A multi-item EPQ model with imperfect production process for time varying demand with shortages

In this paper, economic production quantity(EPQ)models for breakable or deteriorating items are developed with time dependent linear variable demands. Here rate of production and holding cost are time dependent and unit production cost is a function of both production reliability indicator and production rate. Set-up cost is also partially production rate dependent.The production process produces some imperfect quantities which are instantly reworked at a cost to bring back those units to perfect ones. Ultimately, the production process depends on both time and reliability indicator. Here two models are developed in optimal control framework considering the effect of time value of money and inflation. Shortages are allowed for both the models. The first model consists of two items and the second model is of single item. For the first model, a budget constraint is considered. For both the models, total profit function are calculated with the effect of inflation and time value of money in a finite integral over a finite planning horizon.The problems are solved using Eulers-Lagrangian function based on variational calculus and applying generalized reduced gradient method using LINGO 13.0 software to determine the optimal reliability indicator (r) and then corresponding production rates and total profits. From the above two breakable models, other ten models are obtained as particular cases, out of which two are with stock dependent breakability/deterioration and another two are simple EPQ models without breakability.Out of these particular cases two models are of constant demand,two models are of constant holding cost and two models are of constant setup cost. Numerical experiments are performed for both the models to illustrate the models both numerically and graphically.