Application of an effective modified gravitational search algorithm for the coordinated scheduling problem in a two-stage supply chain

  • Jun PeiEmail author
  • Xinbao Liu
  • Panos M. Pardalos
  • Wenjuan Fan
  • Shanlin Yang
  • Ling Wang


This paper investigates a products and vehicles scheduling problem in a two-stage supply chain environment, where jobs first need to be processed on the serial batching machines of multiple manufacturers distributed in various geographic zones and then transported by vehicles to a customer for further processing. The size and processing time of jobs are varying with the difference of types, and each batch takes a setup time before being processed. The problem of minimizing the makespan is formalized as a mixed integer programming model and proved to be NP-hard. In addition, the structural properties and lower bound of the problem are analyzed and inferred. Then a modified gravitational search algorithm (MGSA) is proposed to solve the problem. In the developed MGSA, several improvement strategies and the batching mechanism DP-H are introduced. The effectiveness and efficiency of the proposed MGSA are demonstrated and compared with a particle swarm optimization (PSO) algorithm and a genetic algorithm (GA). Besides, the error ratios between the lower bound and the best found solutions are reported. The experimental results indicate that the proposed MGSA is more robust and outperforms PSO and GA on the studied two-stage supply chain scheduling problem.


Supply chain scheduling Batching Gravitational search algorithm Lower bound Dynamic programming Heuristic algorithm 


