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This paper expands an inventory model for deteriorating items with stock-dependent demand. This model provides time varying backlogging rate as well as time varying deterioration rate. The aim of this model is to determine the optimal cycle length of each product such that the expected total cost (holding, shortage, ordering, deterioration and opportunity cost) is minimized. Further, the necessary and sufficient conditions are provided to show the existence and uniqueness of the optimal solution. Lastly, some numerical examples, sensitivity analysis along with graphical representations are shown to illustrate the practical application of the proposed model.

In recent years, many researchers/scientists have discussed on inventory models for deteriorating items. In daily life, the deterioration of items becomes a common factor. Generally, deterioration indicates the damages of the products. To highlight such type of phenomenon, Ghare and Schrader (1963) first developed a deteriorating inventory model with constant rate of decay. That model was again extended by Covert and Philip (1973) with two-parameter Weibull distribution. Dave and Patel (1981) derived an EOQ (economic order quantity) model for deteriorating items with time-proportional demand and without shortage. Sachan (1984) extended the model of Dave and Patel (1981) by allowing shortages. Later, several related articles were discussed by Wee, 1995 and Hariga, 1996, Wee and Law (1999), Moon et al. (2005), Chung and Wee, 2007 and Sana, 2010aWidyadana et al. (2011), and others. Widyadana and Wee (2011) addressed EOQ models for deteriorating items with different increasing demands. Sarkar et al. (2013) presented an EOQ model for deteriorating items with finite production rate and time dependent increasing demand. In most of the above referred paper, the authors considered constant deterioration rate. But, in real life situation, items may deteriorate due to expiration of their maximum life time i.e., deterioration rate is proportional with time and the maximum life time can be controlled by the production system, i.e., the manufacturer can fix the maximum life time of the product. An inventory model with time dependent deterioration rate, shortage and ramp-type demand rate was discussed by Giri et al. (2003). Manna and Chaudhuri (2006) presented an EOQ model for deteriorating items for both time-dependent demand and deterioration. Loa et al. (2007) represented an integrated production-inventory model for imperfect production with Weibull distribution deterioration under inflation. An inventory model with general ramp type demand rate, Weibull deterioration rate and partial backlogging of unsatisfied demand was considered by Skouri et al. (2009). Sana (2010b) discussed an EOQ model with time varying deterioration and partial backlogging rate. In this model, the deterioration function was considered as functional form of time. Sarkar (2012a) developed an EOQ model for finite replenishment rate where demand and deterioration rate were both time-dependent. Sett et al. (2012) discussed a two-warehouse inventory model with quadratically increasing demand and time varying deterioration. Most recently, Sarkar (2012b) developed a production-inventory model for three different types of continuously distributed deterioration functions. In that paper, Sarkar (2012b) solved the model with the help of algebraical operation and a comparison between the different probabilistic deterioration models was shown by numerical experiments. In the classical economic order quantity model, it is often assumed that the shortages are either completely backlogged or completely lost. In reality, often some customers are willing to wait until replenishment, especially if the wait will be short, while others are more impatient and go elsewhere. To reflect this phenomenon, Padmanabhan and Vrat (1995) considered an EOQ model for perishable items with stock-dependent demand under instantaneous replenishment with zero lead time. Abad (1996) discussed a pricing and lot-sizing problem for a product with a variable rate of deterioration by allowing shortages and partial backlogging. Chang and Dye (1999) recently developed an inventory model in which the backlogging rate was the reciprocal of a linear function of the waiting time. Cárdenas-Barrón (2001) presented an inventory model with shortage by an algebraical approach. An optimal replenishment policy for non-instantaneous deteriorating items with time varying partial backlogging rate was presented by Wu et al. (2006). Cárdenas-Barrón (2007) presented an inventory model on optimal manufacturing batch size with rework process at single-stage production system. A simple derivation on optimal manufacturing batch size with rework process at single-stage production system was presented by Cárdenas-Barrón (2008). An economic production quantity (EPQ) model with rework process at a single-stage