مدل موجودی با وابستگی به نرخ تقاضای وابسته به سهام و نرخ هزینه نگهداری وابسته به سهام

In this paper, we develop an inventory model under a stock-dependent demand rate and stock-dependent holding cost rate with relaxed terminal conditions. Shortages are allowed and partially backlogged in the model. The purpose of this study is to determine the optimal order quantity and the ending inventory level such that the total profit per unit time is maximized for the retailer. We first establish a proper model for a mathematical formulation. Then we develop several theoretical results and provide the decision-maker with an algorithm to determine the optimal solution. Finally, numerical examples are provided to illustrate the solution procedure, and a sensitivity analysis of the optimal solution with respect to major parameters is carried out.

Since Harris (1913) presented an economic order quantity (EOQ) model, many researchers have been made to adjust their assumptions to more realistic situations in inventory management. For instance, it is usually observed in the supermarket that the display of such goods in large quantities attracts more customers and generates a higher demand (Levin et al., 1972). The inventory problem is an issue that has received considerable attention in inventory models with stock-dependent demand rates. Gupta and Vrat (1986) were the first to develop an inventory model for stock-dependent consumption rates. Later, Baker and Urban (1988) also developed an EOQ model for a power-form stock-dependent demand pattern. Mandal and Phaujdar (1989) then proposed an economic production quantity (EPQ) model for deteriorating items based on a constant production rate and linearly stock-dependent demand. Datta and Pal (1990) presented an inventory model where the demand rate is a piecewise function of the inventory level. Pal et al. (1993) extended Baker and Urban׳s (1988) model for deteriorating items. Padmanabhan and Vrat (1995) presented a deteriorating inventory model based on a stock-dependent selling rate and shortage. Wu et al. (2006) developed a replenishment model for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging. Examples of other studies in this area include Sarker et al. (1997), Ray and Chaudhuri (1997), Ray et al. (1998), Dye and Ouyang (2005), Lee and Dye (2012), Min et al. (2012), Avinadav et al. (2013) and Taleizadeh et al. (2013). Furthermore, several inventory models assume that the holding cost per unit time is constant. In real life, the holding cost for perishable goods such as foodstuffs, milk, fruit, vegetables, and meat drops with each passing day, and increasing holding costs are necessary to maintain the freshness of the items and to prevent spoilage. Weiss (1982) supposed that the holding cost per item is a convex potential function of time. Goh (1994) extended Baker and Urban’s (1988) model to relax the assumption of a constant holding cost. Later, Giri and Chaudhuri (1998) extended Goh’s (1994) model and developed an inventory model for deteriorating items. Recently, Pando et al. (2012) formulated an inventory model with both the demand rate and holding cost dependent on the stock level. Other studies in this area include those of Hwang and Hahn (2000), Alfares (2007), Roy (2008), Valliathal and Uthayakumar (2011), Pando et al. (2013), and Tripathi (2013). One of the major assumptions used in the above models is that the replenishment cycle would end with zero stock. However, in real life it may be desirable to order larger quantities, resulting in stock remaining at the end of the cycle, due to the potential profits from the increased demand. Urban (1992) first relaxed the terminal condition of zero ending inventory and suggested that it is more profitable to utilize higher inventory levels resulting in greater demand. Mandal and Maiti (1999) formulated an EPQ model with a stock-dependent consumption rate for damageable items and some units in hand. Furthermore, Giri et al. (1996) extended the model of Urban (1992) for inventory items deteriorating at a constant rate. Chang (2004) amended Giri and Chaudhuri’s (1998) model for deteriorating items by changing the objective to the maximization of profit and relaxing the restriction of a zero ending inventory. Teng et al. (2005) extended Urban’s (1992) model to accommodate not only deteriorating but also non-zero ending inventory, and proposed an algorithm to obtain the optimal replenishment cycle time and ordering quantity. Recently, Chang et al. (2010) extended Wu et al. (2006) model to relax the restriction of zero ending inventory when shortages are not desirable. The major assumptions used in the above-mentioned studies are summarized in Table 1. Table 1. Major characteristics of inventory models on selected researches. Literature EOQ/EPQ model Demand rate Holding cost Deterioration Non-zero ending inventory Shortage Harris (1913) EOQ Constant Constant No No No Weiss (1982) EOQ Constant Nonlinear time-dependent No No No Baker and Urban (1988) EOQ Stock-dependent (power