Advertisement

Evaluation of optimality in the fuzzy single machine scheduling problem including discounted costs

  • Toufik BentrciaEmail author
  • Leila-Hayet Mouss
  • Nadia-Kinza Mouss
  • Farouk Yalaoui
  • Lyes Benyoucef
ORIGINAL ARTICLE

Abstract

The single machine scheduling problem has been often regarded as a simplified representation that contains many polynomial solvable cases. However, in real-world applications, the imprecision of data at the level of each job can be critical for the implementation of scheduling strategies. Therefore, the single machine scheduling problem with the weighted discounted sum of completion times is treated in this paper, where we assume that the processing times, weighting coefficients and discount factor are all described using trapezoidal fuzzy numbers. Our aim in this study is to elaborate adequate measures in the context of possibility theory for the assessment of the optimality of a fixed schedule. Two optimization approaches namely genetic algorithm and pattern search are proposed as computational tools for the validation of the obtained properties and results. The proposed approaches are experimented on the benchmark problem instances and a sensitivity analysis with respect to some configuration parameters is conducted. Modeling and resolution frameworks considered in this research offer promise to deal with optimality in the wide class of fuzzy scheduling problems, which is recognized to be a difficult task by both researchers and practitioners.

Keywords

Single machine scheduling Discounted costs Possibility theory Augmented Lagrangian method Genetic algorithm Pattern search 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahmadizar F, Hosseini L (2011) Single-machine scheduling with a position-based learning effect and fuzzy processing times. Int J Adv Manuf Tech 56(5-8):693–698CrossRefGoogle Scholar
  2. 2.
    Ahmadizar F, Hosseini L (2013) Minimizing makespan in a single-machine scheduling problem with a learning effect and fuzzy processing times. Int J Adv Manuf Tech 65(1-4):581–587CrossRefGoogle Scholar
  3. 3.
    Altomare C, Guglielmann R, Riboldi M, Bellazzi R, Baroni G (2014) Optimal marker placement in hadrontherapy: intelligent optimization strategies with augmented Lagrangian pattern search. J Biomed Inform.  10.1016/j.jbi.2014.09.001
  4. 4.
    Artigues C, Demassey S, Nron E (2008) Resource-constrained project scheduling models, algorithms, extensions and applications. Wiley, New JerseyCrossRefGoogle Scholar
  5. 5.
    Baker KR, Trietsch D (2009) Principles of sequencing and scheduling. Wiley, New JerseyCrossRefzbMATHGoogle Scholar
  6. 6.
    BłaŻewicz J, Ecker KH, Pesch E, Schmidt G, Weglarz J (2007) Handbook on scheduling from theory to applications. Springer, BerlinzbMATHGoogle Scholar
  7. 7.
    Cao C, Gu X, Xin Z (2009) Chance constrained programming models for refinery short-term crude oil scheduling problem. Appl Math Model 33(3):1696–1707MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Castillo O, Melin P (2009) Soft computing models for intelligent control of non-linear dynamical systems. In: Mitkowski W, Kacprzyk J (eds) Modelling dynamics in processes and systems. Springer, Berlin, pp 43–70Google Scholar
  9. 9.
    Chanas S, Kasperski A (2001) Minimizing maximum lateness in a single machine scheduling problem with fuzzy processing times and fuzzy due dates. Eng Appl Artif Intel 14(3):377–386CrossRefGoogle Scholar
  10. 10.
    Chanas S, Kasperski A (2004) Possible and necessary optimality of solutions in the single machine scheduling problem with fuzzy parameters. Fuzzy Set Syst 142(3):359–371MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chryssolouris G (2006) Manufacturing systems: theory and practice, 2nd edn. Springer, New YorkGoogle Scholar
  12. 12.
    Chuang TN (2004) The EDD rule for fuzzy job time. J Inform Optim S 25(1):1–20MathSciNetzbMATHGoogle Scholar
  13. 13.
    Coello Coello CA (2002) Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput Method Appl M 191(11-12):1245–1287MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Conn AR, Gould N, Toint PL (1997) A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Math Comput 66(217):261–288MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Costa L, Esprito Santo IACP, Fernandes EMGP (2012) A hybrid genetic pattern search augmented Lagrangian method for constrained global optimization. Appl Math Comput 218(18):9415–9426MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dahal KP, Tan KC, Cowling PI (2007) Evolutionary scheduling. Springer, BerlinCrossRefzbMATHGoogle Scholar
