Continuous-time scheduling of production, distribution and sales in photovoltaic supply chains with declining prices

  • Wolfgang Albrecht
  • Martin SteinrückeEmail author


Supply chain networks in the photovoltaic sector were faced with a rapid decline in prices during the past few years, which is predicted to go on in the following years. These circumstances force related companies to coordinate production, distribution and transportation planning of all their network sites thoroughly. Moreover, continuous-time scheduling of processes is required to determine times of sales exactly within a multi-day planning horizon. Applying this type of scheduling is possible due to assessable price trends and existing framework agreements with wholesalers, and even necessary due to its impact on the realized sales prices, and thus, the network profit. For this reason, we develop a mixed-integer linear programming model that meets the requirements of fully-integrated photovoltaic supply chains that cover processing of raw materials, manufacturing of intermediate and finished products in two alternative methods, and selling them on international markets. The multi-product approach enables to connect supply chain stages with different product maturities. The modeling is motivated by a real-life case of a global photovoltaic group headquartered in Germany. As it was not possible to optimize this problem with high-performance software and hardware within 3 months, we tested several relax-and-fix decomposition methods. By selecting those algorithms that were able to generate high-quality solutions within acceptable computation times of less than half a day, a satisfying solution was found. The appropriateness of the selected algorithms is additionally demonstrated by analyzing randomly generated scenarios in a numerical study.


Supply chain management Sales and operations planning Continuous-time scheduling Mixed-integer linear programming 



  1. Abdallah T, Diabat A, Rigter J (2013) Investigating the option of installing small scale PVs on facility rooftops in a green supply chain. Int J Prod Econ 146(2):465–477CrossRefGoogle Scholar
  2. Abdeljaouad MA, Bahroun Z, Omrane A, Fondrevelle J (2015) Job-shop production scheduling with reverse flows. Eur J Oper Res 244(1):117–128MathSciNetzbMATHCrossRefGoogle Scholar
  3. Akartunalı K, Miller AJ (2009) A heuristic approach for big bucket multi-level production planning problem. Eur J Oper Res 193(2):396–411MathSciNetzbMATHCrossRefGoogle Scholar
  4. Akartunalı K, Miller AJ (2012) A computational analysis of lower bounds for big bucket production planning problems. Comput Optim Appl 53(3):729–753MathSciNetzbMATHCrossRefGoogle Scholar
  5. Akartunalı K, Fragkos I, Miller AJ, Wu T (2016) Local cuts and two-period convex hull closures for big-bucket lot-sizing problems. Inf J Comput 28(4):766–780MathSciNetzbMATHCrossRefGoogle Scholar
  6. Albrecht W, Steinrücke M (2017) Continuous-time production, distribution and financial planning with periodic liquidity balancing. J Sched 20(3):219–237MathSciNetzbMATHCrossRefGoogle Scholar
  7. Albrecht W, Steinrücke M (2018) Coordinating continuous-time distribution and sales planning of perishable goods with quality grades. Int J Prod Res 56(7):2646–2665CrossRefGoogle Scholar
  8. Amiri A (2006) Designing a distribution network in a supply chain system: formulation and efficient solution procedure. Eur J Oper Res 171(2):567–576zbMATHCrossRefGoogle Scholar
  9. Amrouche B, Sicot L, Guessoum A, Belhamel M (2013) Experimental analysis of the maximum power point’s properties for four photovoltaic modules from different technologies: monocrystalline and polycrystalline silicon, CIS and CdTe. Sol Energy Mater Sol Cells 118:124–134CrossRefGoogle Scholar
