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Set covering-based surrogate approach for solving sup-\({\mathcal{T}}\) equation constrained optimization problems

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Abstract

This work considers solving the sup-\({\mathcal{T}}\) equation constrained optimization problems from the integer programming viewpoint. A set covering-based surrogate approach is proposed to solve the sup-\({\mathcal{T}}\) equation constrained optimization problem with a separable and monotone objective function in each of the variables. This is our first trial of developing integer programming-based techniques to solve sup-\({\mathcal{T}}\) equation constrained optimization problems. Our computational results confirm the efficiency of the proposed method and show its potential for solving large scale sup-\({\mathcal{T}}\) equation constrained optimization problems.

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References

  • Abbasi Molai A., Khorram E. (2007) A modified algorithm for solving the proposed models by Ghodousian and Khorram and Khorram and Ghodousian. Applied Mathematics and Computation 190: 1161V1167

    Google Scholar 

  • Abbasi Molai A., Khorram E. (2008) An algorithm for solving fuzzy relation equations with max-T composition operator. Information Sciences 178(5): 1293–1308

    Article  MathSciNet  MATH  Google Scholar 

  • Balas E., Carrera M. C. (1996) A dynamic subgradient-based branch and bound procedure for set covering. Operations Research 44: 875–890

    Article  MathSciNet  MATH  Google Scholar 

  • Balas E., Ho A. (1980) Set covering algorithms using cutting planes, heuristics and subgradient optimization: A computational study. Mathematical Programming 12: 37–60

    MathSciNet  MATH  Google Scholar 

  • Beasley J. E. (1987) An algorithm for the set covering problem. European Journal of Operational Research 31: 85–93

    Article  MathSciNet  MATH  Google Scholar 

  • Beasley J. E. (1990) A Lagrangian heuristic for the set covering problem. Naval Research Logistics 37: 151–164

    Article  MathSciNet  MATH  Google Scholar 

  • Beasley J. E., Chu P. C. (1996) A genetic algorithm for the set covering problem. European Journal of Operational Research 94: 392–404

    Article  MATH  Google Scholar 

  • Blyth T. S., Janowitz M. F. (1972) Residuation Theory. Pergamon Press, Oxford

    MATH  Google Scholar 

  • Caprara A., Toth P., Fischetti M. (2000) Algorithms for the set covering problem. Annals of Operations Research 98: 353–371

    Article  MathSciNet  MATH  Google Scholar 

  • Di Nola A., Sessa S., Pedrycz W., Sanchez E. (1989) Fuzzy relation equations and their applications to knowledge engineering. Kluwer, Dordrecht, Netherlands

    MATH  Google Scholar 

  • Elbassioni K. M. (2008) A note on systems with max-min and max-product constraints. Fuzzy Sets and Systems 159: 2272–2277

    Article  MathSciNet  MATH  Google Scholar 

  • Fang S.-C., Li G. (1999) Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems 103(1): 107–113

    Article  MathSciNet  MATH  Google Scholar 

  • Fisher M. L., Kedia P. (1990) Optimal solutions of set covering/partitioning problems using dual heuristics. Management Science 36: 674–688

    Article  MathSciNet  MATH  Google Scholar 

  • Ghodousian A., Khorram E. (2006) An algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max-prod composition. Applied Mathematics and Computation 178(2): 502–509

    MathSciNet  MATH  Google Scholar 

  • Golumbic M. C., Hartman I. B.-A. (2005) Graph theory, combinatorics and algorithms: Interdisciplinary applications. Springer, New York

    Book  MATH  Google Scholar 

  • Guu S. M., Wu Y. K. (2002) Minimizing a linear objective function with fuzzy relation equation constraints. Fuzzy Optimization and Decision Making 1(4): 347–360

    Article  MathSciNet  MATH  Google Scholar 

  • Held M., Wolfe P., Crowder H. P. (1974) Validation of subgradient optimization. Mathematical Programming 6: 62–88

    Article  MathSciNet  MATH  Google Scholar 

  • Khachiyan L., Boros E., Elbassioni K., Gurvich V. (2006) An efficient implementation of a quasi-polynomial algorithm for generating hypergraph transversals and its application in joint generation. Discrete Applied Mathematics 154: 2350–2372

    Article  MathSciNet  MATH  Google Scholar 

  • Khorram E., Ghodousian A. (2006) Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition. Applied Mathematics and Computation 173(2): 872–886

    Article  MathSciNet  MATH  Google Scholar 

  • Klement E. P., Mesiar R., Pap E. (2000) Triangular norms. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Li, P., Fang, S.-C. & Zhang, X. (2008). Nonlinear optimization subject to a system of fuzzy relational equations with max-min composition. In Proceedings of the seventh international symposium of operations research and its applications, Lijiang, China, pp. 1–9.