manufacturing system with planned backorders was presented by Cárdenas-Barrón (2009). Most recently, Cárdenas-Barrón, 2010, Cárdenas-Barrón, 2011 and Cárdenas-Barrón, 2012, Roy et al., 2011a and Roy et al., 2011b, Wee and Wang (2012), Yang et al. (2010) done their excellent research in this direction. In the marketing management policy, display stock level plays a very important role in different sectors. Thus, it is very clear that the demand rate increased rapidly if the stored amount is high and vice-versa. Liao et al. (2000) were the first to discuss an inventory model for initial-stock-dependent consumption rate with permissible delay in payment. Dye and Ouyang (2005) extended Padmanabhan and Vrat's (1995) model with linear time-proportional backlogging rate, and then established the unique optimal solution to the problem for non-profitable building up inventory. Alfares (2007) found out an inventory model with stock-level dependent demand rate and variable holding cost. Chung and Wee (2007) developed the scheduling and replenishment plan for an integrated deteriorating inventory model with stock-dependent selling rate. Sana and Chaudhuri (2008) presented a deterministic EOQ model with stock-dependent demand rate where a supplier gives a retailer both a credit period and a price discount on the purchase of merchandise. In this direction, some notable researches were addressed by Goyal and Chang (2009), Roy et al., 2010 and Roy et al., 2011c, Sana, 2011a, Sana, 2011b, Sana, 2011c and Sana, 2012a, Sarkar et al., 2010a and Sarkar et al., 2010b, and others. Sana (2012b) studied a newsboy problem with price-dependent demand and stochastic selling price to maximized the total profit. An EOQ model for perishable item with stock-dependent demand and price discount rate was presented by Sana (2012c). Sarkar (2012c) developed an EOQ model with finite replenishment rate to investigate the retailer's optimal replenishment policy under permissible delay in payment with stock dependent demand. See Table 1 for comparisons. Table 1. Summary of stock dependent demand, time varying deterioration, shortages and partial backlogging related literature. Author/authors Stock dependent demand Other demands Time varying deterioration Other deteriorations Shortages Partial back-logging Ghare and Schrader (1963) √ √ Covert and Philip (1973) √ √ Dave and Patel (1981) √ √ Sachan (1984) √ √ √ Padmanabhan and Vrat (1995) √ √ √ √ Wee (1995) √ √ √ √ Hariga (1996) √ √ √ Chang and Dye (1999) √ √ √ √ Wee and Law (1999) √ √ √ √ Cárdenas-Barrón (2001) √ √ Dye and Ouyang (2005) √ Moon et al. (2005) √ √ √ Loa et al. (2007) √ √ √ √ Goyal and Chang (2009) √ Skouri et al. (2009) √ √ √ √ Cárdenas-Barrón (2009) √ √ √ Sarkar et al. (2010a) √ √ Sarkar et al. (2010b) √ Yang et al. (2010) √ √ √ √ Sana (2010a) √ √ Sana (2010b) √ √ √ Sana (2011a) √ Cárdenas-Barrón (2010) √ √ √ Cárdenas-Barrón (2011) √ √ √ Chung and Wee (2007) √ √ Widyadana and Wee (2011) √ √ √ Widyadana et al. (2011) √ √ √ √ Sarkar et al. (2013) √ √ Sarkar (2012a) √ Sarkar (2012b) √ √ Sarkar (2012c) √ √ Sett et al. (2012) √ √ Wee and Wang (2012) √ √ √ This paper √ √ √ √ Table options In the proposed paper, we develop an inventory model for time varying deteriorating items with stock dependent demand, shortage and partial backlogging. To the authors' knowledge, this type of model has not yet been considered by any of the researchers/scientists in inventory literature. Therefore, this model has a new managerial insight that helps a manufacturing system/industry to gain maximum profit. The rest of the paper is designed as follows: In Section 2, fundamental notation and assumptions are given. In Section 3, mathematical model is shown. Solution procedure of the model is provided in Section 4. Numerical examples and sensitivity are given in 5 and 5.1 respectively. In Section 6, we provide some special cases and their comparison. Finally, conclusions are made in Section 7.

Several papers, discussing the above mentioned research topic have been studied in the literature that investigates inventory problems under varying conditions. The proposed model extends the model of existing literature with infinite replenishment rate, stock-dependent demand, time varying deterioration and partial backlogging. In most of the papers, the researchers have considered constant deterioration rate of items. But, in real life situation, maximum items deteriorate due to expiration of their maximum life time i.e., deterioration rate is proportional with time. In this model, deterioration of items follows a time varying deterioration function. There are two effective and easy to use lemmas to derive the model analytically. Finally, some numerical examples, graphical representations, special cases and their comparisons, and sensitivity analysis are provided to illustrate the proposed model. Further, the model may be extended by considering multi item inventory models, inflations, reliability of the items, etc.