مدل موجودی همراه با تقاضای وابسته به زمان و کمبود تحت سیاست های اعتباری تجاری

This model investigates an economic order quantity (EOQ) model over a finite time horizon for an item with a quadratic time dependent demand by considering shortages in inventory under permissible delay in payments. Shortages are assumed after variable time T1(< T cycle period). The credit period (m) is known and fixed. The model is derived under three different circumstances depending on the time of occurrence of shortages, credit period, and cycle time. The results are illustrated with the help of numerical examples. The sensitivity analysis of key parameters of the optimal solution is studied with respect to changes in different parametric values. Some special features of the model are discussed.

Since the formulation of EOQ in 1913, Harris's (1913) square root formula for the economic order quantity (EOQ) was used in the inventory literature for a pretty long time. This formula was developed on the assumption of constant demand. Thereafter many models were developed in the inventory literature by assuming constant demand. In the real marketing environment, the demand rate of any item may vary with time. Silver and Meal (1969) were the first to suggest a simple modification of the classical square root formula in the case of time varying demand. The classical no-shortage inventory problem for a linear trend in demand over a finite time horizon was analytically solved by Donaldson (1977). However, Donaldson (1977) solution's procedure was computationally complicated. Many researchers like Silver (1979), Ritchie (1985), Dave and Patel (1981), and Goyal (1986) made their valuable contributions in this direction. They did not consider shortages in their models. Deb and Chaudhuri (1987) were the first to extend the model of Silver (1979) to incorporate shortages in inventory. This extension was also studied by Dave (1989), Goyal et al. (1992), Goswami and Chaudhuri, (1991), Giri et al. (1996), and Teng (1996). Some researchers like Wee (1995) and Jalan and Chaudhuri (1999) developed their models by considering exponential time varying demand pattern. Sana and Chaudhuri (2000) extended the EOQ model over a finite time horizon by assuming unequal cycle lengths. From the existing literature, it is clear that while dealing with time varying demand, researchers have studied two types of demand rate, namely linear and exponential. A linearly time varying demand implies uniform change in demand rate of the product per unit time which is rarely seen to occur in real market. On the other hand, exponentially time varying demand indicates a very rapid change in demand which is also rare because the demand rate of any product cannot change with a high rate of change as exponential. Khanra and Chaudhuri (2003) was the first to consider a quadratic demand rate which is more realistic. Ghosh and Chaudhuri (2006) extended the EOQ model over a finite time horizon with shortages in all cycles. In the conventional EOQ model, it was assumed that the customer must pay for the item as soon as it is received. In practice, however the supplier offers the retailer a certain trade credit period, in paying for purchasing cost. During this delay period, the retailer can earn revenue by selling items and by earning interest. An inventory model with permissible delay in payments was first studied by Goyal (1985). Several valuable contributions in this field were studied by Mondal and Phaujder (1989), Aggarwal and Jaggi (1995), Chu et al. (1998), Chung (2000)Sana and Chaudhuri (2008), Khanra et al. (2011), and Sarkar, 2012a, Sarkar, 2012b and Sarkar, 2013. In this paper, an EOQ model is developed for an item with time varying quadratic demand and shortages and permissible delay in payments. However, this type of demand rate is more realistic because it can represent both accelerated growth and retarded growth in demand as it has the general form D(t) = a + bt + ct2. Here c = 0 indicates linear time dependent demand and a = 0 as well as b = 0 simultaneously indicate constant demand. The model has been developed under three circumstances, Case 1: the credit period is less than the time of shortage period, Case 2: the credit period is greater than the time of commencement of shortage period but less than cycle length and Case 3: credit period is greater than the cycle for settling the account. The model is illustrated with numerical examples. Also, the sensitivity analysis of the model is examined for changes in parameters.

The proposed model is based on a quadratic time-varying demand rate. The rationale for considering a quadratic demand instead of a linear demand or an exponential demand has been explained at the beginning of this model. While dealing with time-varying demand patterns, the researchers usually take the demand rate to be a linear function of time. This type of demand of the form R (t ) = a + bt a ≥ 0, b ≠ 0, implies steady increase (b > 0) or decrease (b > 0) in the demand rate, which is rarely seen to occur for any product. Some researchers adopted an exponential functional form like R (t ) = ae bta > 0, b ≠ 0implying exponential increase (b > 0) or decrease (b < 0) in the demand rate. As exponential rate being very high, it is also rarely seen in the real market that the demand of any product can really rise or fall exponentially. A better alternative would be to think of accelerated rise or fall in demand. Accelerated growth in the demand rate takes place in case of the state of the art of aircrafts, computers, machines, and their spare parts. Accelerated decline in the demand rate is found to occur in the case of obsolete aircrafts, computers, machines, and their spare parts. The demand of a seasonal product rises rapidly to a peak in the mid-season and then falls rapidly as the season wanes out. These different types of demand can be better represented by the functional form R (t ) = a + bt + ct 2, a ≥ 0, b ≠ 0, and c ≠ 0. We have View the MathML sourcedRtdt=b+2ct, for b > 0, c > 0, the rate of increase of the demand rate R (t ) is itself an increasing function of time. We call it accelerated growth in demand. For b > 0, c < 0, there is a retarded growth in demand for all times View the MathML sourcet∈0,‐b2c. For b < 0, c > 0, the demand rate falls at an increasing rate for View the MathML sourcet>−b2c. We call it accelerated decline in demand. For b < 0, c < 0, there is retarded decline in the demand rate for all times. Thus, we may have different types of realistic demand patterns from the functional form R(t) = a + bt + ct2 depending on the signs of b and c. This advantage of the time-quadratic demand has motivated authors to adopt it in the present model. Shortages are allowed and are completely backlogged. In many practical situations, stock out is unavoidable due to various uncertainties. Also it is important from the managerial point of view to reduce average total cost. In the real market, we see that suppliers offer their customers a certain credit period without interest during the permissible delay time period. As an outcome, it motivates customer to order more quantities because paying later indirectly reduces the purchase cost. We extend this model by considering inflation, reliability of product, and partial delay in payments.