Production-inventory models for a damageable item with variable demands and inventory costs in an imperfect production process

In this paper, economic production quantity (EPQ) models for breakable or deteriorating item are developed with variable demands, being dependent on time or on-hand stock. Here rate of production and holding cost are time dependent, 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 the perfect ones. The production process ultimately depends on both time and reliability indicator. The models are formulated as optimal control problems and the total profit functions with effect of inflation and time-value of money are expressed as finite integrals over the finite planning horizon. The problems are solved using Euler–Lagrange function based on variational calculus and Newton–Raphson method to determine the optimal production reliability indicator (r) and then corresponding production rates and total profits. In some cases, results of the models for deteriorating item are obtained as particular cases from those of breakable item models. Similarly, results of simple EPQ models (without damageability) are deduced as particular cases. Numerical experiments are performed to illustrate the models both numerically and graphically.