  10. Baumann P, Trautmann N (2013) A continuous-time MILP model for short-term scheduling of make-and-pack production processes. Int J Prod Res 51(6):1707–1727CrossRefGoogle Scholar
  11. Baumeister C, Kilian L (2016) Understanding the decline in the price of oil since June 2014. J Assoc Environ Resource Econ 3(1):131–158Google Scholar
  12. Bhatnagar R, Mehta P, Teo CC (2011) Coordination of planning and scheduling decisions in global supply chains with dual supply modes. Int J Prod Econ 131(2):473–482CrossRefGoogle Scholar
  13. Byrne DM, Oliner SD, Sichel DE (2017) How fast are semiconductor prices falling? Rev Income Wealth. CrossRefGoogle Scholar
  14. Castellano R (2010) Solar panel processing. Old City Publishing, PhiladelphiaGoogle Scholar
  15. Chand S, Hsu VN, Sethi S (2002) Forecast, solution, and rolling horizons in operations management problems: a classified bibliography. Manuf Service Oper Manag 4(1):25–43CrossRefGoogle Scholar
  16. Chang Y-C, Chang K-H, Chang T-K (2013) Applied column generation-based approach to solve supply chain scheduling problems. Int J Prod Res 51(13):4070–4086CrossRefGoogle Scholar
  17. Chen HH, Pang C (2010) Organizational forms for knowledge management in photovoltaic solar energy industry. Knowl-Based Syst 23(8):924–933CrossRefGoogle Scholar
  18. Chen Z, Su S-II (2014) Photovoltaic supply chain coordination with strategic consumers in China. Renewable Energy 68:236–244CrossRefGoogle Scholar
  19. Chien C-F, Kuo R-T (2013) Beyond make-or-buy: cross-company short-term capacity backup in semiconductor industry ecosystem. Flex Services Manuf J 25(3):310–342CrossRefGoogle Scholar
  20. Chien C-F, Wu J-Z, Wu C-C (2013) A two-stage stochastic programming approach for new tape-out allocation decisions for demand fulfillment planning in semiconductor manufacturing. Flex Services Manuf J 25(3):286–309CrossRefGoogle Scholar
  21. Davies J, Joglekar N (2013) Supply chain integration, product modularity, and market valuation: evidence from the solar energy industry. Prod Oper Manag 22(6):1494–1508CrossRefGoogle Scholar
  22. Dillenberger C, Escudero LF, Wollensak A, Zhang W (1994) On practical resource allocation for production planning and scheduling with period overlapping setups. Eur J Oper Res 75(2):275–286zbMATHCrossRefGoogle Scholar
  23. Ekelund M, Persson B (2003) Pharmaceutical pricing in a regulated market. Rev Econ Stat 85(2):298–306CrossRefGoogle Scholar
  24. Erengüç Ş, Simpson NC, Vakharia AJ (1999) Integrated production/distribution planning in supply chains: an invited review. Eur J Oper Res 115(2):219–236zbMATHCrossRefGoogle Scholar
  25. Federgruen A, Meissner J, Tzur M (2007) Progressive interval heuristics for multi-item capacitated lot-sizing problems. Oper Res 55(3):490–502MathSciNetzbMATHCrossRefGoogle Scholar
  26. Ferreira D, Morabito R, Rangel S (2009) Solution approaches for the soft drink integrated production lot sizing and scheduling problem. Eur J Oper Res 196(2):697–706zbMATHCrossRefGoogle Scholar
  27. Floudas CA, Lin X (2004) Continuous-time versus discrete-time approaches for scheduling of chemical processes: a review. Comput Chem Eng 28:2109–2129CrossRefGoogle Scholar
  28. Floudas CA, Lin X (2005) Mixed integer linear programming in process scheduling: modeling, algorithms, and applications. Ann Oper Res 139(1):131–162MathSciNetzbMATHCrossRefGoogle Scholar
  29. Fragkos I, Degraeve Z, De Reyck B (2016) A horizon decomposition approach for the capacitated lot-sizing problem with setup times. Inf J Comput 28(3):465–482MathSciNetzbMATHCrossRefGoogle Scholar
