A Simple and Efficient Algorithm for Solving Type-1 ...

International Journal of Operations Research and Information

Systems

Volume 9 • Issue 3 • July-September 2018 • ISSN: 1947-9328 • eISSN: 1947-9336

An official publication of the Information Resources Management Association

EDITOR-IN-CHIEF John Wang, Montclair State University, USA

MANAGING EDITOR Bin Zhou, University of Houston-Downtown, USA

INTERNATIONAL ADVISORY BOARD Yuval Cohen, Tel-Aviv Afeka College of Engineering, Israel

ASSOCIATE EDITORS Sungzoon Cho, Seould National University, Korea Theodore Glickman, George Washington University, USA Eva K. Lee, Georgia Institute of Technology, USA Panos Pardalos, University of Florida, USA Roman Polyak, George Mason University, USA Jasenkas Rakas, University of California at Berkeley, USA Kathryn E. Stecke, University of Texas at Dallas, USA

EDITORIAL REVIEW BOARD Anil K. Aggarwal, University of Baltimore, USA Sankarshan Basu, Indian Institute of Management Bangalore, India Melike Baykal-Gursoy, Rutgers University, USA Kevin Byrnes, Johns Hopkins University, USA Dean Chatfield, Old Dominion University, USA Chialin Chen, Queen’s University, Canada Jagpreet Chhatwal, Harvard Medical School, USA Wen Chiang, University of Tulsa, USA Louis Anthony Cox Jr., University of Colorado, USA Lauren Davis, North Carolina A&T State University, USA Matt Drake, Duquesne University, USA Ali Elkamel, University of Waterloo, Canada Murat Erkoc, University of Miami, USA Yudi Fernando, Universiti Malaysia Pahang, Malaysia William P. Fox, Naval Postgraduate School, USA Hise Gibson, INFORMS, USA Scott E. Grasman, Rochester Institute of Technology, USA Nalan Gulpinar, Warwick Business School, UK Roger Gung, Response Analytics Inc., USA Zhinling Guo, University of Maryland-Baltimore County, USA Peter Hahn, University of Pennsylvania, USA Mohammed Hajeeh, Kuwait Institute for Scientific Research, Kuwait Steven Harper, James Madison University, USA Michael J Hirsch, Raytheon Inc., USA Xinxin Hu, University of Houston - Downtown, USA Dariusz Jacek Jakóbczak, Koszalin University of Technology, Poland Manoj K. Jha, Morgan State University, USA Rex Kincaid, College of William & Mary, USA Nanda Kumar, University of Texas at Dallas, USA Hyoung-Gon Lee, Massachusetts Institute of Technology, USA Loo Hay Lee, National University of Singapore, Singapore

Editorial Review Board Continued

Fei Li, George Mason University, USA Jian Li, Northeastern Illinois University, USA Jing Li, Arizona State University, USA Leonardo Lopes, University of Arizona, USA Kaye McKinzie, U.S. Army, USA Yefim Haim Michlin, Israel Institute of Technology, Israel Somayeh Moazeni, Princeton University, USA Okesola Moses Olusola, Oludoy Dynamix Consulting Ltd, Nigeria Josefa Mula, Universitat Politècnica de València, Spain B.P.S. Murthi, University of Texas at Dallas, USA Olufemi A Omitaomu, Oak Ridge National Laboratory, USA Kivanc Ozonat, HP Labs, USA Dessislava Pachamanova, Babson College, USA Julia Pahl, University of Hamburg, Germany Francois Pinet, Irstea, France Michael Racer, University of Memphis, USA Marion S. Rauner, University of Vienna, Austria Enzo Sauma Pontificia, Universidad Catolica de Chile, Chile Hsu-Shih Shih, Tamkang University, Taiwan Young-Jun Son, University of Arizona, USA Huaming Song, Nanjing University of Science & Technology, China Yang Sun, California State University - Sacramento, USA Durai Sundaramoorthi, Washington University in St. Louis, USA Pei-Fang Tsai, State University of New York at Binghamton, USA M. Ali Ülkü, Dalhousie University, Canada Bruce Wang, Texas A&M University, USA Yitong Wang, Tsinghua University, China Harris Wu, Old Dominion University, USA Justin Yates, Francis Marion University, USA Xugang Ye, Johns Hopkins University and Microsoft, USA Banu Yukse-Ozkaya, Hacettepe University, Turkey