  • Li P., Fang S.-C. (2008) On the resolution and optimization of a system of fuzzy relational equations with sup-\({\mathcal{T} }\) composition. Fuzzy Optimization and Decision Making 7(2): 169–214

    Article  MathSciNet  MATH  Google Scholar 

  • Li P., Fang S.-C. (2009) A survey on fuzzy relational equations, Part I: classification and solvability. Fuzzy Optimization and Decision Making 8: 179–229

    Article  MathSciNet  MATH  Google Scholar 

  • Li P., Fang S.-C. (2009) Latticized linear optimization on the unit interval. IEEE Transactions on Fuzzy Systems 17: 1353–1365

    Article  Google Scholar 

  • Loetamonphong J., Fang S.-C., Young R. (2002) Multi-objective optimization problems with fuzzy relation equation constraints. Fuzzy Sets and Systems 127(2): 141–164

    Article  MathSciNet  MATH  Google Scholar 

  • Loetamonphong J., Fang S.-C. (2001) Optimization of fuzzy relation equations with max-product composition. Fuzzy Sets and Systems 118(3): 509–517

    Article  MathSciNet  MATH  Google Scholar 

  • Lopes F. B., Lorena L. A. (1994) Surrogate heuristic for set covering problems. European Journal of Operational Research 79: 138–150

    Article  MATH  Google Scholar 

  • Lorena, L. A. N. & Plateau, G. (1988) A monotone decreasing algorithm for the 0-1 multiknapsack dual problem, Rapport de recherche L.I.P.N. 89-1, Université Paris-Nord, France.

  • Lu J., Fang S.-C. (2001) Solving nonlinear optimization problems with fuzzy relation equation constraints. Fuzzy Sets and Systems 119(1): 1–20

    Article  MathSciNet  Google Scholar 

  • Markovskii A. (2005) On the relation between equations with max-product composition and the covering problem. Fuzzy Sets and Systems 153(2): 261–273

    Article  MathSciNet  MATH  Google Scholar 

  • Nemhauser K., Wolsey L. A. (1988) Integer and combinatorial optimization. Wiley, New York

    MATH  Google Scholar 

  • Parker R. G. (1988) Discrete optimization. Academic Press, New York

    MATH  Google Scholar 

  • Pedrycz W. (1985) On generalized fuzzy relational equations and their applications. Journal of Mathematical Analysis and Applications 107: 520–536

    Article  MathSciNet  MATH  Google Scholar 

  • Peeva K., Kyosev Y. (2004) Fuzzy relational calculus: Theory, applications, and software. World Scientific, New Jersey

    MATH  Google Scholar 

  • Sanchez E. (1976) Resolution of composite fuzzy relation equations. Information and Control 30(1): 38–48

    Article  MathSciNet  MATH  Google Scholar 

  • Sanchez E. (1977) Solutions in composite fuzzy relation equations: Application to medical diagnosis in Brouwerian logic. In: Gupta M. M., Saridis G. N., Gaines B. R. (eds) Fuzzy automata and decision processes. North-Holland, Amsterdam, pp 221–234

    Google Scholar 

  • Shieh B. S. (2010) A note on a paper by Molai and Khorram. Applied Mathematics and Computation 216: 3419–3422

    Article  MathSciNet  MATH  Google Scholar 

  • Wang H. F. (1995) A multi-objective mathematical programming problem with fuzzy relation constraints. Journal of Multi-Criteria Decision Analysis 4(1): 23–35

    Article  MATH  Google Scholar 

  • Wang P. Z., Zhang D. Z., Sanchez E., Lee E. S. (1991) Latticized linear programming and fuzzy relation inequalities. Journal of Mathematical Analysis and Applications 159(1): 72–87

    Article  MathSciNet  MATH  Google Scholar 

  • Wu Y. K., Guu S. M., Liu J. Y. C. (2002) An accelerated approach for solving fuzzy relation equations with a linear objective function. IEEE Transactions on Fuzzy Systems 10(4): 552–558

    Article  Google Scholar 

  • Wu Y. K., Guu S. M. (2004) A note on fuzzy relation programming problems with max-strict-t-norm composition. Fuzzy Optimization and Decision Making 3(3): 271–278

    Article  MathSciNet  MATH  Google Scholar 

  • Wu Y. K., Guu S. M. (2005) Minimizing a linear function under a fuzzy max- min relational equation constraint. Fuzzy Sets and Systems 150(1): 147–162

    Article  MathSciNet  MATH  Google Scholar 

  • Yang, J. H. & Cao, B. Y. (2007). Posynomial fuzzy relation geometric programming. In: Melin, P., Castillo, O., Aguilar, L. T., Kacprzyk, J., Pedrycz, W. (eds.), Proceedings of the 12th international fuzzy systems association world congress, Cancun, Mexico, pp. 563–572.

  • Yang, J. H. & Cao, B. Y. (2005). Geometric programming with fuzzy relation equation constraints. in proceedings of the ieee international conference on fuzzy systems, Reno, NV, pp. 557–560.

  • Zimmermann K. (2007) A note on a paper by E. Khorram and A. Ghodousian. Applied Mathematics and Computation 188: 244–245

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Industrial Engineering and Management, I-ShouUniversity, Kaohsiung, 840, Taiwan

    Cheng-Feng Hu

  2. Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27695-7906, USA

    Shu-Cherng Fang

Authors
  1. Cheng-Feng Hu
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  2. Shu-Cherng Fang
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Corresponding author

Correspondence to Cheng-Feng Hu.

Additional information

This work was supported by the National Science Council (NSC) of R.O.C., under Grant NSC 99-2918-I-214-001.

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Hu, CF., Fang, SC. Set covering-based surrogate approach for solving sup-\({\mathcal{T}}\) equation constrained optimization problems. Fuzzy Optim Decis Making 10, 125–152 (2011). https://doi.org/10.1007/s10700-011-9099-0

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