  30. GAMS (2017) GAMS Documentation, GAMS Development Corporation, Washington 2017.
  31. Gan PY, Li Z (2015) Quantitative study on long term global solar photovoltaic market. Renew Sust Energy Rev 46:88–99CrossRefGoogle Scholar
  32. Gomes MC, Barbosa-Póvoa A, Novais AQ (2013) Reactive scheduling in a make-to-order flexible job shop with re-entrant process and assembly: a mathematical programming approach. Int J Prod Res 51(17):5120–5141CrossRefGoogle Scholar
  33. Govindan K, Jafarian A, Khodaverdi R, Devika K (2014) Two-echelon multiple-vehicle location-routing problem with time windows for optimization of sustainable supply chain network of perishable food. Int J Prod Econ 152:9–28CrossRefGoogle Scholar
  34. Günther HO (2014) The block planning approach for continuous time-based dynamic lot sizing and scheduling. Bus Res 7(1):51–76CrossRefGoogle Scholar
  35. Hu J-L, Yeh F-Y (2013) The value migration and innovative capacity of Taiwan’s photovoltaic industry. Energy Sources Part B 8(2):190–199CrossRefGoogle Scholar
  36. IEA (International Energy Agency) (2017) Snapshot of global photovoltaic markets 2016.
  37. IRENA (International Renewable Energy Agency) (2016) The power to change: solar and wind cost reduction potential to 2025.
  38. Jans R, Degraeve Z (2004) Improved lower bounds for the capacitated lot sizing problem with setup times. Oper Res Lett 32(2):185–195MathSciNetzbMATHCrossRefGoogle Scholar
  39. Jayaraman V, Pirkul H (2001) Planning and coordination of production and distribution facilities for multiple commodities. Eur J Oper Res 133(2):394–408zbMATHCrossRefGoogle Scholar
  40. Jula P, Kones I (2013) Continuous-time algorithms for scheduling a single machine with sequence-dependent setup times and time window constraints in coordinated chains. Int J Prod Res 51(12):3654–3670CrossRefGoogle Scholar
  41. Kaldellis JK, Simotas M, Zafirakis D, Kondili E (2009) Optimum autonomous photovoltaic solution for the Greek islands on the basis of energy pay-back analysis. J Clean Prod 17:1311–1323CrossRefGoogle Scholar
  42. Kallrath J, Maindl TI (2006) Real optimization with SAP® APO. Springer, Berlin et alzbMATHGoogle Scholar
  43. Kelly JD, Mann JL (2004) Flowsheet decomposition heuristic for scheduling: a relax-and-fix method. Comput Chem Eng 28(11):2193–2200CrossRefGoogle Scholar
  44. Knoblich K, Heavey C, Williams P (2015) Quantitative analysis of semiconductor supply chain contracts with order flexibility under demand uncertainty: a case study. Comput Ind Eng 87:394–406CrossRefGoogle Scholar
  45. Lan C-W, Hsieh C-K, Hsu W-C (2009) Czochralski silicon crystal growth for photovoltaic applications. In: Nakajima K, Usami N (eds) Crystal growth of Si for solar cells. Springer, Berlin, pp 25–39CrossRefGoogle Scholar
  46. Lee YH (2001) Supply chain model for the semiconductor industry of global market. J Syst Integr 10:189–206zbMATHCrossRefGoogle Scholar
  47. Lee YH, Kim SH (2002) Production-distribution planning in supply chain considering capacity constraints. Comput Ind Eng 43(1–2):169–190CrossRefGoogle Scholar
  48. Li W, Goh M, Wu Y, Petering MEH, de Souza R (2012) A continuous time model for multiple yard crane scheduling with last minute job arrivals. Int J Prod Econ 136(2):332–343CrossRefGoogle Scholar
  49. Liang Z, He Y, Wu T, Zhang C (2015) An informative column generation and decomposition method for a production planning and facility location problem. Int J Prod Econ 170:88–96CrossRefGoogle Scholar
