Single Machine Problems

  • Alessandro Agnetis
  • Jean-Charles Billaut
  • Stanisław Gawiejnowicz
  • Dario Pacciarelli
  • Ameur Soukhal


This chapter is devoted to single-machine agent scheduling problems. We present most of the results for the case of two agents (K = 2), for simplicity and because the most of the results found so far in the literature apply to this case. Whenever it is possible, we illustrate how these results can be extended to scenarios with a larger number of agents.


  1. Agnetis, A., Mirchandani, P., Pacciarelli, D., & Pacifici, A. (2004). Scheduling problems with two competing agents. Operations Research, 52, 229–242.Google Scholar
  2. Agnetis, A., Pacciarelli, D., & Pacifici, A. (2007). Multi-agent single machine scheduling. Annals of Operations Research, 150, 3–15.Google Scholar
  3. Agnetis, A., de Pascale, G., & Pranzo, M. (2009a). Computing the nash solution for scheduling bargaining problems. International Journal of Operational Research, 1, 54–69.Google Scholar
  4. Agnetis, A., Pacciarelli, D., & de Pascale, G. (2009b). A Lagrangian approach to single-machine scheduling problems with two competing agents. Journal of Scheduling, 12, 401–415.Google Scholar
  5. Agnetis, A., Nicosia, G., Pacifici, A., & Pferschy, U. (2013). Two agents competing for a shared machine. Lecture Notes in Computer Science, 8176 LNAI, 1–14.Google Scholar
  6. Aho, A. V., Hopcroft, J. E., & Ullman, J. D. (1974). The design and analysis of computer algorithms. Reading: Addison-Wesley.Google Scholar
  7. Albers, S., & Brucker, P. (1993). The complexity of one-machine batching problems. Discrete Applied Mathematics, 47, 87–107.Google Scholar
  8. Alidaee, B., & Womer, N. K. (1999). Scheduling with time dependent processing times: Review and extensions. Journal of the Operatational Research Society, 50, 711–720.Google Scholar
  9. Angel, E., Bampis, E., & Gourvès, L. (2005). Approximation results for a bicriteria job scheduling problem on a single machine without preemption. Information Processing Letters, 94, 19–27.Google Scholar
  10. Anzanello, M. J., & Fogliatto, F. S. (2011). Learning curve models and applications: Literature review and research directions. International Journal of Industrial Ergonomics, 41, 573–583.Google Scholar
  11. Arbib, C., Flammini, M., & Marinelli, F. (2003). Minimum flow time graph ordering. Lecture Notes on Computer Science, 2880, 23–33.Google Scholar
  12. Bachman, A., & Janiak, A. (2000). Minimizing maximum lateness under linear deterioration. European Journal of Operational Research, 126, 557–566.Google Scholar
  13. Bachman, A., & Janiak, A. (2004). Scheduling jobs with position-dependent processing times. Journal of the Operational Research Society, 55, 257–264.Google Scholar
  14. Baker, K., & Smith, J. C. (2003). A multiple criterion model for machine scheduling. Journal of Scheduling, 6, 7–16.Google Scholar
  15. Balasubramanian, H., Fowler, J., Keha, A., & Pfund, M. (2009). Scheduling interfering job sets on parallel machines. European Journal of Operational Research, 199, 55–67.Google Scholar
  16. Bellman, R. (1957). Dynamic programming. Princeton: Princeton University Press.Google Scholar
  17. Bellman, R., & Dreyfus, S. E. (1962). Applied dynamic programming. Princeton: Princeton University Press.Google Scholar
  18. Biskup, D. (2008). A state-of-the-art review on scheduling with learning effects. European Journal of Operational Research, 188, 315–329.Google Scholar
  19. Blazewicz, J., Ecker, K. H., Pesch, E., Schmidt, G., & Weglarz, J. (2007). Handbook on scheduling: From theory to applications. Berlin/Heidelberg: Springer.Google Scholar
  20. Bowman, E. H. (1959). The schedule sequencing problem. Operations Research, 7, 621–624.Google Scholar
  21. Brewer, P. J., & Plott, C. R. (1996). A binary conflict ascending price (bicap) mechanism for the decentralized allocation of the right to use railroad tracks. International Journal of Industrial Organization, 14(6), 857–886.Google Scholar
  22. Brucker, P. (2007). Scheduling algorithms (5th ed.). Berlin: Springer.Google Scholar
