J. Contemp. Math. Sciences, Vol.8, (2013), no.12,
565-576. doi.10.12988/ijcms.
[16] H. W. Kuhn, The Hungarian method for the assignment problem, Novel Research Logistic Quarterly,
2 (1955), 83-97.
[17] Lin Chi-Jen and Wen Ue-Pyng, An Labeling Algorithm for the fuzzy assignment problem, Fuzzy Sets
and Systems, 142 (2004), 373-391.
[18] L. F. McGinnis, Implementation and testing of a primal-dual algorithm for the assignment problem,
Oper. Res., 31 (2) (1983), 277-291.
[19] Sathi Mukherjee and Kajla Basu, Application of Fuzzy Ranking Method for Solving Assignment
Problems with Fuzzy Costs, Int. Jour. Comp and Appl. Mathematics, Vol 5 Number 3 (2010), pp.359-
368.
[20] Sathi Mukherjee and Kajla Basu, Solution of a class of Intuitionistic Fuzzy Assignment Problem by
using similarity measures, Knowledge-Based Systems, 27 (2012), 170-179.
[21] P. Senthil Kumar, R. Jahir Hussain, A method for solving balanced intuitionistic fuzzy assignment
problem, Int. Journal of Engineering Research and Applications, Vol.4 Issue 3 (Version 1) March
(2014), pp. 897-903.
[22] P. Senthil Kumar and R. Jahir Hussain, New algorithm for solving mixed intuitionistic fuzzy
assignment problem, Elixir Appl.Math., 73 (2014), 25971-25977.
[23] P. Senthil Kumar, R. Jahir Hussain, A method for finding an optimal solution of an assignment
problem under mixed intuitionistic fuzzy environment, Proceedings in International Conference on
Mathematical Sciences Published by ELSEVIER, Sathyabama University, ICMS-(2014), ISBN-978-
93-5107-261-4: 417-421.
[24] P. Senthil Kumar and R. Jahir Hussain, Computationally simple approach for solving fully
intuitionistic fuzzy real life transportation problems, International Journal of System Assurance
Engineering and Management, (2015): 1-12. DOI 10.1007/s13198-014-0334-2.
[25] Y. L. P. Thorani and N. Ravi Sankar, Fuzzy Assignment Problem with Generalized Fuzzy Numbers,
App. Math. Sci, Vol.7, (2013), no.71, 3511-3537.
[26] X. Wang, Fuzzy Optimal Assignment Problem, Fuzzy Math., 3 (1987), 101-108.
[27] L. A. Zadeh, Fuzzy sets, Information and computation, vol.8, (1965), pp.338-353.](https://image.slidesharecdn.com/analgorithmforsolvingunbalancedintuitionisticfuzzyassignmentproblemusingtriangularintuitionisticfuzz-180706171435/85/An-algorithm-for-solving-unbalanced-intuitionistic-fuzzy-assignment-problem-using-triangular-intuitionistic-fuzzy-number-5-320.jpg)
![Instructions to Authors
The Journal of Fuzzy Mathematics seeks to publish articles of worth to the theory of fuzzy sets and
their applications. Authors are required to follow the procedures listed below.
(1) The submission of a manuscript implies that it is not being considered contemporaneously for
publication elsewhere. All papers will be independently refereed.
(2) Contributors are invited to submit three copies of their paper to the editor-in-cheif. The paper
should be written in English and accompanied by an abstract of about 200 words. A list of key words
would be put at the end of the abstract.
(3) The paper should be submitted in its final form. No new material may be inserted in the text
at the time of proof reading unless accepted by the Editor.
(4) The manuscript must be typed double spaced on one side of white paper with wide margins.
Each page should be numbered.
(5) All mathematical symbols should be listed and identified typographically on a separate page.
(6) The original and two copies of each illustration should be provided. In order to make a direct
reproduction possible, all illustrations should be carefully lettered, and drawn large enough in India ink
on a separate page with wide margins. Each illustration must be numbered and the title and name of
the author of the paper must be written on the reverse side.
