Production inventory model for controllable deterioration rate with shortages Umakanta Mishra, Jacobo Tijerina-Aguilera, Sunil Tiwari and Leopoldo Eduardo Cárdenas-Barrón RAIRO-Oper. Res., 55 (2021) S3-S19 Published online: 09 February 2021 DOI: 10.1051/ro/2019042