  23. Brucker, P., & Kovalyov, M. Y. (1996). Single machine batch scheduling to minimize the weighted number of late jobs. Mathematical Methods of Operations Research, 43, 1–8.Google Scholar
  24. Brucker, P., Gladky, A., Hoogeveen, H., Kovalyov, M. Y., Potts, C. N., Tautenhahn, T., & van de Velde, S. L. (1998). Scheduling a batching machine. Journal of Scheduling, 1(1), 31–54.Google Scholar
  25. Bruno, J., Coffman, E. G., & Sethi, R. (1974). Scheduling indepedant tasks to reduce mean finishing time. Communications of the ACM, 17, 382–387.Google Scholar
  26. Chen, Z.-L. (1996). Parallel machine scheduling with time dependent processing times. Discrete Applied Mathematics, 70, 81–93.Google Scholar
  27. Chen, Z.-L. (1997). Erratum to parallel machine scheduling with time dependent processing times. Discrete Applied Mathematics, 75, 103.Google Scholar
  28. Chen, B., Potts, C. N., & Woeginger, G. J. (1998). A review of machine scheduling: Complexity and approximability. In D. Z. Du & P. M. Pardalos (Eds.), Handbook of combinatorial optimization (pp. 21–169). Dordrecht: Kluwer Academic Publishers.Google Scholar
  29. Cheng, T. C. E., & Kovalyov, M. Y. (2001). Single machine batch scheduling with sequential job processing. IIE Transactions, 33, 413–420.Google Scholar
  30. Cheng, T. C. E., Ding, Q., & Lin, B. (2004a). A concise survey of scheduling with time-dependent processing times. European Journal of Operational Research, 152, 1–13.Google Scholar
  31. Cheng, T. C. E., Kovalyov, M. Y., & Chakhlevich, K. N. (2004b). Batching in a two-stage flowshop with dedicated machines in the second stage. IIE Transactions, 36, 87–93.Google Scholar
  32. Cheng, T. C. E., Ng, C., & Yuan, J. J. (2006). Multi-agent scheduling on a single machine to minimize total weighted number of tardy jobs. Theoretical Computer Science, 362, 273–281.Google Scholar
  33. Cheng, T. C. E., Ng, C., & Yuan, J. J. (2008). Multi-agent scheduling on a single machine with max-form criteria. European Journal of Operational Research, 188, 603–609.Google Scholar
  34. Cheng, T. C. E., Cheng, S. R., Wu, W., Hsu, P. H., & Wu, C. C. (2011a). A two-agent single-machine scheduling problem with truncated sum-of-processing-times-based learning considerations. Computers and Industrial Engineering, 60, 534–541.Google Scholar
  35. Cheng, T. C. E., Wu, W., Cheng, S. R., & Wu, C. C. (2011b). Two-agent scheduling with position-based deteriorating jobs and learning effects. Applied Mathematics and Computation, 217, 8804–8824.Google Scholar
  36. Cheng, T. C. E., Chung, Y.-H., Liao, S., & Lee, W.-C. (2013). Two-agent singe-machine scheduling with release times to minimize the total weighted completion time. Computers and Operations Research, 40, 353–361.Google Scholar
  37. Cho, Y., & Sahni, S. (1981). Preemptive scheduling of independent jobs with release and due times on open, flow and job shops. Operations Research, 29, 511–522.Google Scholar
  38. Choi, B., Leung, J.-T., & Pinedo, M. (2009). A note on the complexity of a two-agent, linear combination problem. Technical report, Stern School of Business at New York University, IOMS Department.Google Scholar
  39. Coffman, E. G., Yannakakis, J. M., Magazine, M. J., & Santos, C. A. (1990). Batch sizing and job sequencing on a single machine. Annals of Operations Research, 26, 135–147.Google Scholar
  40. Conway, R., Maxwell, W., & Miller, L. (1967). Theory of scheduling. Reading: Addison-WesleyGoogle Scholar
  41. Cook, S. A. (1971). The complexity of theorem proving procedures. In Third annual ACM symposium on theory of computing (STOC ’71), Shaker Heights (pp. 151–158). New York: ACMGoogle Scholar
  42. Cormen, T. H., Leiserson, C. E., & Rivest, R. L. (1994). Introduction to algorithms. Cambridge: MIT.Google Scholar
  43. Dessouky, M. I., Lageweg, B. J., Lenstra, J. K., & van de Velde, S. L. (1990). Scheduling identical jobs on uniform parallel machines. Statistica Neerlandica, 44, 115–123.Google Scholar
  44. Dileepan, P., & Sen, T. (1988). Bicriterion static scheduling research for a single machine. Omega. The International Journal of Management Science, 16, 53–59.Google Scholar