Call for Articles International Journal of Operations Research and Information Systems Volume 9 • Issue 3 • July-September 2018 • ISSN: 1947-9328 • eISSN: 1947-9336

An official publication of the Information Resources Management Association

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International Journal of Operations Research and Information Systems Volume 9 • Issue 3 • July-September 2018

A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic Fuzzy Solid Transportation Problems P. Senthil Kumar, PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli, India

ABSTRACT This article describes how in solving real-life solid transportation problems (STPs) we often face the state of uncertainty as well as hesitation due to various uncontrollable factors. To deal with uncertainty and hesitation, many authors have suggested the intuitionistic fuzzy (IF) representation for the data. In this article, the author tried to categorise the STP under uncertain environment. He formulates the intuitionistic fuzzy solid transportation problem (IFSTP) and utilizes the triangular intuitionistic fuzzy number (TIFN) to deal with uncertainty and hesitation. The STP has uncertainty and hesitation in supply, demand, capacity of different modes of transport celled conveyance and when it has crisp cost it is known as IFSTP of type-1. From this concept, the generalized mathematical model for type-1 IFSTP is explained. To find out the optimal solution to type-1 IFSTPs, a single stage method called intuitionistic fuzzy min-zero min-cost method is presented. A real-life numerical example is presented to clarify the idea of the proposed method. Moreover, results and discussions, advantages of the proposed method, and future works are presented. The main advantage of the proposed method is that the optimal solution of type-1 IFSTP is obtained without using the basic feasible solution and the method of testing optimality. Keywords Intuitionistic Fuzzy Min-Zero Min-Cost Method, Intuitionistic Fuzzy Set, Optimal Solution, Triangular Intuitionistic Fuzzy Number, Type-1 Intuitionistic Fuzzy Solid Transportation Problem

1. INTRODUCTION The transportation problem (TP) is a special class of linear programming problem, widely used in the areas of inventory control, employment scheduling, aggregate planning, communication network, personal management and so on. In several real life situations, there is a need for shipping the product from different origins (Factories) to different destinations (Warehouses). The transportation problem deals with shipping commodities from different origins to various destinations. The objective of the transportation problem is to determine the optimum amount of a commodity to be transported from various supply points (origins) to different demand points (destinations) so that the total transportation cost is minimum or total transportation profit is maximum. Depending on the nature of objective function, the transportation problem may be classified into two categories, namely, maximization transportation problems and minimization transportation problems. A minimization transportation problem involves cost data. The objective of solution is to minimize the total cost. A maximization transportation problem involves revenue or profit data. DOI: 10.4018/IJORIS.2018070105  Copyright © 2018, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. 

