Soft Computing

, Volume 21, Issue 3, pp 805–816 | Cite as

A two-agent single-machine scheduling problem to minimize the total cost with release dates

  • Du-Juan Wang
  • Yunqiang Yin
  • Wen-Hsiang Wu
  • Wen-Hung Wu
  • Chin-Chia Wu
  • Peng-Hsiang Hsu
Methodologies and Application


This paper considers a two-agent scheduling problem with arbitrary release dates on a single machine. The cost of the first agent is the maximum weighted completion time of its jobs while the cost of the second agent is the total weighted completion time of its jobs. The goal is to schedule the jobs such that the total cost of the two agents is minimized. The problem is known to be strongly NP-hard. Thus, as an alternative, a branch-and-bound algorithm incorporating several dominance properties and a lower bound is provided to derive the optimal solution and a largest- order-value method combined with proposed three initials is developed to derive the near-optimal solutions for the problem. Computational results are also presented to evaluate the performance of the proposed algorithms.


Scheduling Two-agent Largest order value method Maximum weighted completion time 


  1. Agnetis A, Mirchandani PB, Pacciarelli D, Pacifici A (2004) Scheduling problems with two competing agents. Oper Res 52:229–242MathSciNetCrossRefMATHGoogle Scholar
  2. Agnetis A, Pacciarelli D, Pacifici A (2007) Multi-agent single machine scheduling. Ann Oper Res 150:3–15MathSciNetCrossRefMATHGoogle Scholar
  3. Agnetis A, Billaut J-C, Gawiejnovicz S, Pacciarelli D, Soukhal A (2014) Multiagent scheduling. Models and algorithms. Springer, BerlinCrossRefMATHGoogle Scholar
  4. Baker KR, Smith JC (2003) A multiple-criterion model for machine scheduling. J Sched 6:7–16MathSciNetCrossRefMATHGoogle Scholar
  5. Bean JC (1994) Genetic algorithms and random keys for sequencing and optimization. ORSA J Comput 6:154–160CrossRefMATHGoogle Scholar
  6. Belouadah H, Posner ME, Potts CN (1992) Scheduling with release dates on a single machine to minimize total weighted completion time. Discrete Appl Math 36:213–231MathSciNetCrossRefMATHGoogle Scholar
  7. Behnamian J, Fatemi Ghomi SMT (2014) Multi-objective fuzzy multiprocessor flowshop scheduling. Appl Soft Comput 21:139–148CrossRefGoogle Scholar
  8. Cheng SR (2012) A single-machine two-agent scheduling problem by GA approach. Asia-Pac J Oper Res 29(2):1250013MathSciNetCrossRefMATHGoogle Scholar
  9. Cheng TCE, Ng CT, Yuan JJ (2006) Multi-agent scheduling on a single machine to minimize total weighted number of tardy jobs. Theor Comput Sci 362:273–281MathSciNetCrossRefMATHGoogle Scholar
  10. Cheng TCE, Ng CT, Yuan JJ (2008) Multi-agent scheduling on a single machine with max-form criteria. Eur J Oper Res 188:603–609MathSciNetCrossRefMATHGoogle Scholar
  11. Cheng TCE, Cheng SR, Wu WH, Hsu PH, Wu CC (2011a) A two-agent single-machine scheduling problem with truncated sum-of-processing-times-based learning considerations. Comput Ind Eng 60:534–541CrossRefGoogle Scholar
  12. Cheng TCE, Wu WH, Cheng SR, Wu CC (2011b) Two-agent scheduling with position-based deteriorating jobs and learning effects. Appl Math Comput 217:8804–8824MathSciNetMATHGoogle Scholar
  13. Cheng TCE, Wu C-C, Chen J-C, Wu W-H, Cheng S-R (2013) Two-machine owshop scheduling with a truncated learning function to minimize the makespan. Int J Prod Econ 141:79–86CrossRefGoogle Scholar
  14. Dessouky MM (1998) Scheduling identical jobs with unequal ready times on uniform parallel machines to minimize the maximum lateness. Comput Ind Eng 34(4):793–806MathSciNetCrossRefGoogle Scholar
  15. Elvikis D, Kindt VT (2014) Two-agent scheduling on uniform parallel machines with min-max criteria. Ann Oper Res 213(1):79–94MathSciNetCrossRefMATHGoogle Scholar
  16. Feng Q, Yuan JJ (2007) NP-hardness of a multicriteria scheduling on two families of jobs. OR Trans 11(4):121–126Google Scholar
  17. Gerstl E, Mosheiov G (2013) Scheduling problems with two competing agents to minimized weighted earliness-tardiness. Comput Oper Res 40:109–116MathSciNetCrossRefMATHGoogle Scholar
  18. Gerstl E, Mosheiov G (2014) Single machine just-in-time scheduling problems with two competing agents. Naval Res Logist 61:1–16MathSciNetCrossRefGoogle Scholar
  19. Han S, Peng Z, Wang S (2014) The maximum flow problem of uncertain network. Inf Sci 265:167–175MathSciNetCrossRefMATHGoogle Scholar
  20. Ke H, Ma J (2014) Modeling project time-cost trade-off in fuzzy random environment. Appl Soft Comput 19:80–85CrossRefGoogle Scholar
  21. Lee K, Choi BC, Leung JYT, Pinedo ML (2009) Approximation algorithms for multi-agent scheduling to minimize total weighted completion time. Inf Process Lett 109:913–917MathSciNetCrossRefMATHGoogle Scholar
  22. Lee WC, Chen SK, Wu CC (2010) Branch-and-bound and simulated annealing algorithms for a two-agent scheduling problem. Expert Syst Appl 37:6594–6601CrossRefGoogle Scholar