  45. Ding, G., & Sun, S. (2010). Single-machine scheduling problems with two agents competing for makespan. Lecture Notes in Computer Science, 6328, 244–255.Google Scholar
  46. Du, J., & Leung, J. (1990). Minimizing total tardiness on one machine is NP-hard. Mathematics of operations research, 15, 483–495.Google Scholar
  47. Ehrgott, M., Shao, L., & Schobel, A. (2011). An approximation algorithm for convex multi-objective programming problems. Journal of Global Optimization, 50, 397–416.Google Scholar
  48. Elvikis, D., Hamacher, H. W., & T’Kindt, V. (2011). Scheduling two agents on uniform parallel machines with makespan and cost functions. Journal of Scheduling, 14, 471–481.Google Scholar
  49. Fan, B., Cheng, T., Li, S., & Feng, Q. (2013). Bounded parallel-batching scheduling with two competing agents. Journal of Scheduling, 16, 261–271.Google Scholar
  50. Feng, Q., Yu, Z., & Shang, W. (2011). Pareto optimization of serial-batching scheduling problems on two agents. In 2011 international conference on advanced mechatronic systems (ICAMechS) (pp. 165–168). ISBN 978-1-4577-1698-0.Google Scholar
  51. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of \(\mathcal{N}\mathcal{P}\) -completeness. New York: W.H. Freeman and Company.Google Scholar
  52. Garey, M. R., Johnson, D. S., & Sethi, R. (1976). The complexity of flowshop and jobshop scheduling. Mathematics of operations research, 1(2), 117–129.Google Scholar
  53. Gavranovic, H., & Finke, G. (2000). Graph partitioning and set covering for optimal design of production system in the metal industry. In The second conference on management and control of production and logistics – MCPL’00, Grenoble.Google Scholar
  54. Gawiejnowicz, S. (1996). Brief survey of continuous models of scheduling. Foundations of Computing and Decision Sciences, 21, 81–100.Google Scholar
  55. Gawiejnowicz, S. (2008). Time-dependent scheduling: EATCS monographs in theoretical computer science. Berlin/New York: Springer.Google Scholar
  56. Gawiejnowicz, S., & Kononov, A. (2012, in press). Isomorphic scheduling problems. Annals of Operations Research. doi:10.1007/s10479-012-1222-2.Google Scholar
  57. Gawiejnowicz, S., Onak, T., & Suwalski, C. (2006). A new library for evolutionary algorithms. Lecture Notes in Computer Science, 3911, 414–421.Google Scholar
  58. Gawiejnowicz, S., Kurc, W., & Pankowska, L. (2009a). Conjugate problems in time-dependent scheduling. Journal of Scheduling, 12, 543–553.Google Scholar
  59. Gawiejnowicz, S., Kurc, W., & Pankowska, L. (2009b). Equivalent time-dependent scheduling problems. European Journal of Operational Research, 196, 919–929.Google Scholar
  60. Gawiejnowicz, S., Lee, W. C., Lin, C. L., & Wu, C. C. (2011). Single-machine scheduling of proportionally deteriorating jobs by two agents. Journal of the Operational Research Society, 62, 1983–1991.Google Scholar
  61. Geoffrion, A. M. (1968). Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications, 22, 618–630.Google Scholar
  62. Geoffrion, A. M. (1974). Lagrangian relaxation for integer programming. Mathematical Programming Study, 2, 82–114.Google Scholar
  63. Graham, R. L. (1966). Bounds for certain multiprocessor anomalies. Bell System Technical Journals, 17, 1563–1581.Google Scholar
  64. Graham, R. L., Lawler, E. L., Lenstra, J. K., & Rinnooy Kan, A. H. G. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5, 287–326.Google Scholar
  65. He, C., Lin, Y., & Yuan, J. (2007). Bicriteria scheduling on a batching machine to minimize maximum lateness and makespan. Theoretical Computer Science, 381, 234–240.Google Scholar
  66. Hochbaum, D. (1998). Approximation algorithms for NP-hard problems. Boston: PWS Publishing.Google Scholar
  67. Hochbaum, D. S., & Landy, D. (1994). Scheduling with batching: Minimizing the weighted number of tardy jobs. Operations Research Letters, 16, 79–86.Google Scholar
  68. Hoogeveen, J. A. (1996). Single-machine scheduling to minimize a function of two or three maximum cost criteria. Journal of Algorithms, 21, 415–433.Google Scholar
  69. Hoogeveen, H. (2005). Multicriteria scheduling. European Journal of Operational Research, 167, 592–623.Google Scholar
