Research Article  Open Access
Biswajit Sarkar, Shib Sankar Sana, Kripasindhu Chaudhuri, "An Inventory Model with Finite Replenishment Rate, Trade Credit Policy and PriceDiscount Offer", Journal of Industrial Engineering, vol. 2013, Article ID 672504, 18 pages, 2013. https://doi.org/10.1155/2013/672504
An Inventory Model with Finite Replenishment Rate, Trade Credit Policy and PriceDiscount Offer
Abstract
When some suppliers offer trade credit periods and price discounts to retailers in order to increase the demand of their products, retailers have to face different types of discount offers and credits within which they have to take a decision which is the best offer for them to make more profit. The retailers try to buy perfectquality items at a reasonable price, and also they try to invest returns obtained by selling those items in such a manner that their business is not hampered. In this point of view, we consider an economic order quantity (EOQ) model for various types of timedependent demand when delay in payment and price discount are permitted by suppliers to retailers. The models of various demand patterns are discussed analytically. Some numerical examples and graphical representations are considered to illustrate the model.
1. Introduction
Many classical inventory models assume that demand is constant. In present marketing environment, few items follow constant demand. Many productâ€™s demands follow variable timevarying demand. The recent trend of the marketing system is to provide more buy opportunities to the retailer by the supplier by offering different discounts. To take the discount opportunity, retailers prefer to buy more beyond their capacity of buying. As a result, the supplier has the opportunity to sell more for better earning. This is the benefit of the supplier. The classical inventory model does not consider the delay time concept or variable demand. The proposed model considers timevarying demand and delay in payments along with finite replenishment rate.
The basic wellknown square root formula for the EOQ of the item was formulated by Harris [1] based on constant demand. Donaldson [2] extended the constant demand to linear timedependent demand model analytically with finite time horizon. Following Donaldson [2], significant contribution in this direction came out from researchers like Goyal [3], Goswami and Chaudhuri [4], Goyal et al. [5], and others. Hariga and Benkherouf [6] discussed an optimal and heuristic replenishment model for deteriorating items with an exponentially timevarying demand. Wee [7] studied a deterministic lot size inventory model for deteriorating items with shortage and decline market. Khanra and Chaudhuri [8] extended an inventory model with quadratic increasing demand over a finite time horizon and shortages. Sana and Chaudhuri [9] studied an inventory model with linear trend demand incorporating shortages. CÃ¡rdenasBarrÃ³n [10] discussed the derivation of inventory models by using analytic geometry and algebra. Sarkar et al. [11] explained an inventory model with quadratic timevarying demand by considering EulerLagrange method.
It is common to all that every customer prefers to buy more at reduced price. Some researchers like Abad [12], and Kim and Hwang [13] developed the traditional quantity discount model. In the traditional EOQ model, it was assumed that the retailer pays the purchasing cost when he received the items from a supplier. In tradecredit policy, the supplier allows a certain fixed period to pay the purchasing cost. This fixed period which is settled by the supplier is called the credit period to the retailer. During this credit period, the supplier sells items to the retailer with different types of discounts to obtain more profit as early as possible during the credit period. Depending on this policy, Goyal [14] first developed an inventory model with permissible delay in payments. Aggarwal and Jaggi [15] developed an inventory model with an exponentially deteriorating rate by considering permissible delay in payments. Chu et al. [16] extended Goyalâ€™s [14] model by considering the case of deterioration. Jamal et al. [17] extended an inventory model with shortages.
Teng [18] developed an EOQ model for a retailer to order small lot size in order to take the benefit of permissible delay in payments. Arcelus et al. [19] developed an inventory model by considering the retailerâ€™s maximizing profit and inventory policies for vendorâ€™s trade promotion offer of price/credit on the purchase of perishable items. Huang [20] extended an inventory model of retailerâ€™s inventory system as a cost minimization model to determine the retailerâ€™s optimum inventory cycle time and optimal order quantity. Huang [21] developed an economic production quantity (EPQ) model of retailerâ€™s inventory system to investigate the optimal retailerâ€™s decisions under two levels of trade credit policy. CÃ¡rdenasBarrÃ³n [22] extended optimal ordering policies in response to a discount offer. Teng et al. [23] explained optimal ordering decisions with returns and excess inventory. Sarkar [24] discussed an inventory model with delay in payments in the presence of imperfect production. Sarkar [25] developed an inventory model with delay in payments and timevarying deterioration rate. Forghani et al. [26] explained an inventory model in the single period inventory system with price adjustment.
