Inventory of differential items selling from two shops under a single management with periodically increasing demand over a finite time-horizon

Inventories of differential items including the defective ones purchased/produced in a lot and sold from two shops (primary and secondary shops) under a single management are considered here over a finite time-horizon. A primary shop receives the differential units in a lot but sells only the non-defective ones whose demand periodically increases with time and decreases during the shortage period in such a way that it becomes back to the initial value at the beginning of the next cycle. Hence in this shop, shortages are allowed and fully backlogged. Moreover, at the beginning of the next cycle, the retailer purchases purely non-defective units at a higher price to meet up the shortage amount along with the usual lot of differential units for regular sale. The defective units identified at the time of selling at the primary shop are continuously transferred to the adjacent secondary shop from which the defective ones are sold at a reduced price after some rework. Normally, the price of a defective item is fixed depending upon the quantum of its defect and people go for these items if they are cheap. Hence, demand for these units is dependent on the selling price, which is again inversely proportional to the rate of defectiveness. There may be five scenarios for dealing with defective units depending upon the coincidence of the time periods at two shops. For all scenarios, problems have been mathematically formulated and solved by the use of both parametric study and a gradient-based non-linear optimisation method. The models are illustrated with the help of numerical examples.