We develop here an economic order quantity (EOQ) model over a finite time-horizon for a deteriorating item with a quadratic, time-dependent demand, allowing shortages in inventory. The rate of deterioration is taken to be time-proportional and it is assumed that shortage occurs in every cycle. The time-horizon is divided into a finite number of equal replenishment cycles. The reorder number, the interval between two successive reorders and the shortage intervals are all determined in an optimal manner so as to minimize the average system cost. The results are illustrated with the help of a numerical example. Sensitivity of the optimal solution is also studied with respect to changes in different parameter values.

Considering the increasing number of end-of-life goods in the context of improving the ambience and health of a population and their destructive impacts, recycling strategies are important for industries and organizations. In this article, a closed-loop supply chain management containing a single manufacturer, a single retailer, and a third party is introduced in which the manufacturer first propagates newly finished goods and then dispatches some of the finished goods to the retailer considering a single-setup multi-delivery policy. Due to shipping, carbon emission is taken into account as well as a carbon emission trading mechanism to curb the amount of carbon emissions by the retailer. For recycling through collection, inspection, remanufacturing, and landfill, the third party collects the end-of-life goods from its customers and ships perfect products to the manufacturer after a two-stage inspection. In this model, major sources of emissions such as shipping, replenishment orders, and inventory have been taken care of. The minimizing of the total cost relating to the container capacity, shipment numbers, and replenishment cycle length is the main objective of the closed-loop supply chain management for making the system more profitable. Expository numerical explorations, analysis, and graphic representations are conferred to elucidate this model, and it is observed that this model saves some percentage of the cost compared to the existing literature.

The waste of energy in the present era is a dangerous signal for the future. All categories of consumers should come forward to moderate energy use and prevent wastage. This study focuses on a controllable energy consumption-based sustainable inventory model incorporating variable production rates, improved service, partial outsourcing planning, defective production, restoring reworkable items, disposing of non-reworkable items, and energy-saving steps. Reducing unusual energy consumption in production systems reduces carbon emissions and maximizes the system’s profit. An improved service level attracts customers, increases demand, and improves product reputation. Separate holding costs of reworked, defective, and perfect-quality items are considered for every lot delivered and reworked. The demand in the market is related to price and service. A traditional optimization technique examines the global optimization for the profit function and decision variables. Numerical illustrations as well as concave 3D graphs validate the analytical results and provide a sensitivity analysis for different parameters. The model is validated through special cases and comparison graphs.