  70. Hoogeveen, J. A., & van de Velde, S. L. (1995). Minimizing total completion time and maximum cost simultaneously is solvable in polynomial time. Operations Research Letters, 17, 205–208.Google Scholar
  71. Hopcroft, J. E., & Karp, R. M. (1973). An \({n}^{frac52}\) algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 4, 225–231.Google Scholar
  72. Hopcroft, J., & Ullman, J. (1979). Introduction to automata theory, languages and computation. Reading: Addison-Wesley.Google Scholar
  73. Horn, W. A. (1973). Minimizing average flow time with parallel machines. Operations Research, 21, 846–847.Google Scholar
  74. Huo, Y., Leung, J. Y.-T., & Zhao, H. (2007a). Bi-criteria scheduling problems: Number of tardy jobs and maximum weighted tardiness. European Journal of Operational Research, 177, 116–134.Google Scholar
  75. Huo, Y., Leung, J. Y.-T., & Zhao, H. (2007b). Complexity of two dual criteria scheduling problems. Operations Research Letters, 35, 211–220.Google Scholar
  76. Jackson, J. R. (1955). Scheduling a production line to minimize maximum tardiness. In Management Science Research (Vol. 43). Los Angeles: University of California.Google Scholar
  77. Johnson, S. M. (1954). Optimal two and three-stage production schedules with setup times included. Naval Research Logistic Quarterly, 1, 61–67.Google Scholar
  78. Johnson, D. (1982). The NP-completeness column: An ongoing guide. Journal of Algorithms, 2, 393–405.Google Scholar
  79. Johnson, D. S. (1990). A catalog of complexity classes. In J. van Leeuwen (Ed.), Handbook of theoretical computer science: Algorithms and complexity (pp. 67–161). Elsevier/MIT: Amsterdam/Cambridge.Google Scholar
  80. Jozefowska, J. (2007). Just-in-time scheduling: Models and algorithms for computer and manufacturing systems. Berlin: Springer.Google Scholar
  81. Karp, R. M. (1972). Reducibility among combinatorial problems. In R. E. Miller & J. W. Thatcher (Eds.), Complexity of computer computations (pp. 85–104). New York: Plenum Press.Google Scholar
  82. Kellerer, H., & Strusevich, V. A. (2010). Fully polynomial approximation schemes for a symmetric quadratic knapsack problem and its scheduling applications. Algorithmica, 57, 769–795.Google Scholar
  83. Khowala, K., Fowler, J., Keha, A., & Balasubramanian, H. (2009). Single machine scheduling with interfering job sets. In Multidisciplinary international conference on scheduling: Theory and applications (MISTA 2009), 10–12 Aug 2009, Dublin (pp. 357–365).Google Scholar
  84. Knotts, G., Dror, M., & Hartman, B. C. (2000). Agent-based project scheduling. IIE Transactions, 32, 387–401.Google Scholar
  85. Knuth, D. E. (1967–1969). The art of computer programming (Vols. 1–3). Reading: Addison-Wesley.Google Scholar
  86. Kononov, A. (1997). Scheduling problems with linear increasing processing times. In Operations research September 3–6, 1996, Braunschweig (pp. 208–212). Springer.Google Scholar
  87. Kononov, A. (1998). Single machine scheduling problems with processing times proportional to an arbitrary function. Discrete Analysis and Operations Research, 5, 17–37.Google Scholar
  88. Kononov, A., & Gawiejnowicz, S. (2001). NP-hard cases in scheduling deteriorating jobs on dedicated machines. Journal of the Operational Research Society, 52, 708–718.Google Scholar
  89. Kovalyov, M. Y., Oulamara, A., & Soukhal, A. (2012). Two-agent scheduling on an unbounded serial batching machine. Lecture Notes in Computer Science, 7422 LNCS, 427–438.Google Scholar
  90. Kovalyov, M. Y., Oulamara, A., & Soukhal, A. (2012b). Two-agent scheduling with agent specific batches on an unbounded serial batching machine. In The 2nd international symposium on combinatorial optimization, ISCO 2012: Vol. 7422. Lecture Notes in Computer Science, Athens.Google Scholar
  91. Laumanns, M., Thiele, L., & Zitzler, E. (2006). An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. European Journal of Operational Research, 169(3), 932–942.Google Scholar
  92. Lawler, E. L. (1973). Optimal sequencing of a single machine subject to precedence constraints. Management Science, 19(8), 544–546.Google Scholar
