This paper presents an integrated inventory model for a supply chain system that consists of a vendor and a buyer under stochastic demand and imperfect production. The vendor operates a hybrid system combining a regular production and a green production. Green production is cleaner than the regular production, but it is more costly. The production rate is adjustable and influences the production cost and emissions resulted from production and reworking processes. The production system is imperfect and thus produces a certain percentage of defective items, but the vendor is willing to invest money to reduce the defect rate. The objective of the model is to find the optimal shipment quantity, production allocation, number of shipments, safety factor, defective rate and production rate so that the supply chain cost is minimized. An iterative procedure is proposed to obtain the solution and a numerical example is provided to show the application of the model. The results show that by making an investment, controlling the production rate and setting the production allocation, the system can reduce the defective items. In addition, the last two actions can control the emissions and manage the trade-off. Finally, the sensitivity analysis is also performed to investigate the effect of the changes in key parameters’ values on the behaviour of the model.

Original languageEnglish
Pages (from-to)543-572
Number of pages30
JournalAnnals of Operations Research
Issue number1-2
Publication statusPublished - May 2023


  • Emission
  • Hybrid production
  • Imperfect production
  • Lot-sizing
  • Rework


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