(7) Reference should be listed alphabetically at the end of the paper, For example:
[1] L.A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl., 23(1968),
421-427.
[2] C. V. Negoita and D. A. Ralescu, Fuzzy sets and their applications, Technical
Press, Bucharest, 1974.
(8) No page charge is made. Reprints may be ordered by using the reprint order form which
accompanies the proofs, prior to publication. Post-publication orders cannot be filled at regular reprint
prices.
(9) If an article is accepted for publication, the author will be asked to transfer the copyright of the
article to the publisher.](https://image.slidesharecdn.com/analgorithmforsolvingunbalancedintuitionisticfuzzyassignmentproblemusingtriangularintuitionisticfuzz-180706171435/85/An-algorithm-for-solving-unbalanced-intuitionistic-fuzzy-assignment-problem-using-triangular-intuitionistic-fuzzy-number-6-320.jpg)
An algorithm for solving unbalanced intuitionistic fuzzy assignment problem using triangular intuitionistic fuzzy number
- 2. The Journal of Fuzzy Mathematics Vol. 24, No. 2, 2016 289 Los Angeles __________________________ Received February, 2015 1066-8950/16 $8.50 © 2016 International Fuzzy Mathematics Institute Los Angeles An Algorithm for Solving Unbalanced Intuitionistic Fuzzy Assignment Problem Using Triangular Intuitionistic Fuzzy Number P. Senthil Kumar Assistant Professor, PG and Research Department of Mathematics, Jamal Mohamed College, Tiyuchirappalli-620020, Tamil Nadu, Corresponding author Email: [email protected] R. Jahir Hussain Associate Professor PG and Research Department of Mathematics, Jamal Mohamed College, Tiruchirappalli-620020, India. Email: [email protected] Abstract: In solving real life assignment problem, we often face the state of uncertainty as well as hesitation due to varies uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representation for the data. In this paper, computationally a simple method is proposed to find the optimal solution for an unbalanced assignment problem under intuitionistic fuzzy environment. In conventional assignment problem, cost is always certain. This paper develops an approach to solve the unbalanced assignment problem where the time/cost/profit is not in deterministic numbers but imprecise ones. In this assignment problem, the elements of the cost matrix are represented by the triangular intuitionistic fuzzy numbers. The existing Ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the AP . Finally the method is illustrated by a numerical example which is followed by graphical representation and discussion of the finding. Keywords: Intuitionistic Fuzzy Set, Triangular Intuitionistic Fuzzy Number, Unbalanced Intuitionistic Fuzzy Assignment Problem, Optimum Schedule. 1. Introduction
- 3. 11 more pages are available in the full version of this document, which may be purchased using the "Add to Cart" button on the product's webpage: http://maths.swjtu.edu.cn/fuzzy/2016_6_details.htm
- 4. An Algorithm for Solving Unbalanced Intuitionistic Fuzzy Assignment 301 Problem Using Triangular Intuitionistic Fuzzy Number The degree of acceptance of the production of toys for the DM increases if the production of toys increases from 148 to 182; while it decreases if the production of toys increases from 182 to 262. Beyond (182, 342), the level of acceptance or the level of satisfaction for the DM is zero. The DM is totally satisfied or the production of toys is totally acceptable if the production is 262. The degree of non-acceptance of the production for the DM decreases if the production increases from 148 to 262 while it increases if the production increases from 262 to 376. Beyoned (148, 376), the production of toys is totally un-acceptable. Assuming that ( )I Z pm is membership value (degree of acceptance) and ( )I Z pJ is non- membership value (degree of non-acceptance) of production