مدل موجودی با تقاضای متغیر، هزینه جزء و قیمت فروش برای به آیتم های در حال وخامت

In this paper, we develop an economic order quantity (EOQ) model for finite production rate and deteriorating items with time dependent increasing demand. The component cost and the selling price are considered at a continuous rate of time. The objective of this model is to maximize the total profit over the finite planning horizon. We also want to find the integral number of orders in the finite planning horizon. A numerical example, graphical representations and sensitivity analysis are given to illustrate the model.

In the classical inventory model, Harris (1915) considered an EOQ model with constant demand rate. Later, this model was discussed by Wilson (1934). Silver and Meal (1969) extended the EOQ model for varying demand rate. Donaldson (1977) derived an inventory model for linear trend in demand. Other researchers such as Buchanan (1980), Silver and Peterson (1985), Goyal et al. (1992), Teng (1996) and Lo et al. (2002) studied an inventory model with a linear trend in demand. In most of the above papers linear trend in demand or exponentially increasing demand rate were assumed. In reality, the demand rate for items may depend on time. Hsu and Li (2006) presented an optimal delivery service with time dependent consumer demand. Later, many researchers such as Banerjee and Sharma (2010) and Sarkar et al. (2011) developed the inventory model with different types of time dependent demand. All of the above models were derived without deterioration. Deterioration is defined as decay, evaporation and loss of utility of a commodity that results in the decreasing usefulness from the original condition. Also, it is considered by the damages when the items are broken or loose their marginal value due to accumulated stress, bad handling etc. Ghare and Schrader (1963) formulated a model with exponentially decaying inventory. Shah and Jaiswal (1977) mentioned an order-level inventory model for a system with constant rate of deterioration. Aggarwal (1978) derived a note on an order-level inventory model for a system with constant rate of deterioration. Dave and Patel (1981) discussed inventory models with time proportional demand and deterioration. Dave (1986) addressed an order level inventory model for deteriorating items with variable instantaneous demand and discrete opportunities for replenishment. Bahari-Kashani (1989) discussed a replenishment schedule for deteriorating items with time proportional demand. Several researchers like Goswami and Chaudhuri (1991), Hariga (1995), Sarker et al. (1997), Jamal et al. (1997), Yan and Cheng (1998), Mandal and Pal (1998), Giri et al. (2000), Liao et al. (2000), Chang et al. (2001), Goyal and Giri (2003), Khanra and Chaudhuri (2003), Arcelus et al. (2003), Sana and Chaudhuri (2004) and Ouyang et al. (2005) developed different types of inventory models for different forms of deteriorating items with time dependent demand. Based on their models, Roy (2008), Sana (2008) and Lee and Hsu (2009) investigated order level inventory models with different forms of deteriorating items for time varying demand. Khanra et al. (2011) investigated an EOQ model for deteriorating items with time dependent quadratic demand. Widyadana et al. (2011) formulated an economic order quantity model for deteriorating items and planned back order level. Later, Sarkar (2012) obtained an EOQ model with time varying deterioration rate. Thursby et al. (1986) addressed an inventory model with selling price. Lev and Weiss (1990) produced an inventory model with cost changes. Erel (1992) considered an EOQ model with price changes. Gascon (1995) surveyed an inventory model with cost changes for finite horizon. Wee (1995), Khouja and Park (2003), Goyal and Cárdenas-Barrón (2003), Teunter (2005) derived economic order quantity model under continuous price decrease. Several researchers like Cárdenas-Barrón, 2006a and Cárdenas-Barrón, 2006b, Smith et al. (2007), Cárdenas-Barrón (2007), Mishra and Mishra (2008), Cárdenas-Barrón (2008), Smith et al. (2009), Cárdenas-Barrón, 2009a and Cárdenas-Barrón, 2009b, Dye and Hsieh (2010), Sana (2011), Cárdenas-Barrón (2011), Yang et al. (2011), Cárdenas-Barrón et al. (2011), Lu et al. (2011), Naimzada and Ricchiuti (2011), Pal et al. (2012), and Sana, 2012a and Sana, 2012b obtained EOQ models with different types of practical assumptions. In this paper, an effort has been made to develop an EOQ model with time dependent demand, component cost and product selling price for deteriorating items. This model is an extension of the concept of the model of Yang et al. (2011). The component cost and product selling price are increased day by day due to high design technology, reliability, inflations. Therefore, it is better to consider them as the increasing value rather than the decreasing value. Hence, the cost function and selling price are assumed inclined at a continuous rate per unit time which differs our paper from Yang et al. (2011) and many previous research models. Most of them considered reduced price and the corresponding demand is either constant or exponential dependent. But, in our model the demand is assumed as continuously increasing with respect to time quadratically. We consider deterioration of the product to make this model more realistic. The associated profit function is calculated and is optimized with respect to the integer number of orders in the entire planning horizon (Fig. 1). A numerical example, graphical illustration and sensitivity analysis are used to illustrate the model (Table 1). Full-size image (6 K) Fig. 1. Graphical representation of inventory model. Figure options Table 1. Numerical analysis: to determine the net profit and also for integer values n we use Mathematica software. Integral number of orders (n) Net profit (NP) n = 4 10187.1 n = 5 57717.8 n = 6 65297.9 n = 7 67215.7 n = 8 67703.3 n = 9 67655.3 n = 10 67344. n = 11 66877.7 n = 12 66309.2

This model is an extension of the idea of Yang et al. (2011) with finite replenishment rate, increasing quadratic demand and deterioration factor. From Yang et al. (2011), the fixed replenishment interval has produced a significant result, therefore we derive only the fixed replenishment interval case for increasing component cost and also increasing selling price as demand is increasing with time. We assume the case of deterioration of the product at a constant rate which is more realistic than Yang et al. (2011). Therefore, the model is more realistic than the model of Yang et al. (2011) and it allows more profit at the optimum level. To the authors' best knowledge, such type of model has not yet been discussed in inventory literature. Some possible extensions of this paper are to consider the multi-item EOQ model with variable lead time by considering shortage, inflation, reliability and delay in payments. These are, among others some subset of ongoing future research.