  23. Lee WC, Chen SK, Chen WC, Wu CC (2011) A two-machine flowshop problem with two agents. Comput Oper Res 38:98–104MathSciNetCrossRefMATHGoogle Scholar
  24. Leung JYT, Pinedo M, Wan G (2010) Competitive two-agent scheduling and its applications. Oper Res 58:458–469Google Scholar
  25. Liu G, Zeng Y, Li D, Chen Y (2014) Schedule length and reliability-oriented multi- objective scheduling for distributed computing. Soft Comput. doi:10.1007/s00500-014-1360-3
  26. Luo W, Chen L, Zhang G (2012) Approximation schemes for two-machine flow shop scheduling with two agents. J Comb Optim 24(3):229–239MathSciNetCrossRefMATHGoogle Scholar
  27. Mor B, Mosheiov G (2010) Scheduling problems with two competing agents to minimize minmax and minsum earliness measures. Eur J Oper Res 206:540–546MathSciNetCrossRefMATHGoogle Scholar
  28. Marichelvam MK, Prabaharan T, Yang XS (2014) Improved cuckoo search algorithm for hybrid flow shop scheduling problems to minimize makespan. Appl Soft Comput 19:93–101CrossRefGoogle Scholar
  29. Ng CT, Cheng TCE, Yuan JJ (2006) A note on the complexity of the problem of two-agent scheduling on a single machine. J Comb Optim 12:387–394MathSciNetCrossRefMATHGoogle Scholar
  30. Nong QQ, Cheng TCE, Ng CT (2011) Two-agent scheduling to minimize the total cost. Eur J Oper Res 215:39–44MathSciNetCrossRefMATHGoogle Scholar
  31. Ou Z-H, Chen L-H (2014) A steganographic method based on tetris games. Inf Sci 276:343–353MathSciNetCrossRefGoogle Scholar
  32. Reeves C (1995) Heuristics for scheduling a single machine subject to unequal job release times. Eur J Oper Res 80:397–403CrossRefGoogle Scholar
  33. Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359Google Scholar
  34. Tamura Y, Iizuka H, Yamamoto M, Furukawa M (2015) Application of local clustering organization to reactive job-shop scheduling. Soft Comput 29(4):891–899Google Scholar
  35. Wan G, Vakati RS, Leung JYT, Pinedo M (2010) Scheduling two agents with controllable processing times. Eur J Oper Res 205:528–539MathSciNetCrossRefMATHGoogle Scholar
  36. Wu W-H, Cheng S-R, Wu C-C, Yin Y (2012) Ant colony algorithms for two-agent scheduling with sum-of-processing-times-based learning and deteriorating considerations. J Intell Manuf 23:1985–1993CrossRefGoogle Scholar
  37. Wu C-C, Wu W-H, Chen J-C, Yin Y, Wu W-H (2013) A study of the single-machine two-agent scheduling problem with release times. Appl Soft Comput 13(2):998–1002CrossRefGoogle Scholar
  38. Wu W-H, Yin Y, Wu W-H, Wu C-C, Hsu P-H (2014a) A time-dependent scheduling problem to minimize the sum of the total weighted tardiness among two agents. J Ind Manag Optim 10(2):591–611MathSciNetCrossRefMATHGoogle Scholar
  39. Wu C-C, Wu W-H, Wu W-H, Hsu P-H, Yin Y, Xu J (2014b) A single-machine scheduling with a truncated linear deterioration and ready times. Inf Sci 256:109–125MathSciNetCrossRefMATHGoogle Scholar
  40. Yin Y, Cheng S-R, Cheng TCE, Wu C-C, Wu W-H (2012a) Two-agent single-machine scheduling with assignable due dates. Appl Math Comput 219:1674–1685MathSciNetMATHGoogle Scholar
  41. Yin Y, Wu W-H, Cheng S-R, Wu C-C (2012b) An investigation on a two-agent single-machine scheduling problem with unequal release dates. Comput Oper Res 39:3062–3073MathSciNetCrossRefMATHGoogle Scholar
  42. Yin Y, Cheng S-R, Wu C-C (2012c) Scheduling problems with two agents and a linear non-increasing deterioration to minimize earliness penalties. Inf Sci 189:282–292MathSciNetCrossRefMATHGoogle Scholar
  43. Yin Y, Cheng S-R, Cheng TCE, Wu W-H, Wu C-C (2013a) Two-agent single-machine scheduling with release times and deadlines. Int J Shipp Transport Logist 5(1):75–94CrossRefGoogle Scholar
  44. Yin Y, Wu C-C, Wu W-H, Hsu C-J, Wu W-H (2013b) A branch-and-bound procedure for a single-machine earliness scheduling problem with two agents. Appl Soft Comput 13:1042–1054CrossRefGoogle Scholar
  45. Yin Y, Wu W-H, Wu W-H, Wu C-C (2014) A branch-and-bound algorithm for a single machine sequencing to minimize the total tardiness with arbitrary release dates and position-dependent learning effects. Inf Sci 256:91–108MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Du-Juan Wang
    • 1
  • Yunqiang Yin
    • 2
  • Wen-Hsiang Wu
    • 3
  • Wen-Hung Wu
    • 4
  • Chin-Chia Wu
    • 5
  • Peng-Hsiang Hsu
    • 4
  1. 1.School of Management Science and EngineeringDalian University of TechnologyDalianChina
  2. 2.Faculty of ScienceKunming University of Science and TechnologyKunmingChina
  3. 3.Department of Healthcare ManagementYuanpei UniversityHsinchuTaiwan
  4. 4.Department of Business AdministrationKang-Ning Junior College of Medical Care and ManagementTaipeiTaiwan
  5. 5.Department of StatisticsFeng Chia UniversityTaichungTaiwan

Personalised recommendations