p . Then the degree of acceptance of the production p is ( )100 %I Z pm for the DM and the degree of non acceptance is ( )100 %Z pn for the DM . DM is not sure by ( ) ( )( )100 1 %I I Z Z p pm J- - that he/she should accept the production p or not. Values of ( )I Z pm and ( )I Z pJ at different values of p can be determined using equations given below. ( ) 0 for 182 182 for 182 262 80 1 for 262 342 for 262 342 80 0 for 342 Z p p p p p p p p m <Ï Ô - Ô £ £ Ô Ô = =Ì Ô - Ô £ £ Ô Ô >Ó ( ) 1 for 148 262 for 148 262 114 0 for 262 262 for 262 376 114 1 for 376 Z p p p p p p p p J <Ï Ô - Ô £ £ Ô Ô = =Ì Ô - Ô £ £ Ô Ô >Ó 6. Conclusion In this paper we have proposed a simple computational method for solving unbalanced intuitionistic fuzzy assignment problem. This proposed method gives the opportunity to find the optimum schedule and optimum objective value of UIFAP directly. On the basis of the present study, it can be concluded that the UIFAP , BIFAP and MIFAP which can be solved by the existing methods (Kumar and Hussain (2014)) can also be solved by the proposed method. Hence the proposed method is computationally very simple and easy to understand and it can be easily applied by decision maker to solve real life UIFAP . This technique can also be used in solving other types of problems like, project schedules, transportation problems and network flow problems. References [1] Amit Kumar and Anila Gupta, Assignment and Travelling Salesman Problems with Coefficients as LR Fuzzy Parameters, International Journal of Applied Science and Engineering, (2012) 10, 3:155- 170. [2] Amit Kumar, Amarpreet Kaur and Anila Gupta, Fuzzy Linear Programming Approach for Solving Fuzzy Transportation probles with Transshipment, J Math Model Algor, (2011), 10:163-180. [3] Amit Kumar, Anila Gupta and Amarpreet Kumar, Method for Solving Fully Fuzzy Assignment Problems Using Triangular Fuzzy Numbers, International Journal of Computer and Information Engineering, 3: 4 (2009). [4] Annie Varghese and Sunny Kuriakose, Centroid of an intuitionistic fuzzy number, Notes on Intuitionistic Fuzzy Sets, Vol.18, (2012), No.1, 19-24.
- 5. 302 P. Senthil Kumar and R. Jahir Hussain [5] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems, Vol.20, no.1 (1986), pp.87-96. [6] D. Avis, L. Devroye, An analysis of a decomposition heuristic for the assignment problem, Oper. Res. Lett., 3 (6) (1985), 279-283. [7] R. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, management sci.,17 (B) (1970), 141-164. [8] M. L. Balinski, A Competitive (dual) simplex method for the assignment problem, Math. Program, 34 (2) (1986), 125-141. [9] R. S. Barr, F. Glover and D. Klingman, The alternating basis algorithm for assignment problems, Math. Program, 13 (1) (1977), 1-13. [10] M. S. Chen, On a Fuzzy Assignment Problem, Tamkang Journal, 22 (1985), 407-411. [11] P. K. De and Bharti Yadav, A General Approach for Solving Assignment Problems Involving with Fuzzy Costs Coefficients, Modern Applied Science, Vol.6, No.3; March (2012). [12] S. P. Eberhardt, T. Duad, A. Kerns, T. X. Brown and A. S. Thakoor, Competitive neural architecture for hardware solution to the assignment problem, Neural Networks, 4 (4) (1991), 431-442. [13] M. S. Hung and W. O. Rom, Solving the assignment problem by relaxation, Oper. Res., 24 (4) (1980), 969-982. [14] R. Jahir Hussain and P. Senthil Kumar, The Transportation Problem in an Intuitionistic Fuzzy Environment, International Journal of Mathematics Research, Vol.4 Number 4 (2012), pp.411-420. [15] R. Jahir Hussain and P. Senthil Kumar, An Optimal More-for-Less Solution of Mixed Constrains Intuitionistic Fuzzy Transportation Problems, Int. J. Contemp. Math. Sciences, Vol.8, (2013), no.12, 565-576. doi.10.12988/ijcms. [16] H. W. Kuhn, The Hungarian method for the assignment problem, Novel Research Logistic Quarterly, 2 (1955), 83-97. [17] Lin