Production, Manufacturing and LogisticsA deterministic EOQ model with delays in payments and price-discount offers
Introduction
That the study of inventory problems dates back to 1915, by Harris (1915), was quite natural for the inventory problem to be among the first selected for mathematical analysis. He established the simple but famous EOQ (Economic Order Quantity) formula that was also derived, apparently independently, by Wilson (1934). Generally speaking, the demand rate of any product is always in a dynamic state. Demand of goods may vary with time or with price or even with the instantaneous level of inventory displayed in a supermarket. In recent years, there is a spate of interest in the problem of finding the economic replenishment policy for an inventory system having a time-dependent demand pattern. It started with the work of Silver and Meal (1969) who developed a heuristic approach to determine EOQ in the general case of a time-varying demand rate. Donaldson (1977) came out with a full analytic solution of the inventory replenishment problem with a linear trend in demand over a finite-time horizon. The discrete version of this problem was discussed by Wagner and Whitin (1958). Other noteable works in this direction came from Ritchie, 1980, Ritchie, 1984, Ritchie, 1985, Kicks and Donaldson, 1980, Buchanan, 1980, Mitra et al., 1984, Ritchie and Tsado, 1986, Goyal, 1986, Goyal et al., 1986, among others. All these models were developed on the assumption of no shortages in inventory. Deb and Chaudhuri (1987) were the first to extend the model of Donaldson (1977) to incorporate shortages in inventory. This extension was further studied by Goyal, 1988, Dave, 1979, Dave, 1988, Murdeshwar, 1988. Inventory models with a linear trend in demand incorporating shortages were studied by Hariga, 1996, Goyal et al., 1992, Goswami and Chaudhuri, 1991, Giri et al., 2000, Teng, 1996, etc. Hariga and Bankherouf (1994) discussed some optimal and heuristic replenishment models for deteriorating items with an exponentially time-dependent demand without allowing shortages in inventory. Wee (1995) studied an EOQ model with shortages, assuming that the demand declines exponentially over time. Sana and Chaudhuri (2000) developed an EOQ model over a finite-time horizon for perishable items, considering unequal cycle lengths. Khanra and Chaudhuri (2003) developed an EOQ model with shortages over a finite-time horizon, assuming a quadratic demand pattern.
It is a common belief that large piles of goods displayed in a supermarket will lead the customer to buy more. According to Levin et al. (1972), one of the functions of inventories is that of a motivator, as indicated in the statement that ‘At times, the presence of inventory has a motivational effect on the people around it’. Silver and Peterson (1985) have also noted that sales at the retail level tend to be proportional to inventory displayed. Gupta and Vrat, 1986, Baker and Urban, 1988, Mondal and Phaujdar, 1989a, Mondal and Phaujdar, 1989b, Datta and Pal, 1990, Urban, 1992, Pal et al., 1993 have observed that sales of some items at the retail level is directly related to the amount of inventory displayed. Recently, Urban (2005) conducted a comprehensive review of this literature, distinguishing between Type I models in which the demand rate of an item is a function of the initial inventory level and Type II models in which it is dependent on the instantaneous inventory level. A periodic-review model is then developed, first solving the general Type I problem, then illustrating how the more complex Type II model can be solved.
In a competitive market, price of goods plays an important factor to a customer. Generally, a reduced price encourages a customer to buy more. In this point of view, Abad, 1988, Kim and Hwang, 1988, Hwang et al., 1990, Burwell et al., 1991 developed the traditional quantity discount models. Urban and Baker (1997) generalized the EOQ model in which the demand is a multivariate function of price, time and level of inventory. Datta and Pal (2001) analyzed a multi-period EOQ model with stock-dependent and price-sensitive demand rate. Recently, Teng and Chang (2005) extended an EPQ (economic production quantity) model for perishable items, considering the demand rate as the sum of two terms: first term is inversely proportional to the price and second term is directly proportional to the stock-level of inventory displayed.
The traditional EOQ model considers that retailer pays purchasing cost for the items as soon as the items are received. In practice, the supplier offers the retailer a delay period, known as trade credit period, in paying for purchasing cost. During trade credit period, the retailer can accumulate revenues by selling items and by earning interests. In a competitive market, the supplier offers different delay periods with different price discounts to encourage the retailer to order more quantities. Goyal (1985) is the first person who developed the EOQ model under conditions of permissible delay in payments. Shah et al. (1988) studied the same model, incorporating shortages. Mondal and Phaujdar (1989c) studied the same situation by considering the interest earned from the sales revenue. Shah, 1993a, Shah, 1993b also developed EOQ models for perishable items where delay in payment is permissible. Shah and Sreehari (1996) extended an EOQ model when the delay in payment is permitted and the capacity of own ware-house is limited. Other noteable works in this direction are those of Aggarwal and Jaggi, 1995, Chang and Dye, 2001, Chang and Dye, 2002, Chang et al., 2001, Chen and Chuang, 1999, Chu et al., 1998, Chung, 1998a, Chung, 1998b, Chung, 2000, Jamal et al., 1997, Jamal et al., 2000, Liao et al., 2000, Sarker et al., 2000, Sarker et al., 2001, Salameh et al., 2003, Chung and Huang, 2003, etc.
In this paper, we consider an EOQ model for various types of deterministic demand (namely, constant, linear increasing demand, quadratic increasing demand, exponential time varying demand, stock-dependent demand, price-dependent demand, both price and stock-dependent demand) when delay in payment is permitted by retailer to supplier . Moreover, ‘’ is introduced in the proposed model meaning thereby that the supplier offers percent discount on the purchasing cost to the retailer, when delay period is .
Section snippets
Assumptions
- (i)
The inventory system involves a single type of items.
- (ii)
Shortages are not permitted.
- (iii)
Replenishment is instantaneous.
- (iv)
Lead time is neglected.
- (v)
Permissible delay in payment to the supplier by the retailer is considered. The supplier offers different discount rates of price at different delay periods.
- (vi)
Planning horizon is infinite.
Notations
varying demand rate
on-hand inventory at time ‘t’ ⩾ 0’
replenishment lot size of a cycle
ordering cost per order
holding cost per unit time, excluding interest charges
Formulation of the model
The cycle starts with initial lot-size and ends with zero inventory at time . Then the governing differential equation of the on-hand inventory isThe purchasing costs at different delay periods arewhere Cr = maximum retail price per unit, Mi (i = 1, 2, 3) = decision point in settling the account to the supplier at which supplier offers % discount to the retailer. M3 is the maximum delay period after which the
Numerical examples
Example A1 We consider the values of the parameters as follows: , per order, , , , , , , , year, , , . Then the optimal solutions are: , , , ,
Conclusion
A widespread approach to inventory modeling is to associate costs and profits with measures of system performance and determine the control policy which maximizes the average profit per unit time. In this paper, we analyze the EOQ model of deterministic demand (namely, constant, linearly varying with time, quadratically varying with time, exponentially varying with time, stock-dependent, price-dependent, both stock and price-sensitive) for a retailer where supplier’s trade offer gives the
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