  17. 17.
    Deb K, Srivastava S (2012) A genetic algorithm based augmented Lagrangian method for constrained optimization. Comput Optim Appl 53(3):869–902MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dong Y (2003) One machine fuzzy scheduling to minimize total weighted tardiness, earliness, and recourse cost. Int J Smart Eng Sys Des 5(3):135–147CrossRefGoogle Scholar
  19. 19.
    Dubois D, Prade H (1988) Possibility theory an approach to computerized processing of uncertainty. Plenum Press, New YorkzbMATHGoogle Scholar
  20. 20.
    Duenas A, Petrovic D (2008) Multi-objective genetic algorithm for single machine scheduling problem under fuzziness. Fuzzy Optim Decis Ma 7(1):87–104MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gawiejnowicz S (2008) Time-dependent scheduling. Springer, BerlinzbMATHGoogle Scholar
  22. 22.
    Georgescu I (2012) Possibility theory and the risk. Springer, BerlinCrossRefzbMATHGoogle Scholar
  23. 23.
    Glover F, Kochenberger GA (2003) Handbook of metaheurustics. Kluwer, DordrechtGoogle Scholar
  24. 24.
    Gupta SK, Kyparisis J (1987) Single machine scheduling research. OMEGA-Int J Manage S15(3):207–227CrossRefGoogle Scholar
  25. 25.
    Han S, Ishii H, Fuji S (1994) One machine scheduling problem with fuzzy due dates. Eur J Oper Res 79(1):1–12CrossRefGoogle Scholar
  26. 26.
    Harikrishnan KK, Ishii H (2005) Single machine batch scheduling problem with resource dependent setup and processing time in the presence of fuzzy due date. Fuzzy Optim Decis Ma 4(2):141–147MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Jamison KD (1998) Modeling uncertainty using probabilistic based possibility theory with applications to optimization. PhD Dissertation, University of Colorado, DenverGoogle Scholar
  28. 28.
    Kasperski A (2005) A possibilistic approach to sequencing problems with fuzzy parameters. Fuzzy Set Syst 150(1):77–86MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kasperski A, Ziełiński P (2011) Possibilistic minmax regret sequencing problems with fuzzy parameters. IEEE T Fuzzy Syst 19(6):1072–1082CrossRefGoogle Scholar
  30. 30.
    Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic theory and applications. Prentice Hall, New JerseyzbMATHGoogle Scholar
  31. 31.
    Kroll A, Schulte H (2014) Benchmark problems for nonlinear system identification and control using soft computing methods: need and overview. Appl Soft Comput 25:496–513CrossRefGoogle Scholar
  32. 32.
    Leung JYT (2004) Handbook of scheduling algorithms, models, and performance analysis. CRC, FloridazbMATHGoogle Scholar
  33. 33.
    Lewis RM, Torczon (1999) Pattern search algorithms for bound constrained minimization. SIAM J Optimiz 9(4):1082–1099MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lewis RM, Torczon V, Trosset MW (2000) Direct search methods: then and now. J Comput Appl Math 124(1-2):191–207MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Li J, Yuan X, Lee ES, Xu D (2011) Setting due dates to minimize the total weighted possibilistic mean value of the weighted earliness-tardiness costs on a single machine. Comput Math Appl 62(11):4126–4139MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Liao LM, Liao CJ (1998) Single machine scheduling problem with fuzzy due date and processing time. J Chinese Inst Eng 21(2):189–196MathSciNetCrossRefGoogle Scholar
  37. 37.
    Liu HC, Yih Y (2013) A fuzzy-based approach to the liquid crystal injection scheduling problem in a TFT-LCD fab. Int J Prod Res 51(20):6163–6181CrossRefGoogle Scholar
  38. 38.
    Lodwick WA, Kacprzyk J (2010) Fuzzy optimization recent advances and applications. Springer, BerlinzbMATHGoogle Scholar
  39. 39.
    Lopez P, Roubellat F (2001) Ordonnancement de la production. HermsGoogle Scholar
  40. 40.
    Mehrabad MS, Pahlavani A (2009) A fuzzy multi-objective programming for scheduling of weighted jobs on a single machine. Int J Adv Manuf Tech 45(1-2):122–139CrossRefGoogle Scholar
  41. 41.