Chi-Jen and Wen Ue-Pyng, An Labeling Algorithm for the fuzzy assignment problem, Fuzzy Sets and Systems, 142 (2004), 373-391. [18] L. F. McGinnis, Implementation and testing of a primal-dual algorithm for the assignment problem, Oper. Res., 31 (2) (1983), 277-291. [19] Sathi Mukherjee and Kajla Basu, Application of Fuzzy Ranking Method for Solving Assignment Problems with Fuzzy Costs, Int. Jour. Comp and Appl. Mathematics, Vol 5 Number 3 (2010), pp.359- 368. [20] Sathi Mukherjee and Kajla Basu, Solution of a class of Intuitionistic Fuzzy Assignment Problem by using similarity measures, Knowledge-Based Systems, 27 (2012), 170-179. [21] P. Senthil Kumar, R. Jahir Hussain, A method for solving balanced intuitionistic fuzzy assignment problem, Int. Journal of Engineering Research and Applications, Vol.4 Issue 3 (Version 1) March (2014), pp. 897-903. [22] P. Senthil Kumar and R. Jahir Hussain, New algorithm for solving mixed intuitionistic fuzzy assignment problem, Elixir Appl.Math., 73 (2014), 25971-25977. [23] P. Senthil Kumar, R. Jahir Hussain, A method for finding an optimal solution of an assignment problem under mixed intuitionistic fuzzy environment, Proceedings in International Conference on Mathematical Sciences Published by ELSEVIER, Sathyabama University, ICMS-(2014), ISBN-978- 93-5107-261-4: 417-421. [24] P. Senthil Kumar and R. Jahir Hussain, Computationally simple approach for solving fully intuitionistic fuzzy real life transportation problems, International Journal of System Assurance Engineering and Management, (2015): 1-12. DOI 10.1007/s13198-014-0334-2. [25] Y. L. P. Thorani and N. Ravi Sankar, Fuzzy Assignment Problem with Generalized Fuzzy Numbers, App. Math. Sci, Vol.7, (2013), no.71, 3511-3537. [26] X. Wang, Fuzzy Optimal Assignment Problem, Fuzzy Math., 3 (1987), 101-108. [27] L. A. Zadeh, Fuzzy sets, Information and computation, vol.8, (1965), pp.338-353.
- 6. Instructions to Authors The Journal of Fuzzy Mathematics seeks to publish articles of worth to the theory of fuzzy sets and their applications. Authors are required to follow the procedures listed below. (1) The submission of a manuscript implies that it is not being considered contemporaneously for publication elsewhere. All papers will be independently refereed. (2) Contributors are invited to submit three copies of their paper to the editor-in-cheif. The paper should be written in English and accompanied by an abstract of about 200 words. A list of key words would be put at the end of the abstract. (3) The paper should be submitted in its final form. No new material may be inserted in the text at the time of proof reading unless accepted by the Editor. (4) The manuscript must be typed double spaced on one side of white paper with wide margins. Each page should be numbered. (5) All mathematical symbols should be listed and identified typographically on a separate page. (6) The original and two copies of each illustration should be provided. In order to make a direct reproduction possible, all illustrations should be carefully lettered, and drawn large enough in India ink on a separate page with wide margins. Each illustration must be numbered and the title and name of the author of the paper must be written on the reverse side. (7) Reference should be listed alphabetically at the end of the paper, For example: [1] L.A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl., 23(1968), 421-427. [2] C. V. Negoita and D. A. Ralescu, Fuzzy sets and their applications, Technical Press, Bucharest, 1974. (8) No page charge is made. Reprints may be ordered by using the reprint order form which accompanies the proofs, prior to publication. Post-publication orders cannot be filled at regular reprint prices. (9) If an article is accepted for publication, the author will be asked to transfer the copyright of the article to the publisher.