Optimal production run in an imperfect production process with maintenance under warranty and product insurance

In the business world, product warranty and insurance plays an important role by increasing the demand of the product to maximizing the average profit or minimizing the average cost in a manufacturing system. Also, the customers' demand depends on several factors, viz. quality, selling price, payment strategies, warranty and insurance of the product. This paper develops an imperfect production inventory model with preventive machine maintenance, rework, warranty and insurance coverage on the accidentally damaged part(s) of the product. The concerned production system can randomly shift from an in-control state to an out-of-control state at any time during a production run, and the time elapsed while the production process stays in the in-control state is assumed to be exponentially distributed. The rates of producing reworkable defective items in the in-control state and out-control states are different. The demand rate of the customers is considered as the product's selling price and warranty period dependent. To attract customers and increase the demand for the product, the manufacturer offers an insurance policy on the damageable part(s) of the product within the warranty period. The main aim of this study is to find the optimal production time, selling price for the different warranty periods of the product, which maximizes the profit per unit business period. To examine the validity of the proposed model, two numerical examples are considered and solved with the help of MATHEMATICA software. Finally, the effects of warranty and insurance policies on the products are analyzed, based on the numerical examples.