An inventory model with shortages for imperfect items using substitution   of two products
  Inventory models with imperfect quality items are studied by researchers in past two decades. Till now none of them have considered the effect of substitutions to cope up with shortage and avoid lost sales. This paper presents an EOQ approach for inventory system with shortages and two types of products with imperfect quality by one way substitution. Our model provides significant advantage for substitution case while maintaining its inherent simplicity. We have provided numerical example and sensitivity analysis to justify the effectiveness of our model. It is observed that, presence of imperfect items affect the lot size of minor and major products differently. Under certain conditions, our model generalizes the previous existing models in this direction. 
Arindum Mukhopadhyay, A. Goswami
math.OC