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Just-in-time scheduling with controllable processing times on parallel machines

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Abstract

We study scheduling problems with controllable processing times on parallel machines. Our objectives are to maximize the weighted number of jobs that are completed exactly at their due date and to minimize the total resource allocation cost. We consider four different models for treating the two criteria. We prove that three of these problems are \(\mathcal{NP}\) -hard even on a single machine, but somewhat surprisingly, the problem of maximizing an integrated objective function can be solved in polynomial time even for the general case of a fixed number of unrelated parallel machines. For the three \(\mathcal{NP}\) -hard versions of the problem, with a fixed number of machines and a discrete resource type, we provide a pseudo-polynomial time optimization algorithm, which is converted to a fully polynomial time approximation scheme.

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References

  • Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows—theory, algorithms and applications. Englewood Cliffs, Prentice-Hall

    Google Scholar 

  • Alidaee B, Ahmadian A (1993) Two parallel machine sequencing problems involving controllable job processing times. Eur J Oper Res 70:335–341

    Article  MATH  Google Scholar 

  • Arkin EM, Silverberg EL (1987) Scheduling jobs with fixed start and finish times. Discrete Appl Math 18:1–8

    Article  MATH  MathSciNet  Google Scholar 

  • Armstrong R, Gu S, Lei L (1995) An algorithm for the two-resource allocation problem with a non-differentiable convex objective function. J Oper Res Soc 46:116–122

    MATH  Google Scholar 

  • Armstrong R, Gu S, Lei L (1997) Solving a class of two-resource allocation problems by equivalent load method. J Oper Res Soc 48:818–825

    MATH  Google Scholar 

  • Baker KR, Scudder GD (1990) Sequencing with earliness and tardiness penalties: a review. Oper Res 38:22–36

    Article  MATH  MathSciNet  Google Scholar 

  • Bouzina KI, Emmons H (1996) Interval scheduling on identical machines. J Glob Optim 9:379–393

    Article  MATH  MathSciNet  Google Scholar 

  • Carlisle MC, Lloyd EL (1995) On the k-coloring of intervals. Discrete Appl Math 59:225–235

    Article  MATH  MathSciNet  Google Scholar 

  • Cheng TCE, Janiak A, Kovalyov MY (1998) Bicriterion single machine scheduling with resource dependent processing times. SIAM J Optim 8(2):617–630

    Article  MATH  MathSciNet  Google Scholar 

  • Čepek O, Sung SC (2005) A quadratic time algorithm to maximize the number of just-in-time jobs on identical parallel machines. Comput Oper Res 32:3265–3271

    Article  MATH  MathSciNet  Google Scholar 

  • Cheng TCE, Oğuz C, Qi XD (1996) Due-date assignment and single machine scheduling with compressible processing times. Int J Prod Econ 43:29–35

    Article  Google Scholar 

  • Chudzik K, Janiak A, Lichtenstein M (2006) Scheduling problems with resource allocation. In: Janiak A (ed) Scheduling in computer and manufacturing systems. WKL, Warszawa

    Google Scholar 

  • Frank A (1980) On chains and antichains families of a partially ordered set. J Combin Theory Ser B 29:176–184

    Article  MATH  MathSciNet  Google Scholar 

  • Graham RL, Lawler EL, Lenstra JK, Rinnooy Kan AHG (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discrete Math 5:287–326

    Article  MATH  MathSciNet  Google Scholar 

  • Hiraishi K, Levner E, Vlach M (2002) Scheduling of parallel identical machines to maximize the weighted number of just-in-time jobs. Comput Oper Res 29(7):841–848

    Article  MATH  MathSciNet  Google Scholar 

  • Hoogeveen H, Woeginger GJ (2002) Some comments on sequencing with controllable processing times. Computing 68:181–192

    Article  MATH  MathSciNet  Google Scholar 

  • Hsiao JY, Tang CY, Chang RS (1992) An efficient algorithm for finding a maximum weight 2-independent set of interval graphs. Inf Process Lett 43:229–235

    Article  MATH  MathSciNet  Google Scholar 

  • Janiak A (1987) One-machine scheduling with allocation of continuously-divisible resource and with no precedence constraints. Kybernetika 23(4):289–293

    MATH  MathSciNet  Google Scholar 

  • Janiak A (1988) General flow-shop scheduling with resource constraints. Int J Prod Res 26(6):1089–1102

    Article  MATH  Google Scholar 

  • Janiak A (1998) Minimization of the makespan in a two-machine problem under given resource constraints. Eur J Oper Res 107:325–337

    Article  MATH  Google Scholar 

  • Janiak A, Janiak W, Lichtenstein M (2007) Resource management in machine scheduling problems: a survey. Decis Making Manuf Serv 1:59–89

    MathSciNet  Google Scholar 

  • Janiak A, Kovalyov MY (1996) Single machine scheduling subject to deadlines and resource dependent processing times. Eur J Oper Res 94:284–291

    Article  MATH  Google Scholar 

  • Janiak A, Kovalyov MY, Kubiak W, Werner F (2005) Positive half-products and scheduling with controllable processing times. Eur J Oper Res 165:416–422

    Article  MATH  MathSciNet  Google Scholar 

  • Janiak A, Portmann C (1998) Genetic algorithm for the permutation fow-shop scheduling problem with linear models of operation. Ann Oper Res 83:95–114

    Article  MATH  MathSciNet  Google Scholar 

  • Janiak A, Szkodny T (1994) Job-shop scheduling with convex models of operations. Math Comput Model 20(2):59–68

    Article  MATH  Google Scholar 

  • Kaspi M, Shabtay D (2003) Optimization of machining economics problem for a multi-stage transfer machine under failure, opportunistic and integrated replacement strategies. Int J Prod Res 41(10):2229–2248