This paper considers an inventory model for credit periods and pricediscount offers with different types of time varying demand and constant supply rate up to time . During , inventory piles up by adjusting the demand. The accumulated inventory level at time depletes gradually to meet the demand, and the level reaches zero level at time . The agreement between the supplier and the retailer is such that total purchasing cost of whole amount would be paid within the time with purchasing cost at discount rate . The different delay periods with different discount rates on the purchasing cost are permitted by the supplier to the retailer. During the credit period, the retailer can earn interest by selling items whereas interest of purchasing cost is charged against the delay of excess time of credit of payment period by the retailer to the supplier.
The rest of the paper is designed as follows. The mathematical model is presented in Section 2. In Section 3, numerical examples are given. Finally, concluding remarks are explained in Section 4. See Table 1 for the comparison of this model with previous works.

2. Mathematical Model
We consider the following notation to develop the model.: the optimal length of inventory cycle (decision variable):the optimal duration of replenishment (decision variable):onhand inventory at time :onhand inventory at time :timevarying demand rate:constant replenishment/supply rate: variable delay period:th permissible delay period:discount rate on purchasing cost at th permissible delay period:ordering cost per order:unit holding cost per unit time, excluding interest charge:purchasing cost per unit:maximum retail price per unit:selling price per unit: rate of interest gaining due to the credit balance:rate of interest due to financing inventory:length of the inventory cycle: duration of the replenishment:average profit of the system when :average profit of the system when .
The following assumptions are considered to develop this model.(1)The inventory system involves only single type of product.(2)The demand rate is constant or time dependent (quadratic, linear, and exponential).(3)Different discount rates on the purchasing cost for different delay periods are considered.(4)Replenishment rate is instantaneously infinite, but its size is finite.(5)Time horizon is infinite.(6)Lead time is negligible.(7)Neither shortage nor backlogging is considered.
The cycle starts with zero inventory at supply rate . The replenishment or supply continues up to time . During the time span , inventory piles up by adjusting the demand in the market. This accumulated inventory level at time depletes gradually to meet the demand and it reaches zero level at time . Generally, the supplier offers delay period to the retailer to pay the total purchasing cost of items. For different delay periods , different discounts of purchasing cost are offered to the retailer by the supplier. In this direction, we consider the purchasing cost of different delay periods as follows: where s are the th permissible delay to settle the purchasing cost at which the discount rate to the retailer is . Also tends to be at . That is, at infinite purchasing cost, the retailer never purchases any item from the supplier. Indirectly, the supplier would not supply the product to the retailer while delay period exceeds . In our model two cases may arise.
Case 1 (). That is, inventory cycle length is larger or equal to the credit period (see Figure 1). When , there are some profits based on credit balance during the delay period and there is some interest charged due to financing inventory during .
Case 2 (). That is, inventory cycle length is smaller or equal to the credit period (see Figure 2). When , there are some profits based on credit balance during the delay period and there is no interest charged due to financing inventory.
The governing differential equations of this model are
From (2), we obtain
By utilizing the continuity at , , we have
that is,
Now we formulate Cases 1 and 2 using (3) to (6).
Case 1 (). When the inventory cycle is larger or equal to the credit period , the holding cost excluding the interest charges is .
The profit gains due to credit balance during the delay period are .
The interest charged for financing inventory during is .
Therefore, the total profit is (see Figure 1)
Hence, the average profit is
Case 2 (). When the inventory cycle () is smaller or equal to the credit period , the holding cost excluding the interest charges is .
The profit gains due to credit balance during the time are .
Therefore, the total profit is (see Figure 2)
Hence, the average profit is
Now our objective is to maximize for and obtain the optimal replenishment period and inventory cycle length . We discuss various timedependent or constant demands by utilizing this general formula.
2.1. Quadratic Demand Pattern
The quadratic timedependent demand is of the form where , , and . This trend of demand is applied to the products like essential commodities and seasonal goods.
From (2), we have Using these and in (8) and (10), we obtain Using (6) in the above objective functions, we obtain
To obtain the optimal cycle length, we construct two lemmas as follows.
Lemma 1. When exists for , then has a maximum value at if . Otherwise, has a maximum value at unique if holds.
Proof. See Appendix A.
Lemma 2. When exists for , then has a maximum value at if . Otherwise, has a maximum value at if .
Proof. See Appendix A. â€‰
2.2. Linear TimeDependent Demand Pattern
We consider linear timedependent demand which can be obtained if we substitute in the form of quadratic demand. We get , , (see for instances Donaldson [2], Goyal [3], and Goswami and Chaudhuri [4]). Generally, in some computer games, computer android applications, we obtain this type of linear timedependent demand.