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  1. 1.
    Hall NG, Potts CN (2003) Supply chain scheduling: batching and delivery. Oper Res 51(4):566–584CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Dayou L, Pu Y, Ji Y (2009) Development of a multiobjective GA for advanced planning and scheduling problem. Int J Adv Manuf Technol 42(9–10):974–992CrossRefGoogle Scholar
  3. 3.
    Delavar MR, Hajiaghaei-Keshteli M, Molla-Alizadeh-Zavardehi S (2010) Genetic algorithms for coordinated scheduling of production and air transportation. Expert Syst Appl 37(12):8255–8266CrossRefGoogle Scholar
  4. 4.
    Yimer AD, Demirli K (2010) A genetic approach to two-phase optimization of dynamic supply chain scheduling. Comput Ind Eng 58(3):411–422CrossRefGoogle Scholar
  5. 5.
    Liao C-J, Tsai Y-L, Chao C-W (2011) An ant colony optimization algorithm for setup coordination in a two-stage production system. Appl Soft Comput 11(8):4521–4529CrossRefGoogle Scholar
  6. 6.
    Cakici E, Mason SJ, Kurz ME (2012) Multi-objective analysis of integrated an supply chain scheduling problem. Int J Prod Res 50(10):2624–2638CrossRefGoogle Scholar
  7. 7.
    Amin-Naseri MR, Afshari AJ (2012) A hybrid genetic algorithm for integrated process planning and scheduling problem with precedence constraints. Int J Adv Manuf Technol 59(1–4):273–287CrossRefGoogle Scholar
  8. 8.
    Hamidinia A, Khakabimamaghani S, Mazdeh MM, Jafari M (2012) A genetic algorithm for minimizing total tardiness/earliness of weighted jobs in a batched delivery system. Comput Ind Eng 62(1):29–38CrossRefGoogle Scholar
  9. 9.
    Zegordi SH, Kamal Abadi IN, Beheshti Nia MA (2012) A novel genetic algorithm for solving production and transportation scheduling in a two-stage supply chain. Comput Ind Eng 58(3):373–381CrossRefGoogle Scholar
  10. 10.
    Chandra P (1993) A dynamic distribution model with warehouse and customer replenishment requirements. J Oper Res Soc 44(7):681–692zbMATHGoogle Scholar
  11. 11.
    Chi-Shiang S, Panb JC-H, Hsua T-S (2009) A new heuristic algorithm for the machine scheduling problem with job delivery coordination. Theor Comput Sci 410(27–29):2581–2591zbMATHGoogle Scholar
  12. 12.
    Cheng TCE, Wang X (2010) Machine scheduling with job class setup and delivery considerations. Comput Oper Res 37(6):1123–1128CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Pardalos PM, Shylo OV, Vazacopoulos A (2010) Solving job shop scheduling problems utilizing the properties of backbone and "big valley". Comput Optim Appl 47(1):61–76CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Steinruecke M (2011) An approach to integrate production-transportation planning and scheduling in an aluminium supply chain network. Int J Prod Res 49(21):6559–6583CrossRefGoogle Scholar
  15. 15.
    Mehravaran Y, Logendran R (2012) Non-permutation flowshop scheduling in a supply chain with sequence-dependent setup times. Int J Prod Econ 135(2):953–963CrossRefGoogle Scholar
  16. 16.
    Aldowaisan TA, Ali A (2012) No-wait flowshop scheduling problem to minimize the number of tardy jobs. Int J Adv Manuf Technol 61(1–4):311–323CrossRefGoogle Scholar
  17. 17.
    You P-S, Hsieh Y-C (2012) A heuristic approach to a single stage assembly problem with transportation allocation. Appl Math Comput 218(22):11100–11111CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Lee C-H, Liao C-J, Chao C-W (2012) Scheduling with multi-attribute setup times. Comput Ind Eng 63(2):494–502CrossRefGoogle Scholar
  19. 19.
    E. Alcali, J. Geunes, P.M.Pardalos, H.E. Romeijn, and Z.J. Shen (2004). Applications of supply chain management and E-commerce research in industry. Kluwer, Norwell, MAGoogle Scholar
  20. 20.
    Mazdeh MM, Sarhadi M, Hindi KS (2008) A branch-and-bound algorithm for single-machine scheduling with batch delivery and job release times. Comput Oper Res 35(4):1099–1111CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Gordon VS, Strusevich VA (2009) Single machine scheduling and due date assignment with positionally dependent processing times. Eur J Oper Res 198(1):57–62CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Bard JF, Nananukul N (2010) A branch-and-price algorithm for an integrated production and inventory routing problem. Comput Oper Res 37(12):2202–2217CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Chrétienne P, Hazır Ö, Kedad-Sidhoum S (2011) Integrated batch sizing and scheduling on a single machine. J Sched 14(6):541–555CrossRefMathSciNetGoogle Scholar
  24. 24.
    Mazdeh MM, Shashaani S, Ashouri A, Hindi KS (2011) Single-machine batch scheduling minimizing weighted flow times and delivery costs. Appl Math Model 35(1):563–570CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Rashedi E, Nezamabadi-pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179(13):2232–2248CrossRefzbMATHGoogle Scholar
  26. 26.
    Yin M, Yanmei H, Yang F, Li X, Wenxiang G (2011) A novel hybrid K-harmonic means and gravitational search algorithm approach for clustering. Expert Syst Appl 38(8):9319–9324CrossRefGoogle Scholar
  27. 27.
    Behrang MA, Assareh E, Ghalambaz M, Assari MR, Noghrehabadi AR (2011) Forecasting future oil demand in Iran using GSA (gravitational search algorithm). Energy 36(9):5649–5654CrossRefGoogle Scholar
  28. 28.
    Han X, Chang X (2012) Chaotic secure communication based on a gravitational search algorithm filter. Eng Appl Artif Intell 25(4):766–774CrossRefGoogle Scholar
  29. 29.
    Mirjalili S, Hashim S, Sardroudi H (2012) Training feedforward neural networks using hybrid particle swarm optimization and gravitational search algorithm. Appl Math Comput 218(22):11125–11137CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Han X, Chang X (2012) A chaotic digital secure communication based on a modified gravitational search algorithm filter. Inf Sci 208(15):14–27CrossRefGoogle Scholar
  31. 31.
    Precup R-E, David R-C, Petriu EM, Preitl S, Radac M-B (2012) IEEE Trans Ind Inf 8(4):791–800CrossRefGoogle Scholar
  32. 32.
    Bahrololoum A, Nezamabadi-pour H, Bahrololoum H, Saeed M (2012) A prototype classifier based on gravitational search algorithm. Appl Soft Comput 12(2):819–825CrossRefGoogle Scholar
  33. 33.
    Graham RL, Lawler EL, Lenstra JK, Rinnooy Kan AHG (1979) Optimization and approximation in deterministic machine scheduling: a survey. Ann Discrete Math 5:287–326CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Garey MR, Johnson DS (1978) “Strong” NP-completeness results: motivation, examples, and implications. J Assoc Comput Mach 25(3):499–508CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Heo J-H, Lyu J-K, Kim M-K, Park J-K (2012) Application of particle swarm optimization to the reliability centered maintenance method for transmission systems. J Electr Eng Technol 7(6):814–823CrossRefGoogle Scholar
  36. 36.
    Mazloumi E, Mesbah M, Ceder A, Moridpour S, Currie G (2012) Efficient transit schedule design of timing points: a comparison of ant colony and genetic algorithms. Transp Res B 46(1):217–234CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Jun Pei
    • 1
    • 2
    Email author
  • Xinbao Liu
    • 1
    • 3
  • Panos M. Pardalos
    • 2
  • Wenjuan Fan
    • 1
    • 4
  • Shanlin Yang
    • 1
    • 3
  • Ling Wang
    • 5
  1. 1.School of ManagementHefei University of TechnologyHefeiChina
  2. 2.Center for Applied Optimization, Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA
  3. 3.Key Laboratory of Process Optimization and Intelligent Decision-making of Ministry of EducationHefeiChina
  4. 4.Department of Computer ScienceNorth Carolina State UniversityRaleighUSA
  5. 5.Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronics and AutomationShanghai UniversityShanghaiChina

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