function) Constant No No No Mandal and Phaujdar (1989) EPQ Stock-dependent (linear function) Constant Constant No No Datta and Pal (1990) EOQ Stock-dependent (piecewise function) Constant Constant No No Urban (1992) EOQ Stock-dependent (piecewise function) Constant No Yes No Pal et al. (1993) EOQ Stock-dependent (power function) Constant Constant No No Goh (1994) EOQ Stock-dependent (power function) Nonlinear stock dependent No No No Padmanabhan and Vrat (1995) EOQ Stock-dependent (linear function) Constant Constant No Completely/partially backlogged Giri et al. (1996) EOQ Stock-dependent (power function) Constant Constant Yes No Ray and Chaudhuri (1997) EOQ Stock-dependent (power function) Constant No No Completely backlogged Sarker et al. (1997) EPQ Stock-dependent (linear function) Constant Constant No Completely backlogged Giri and Chaudhuri (1998) EOQ Stock-dependent (power function) Nonlinear time/stock-dependent Constant No No Ray et al. (1998) EOQ Stock-dependent (power function) Constant (two warehouses) No No No Mandal and Maiti (1999) EPQ Stock-dependent (power function) Constant Stock-dependent Yes No Hwang and Hahn (2000) EOQ Stock-dependent (power function) Constant Fixed lifetime Yes No Chang (2004) EOQ Stock-dependent (power function) Nonlinear time/stock-dependent Constant Yes No Dye and Ouyang (2005) EOQ Stock-dependent (linear function) Constant Constant No Partially backlogged Teng et al. (2005) EOQ Stock-dependent (piecewise function) Constant Constant Yes No Wu et al. (2006) EOQ Stock-dependent (linear function) Constant Non-instantaneous No Partially backlogged Alfares (2007) EOQ Stock-dependent (linear function) Nonlinear time-dependent No No No Roy (2008) EOQ Price-dependent (linear function) Linear time-dependent Time-dependent No Completely backlogged Chang et al. (2010) EOQ Stock-dependent (linear function) Constant Non-instantaneous Yes Partially backlogged Valliathal and Uthayakumar (2011) EPQ Stock and time-dependent Nonlinear stock-dependent Time-dependent No Partially backlogged Pando et al. (2012) EOQ Stock-dependent (power function) Nonlinear stock-dependent No No No Lee and Dye (2012) EOQ Stock-dependent (linear function) No Controllable No Partially backlogged Min et al. (2012) EPQ Stock-dependent (linear function) Constant (excluding interest charges) No No No Avinadav et al. (2013) EOQ Price and Time-depend (power function) Constant Constant No No Pando et al. (2013) EOQ Stock-dependent (power function) Nonlinear stock and time –dependent No No No Taleizadeh et al. (2013) EOQ Constant Constant Constant No Completely backlogged Tripathi (2013) EOQ Time-dependent (power function) Linear time-dependent No No No This paper EOQ Stock-dependent (power function) Nonlinear stock-dependent No Yes Partially backlogged Table options Thus, the problem of determining the optimal replenishment policy with a stock-dependent demand and stock-dependent holding cost rate is addressed in this paper in a manner that reflects realistic circumstances. The terminal condition of a zero ending inventory level is relaxed and shortages are allowed and partially backlogged. The rest of the paper is organized as follows. In the next section, we describe the notations and assumptions used throughout this paper. In Section 3, we establish the mathematical model. We then develop several theoretical results in Section 4 and provide the decision-maker with an algorithm for finding the optimal solution. A couple of numerical examples are provided in Section 5 to illustrate the solution procedure. In addition, a sensitivity analysis of the optimal solution with respect to major parameters is also carried out. Finally, we summarize our findings in Section 6 and provide some suggestions for future research.

This paper develops an inventory model under a stock-dependent demand rate and stock-dependent holding cost rate with relaxed terminal conditions. Shortages are allowed and partially backlogged. We have developed several theoretical results and provided the decision-maker with an algorithm to find the optimal solution. Furthermore, we present several numerical examples to demonstrate our model and solution. The numerical examples reveal that (1) the retailer should maintain high levels of initial and ending inventories to raise the demand rate when the elasticity of the demand rate for the inventory level is high, (2) when the holding cost elasticity, γγ, rises as the cost from an additional unit of inventory increases, the retailer should lower the ending inventory as far as possible or even permit shortages, and (3) when ββ is equal to zero, this implies that the demand rate is constant, the retailer should maintain negative inventory at the end of the replenishment cycle. In contrast, when ββ is large enough, the retailer should always keep positive inventory at the end of the replenishment cycle. We believe that our work provides a foundation for the further study of this type of inventory model under a stock-dependent demand rate and holding cost rate.