  93. Lawler, E. L. (1977). A pseudopolynomial algorithm for sequencing jobs to minimize total tardiness. Annals of Discrete Mathematics, 1, 331–342.Google Scholar
  94. Lawler, E. L. (1982). Scheduling a single machine to minimize the number of late jobs (Vol. 1, pp. 331–342). Berkeley: Computer Science Division, University of California. (preprint)Google Scholar
  95. Lawler, E. L. (1990). A dynamic programming algorithm for preemptive scheduling of a single machine to minimize the number of late jobs. Annals of Operations Research, 26, 125–133.Google Scholar
  96. Lawler, E. L., & Moore, J. (1969). A functional equation and its application to resource allocation and sequencing problems. Management Science, 16(1), 77–84.Google Scholar
  97. Lee, C. (1991). Parallel machines scheduling with nonsimultaneous machine available time. Discrete Applied Mathematics, 20, 53–61.Google Scholar
  98. Lee, C. Y., & Vairaktarakis, G. (1993). Complexity of single machine hierarchical scheduling: A survey. In P. M. Pardalos (Ed.), Complexity in numerical optimization (pp. 269–298). Singapore: World Scientific.Google Scholar
  99. Lee, K., Choi, B.-C., Leung, J. Y.-T., & Pinedo, M. L. (2009). Approximation algorithms for multi-agent scheduling to minimize total weighted completion time. Information Processing Letters, 109, 913–917.Google Scholar
  100. Lee, W. C., Wang, W. J., Shiau, Y. R., & Wu, C. C. (2010). A single-machine scheduling problem with two-agent and deteriorating jobs. Applied Mathematical Modelling, 34(10), 3098–3107.Google Scholar
  101. Lee, W. C., Chung, Y., & Hu, M. (2012). Genetic algorithms for a two-agent single-machine problem with release time. Applied Soft Computing, 12, 3580–3589.Google Scholar
  102. Lenstra, J. K., Rinnooy Kan, A. H. G., & Brucker, P. (1977). Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1, 343–362.Google Scholar
  103. Leung, J. Y.-T., & Young, G. H. (1989). Minimizing schedule length subject to minimum flow time. SIAM Journal on Computing, 18(2), 314–326.Google Scholar
  104. Leung, J. Y.-T., Yu, V. K. M., & Wei, W.-D. (1994). Minimizing the weighted number of tardy task units. Discrete Applied Mathematics, 51, 307–316.Google Scholar
  105. Leung, J. Y.-T., Pinedo, M. L., & Wan, G. (2010). Competitive two-agent scheduling and its applications. Operations Research, 58, 458–469.Google Scholar
  106. Levin, A., & Woeginger, G. J. (2006). The constrained minimum weighted sum of job completion times problem. Mathematical Programming Series A, 108, 115–126.Google Scholar
  107. Lew, A., & Mauch, H. (2007). Dynamic programming: A computational tool. Berlin/Heidelberg: Springer.Google Scholar
  108. Lewis, H. R., & Papadimitriou, C. H. (1998). Elements of the theory of computation (2nd ed.). Upper Saddle River: Prentice-Hall.Google Scholar
  109. Li, D. C., & Hsu, P. H. (2012). Solving a two-agent single-machine scheduling problem considering learning effect. Computers and Operations Research, 39, 1644–1651.Google Scholar
  110. Li, S., & Yuan, J. (2012). Unbounded parallel-batching scheduling with two competitive agents. Journal of Scheduling, 15, 629–640.Google Scholar
  111. Liu, P., & Tang, L. (2008). Two-agent scheduling with linear deteriorating jobs on a single machine. Lecture Notes in Computer Science, 5092, 642–650.Google Scholar
  112. Liu, P., Tang, L., & Zhou, X. (2010a). Two-agent group scheduling with deteriorating jobs on a single machine. International Journal of Advanced Manufacturing Technology, 47, 657–664.Google Scholar
  113. Liu, P., Zhou, X., & Tang, L. (2010b). Two-agent group single-machine scheduling with position-dependent processing times. International Journal of Advanced Manufacturing Technology, 48, 325–331.Google Scholar
  114. Liu, P., Yi, N., & Zhou, X. Y. (2011). Two-agent single-machine scheduling problems under increasing linear deterioration. Applied Mathematical Modelling, 35, 2290–2296.Google Scholar
  115. Liu, P., Yi, N., Zhou, X., & Gong, H. (2013). Scheduling two agents with sum-of-processing-times-based deterioration on a single machine. Applied Mathematics and Computation, 219, 8848–8855.Google Scholar