  50. Low C, Li R-K, Chang C-M (2013) Integrated scheduling of production and delivery with time windows. Int J Prod Res 51(3):897–909CrossRefGoogle Scholar
  51. Mockus L, Reklaitis GV (1999) Continuous time representation approach to batch and continuous process scheduling. 1. MINLP formulation. Ind Eng Chem Res 38:197–203CrossRefGoogle Scholar
  52. Mohammadi M, Fatemi Ghomi SMT, Karimi B, Torabi SA (2010) Rolling-horizon and fix-and-relax heuristics for the multi-product multi-level capacitated lotsizing problem with sequence-dependent setups. J Intell Manuf 21(4):501–510CrossRefGoogle Scholar
  53. Mohammadi G, Karampourhaghghi A, Samaei F (2012) A multi-objective optimisation model to integrating flexible process planning and scheduling based on hybrid multi-objective simulated annealing. Int J Prod Res 50(18):5063–5076CrossRefGoogle Scholar
  54. Mokhtari H, Abadi INK, Amin-Naseri MR (2012) Production scheduling with outsourcing scenarios: a mixed integer programming and efficient solution procedure. Int J Prod Res 50(19):5372–5395CrossRefGoogle Scholar
  55. Mönch L, Fowler JW, Mason SJ (2013) Production planning and control for semiconductor wafer fabrication facilities. Springer, New York et alCrossRefGoogle Scholar
  56. Mönch L, Uzsoy R, Fowler JW (2018) A survey of semiconductor supply chain models part III: master planning, production planning, and demand fulfilment. Int J Prod Res 56(13):4565–4584CrossRefGoogle Scholar
  57. Øvrelid EJ, Tang K, Engh T, Tangstad M (2009) Feedstock. In: Nakajima K, Usami N (eds) Crystal growth of Si for solar cells. Springer, Berlin, pp 1–23Google Scholar
  58. Park YB (2005) An integrated approach for production and distribution planning in supply chain management. Int J Prod Res 43(6):1205–1224zbMATHCrossRefGoogle Scholar
  59. Pegels A, Vidican-Auktor G, Lütkenhorst W, Altenburg T (2018) Politics of green energy policy. J Environ Dev 27(1):26–45CrossRefGoogle Scholar
  60. Relvas S, Boschetto Magatão SN, Barbosa-Póvoa APFD, Neves F Jr (2013) Integrated scheduling and inventory management of an oil products distribution system. Omega 41:955–968CrossRefGoogle Scholar
  61. Rodrigo P, Fernández EF, Almonacid F, Pérez-Higueras PJ (2013) Models for the electrical characterization of high concentration photovoltaic cells and modules: a review. Renew Sust Energy Rev 26:752–760CrossRefGoogle Scholar
  62. Roslöf J, Harjunkoski I, Westerlund T, Isaksson J (2002) Solving a large-scale industrial scheduling problem using MILP combined with a heuristic procedure. Eur J Oper Res 138(1):29–42MathSciNetzbMATHCrossRefGoogle Scholar
  63. Sawik T (2009) Coordinated supply chain scheduling. Int J Prod Econ 120(2):437–451CrossRefGoogle Scholar
  64. Solanki CS (2011) Solar photovoltaics, 2nd edn. PHI, New DelhiGoogle Scholar
  65. Steinrücke 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
  66. Steinrücke M (2015) Integrated production, distribution and scheduling in the aluminium industry: a continuous-time MILP model and decomposition method. Int J Prod Res 53(19):5912–5930CrossRefGoogle Scholar
  67. Steinrücke M, Albrecht W (2016) A flow-to-equity approach to coordinate supply chain network planning and financial planning with annual cash outflows to an institutional investor. Bus Res 9:297–333CrossRefGoogle Scholar
  68. Steinrücke M, Albrecht W (2018) Integrated supply chain network planning and financial planning respecting the imperfection of the capital market. J Bus Econ 88(6):799–825CrossRefGoogle Scholar
