EOQ Model for Deteriorating Items with Stock-Level-Dependent Demand Rate and Order-Quantity-Dependent Trade Credit

This paper develops a generalized inventory model for exponentially deteriorating items with current-stock-dependent demand rate and permissible delay in payments. In the model, the payment for the item must be made immediately if the order quantity is less than the predetermined quantity; otherwise, a fixed trade credit period is permitted.Themaximization of the average profit per unit of time is taken as the inventory system’s objective.The necessary and sufficient conditions and some properties of the optimal solution to themodel are developed. Simple solution procedures are proposed to efficiently determine the optimal ordering policies of the considered problem. Numerical example is also presented to illustrate the solution procedures obtained.


Introduction
In classical inventory models, the demand rate of items was often assumed to be either constant or time dependent.However, many marketing researchers and practitioners have recognized that the demand rate of many retail items is usually influenced by the amount of inventory displayed.For example, Whitin [1] stated, "For retail stores the inventory control problem for style goods is further complicated by the fact that inventory and sales are not independent of one another.An increase in inventories may bring about increased sales of some items." Levin et al. [2] and Sliver and Peterson [3] also observed that sales at the retail level tend to be proportional to the inventory displayed and that large piles of goods displayed in a supermarket will lead customers to buy more.The reason behind this phenomenon is a typical psychology of customers.They may have the feeling of obtaining a wide range for selection when a large amount is stored or displayed.Conversely, they may have doubt about the freshness or quality of the product when a small amount is stored.Based on the observed phenomenon, it is clear that in real life the demand rate of items may be influenced by the stock levels.Many researchers have developed lots of inventory models to cover this phenomenon.Gupta and Vrat [4] were the first to develop an inventory model with the initial-stock-dependent consumption rate.Padmanabhan and Vrat [5] proposed an EOQ model for initial-stock-dependent demand and exponential decay items.Giri et al. [6] presented a model under inflation for initial-stock-dependent consumption rate and exponential decay.In addition, Baker and Urban [7] investigated another kind of inventory model in which the demand of item is a polynomial function of the instantaneous stock level.Mandal and Phaujdar [8] developed an inventory model for deteriorating items with linearly current-stock-dependent demand.Pal et al. [9] extended the model of Baker and Urban [7] to the case of deteriorating items.Padmanabhan and Vrat [10] further developed three models for deteriorating items under current-stock-level-dependent demand rate: without backorder, with complete backorder, and with partial backorder.Chung et al. [11] modified Padmanabhan and Vrat's [10] models by showing the necessary and sufficient conditions of the existing optimal solution for models with no backorder and complete backorder.Zhou and Yang [12] developed an inventory model for a general inventory-level-dependent demand with limited storage space.Moreover, many further extensive models were developed by researchers such as Balkhi and Benkherouf [13], Dye and Ouyang [14], Zhou and Yang [15], Wu et al. [16], and Alfares [17].
In all the above inventory models with stock-leveldependent demand rate assumed that the retailer must pay for the items as soon as the items are received.However, in the real life, the supplier sometimes will offer the retailer a delay period, that is, a trade credit period, in paying for the amount of purchasing cost.Before the end of the trade credit period, the retailer can sell the goods and accumulate the revenue to earn interest.A higher interest is charged if the payment is not settled by the end of trade credit period.In real world, the supplier often makes use of this policy to promote their commodities.During the recent two decades, the effect of a permissible delay in payments on the optimal inventory systems has received more attention from numerous researchers.Goyal [18] explored a single item EOQ model under permissible delay in payments.Aggarwal and Jaggi [19] extended Goyal's [18] model to the case with deteriorating items.Jamal et al. [20] further generalized the above inventory models to allow for shortages.Other related articles can be found in Sarker et al. [21], Teng [22], Huang [23], Chang et al. [24,25], Chung and Liao [26], Teng et al. [27], Liao [28] and Teng and Goyal [29], and their references.More recently, Chen et al. [30] studied an economic order quantity model under conditionally permissible delay in payments, in which the supplier offers the retailer a fully permissible delay periods if the retailer orders more than or equal to a predetermined quantity, otherwise partial payments will be provided.Chang et al. [31] developed an appropriate inventory model for noninstantaneously deteriorating items where suppliers provide a permissible payment delay schedule linked to order quantity.However, all of the abovementioned papers with trade credit financing did not address inventory-level-dependent demand rate.
The present paper establishes an EOQ model for deteriorating items with current-inventory-level-dependent demand rate and permissible delay in payments which is dependent on the retailer's order quantity.That is, if the order quantity is less than the predetermined quantity, the payment for the item must be made immediately; otherwise, a fixed trade credit period is permitted.We then establish the proper mathematical model and study the necessary and sufficient conditions of the optimal solution to the model.Some properties of the optimal solution to the model are proposed for obtaining the optimal replenishment quantity and replenishment cycle, which make the average profit of the system maximized.
The remainder of the paper is organized as below.The next section introduces some notations and assumptions used in this paper.We then present in Section 3 the formulation of the inventory model with current-stock-leveldependent demand rate and order-quantity-dependent trade credit.Section 4 examines the existence and uniqueness of the optimal solution to the considered inventory system and shows some properties of the optimal ordering policies.In Section 5, numerical example is given to illustrate the solution procedure obtained in this paper.Some conclusions are included in Section 6.