  116. Manne, A. S. (1960). On the job-shop scheduling problem. Operations Research, 8, 219–223.Google Scholar
  117. Mavrotas, G. (2009). Effective implementation of the epsilon-constraint method in multi-objective mathematical programming problems. Applied Mathematics and Computation, 213(2), 455–465.Google Scholar
  118. Mc Naughton, R. (1959). Scheduling with deadlines and loss functions. Management Science, 6, 1–12.Google Scholar
  119. Meiners, C. R., & Torng, E. (2007). Mixed criteria packet scheduling. In M. Y. Kao & X.-Y. Li (Eds.), AAIM 2007: Vol. 4508. Lecture Notes on Computer Science (pp. 120–133). Berlin/Heidelberg: Springer.Google Scholar
  120. Mohri, S., Masuda, T., & Ishii, H. (1999). Bi-criteria scheduling problem on three identical parallel machines. International Journal of Production Economics, 60–61, 529–536.Google Scholar
  121. Moore, J. M. (1968). An n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15, 102–109.Google Scholar
  122. Mor, B., & Mosheiov, G. (2010). Scheduling problems with two competing agents to minimize minmax and minsum earliness measures. European Journal of Operational Research, 206(3), 540–546.Google Scholar
  123. Mor, B., & Mosheiov, G. (2011). Single machine batch scheduling with two competing agents to minimize total flowtime. European Journal of Operational Research, 215(3), 524–531.Google Scholar
  124. Mosheiov, G. (1994). Scheduling jobs under simple linear deterioration. Computers and Operations Research, 21, 653–659.Google Scholar
  125. Mosheiov, G. (2002). Complexity analysis of job-shop scheduling with deteriorating jobs. Discrete Applied Mathematics, 117, 195–209.Google Scholar
  126. Nagar, A., Haddock, J., & Heragu, S. (1995). Multiple and bicriteria scheduling: A literature survey. European Journal of the Operational Research, 81, 88–104.Google Scholar
  127. Ng, C. T., Cheng, T. C. E., & Yuan, J. J. (2006). A note on the complexity of the problem of two-agent scheduling on a single machine. Journal of Combinatorial Optimization, 12, 387–394.Google Scholar
  128. Nong, Q., Ng, C., & Cheng, T. (2008). The bounded single-machine parallel-batching scheduling problem with family jobs and release dates to minimize makespan. Operations Research Letters, 36(1), 61–66.Google Scholar
  129. Nong, Q., Cheng, T., & Ng, C. (2011). Two-agent scheduling to minimize the total cost. European Journal of Operational Research, 215, 39–44.Google Scholar
  130. Nowicki, E., & Zdrzalka, S. (1990). A survey of results for sequencing problems with controllable processing times. Discrete Applied Mathematics, 26, 271–287.Google Scholar
  131. Oulamara, A., Kovalyov, M. Y., & Finke, G. (2005). Scheduling a no-wait flowshop with unbounded batching machines. IIE Transactions on Scheduling and Logistics, 37, 685–696.Google Scholar
  132. Oulamara, A., Finke, G., & Kuiten, A. K. (2009). Flowshop scheduling problem with batching machine and task compatibilities. Computers & Operations Research, 36, 391–401.Google Scholar
  133. Papadimitriou, C. M. (1994). Computational complexity. Reading: Addison Wesley.Google Scholar
  134. Papadimitriou, C. H., & Steiglitz, K. (1982). Combinatorial optimization: Algorithms and complexity. Englewood Cliffs: Prentice-Hall.Google Scholar
  135. Peha, J. M. (1995). Heterogeneous-criteria scheduling: Minimizing weighted number of tardy jobs and weighted completion time. Journal of Computers and Operations Research, 22, 1089–1100.Google Scholar
  136. Pessan, C., Bouquard, J.-L., & Neron, E. (2008). An unrelated parallelmachines model for an industrial production resetting problem. European Journal of Industrial Engineering, 2, 153–171.Google Scholar
  137. Pinedo, M. (2008). Scheduling: Theory, algorithms, and systems (3rd ed.). Berlin: Springer.Google Scholar
  138. Potts, C., & Kovalyov, M. (2000). Scheduling with batching: A review. European Journal of Operational Research, 120(2), 228–249.Google Scholar
  139. Potts, C., Strusevich, V., & Tautenhahn, T. (2001). Scheduling batches with simultaneous job processing for two-machine shop problems. Journal of Scheduling, 4(1), 25–51.Google Scholar