  69. Steinrücke M, Jahr M (2012) Tactical planning in supply chain networks with customer oriented single sourcing. Int J Logist Manag 23(2):259–279CrossRefGoogle Scholar
  70. Steinrücke M, Jahr M (2018) Simultaneous optimisation of forward and reverse distribution processes with multiple types of reuse within an industrial tool supply chain. Int J Oper Res 32(4):397–420MathSciNetCrossRefGoogle Scholar
  71. Sun C, Rose T (2015) Supply chain complexity in the semiconductor industry: assessment from system view and the impact of changes. IFAC-PapersOnLine 48:1210–1215CrossRefGoogle Scholar
  72. Tempelmeier H (2002) A simple heuristic for dynamic order sizing und supplier selection with time-varying data. Prod Oper Manag 11(4):499–515CrossRefGoogle Scholar
  73. Tian Z, Kouvelis P, Munson CL (2015) Understanding and managing product line complexity: applying sensitivity analysis to a large-scale MILP model to price and schedule new customer orders. IIE Trans 47:307–328CrossRefGoogle Scholar
  74. Ullrich CA (2013) Integrated machine scheduling and vehicle routing with time windows. Eur J Oper Res 227(1):152–165MathSciNetzbMATHCrossRefGoogle Scholar
  75. Umetani S, Fukushima Y, Morita H (2017) A linear programming based heuristic algorithm for charge and discharge scheduling of electric vehicles in a building energy management system. Omega 67:115–122CrossRefGoogle Scholar
  76. Uzsoy R, Lee C-Y, Martin-Vega LA (1992) A review of production planning and scheduling models in the semiconductor industry. IIE Trans 24(4):47–60CrossRefGoogle Scholar
  77. Uzsoy R, Lee C-Y, Martin-Vega LA (1994) A review of production planning and scheduling models in the semiconductor industry. Part II: Shop-floor control. IIE Trans 26(5):44–55CrossRefGoogle Scholar
  78. Uzsoy R, Fowler JW, Mönch L (2018) A survey of semiconductor supply chain models part II: demand planning, inventory management, and capacity planning. Int J Prod Res 56(13):4546–4564CrossRefGoogle Scholar
  79. Van Vyve M, Wolsey LA, Yaman H (2014) Relaxations for two-level multi-item lot-sizing problems. Math Program 146(1–2):495–523MathSciNetzbMATHCrossRefGoogle Scholar
  80. Wang W, Rivera DE, Kempf KG (2007) Model predictive control strategies for supply chain management in semiconductor manufacturing. Int J Prod Econ 107:56–77CrossRefGoogle Scholar
  81. Wang I-L, Wang Y-C, Chen C-W (2013) Scheduling unrelated parallel machines in semiconductor manufacturing by problem reduction and local search heuristics. Flex Services Manuf J 25(3):343–366CrossRefGoogle Scholar
  82. West J (2014) Too little, too early: california’s transient advantage in the photovoltaic solar industry. J Technol Transf 39(3):487–501CrossRefGoogle Scholar
  83. Wu T, Shi L, Song J (2012) An MIP-based interval heuristic for the capacitated multi-level lot-sizing problem with setup times. Ann Oper Res 196(1):635–650MathSciNetzbMATHCrossRefGoogle Scholar
  84. Yeung W-K, Choi T-M, Cheng TCE (2011) Supply chain scheduling and coordination with dual delivery modes and inventory storage cost. Int J Prod Econ 132(2):223–229CrossRefGoogle Scholar
  85. Zhang M, Küçükyavuz S, Yaman H (2012) A polyhedral study of multiechelon lot sizing with intermediate demands. Oper Res 60(4):918–935MathSciNetzbMATHCrossRefGoogle Scholar
  86. Zipp A (2015) Revenue prospects of photovoltaic in Germany—Influence opportunities by variation of the plant orientation. Energy Policy 81:86–97CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Faculty of Law and EconomicsUniversity of GreifswaldGreifswaldGermany

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