Now from (2), we have and , respectively.
Using and in (8) and (10), we obtain
Using (6) in the above objective functions, we have
To obtain the optimal cycle length, we construct the following lemmas.
Lemma 3. When exists for , then has a maximum value at if . Otherwise has a global maximum value at if holds.
Proof. See Appendix B.
Lemma 4. When exists for , then has a maximum value at if . Otherwise has a maximum value at if .
Proof. See Appendix B. â€‰
2.3. Constant Demand Pattern
We consider the demand as constant which can be found by substituting in the linear demand. We obtain , . See for instances Harris [1] and CÃ¡rdenasBarrÃ³n [10, 22]. This type of demand is usually found in productâ€™s life cycle.
Now from (2), we have and , respectively.
Using and in (8) and (10), we get respectively. Using (6) in the above functions, we obtain To obtain optimal cycle length, we formulate the following lemmas.
Lemma 5. If the conditions and hold, then has a global maximum at .
Proof. See Appendix C.
Lemma 6. If hold, then has a global maximum at .
Proof. See Appendix C. â€‰
2.4. Exponential Demand Pattern
We consider another important type of demand as exponential type, that is, varies exponentially with time . In this case, the demand rate increases very fast as in seasonal goods, new computer parts like RAM and data storage device. For this type of demand rate, we consider where , .
From (2), we have Using these and in (8) and (10), we obtain From (6), we have
Substituting this in and , we have
Lemma 7. When exists for , then has a maximum value at if . Otherwise, has a unique maximum value , if .
Proof. See Appendix D.
Lemma 8. When exists for , then has a maximum value at if . Otherwise, has a maximum value , if
Proof. See Appendix D.â€‰
3. Numerical Examples
Exampleâ€‰â€‰A. We consider the following parametric values in appropriate units: per order, , year, year, year, , , , units, units, units, , , , , , and units.
Then the optimal solutions are , year, year}, , year, year},â€‰â€‰, year, year}, , year, year}, , year, year}, , year, year}.
Among the above optimal solutions, the optimal solution is , year, year} global maximum. The average profit function is highly nonlinear. Figure 3 shows the concavity of the functions and comparisons of the cost functions.
Exampleâ€‰â€‰B. We consider the following parametric values in appropriate units: per order, , year, year, year, , , , units, units, , , , , , and units.
Then the optimal solutions are , year, year}, , year, year}, , year, year}, , year, year}, , year, year}, and , year, year}.
Among the above optimal solutions, the optimal solution is , year, year} global maximum. The average profit function is highly nonlinear. Figure 4 shows the concavity of the functions and comparisons of the cost functions.
Exampleâ€‰â€‰C. We consider the following parametric values in appropriate units: per order, , year, year, year, , , , units, , ,â€‰â€‰,â€‰â€‰,â€‰â€‰, and .
Then the optimal solutions are , year,â€‰â€‰year}, , year,â€‰â€‰ year},â€‰â€‰, year,â€‰â€‰year}, , year,â€‰â€‰year}, , year,â€‰â€‰year}, and , year,â€‰â€‰year}.
From the above optimal solutions, the optimal solution is , year,â€‰â€‰year} global maximum. We obtain a closed type formula for , and Figure 5 shows the concavity of the functions and comparisons of the cost functions.
Exampleâ€‰â€‰D. We consider the following parametric values in appropriate units: per order, , month, â€‰â€‰months, â€‰â€‰months, , , , units, units, , , , , , and units.
Then the optimal solutions are , â€‰â€‰months, month}, , â€‰â€‰months, month}, , â€‰â€‰months, month}, , month, month}, , â€‰â€‰months, month}, and , â€‰â€‰months, month}.
From the above optimal solutions, the optimal solution is , â€‰â€‰months, month} global maximum. The average profit function is highly nonlinear. Figure 6 shows the concavity of the functions and comparisons of the cost functions.
Sensitivity Analysis. The sensitivity analysis of the key parameters is given in Table 2 for quadratic demand, Table 3 for linear demand, Table 4 for constant demand, and Table 5 for exponential demand. (i)If ordering cost increases, then material handlling cost, shipping cost, and placing orderâ€™s cost increase; as a result the total relevant profit decreases. (ii)If the unit holding cost per unit item increases, the total profit of the system decreases. (iii)Increasing value of maximum retailer price increases the total purchasing cost of the whole system which decreases the total profit. (iv)Increasing value of selling price per item increases the total selling price per lot which reduces the total cost of the whole system. Therefore, the total profit of the system increases. (v)If we increase the replenishment rate then the holding cost increases which results the decreasing value of the total profit.