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The objective of solution is maximization of total profit. These problems are occurring either in corporations or in industry. The unit costs, that is, the cost of transporting one unit from a particular supply point to a particular demand point, the amounts available at the supply points and the amounts required at the demand points are the parameters of the transportation problem. In literature, Hitchcock (1941) developed a basic transportation problem. Koopmans (1949) presented optimum utilization of the transportation system. Charnes and Cooper (1954) developed the Stepping Stone Method (SSM), which provides an alternative way of determining the simplex method information. The transportation algorithm for solving transportation problems with equality constraints introduced by Dantzig (1963) is the simplex method specialized to the format of a table called transportation table. It involves two steps. First, we compute an initial basic feasible solution for the transportation problem and then, we test optimality and look at improving the basic feasible solution to the transportation problem. Dalman et al. (2013) designed a solution proposal to indefinite quadratic interval transportation problem. The solid transportation problem is a generalization of the classical transportation problem in which three-dimensional properties are taken into account in the objective and constraint set instead of source (origin) and destination. Shell (1955) stated an extension of well-known transportation problem is called a solid transportation problem in which bounds are given on three items, namely, supply, demand and conveyance. In many industrial problems, a homogeneous product is transported from an origin to a destination by means of different modes of transport called conveyances, such as trucks, cargo flights, goods trains, ships and so on. Haley (1962) proposed the solution procedure for solving solid transportation problem, which is an extension of the modified distribution method. Patel and Tripathy (1989) presented a computationally superior method for a solid transportation problem with mixed constraints. Basu et al. (1994) studied an algorithm for finding the optimum solution of a solid fixed charge linear transportation problem. For finding an optimal solution, the solid transportation problem requires m  n  l  2 nonnegative values of the decision variables to start with a basic feasible solution. Jimenez and Verdegay (1996) investigated interval multiobjective solid transportation problem via genetic algorithms. Li et al. (1997a) designed a neural network approach for a multicriteria solid transportation problem. Efficient algorithms have been developed for solving transportation problems when the coefficient of the objective function, demand, supply and conveyance values are known precisely. Many of the distribution problems are imprecise in nature in today’s world such as in corporate or in industry due to variations in the parameters. The TP is a distribution-type problem, the main objective of which is to decide how to transfer goods from various sending locations to various receiving locations with least costs or maximum profit. The sending locations is also known as origins or sources or factories. Similarly, the receiving locations is also known as destinations or warehouses or retail stores. In classical transportation problem it is assumed that the transportation costs and values of supplies and demands are exactly known. But in real life, the transportation parameters may not be precise always due to lack of information, environmental factors, changing weather, imprecision in judgment, social, or economic conditions and so on. Therefore, it is very interesting to deal with TPs under uncertainty. The best way to denote the imprecise data is fuzzy number. In literature, to deal quantitatively with imprecise information in making decision, Zadeh (1965) introduced the fuzzy set theory and has applied it successfully in various fields. The use of fuzzy set theory becomes very rapid in the field of optimization after the pioneering work done by Bellman and Zadeh (1970). The fuzzy set deals with the degree of membership (belongingness) of an element in the set but it does not consider the non-membership (non-belongingness) of an element in the set. In a fuzzy set the membership value (level of acceptance or level of satisfaction) lies between 0 and 1 where as in crisp set the element belongs to the set represent 1 and the element not in the set represent 0. Due to the lack of certainty in the parameters of a crisp transportation problem, several authors Dinagar and Palanivel (2009), Mohideen and Kumar (2010), Pandian and Natarajan (2010) have 91

A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic Fuzzy Solid Transportation Problems P. Senthil Kumar (PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli, India) Source Title: International Journal of Operations Research and Information Systems (IJORIS) 9(3) Copyright: © 2018 |Pages: 33 DOI: 10.4018/IJORIS.2018070105 OnDemand PDF $30.00 Download: List Price: $37.50

Reference to this paper should be made as follows: MLA Kumar, P. Senthil. "A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic Fuzzy Solid Transportation Problems." IJORIS 9.3 (2018): 90-122. Web. 2 Jun. 2018. doi:10.4018/IJORIS.2018070105 APA Kumar, P. S. (2018). A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic Fuzzy Solid Transportation Problems. International Journal of Operations Research and Information Systems (IJORIS), 9(3), 90-122. doi:10.4018/IJORIS.2018070105 Chicago Kumar, P. Senthil. "A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic Fuzzy Solid Transportation Problems," International Journal of Operations Research and Information Systems (IJORIS) 9 (2018): 3, accessed (June 02, 2018), doi:10.4018/IJORIS.2018070105 Harvard Kumar, P.S., 2018. A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic Fuzzy Solid Transportation Problems. International Journal of Operations Research and Information Systems (IJORIS), 9(3), pp.90-122. Vancouver Kumar PS. A Simple and Efficient Algorithm for Solving Type-1 Intuitionistic Fuzzy Solid Transportation Problems. International Journal of Operations Research and Information Systems (IJORIS). 2018 Jun 2;9(3):90-122.

