All-units discount facilities are one of the attractive features in the competitive business situation. Due to the globalization of the marketing policy, all-units discount facilities play an important role in the competitive business. Typical economic order quantity (EOQ) models are cloistered by considering as constant not only the purchase cost (irrespective of the order size of the product) but also the carrying cost during the entire cycle period. However, the unit purchase cost has an antagonistic relationship with the order size, and the carrying cost has a commensurate relationship with the storage time-period of the product, that is, the higher the order size, the lower the unit purchase cost, and the longer the storage time-period, the greater carrying cost per unit. Also deterioration is another imperative issue in inventory analysis as it has a huge impact on profit or cost of the inventory system. Considering all of the above-mentioned factors, we study two different inventory models, namely (a) inventory model for zero-ending case and (b) inventory model for shortages case. The demand for both models is considered as price and stock dependent, whereas shortages are partially backlogged at a rate with the length of the waiting time to the arrivals of the next lot. The existence and uniqueness of the optimal solution for both models are examined theoretically and the solution procedures are discussed along with two proposed algorithms for minimizing the total cost. Finally, we perform sensitivity analyses for both models and make a fruitful conclusion regarding the proposed work.

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Generally, in the business world, it is observed that suppliers give different kinds of benefits to retailers due to advance payment. One of the popular benefits is instant cash discount due to advance payment. If a retailer pays off his total purchase cost before receiving the products, then he receives a certain percentage of cash discount instantly. However, if the retailer pays off a certain fraction of the total purchasing cost, then price discount is given only at the time of receiving the products while paying the remaining amount of the total purchasing cost. Using this concept, this paper formulates, under both cases of advance payment (full or partial), an inventory model for deteriorating products where shortages are allowed and demand function is considered as price and stock dependent. The closed‐form solutions for each case are presented and two numerical examples are solved. In addition, a sensitivity analysis is also performed to show the effects of advance payment with discount facility.

Advertisement of the product is an important factor in inventory analysis. Also, price and stock have an important role to attract more customers in the competitive business situations. Trade credit policy is another important role in inventory analysis. We have combined these three factors together in a two-warehouse inventory model and represented it mathematically. In addition, we have added deteriorating factor of our proposed problem with price-and stock-dependent demand under partial backlogged shortage and solved. The frequency of advertisement is considered constant for a year in this paper. The proposed model is highly nonlinear in nature. Due to highly nonlinearity, we could not find the closed form solution. In this paper, trade credit facility is taken in the perspective of retailer, and all the possible cases and subcases of the model are discussed and solved using lingo 10.0 software. The results of sensitivity analysis prove the effectiveness of the proposed model.

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