  140. Qi, F., Yuan, J. J., Liu, H., & He, C. (2013). A note on two-agent scheduling on an unbounded parallel-batching machine with makespan and maximum lateness objectives. Applied Mathematical Modelling, 37, 7071–7076.Google Scholar
  141. Queyranne, M. (1993). Structure of a simple scheduling polyhedron. Mathematical Programming, 58, 263–285.Google Scholar
  142. Rustogi, K., & Strusevich, V. A. (2012). Simple matching vs linear assignment in scheduling models with positional effects: A critical review. European Journal of Operational Research, 222, 393–407.Google Scholar
  143. Ruzika, S., & Wiecek, M. M. (2005). Approximation methods in multiobjective programming. Journal of Optimization Theory and Applications, 126(3), 473–501.Google Scholar
  144. Sabouni, M. Y., & Jolai, F. (2010). Optimal methods for batch processing problem with makespan and maximum lateness objectives. Applied Mathematical Modelling, 34(2), 314–324.Google Scholar
  145. Sadi, F., Soukhal, A., & Billaut, J.-C. (2013, to appear). Solving multi-agent scheduling problems on parallel machines with a global objective function. RAIRO Operations Research.Google Scholar
  146. Saule, E., & Trystram, D. (2009). Multi-users scheduling in parallel systems. In Proceedings of the 23rd international symposium on parallel & distributed computing 2009, Rome (pp. 1–9). IEEE Computer Society.Google Scholar
  147. Schuurman, P., & Woeginger, G. J. (2011). Approximation schemes – A tutorial. In R. H. Mohring, C. N. Potts, A. S. Schulz, G. J. Woeginger, & L. A. Wolsey (Eds.), Lectures on scheduling.Google Scholar
  148. Sedeño-Noda, A., Alcaide, D., & González-Martín, C. (2006). Network flow approaches to pre-emptive open-shop scheduling problems with time-windows. European Journal of Operational Research, 174(3), 1501–1518.Google Scholar
  149. Shabtay, D., & Steiner, G. (2007). A survey of scheduling with controllable processing times. Discrete Applied Mathematics, 155, 1643–1666.Google Scholar
  150. Smith, W. E. (1956). Various optimizer for single-stage production. Naval Research Logistics Quarterly, 3, 59–66.Google Scholar
  151. Su, L.-H. (2009). Scheduling on identical parallel machines to minimize total completion time with deadline and machine eligibility constraints. The International Journal of Advanced Manufacturing Technology, 40, 572–581.Google Scholar
  152. Tan, Q., Chen, H.-P., Du, B., & Li, X.-L. (2011). Two-agent scheduling on a single batch processing machine with non-identical job sizes. In Proceedings of the 2nd international conference on artificial intelligence, management science and electronic commerce, AIMSEC 2011, Art. No. 6009883 (pp. 7431–7435).Google Scholar
  153. T’kindt, D. E. V. (2012, in press). Two-agent scheduling on uniform parallel machines with min-max criteria. Annals of Operations Research, 1–16.Google Scholar
  154. T’Kindt, V., & Billaut, J.-C. (2006). Multicriteria scheduling: Theory, models and algorithms (2nd ed.). Berlin/Heildelberg/New York: Springer.Google Scholar
  155. Tuong, N. H. (2009). Complexité et Algorithmes pour l’Ordonnancement Multicritere de Travaux Indépendants: Problèmes Juste-À-Temps et Travaux Interférants (in French). PhD thesis, Université François-Rabelais de Tours, Tours.Google Scholar
  156. Tuong, N. H., Soukhal, A., & Billaut, J.-C. (2012). Single-machine multi-agent scheduling problems with a global objective function. Journal of Scheduling, 15, 311–321.Google Scholar
  157. Tuzikov, A., Makhaniok, M., & Manner, R. (1998). Bicriterion scheduling of identical processing time jobs by uniform processors. Computers and Operations Research, 25, 31–35.Google Scholar
  158. Uzsoy, R., & Yang, Y. (1997). Minimizing total weighted completion time on a single batch processing machine. Production and Operations Management, 6, 57–73.Google Scholar
  159. Van de Velde, S. (1991). Machine scheduling and Lagrangian relaxation. PhD thesis, CWI Amsterdam.Google Scholar
  160. Van Wassenhove, L. N., & Gelders, L. F. (1980). Solving a bicriterion problem. European Journal of Operational Research, 4(1), 42–48.Google Scholar
  161. Vazirani, V. V. (2003). Approximation algorithms (2nd ed.). Berlin/Heidelberg: Springer.Google Scholar