International Journal of Operations Research and Information Systems Volume 9 • Issue 3 • July-September 2018

6.2. Advantages of the Proposed Method By using the proposed method a decision maker has the following advantages: 1. We need not find out the basic feasible solution and we need not apply the optimality test because the solution obtained by proposed method is always optimal; 2. The proposed method is a single step method. So, the use of intuitionistic fuzzy modified distribution is not required. 7. CONCLUSION AND FUTURE WORK In this article, new problem called type-1 IFSTP is introduced. Different types of IFSTPs is identified. Intuitionistic fuzzy min-zero min-cost method for finding the intuitionistic fuzzy optimal solution to type-1 IFSTP is proposed. The proposed methodology is illustrated with the help of real life numerical example and the optimal solution is obtained in terms of triangular intuitionistic fuzzy numbers. Results, discussion and advantages of the proposed method is also devoted. The type-1 IFSTPs are solved by the proposed method, which differs from the existing methods namely, the extended version of modified distribution method (Haley (1962)) and intuitionistic fuzzy modified distribution method (Kumar and Hussain (2015)). The extended version of modified distribution method and intuitionistic fuzzy modified distribution method both are depends on its basic feasible solution. But, the main advantage of the proposed method is that the obtained solution is always optimal. To apply this method, there is no necessity to have (m + n + l − 2) number of non-negative allotted entries (i.e., basic feasible solution). Also, we need not test the optimality condition. It is applicable to type-1, type-2, type-3 and type-4 IFSTPs. The proposed method can help decision-makers in the logistics related issues of real-life problems by aiding them in the decisionmaking process and providing an optimal solution in a simple and effective manner. Further, it can be served as an important tool for a decision-maker when he/she handles various types of logistic problems having different types of parameters. In future this approach can be applied in solving solid transportation problems having uncertainty and hesitation in costs. In future our algorithm can be extended for solving solid transportation problems having all parameters as TIFNs. In addition, we will attempt to develop a linear programming problem approach for solving type-1 IFSTP to reduce both the computation time and computation complexity of the proposed method. ACKNOWLEDGMENT The author sincerely thanks the anonymous reviewers and Editor-in-Chief Professor John Wang for their careful reading, constructive comments and fruitful suggestions. The author would also like to acknowledge Dr. S. Ismail Mohideen, Additional Vice Principal, My Guide and Associate Professor Dr. R. Jahir Hussain, Dr. A. Nagoor Gani, Associate Professor, Dr. K. Ramanaiah, Associate Professor (retired), Mr. N. Shamsudeen, Associate Professor (retired), Jamal Mohamed College (Autonomous), Tiruchirappalli, Tamil Nadu, India for their motivation and kind support.