    Article  MATH  Google Scholar 

  • Kayan RK, Akturk MS (2005) A new bounding mechanism for the CNC machine scheduling problem with controllable processing times. Eur J Oper Res 167:624–643

    Article  MATH  MathSciNet  Google Scholar 

  • Kovalyov MY, Ng CT, Cheng TCE (2007) Fixed interval scheduling: models, applications, computational complexity and algorithms. Eur J Oper Res 178:331–342

    Article  MATH  MathSciNet  Google Scholar 

  • Lann A, Mosheiov G (1996) Single machine scheduling to minimize the number of early and tardy job. Comput Oper Res 23:765–781

    Article  Google Scholar 

  • Lee CY, Lei L (2001) Multiple-project scheduling with controllable project duration and hard resource constraint: some solvable cases. Ann Oper Res 102:287–307

    Article  MATH  MathSciNet  Google Scholar 

  • Ng CTD, Cheng TCE, Janiak A, Kovalyov MY (2005) Group scheduling with controllable setup and processing times: minimizing total weighted completion time. Ann Oper Res 133:163–174

    Article  MATH  MathSciNet  Google Scholar 

  • Monma CL, Schrijver A, Todd MJ, Wei VK (1990) Convex resource allocation problems on directed acyclic graphs: duality, complexity, special cases and extensions. Math Oper Res 15:736–748

    Article  MATH  MathSciNet  Google Scholar 

  • Nowicki E, Zdrzalka S (1990) A survey of results for sequencing problems with controllable processing times. Discrete Appl Math 26:271–287

    Article  MATH  MathSciNet  Google Scholar 

  • Pal M, Bhattacharjee GP (1996) A sequential algorithm for finding a maximum weight K-independent set on interval graphs. Int J Comput Math 60:205–214

    Article  MATH  Google Scholar 

  • Saha A, Pal M (2003) Maximum weight k-independent set problem on permutation graphs. Int J Comput Math 80:1477–1487

    Article  MATH  MathSciNet  Google Scholar 

  • Sahni S (1976) Algorithms for scheduling independent tasks. J Assoc Comput Mach 23(1):116–127

    MATH  MathSciNet  Google Scholar 

  • Sarrafzadeh M, Lou RD (1993) Maximum k-covering of weighted transative graph with applications. Algorithmica 9:84–100

    Article  MATH  MathSciNet  Google Scholar 

  • Scott SC, Jefferson TR (1995) Allocation of resources in project management. Int J Syst Sci 26(2):413–420

    Article  MATH  MathSciNet  Google Scholar 

  • Shabtay D (2004) Single and a two-resource allocation algorithms for minimizing the maximal lateness in a single machine-scheduling problem. Comput Oper Res 31(8):1303–1315

    Article  MATH  Google Scholar 

  • Shabtay D, Kaspi M (2004) Minimizing the total weighted flow time in a single machine with controllable processing times. Comput Oper Res 31(13):2279–2289

    Article  MATH  MathSciNet  Google Scholar 

  • Shabtay D, Kaspi M, Steiner G (2007) The no-wait two-machine flow-shop scheduling problem with convex resource-dependent processing times. IIE Trans 39(5):539–557

    Article  Google Scholar 

  • Shabtay D, Steiner G (2007) A survey of scheduling with controllable processing times. Discrete Appl Math 155(13):1643–1666

    Article  MATH  MathSciNet  Google Scholar 

  • Shakhlevich NV, Strusevich VA (2005) Pre-emptive scheduling problems with controllable processing times. J Sched 8:233–253

    Article  MATH  MathSciNet  Google Scholar 

  • Shakhlevich NV, Strusevich VA (2006) Single machine scheduling with controllable release and processing times. Discrete Appl Math 154:2178–2199

    Article  MATH  MathSciNet  Google Scholar 

  • Steiner G (1992) Finding the largest suborder of fixed width. Order 9:357–360

    Article  MATH  MathSciNet  Google Scholar 

  • Sung SC, Vlach M (2005) Maximizing weighted number of just-in-time jobs on unrelated parallel machines. J Sched 8:453–460

    Article  MATH  MathSciNet  Google Scholar 

  • Trick M (1994) Scheduling multiple variable-speed machines. Oper Res 42:234–248

    Article  MATH  Google Scholar 

  • Van Wassenhove L, Baker KR (1982) A bicriterion approach to time/cost trade-offs in sequencing. Eur J Oper Res 11:48–54

    Article  MATH  Google Scholar 

  • Yannakakis M, Gavril F (1987) The maximum k-colorable subgraph problem for chordal graphs. Inf Process Lett 24:133–137

    Article  MATH  MathSciNet  Google Scholar 

  • Yedidsion L, Shabtay D, Kaspi M (2007) A bicriteria approach to minimize maximal lateness and resource consumption for scheduling a single machine. J Sched 10(8):341–352

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Yaron Leyvand & Dvir Shabtay

  2. Operations Management Area, DeGroote School of Business, McMaster University, Hamilton, Ontario, Canada

    George Steiner

  3. Faculty of Industrial Engineering and Management, Technion—Israel Institute of Technology, Haifa, Israel

    Liron Yedidsion

Authors
  1. Yaron Leyvand
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  2. Dvir Shabtay
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  3. George Steiner
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  4. Liron Yedidsion
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Correspondence to George Steiner.

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Leyvand, Y., Shabtay, D., Steiner, G. et al. Just-in-time scheduling with controllable processing times on parallel machines. J Comb Optim 19, 347–368 (2010). https://doi.org/10.1007/s10878-009-9270-5

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