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### Pareto Optimal Solutions of the Fuzzy Bicriteria Sheet Metal Problem

#### Affiliations

• Department of Mathematics, Amity School of Engineering and Technology, Bijwasan – 110061, New Delhi, India
• Department of Mathematics, Kalindi College, Delhi University, Central Delhi – 110008, New Delhi, India

#### Abstract

Objectives: An algorithm has been developed to find Pareto Optimal solutions of the fuzzy bicriteria sheet metal problem with pairwise nesting of designs. Methods and Statistical Analysis: The sheet metal problem has been solved by many workers, all of whom have considered the entities of cost and time as crisp numbers. However, in practical situations since cost and time are imprecise, the present work considers them as interval fuzzy numbers. Ordering between overlapping interval numbers is obtained by applying a fuzzy membership approach and a modified Hungarian algorithm is developed to obtain fuzzy Pareto Optimal solutions of the bicriteria problem. The newly developed algorithm is explained by a numerical example. Findings and Results: The set of both fuzzy Pareto optimal and other solutions obtained by applying the proposed algorithm, provide the Decision maker a lot of flexibility in making decisions. He can select the solution according to his priority. From amongst the fuzzy Pareto Optimal solutions obtained, he can select the solution which minimizes the cost or the solution which minimizes the time or take the middle path and select the solution which minimizes both cost and time as much as possible. Apart from the three fuzzy Pareto Optimal solutions, other solutions obtained by the proposed method can also be selected by the decision maker as per requirement and conditions. The problem being NP hard, it is very difficult and expensive to find fuzzy Pareto Optimal solutions of the bicriteria problem by analytical methods. The newly developed algorithm is not only easy to understand and implement but also gives good fuzzy Pareto optimal solutions. Improvements: The method can also be applied to costs and times being triangular and trapezoidal fuzzy numbers and it can be extended to nesting of up to three designs on a sheet.

#### Keywords

Interval Number, Nesting, Pareto Optimal, Sheet Metal

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#### References

• Herrman JW, Delalio DR. Evaluating Sheet Metal Nesting Decisions. Technical Research report, T.R. 98-8. United States: University of Maryland; 1998.
• Herrman JW, Delalio DR. Algorithms for sheet metal nesting. IEEE Transactions on Robotics and Automation. 2001; 17(2):183–90. Crossref
• Prasad YKDV, Somasundaram S, Rao KP. A sliding algorithm for optimal nesting or arbitrarily shaped sheet metal blanks. International Journal of Production Research. 1995; 33(6):1505–20. Crossref
• Zoran D, Moidrag M. Intelligent Nesting System. Yugoslav Journal of Operations Research. 2003; 13(2):229–43. Crossref
• Cheng SK, Rao KP. Large-scale nesting of irregular patterns using compact neighborhood algorithm. Journal of Materials Processing Technology. 2000; 103:135–40. Crossref
• Babu AR, Babu NR. A generic approach for nesting of 2-D parts in 2-D sheets using genetic and heuristic algorithms. Computer-Aided Design. 2001; 33:879–91. Crossref
• Tay FEH, Chong TY, Lee FC. Pattern nesting on irregular shaped stock using Genetic Algorithms. Engineering Applications of Artificial Intelligence. 2002; 15:551–8. Crossref
• Sakaguchi T, Ohtani H, Shimizu Y. Genetic Algorithm Based Nesting Method with Considering Schedule for Sheet Metal Processing. Transactions of the Institute of Systems, control and Information Engineering. 2015; 28(3):99–106. Crossref
• Reddy GHK. Genetic Algorithm Based 2D Nesting of Sheet Metal Parts. International Research Journal of Engineering and Technology. 2016; 3(6):1367–75.
• Chauhan SK, Tuli R, Singh PK. A Heuristic Technique to find Pareto Optimal Solutions of the Bicriteria Sheet Metal problem. International Journal of Advances in Engineering Sciences. 2014; 4(4):27–34.
• Zadeh LA. Fuzzy Sets. Information and Control. 1965; 8:338–53. Crossref
• Moore RE. Methods and Application of Interval Analysis. Philadelphia: SIAM; 1979. Crossref
• Ishibuchi H, Tanaka H. Multi-objective Programming in Optimization of the Interval Objective Function.
• European Journal of Operational Research. 1990; 48:219– 25. Crossref
• Okada S, Gen M. Order Relation between Intervals and its Application to Shortest Path Problem. Computers and Industrial Engineering. 1993; 25:147–50. Crossref
• Kundu S. Min-transitivity of Fuzzy Leftness Relationship and its application to Decision Making. Fuzzy sets and systems. 1997; 86:357–67. Crossref
• Sengupta A, Pal TK. On Comparing Interval Numbers. European Journal of Operational Research. 2000; 127:28– 43. Crossref
• Hu BQ, Wang S. A Novel Approach in Uncertain Programming Part I: New Arithmetic and Order Relation for Interval Numbers. Journal of Industrial and Management Optimization. 2006; 2(4):351–71 Crossref
• Mahato SK, Bhunia AK. Interval-arithmetic-Oriented Interval Computing Technique for Global Optimization. Applied Mathematics Research Express. 2006; 1–19. Crossref
• Tuli R, Sharma S. A Fuzzy Membership Approach to obtain Efficient Solutions of the Interval Valued Bicriteria Shortest Path Problem. Turkish journal of fuzzy system. 2015; 6(1):1–16
• Ignizio JP. Linear Programming in Single- and MultipleObjective Systems. Prentice Hall, Englewood Cliffs: N.J. 07632. 1982. p. 374–6.
• Steuer RE. Multiple Criteria Optimization: Theory, Computation, and Application. New York: Wiley; 1986. p. 138–59.

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