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Kumar, P. S. (2016a). PSK method for solving type-1 and type-3 fuzzy transportation problems. International Journal of Fuzzy System Applications, 5(4), 121-146, doi:10.4018/IJFSA.2016100106 Kumar, P. S. (2016b). A simple method for solving type-2 and type-4 fuzzy transportation problems. International Journal of Fuzzy Logic and Intelligent Systems, 16(4), 225–237. doi:10.5391/IJFIS.2016.16.4.225 Kumar, P. S. (2017a). PSK method for solving type-1 and type-3 fuzzy transportation problems. In Fuzzy Systems: Concepts, Methodologies, Tools, and Applications (pp. 367–392). Hershey, PA: IGI Global; doi:10.4018/9781-5225-1908-9.ch017 Kumar, P. S. (2017b). Algorithmic approach for solving allocation problems under intuitionistic fuzzy environment [PhD thesis]. Jamal Mohamed College, Tiruchirappalli, India. Kumar, P. S. (2018a). A note on ‘a new approach for solving intuitionistic fuzzy transportation problem of type2’, Int. J. Logistics Systems and Management, 29(1), 102–129. doi:10.1504/IJLSM.2018.10009204 Kumar, P. S. (In pressa). Linear programming approach for solving balanced and unbalanced intuitionistic fuzzy transportation problems. International Journal of Operations Research and Information Systems. Kumar, P. S. (In pressb). PSK method for solving intuitionistic fuzzy solid transportation problems. International Journal of Fuzzy System Applications. Kumar, P. S. (In pressc). Intuitionistic fuzzy zero point method for solving type-2 intuitionistic fuzzy transportation problem. International Journal of Operational Research. Kumar, P. S., & Hussain, R. J. (2014a). A systematic approach for solving mixed intuitionistic fuzzy transportation problems. International Journal of Pure and Applied Mathematics, 92(2), 181–190. doi:10.12732/ijpam.v92i2.4 Kumar, P. S., & Hussain, R. J. (2014b). A method for finding an optimal solution of an assignment problem under mixed intuitionistic fuzzy environment. In Proceedings in International Conference on Mathematical Sciences (ICMS-2014), Sathyabama University, Chennai (pp. 417-421). Elsevier. Kumar, P. S., & Hussain, R. J. (2014c). New algorithm for solving mixed intuitionistic fuzzy assignment problem. In Elixir Applied Mathematics (pp. 25971–25977). Salem, Tamilnadu, India: Elixir publishers. Kumar, P. S., & Hussain, R. J. (2014d). A method for solving balanced intuitionistic fuzzy assignment problem. International Journal of Engineering Research and Applications, 4(3), 897–903. Kumar, P. S., & Hussain, R. J. (2015). A method for solving unbalanced intuitionistic fuzzy transportation problems. Notes on Intuitionistic Fuzzy Sets, 21(3), 54–65. Kumar, P. S., & Hussain, R. J. (2016a). Computationally simple approach for solving fully intuitionistic fuzzy real life transportation problems. International Journal of System Assurance Engineering and Management, 7(1), 90–101. doi:10.1007/s13198-014-0334-2 Kumar, P. S., & Hussain, R. J. (2016b). A simple method for solving fully intuitionistic fuzzy real life assignment problem. International Journal of Operations Research and Information Systems, 7(2), 39–61. doi:10.4018/ IJORIS.2016040103 Kumar, P. S., & Hussain, R. J. (2016c). An algorithm for solving unbalanced intuitionistic fuzzy assignment problem using triangular intuitionistic fuzzy number. The Journal of Fuzzy Mathematics, 24(2), 289–302. Li, Y., Ida, K., & Gen, M. (1997b). Improved genetic algorithm for solving multiobjective solid transportation problem with fuzzy numbers. Computers & Industrial Engineering, 33(3), 589–592. Li, Y., Ida, K., Gen, M., & Kobuchi, R. (1997a). Neural network approach for multicriteria solid transportation problem. Computers & Industrial Engineering, 33(3), 465–468. Liu, S. T. (2006). Fuzzy total transportation cost measures for fuzzy solid transportation problem. Applied Mathematics and Computation, 174(2), 927–941. Mahapatra, B. S., & Mahapatra, G. S. (2010). Intuitionistic fuzzy fault tree analysis using intuitionistic fuzzy numbers. International Mathematical Forum, 5(21), 1015-1024. Mahapatra, G. S., & Roy, T. K. (2009). Reliability evaluation using triangular intuitionistic fuzzy numbers, arithmetic operations. International Scholarly and Scientific Research & Innovation, 3(2), 422–429. 120