  162. Vickson, R. G. (1980a). Choosing the job sequence and processing times to minimize total processing plus flow cost on a single machine. Operations Research, 28, 1155–1167.Google Scholar
  163. Vickson, R. G. (1980b). Two single machine sequencing problems involving controllable job processing times. AIIE Transactions, 12, 258–262.Google Scholar
  164. Wagner, H. M. (1959). An integer linear programming model for machine scheduling. Naval Research Logistic Quarterly, 6, 131–140.Google Scholar
  165. Walukiewicz, S. (1991). Integer programming. Warszawa: Polish Scientific Publishers.Google Scholar
  166. Wan, G., Yen, B. P. C., & Li, C. L. (2001). Single machine scheduling to minimize total compression plus weighted flow cost is NP-hard. Information Processing Letters, 79, 273–280.Google Scholar
  167. Wan, G., Leung, J.-Y., & Pinedo, M. (2010). Scheduling two agents with controllable processing times. European Journal of Operational Research, 205, 528–539.Google Scholar
  168. Wan, L., Yuan, J., & Geng, Z. (2013, to appear). A note on the preemptive scheduling to minimize total completion time with release and deadline constraints. Journal of Scheduling.Google Scholar
  169. Webster, S., & Baker, K. (1995). Scheduling groups of jobs on a single machine. Operations Research, 43, 692–704.Google Scholar
  170. Woeginger, G. J. (2003). Exact algorithms for NP-hard problems: A survey. Lecture Notes in Computer Science, 2570, 187–205.Google Scholar
  171. Wu, W. H. (2013). An exact and meta-heuristic approach for two-agent single-machine scheduling problem. Journal of Marine Science and Technology, 21, 215–221.Google Scholar
  172. Wu, C. C., Huang, S. K., & Lee, W. C. (2011). Two-agent scheduling with learning consideration. Computers and Industrial Engineering, 61, 1324–1335.Google Scholar
  173. Wu, W. H., Cheng, S. R., Wu, C. C., & Yin, Y. Q. (2012). Ant colony algorithms for a two-agent scheduling with sum-of processing times-based learning and deteriorating considerations. Journal of Intelligent Manufacturing, 23, 1985–1993.Google Scholar
  174. Wu, C.-C., Wu, W.-H., Chen, J.-C., Yin, Y., & Wu, W.-H. (2013a). A study of the single-machine two-agent scheduling problem with release times. Applied Soft Computing, 13, 998–1006.Google Scholar
  175. Wu, W. H., Xu, J., Wu, W., Yin, Y., Cheng, I., & Wu, C. C. (2013b). A tabu method for a two-agent single-machine scheduling with deterioration jobs. Computers & Operations Research, 40, 2116–2127.Google Scholar
  176. Yin, Y. Q., Cheng, S. R., Cheng, T., Wu, C. C., & Wu, W.-H. (2012a). Two-agent single-machine scheduling with assignable due dates. Applied Mathematics and Computation, 219, 1674–1685.Google Scholar
  177. Yin, Y. Q., Cheng, S. R., & Wu, C. C. (2012b). Scheduling problems with two agents and a linear non-increasing deterioration to minimize earliness penalties. Information Sciences, 189, 282–292.Google Scholar
  178. Yin, Y. Q., Wu, W., Cheng, S. R., & Wu, C. C. (2012c). An investigation on a two-agent single-machine scheduling problem with unequal release dates. Computers & Operations Research, 39, 3062–3073.Google Scholar
  179. Yuan, J. J., Shang, W. P., & Feng, Q. (2005). A note on the scheduling with two families of jobs. Journal of Scheduling, 8, 537–542.Google Scholar
  180. Zhao, K., & Lu, X. (2013). Approximation schemes for two-agent scheduling on parallel machines. Theoretical Computer Science, 468, 114–121.Google Scholar
  181. Zitzler, E., Knowles, J., & Thiele, L. (2008). Quality assessment of Pareto set approximations. Lecture Notes in Computer Science, 5252 LNCS, 373–404.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Alessandro Agnetis
    • 1
  • Jean-Charles Billaut
    • 2
  • Stanisław Gawiejnowicz
    • 3
  • Dario Pacciarelli
    • 4
  • Ameur Soukhal
    • 2
  1. 1.Dipartimento di Ingegneria dell’Informazione e Scienze MatematicheUniversità di SienaSienaItaly
  2. 2.Laboratoire d’InformatiqueUniversité François Rabelais ToursToursFrance
  3. 3.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland
  4. 4.Dipartimento di IngegneriaUniversità Roma TreRomaItaly

Personalised recommendations