    Nguyen HT (1978) A note on the extension principle for fuzzy sets. J Math Anal Appl 64(2):369–380MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Nikulin Y, Drexl A (2010) Theoretical aspects of multicriteria flight gate scheduling: deterministic and fuzzy models. J Sched D 13(3):261–280MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Nocedal J, Wrigh SJ (1999) Numerical optimization. Springer, New YorkCrossRefGoogle Scholar
  44. 44.
    Olaru D, Smith B (2005) Modelling behavioural rules for daily activity scheduling using fuzzy logic. Transportation 32(4):423–441CrossRefGoogle Scholar
  45. 45.
    Özelkan EC, Duckstein L (1999) Optimal fuzzy counterparts of scheduling rules. Eur J Oper Res 113 (3):593–609CrossRefzbMATHGoogle Scholar
  46. 46.
    Petrovic S, Petrovic D, Burke E (2011) Fuzzy logic-based production scheduling and rescheduling in the presence of uncertainty. In: Kempf KG (ed) Planning production and inventories in the extended enterprise. Springer, Berlin, pp 531–562Google Scholar
  47. 47.
    Pinedo ML (2012) Scheduling theory, algorithms, and systems, 4th edn. Springer, New YorkzbMATHGoogle Scholar
  48. 48.
    Prade H (1979) Using fuzzy set theory in a scheduling problem: a case study. Fuzzy Set Syst 2(2):153–165CrossRefzbMATHGoogle Scholar
  49. 49.
    Rahim MA, Khalid HM, Khoukhi A (2012) Nonlinear constrained optimal control problem: a PSO-GA-based discrete augmented Lagrangian approach. Int J Adv Manuf Tech 62(1-4):183–203CrossRefGoogle Scholar
  50. 50.
    Rocha AMAC, Martins TFMC, Fernandes EMGP (2011) An augmented Lagrangian fish swarm based method for global optimization. J Comput Appl Math 235(16):4611–4620MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Schultmann F, Fröhling M, Rentz O (2006) Fuzzy approach for production planning and detailed scheduling in paints manufacturing. Int J Prod Res 44(8):1589–1612CrossRefzbMATHGoogle Scholar
  52. 52.
    Sivanandam SN, Sumathi S, Deepa SN (2007) Introduction to fuzzy logic using MATLAB. Springer, BerlinCrossRefzbMATHGoogle Scholar
  53. 53.
    Srivastava S, Deb K (2010) A genetic algorithm based augmented Lagrangian method for computationally fast constrained optimization. In: Panigrahi BK (ed) Swarm, evolutionary, and memetic computing. Springer, Berlin, pp 330–337Google Scholar
  54. 54.
    Stanfield PM, King RE, Joines JA (1996) Scheduling arrivals to a production system in a fuzzy environment. Eur J Oper Res 93(1):75–87CrossRefzbMATHGoogle Scholar
  55. 55.
    Talbi EG (2013) Combining metaheuristics with mathematical programming, constraint programming and machine learning. 4OR-Q J Oper Res 11(2):101–150CrossRefzbMATHGoogle Scholar
  56. 56.
    Torczon V (1997) On the convergence of pattern search algorithms. SIAM J Optimiz 7(1):1–25MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Zadeh LA (1965) Fuzzy sets. Inform Control 3(3):338–353MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Set Syst 1:3–28MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Zimmermann HJ (1985) Fuzzy sets theory and applications. Kluwer, DorrechtCrossRefGoogle Scholar
  60. 60.
    Wang C, Wang D, Ip WH, Yuen DW (2002) The single machine ready time scheduling problem with fuzzy processing times. Fuzzy Set Syst 127(2):117–129MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Wonga BK, Lai VS (2011) A survey of the application of fuzzy set theory in production and operations management: 1998–2009. Int J Prod Econ 129(1):157–168CrossRefGoogle Scholar
  62. 62.
    Wu CC, Lee WC (2005) A single-machine group schedule with fuzzy setup and processing times. J Inform Optim S 26(3):683–691MathSciNetzbMATHGoogle Scholar
  63. 63.
    Wu HC (2010) Solving the fuzzy earliness and tardiness in scheduling problems by using genetic algorithms. Expert Syst Appl 37(7):4860–4866CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2015

Authors and Affiliations

  • Toufik Bentrcia
    • 1
    Email author
  • Leila-Hayet Mouss
    • 1
  • Nadia-Kinza Mouss
    • 1
  • Farouk Yalaoui
    • 2
  • Lyes Benyoucef
    • 3
  1. 1.LAP, Industrial Engineering DepartmentUniversity of BatnaBatnaAlgeria
  2. 2.Charles Delaunay Institute, LOSI, UMR STMR 6279University of Technology of TroyesTroyesFrance
  3. 3.Aix-Marseille University, LSIS UMR 7296Marseille Cedex 20France

Personalised recommendations