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Mahapatra, G. S., & Roy, T. K. (2013). Intuitionistic fuzzy number and its arithmetic operation with application on system failure. Journal of Uncertain Systems, 7(2), 92–107. Mitchell, H. B. (2004). Ranking intuitionistic fuzzy numbers. International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, 12(3), 377–386. Mohideen, S. I., & Kumar, P. S. (2010). A comparative study on transportation problem in fuzzy environment. International Journal of Mathematics Research, 2(1), 151–158. Nayagam, G., Lakshmana, V., Venkateshwari, G., & Sivaraman, G. (2008, June). Ranking of intuitionistic fuzzy numbers. In Proceedings of the IEEE International Conference on Fuzzy Systems FUZZ-IEEE ’08 (pp. 1971-1974). IEEE. Nehi, H. M. (2010). A new ranking method for intuitionistic fuzzy numbers. International Journal of Fuzzy Systems, 12(1), 80–86. Nehi, H. M., & Maleki, H. R. (2005, July). Intuitionistic fuzzy numbers and it’s applications in fuzzy optimization problem. In Proceedings of the Ninth WSEAS International Conference on Systems, Athens, Greece. Ojha, A., Das, B., Mondal, S., & Maiti, M. (2009). An entropy based solid transportation problem for general fuzzy costs and time with fuzzy equality. Mathematical and Computer Modelling, 50(1), 166–178. Pandian, P., & Anuradha, D. (2010). A new approach for solving solid transportation problems. Applied Mathematical Sciences, 4(72), 3603–3610. Pandian, P., & Natarajan, G. (2010). A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Applied Mathematical Sciences, 4(2), 79–90. Parvathi, R., & Malathi, C. (2012). Intuitionistic fuzzy linear programming problems. World Applied Sciences Journal, 17(12), 1802–1807. Patel, G., & Tripathy, J. (1989). The solid transportation problem and its variants. International Journal of Management and Systems, 5(5), 17–36. Shabani, A., & Jamkhaneh, E. B. (2014). A new generalized intuitionistic fuzzy number. Journal of Fuzzy Set Valued Analysis, 24. doi:10.5899/2014/jfsva-00199 Shaw, A. K., & Roy, T. K. (2012). Some arithmetic operations on triangular intuitionistic fuzzy number and its application on reliability evaluation. International Journal of Fuzzy Mathematics and Systems, 2(4), 363–382. Shell, E. (1955). Distribution of a product by several properties. Directorate of Management Analysis. In Proceedings of the Second Symposium in Linear Programming (Vol. 2, pp. 615-642). Singh, S. K., & Yadav, S. P. (2014). Efficient approach for solving type-1 intuitionistic fuzzy transportation problem. Int. J. Syst. Assur. Eng. Manag. doi:10.1007/s13198-014-0274-x Singh, S. K., & Yadav, S. P. (2015). Fuzzy Programming Approach for Solving Intuitionistic Fuzzy Linear Fractional Programming Problem. International Journal of Fuzzy Systems. Singh, S. K., & Yadav, S. P. (2016). A novel approach for solving fully intuitionistic fuzzy transportation problem. International Journal of Operational Research, 26(4), 460–472. doi:10.1504/IJOR.2016.077684 Varghese, A., & Kuriakose, S. (2012). Centroid of an intuitionistic fuzzy number. Notes on Intuitionistic Fuzzy Sets, 18(1), 19–24. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. doi:10.1016/S0019-9958(65)90241-X

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P. Senthil Kumar is an Assistant Professor in PG and Research Department of Mathematics at Jamal Mohamed College (Autonomous), Tiruchirappalli, Tamil Nadu, India. He has seven years of teaching experience. He received his BSc, MSc and MPhil degrees from Jamal Mohamed College, Tiruchirappalli in 2006, 2008 and 2010, respectively. He completed his BEd in Jamal Mohamed College of Teacher Education in 2009. He completed PGDCA in 2011 in the Bharathidasan University and PGDAOR in 2012 in the Annamalai University, Tamil Nadu, India. He completed his PhD in the area of intuitionistic fuzzy optimization technique at Jamal Mohamed College in 2017. He has published many research papers in referred national and international journals like Springer, Korean Institute of Intelligent Systems (KIIS), IGI Global, Inderscience, etc. He also presented his research papers in Elsevier Conference Proceedings (ICMS-2014), MMASC-2012, etc. His areas of interest include operations research, fuzzy optimization, intuitionistic fuzzy optimization, numerical analysis and graph theory, etc. 122