The constant demand rate is the most common assumption of the basic economic production quantity model, which is not very frequent in practice. In real world situations, demand usually varies with time. With regard to the widespread necessity of power demand pattern, demand is supposed to follow a power law.Another unrealistic assumption is perfect quality of all items. This paper presents a production system with defective items to determine the optimal replenishment quantity, cycle length and backordered size with a power demand rate dependent production rate. We assume that a manufacturer may be faced with three different cases regarding to the date that defective items are drawn from inventory. The set-up, backordering, inspection, and production costs, as well as holding cost of both perfect and imperfect items are accounted in the inventory system. An algorithm is offered to optimize total inventory cost and then numerical analyses are presented to demonstrate the applicability of the proposed models. Finally, some sensitivity analyses and managerial insights are provided.

This paper addresses a new location-allocation-pricing problem in designing a three-level uncertain supply chain network with stochastic price-sensitive demands. Using the market segmentation problems, a supply-chain network is developed with two distribution channels that consist of Brick & Mortar and onlinemarkets, when some demand leakages occur from the market with a higher price. Due to the lack of physical observation of products in online markets, a return policy is used. So, demand behavior is analyzed in terms of the pricing and return policy. The problem established location, allocation, order quantities, pricing and refundprice decisions to optimize the total profit of the chain. Furthermore, it is formulated as a mixed-integer nonlinearprogramming (MINLP) model and solved by a Lagrangian relaxation algorithm. The numerical study andcomputational results indicate the efficiency and effectiveness of the proposed algorithm