Notations and Assumptions
The following notations and assumptions are used in the paper.

Notations
: ordering cost one order in dollars, : purchase cost per unit in dollars, : selling price per unit in dollars ( > ), ℎ: stock holding cost per year per unit in dollars (excluding interest charges), : deteriorating rate of items (0 <  < 1),   : the qualified quantity in units at or beyond which the delay in payments is permitted,   : the time interval in years in which   units are depleted to zero due to both demand and deterioration,   : interest which can be earned per $ per year by the retailer,   : interest charges per $ in stocks per year by the supplier, : the retailer's trade credit period offered by supplier in years, : inventory cycle length in years (decision variable), : the retailer's order quantity in units per cycle, Π(): the retailer's average profit function per unit time in dollars.

Assumptions
(1) The demand rate () is a known function of retailer's instantaneous stock level (), which is given by where  and  are positive constants.
(2) The time horizon of the inventory system is infinite.
(3) The lead time is negligible.
(4) Shortages are not allowed to occur.
(5) When the order quantity is less than the qualified quantity   , the payment for the item must be made immediately.Otherwise, the fixed trade credit period  is permitted. (

Mathematical Formulation of the Model
Based on the above assumptions, the considered inventory system goes like this: at the beginning (say, the time  = 0) of each replenishment cycle, items of  units are held, and the items are depleted gradually in the interval [0, ] due to the combined effects of demand and deterioration.At time  = , the inventory level reaches zero, and the whole process is repeated.
The variation of inventory level, (), with respect to time can be described by the following differential equation: with the boundary condition () = 0.
The solution to the above differential equation is So the retailer's order size per cycle is We observe that if the order quantity  <   , then the payment must be made immediately.Otherwise, the retailer will get a certain credit period, .Hence, from (4) one has the fact that the inequality  <   holds if and only if  <   , where   is given as below: The elements comprising the profit function per cycle of the retailer are listed below.(e) According to the above assumption, there are two cases to occur in interest payable in each order cycle.
Case 1 ( ≤   ).In this case, the delay in payments is not permitted.
(f) Similarly, there are also two cases to occur in interest earned in each order cycle.
Case 1 ( ≤   ).In this case, the delay in payments is not permitted, so the interest earned in each order cycle = 0.
Case 2 ( ≥   ).In this case, the delay in payments is permitted, which causes two subcases as below.
Subcase 2.2 ( ≥ ).In this case, the delay in payments is permitted, and during time 0 through  the retailer sells the goods and continues to accumulate sales revenue and earns the interest with rate   .Therefore, the interest earned from time 0 to  per cycle From the above, the profit function for the retailer can be expressed as On simplification and summation, we get the profit function as the following three forms: (C) if  >   and  > , then where the meaning of notations  and  in the above expressions is For convenience, we will consider the problem through the two following situations: (1)  ≥   and (2)  <   .
(1) Suppose That  ≥   .In the situation of  ≥   , Π() has three different expressions as follows: It is easy to verify that, in the case of  ≥   , Π() (2) Suppose That  <   .If  <   , then Π() has two different expressions as follows: Here, Π 1 () and Π 3 () are given by ( 11) and ( 13), respectively.Π() continues except at  =   .In fact, we could prove that In the next section, we will determine the retailer's optimal cycle time  * for the above two situations using some algebraic method.
Then we have the following lemma to describe the property of Π 1 () on (0, +∞).
Similarly, the first-order necessary condition for Π 2 () in ( 12) to be maximized is Π 2 ()/ = 0, which leads to We set the left-hand expression of (20) as  2 ().Taking the first derivative of  2 () with respect to  gives Consequently, from the sign of  there are two distinct cases for finding the property of Π 2 () as follows.
Then we have the following lemma to describe the property of Π 2 () on (0, +∞).
Likewise, the first-order necessary condition for Π 3 () in (13) to be maximum is Π 3 ()/ = 0, which implies that  ( (+) − 1) We set the left-hand expression of (24) as  3 ().Taking the first derivative of  3 () with respect to  gives We consider the following two cases.
Summarizing the above arguments, the following lemma to describe the property of Π 3 () on (0, +∞) will be obtained.
Based upon Lemmas 1-3, we now will decide the optimal solution  * to Π() from the below two situations.
For convenience, we rewrite the notation  as  =  −   +   /( + ) −   , and the following two theorems would be obtained to decide the optimal solution  * when  ≥   .
1 or   associated with maximum profit.
Proof.See Appendix E for detail.
Proof.See Appendix E for detail.

Decision Rule of the Optimal
Cycle Time  * When  <   .When  <   , the piecewise profit function Π() has only two different expressions, that is, Π 1 () and Π 3 (), respectively.From ( 24), we have As shown in Appendix D, we will find that  1 (  ) >  3 (  ).The following theorem would be given to decide the system's optimal cycle time  * when  <   .Theorem 6.If  <   , then one has the following results.
associated with the maximum profit.
Proof.See Appendix F for detail.

Numerical Example
In this section, we will provide the following numerical example to illustrate the present model.
Next, we further study the effects of changes of parameters , ,   , and  on the optimal solutions.The values of other parameters keep the same as in the above example when each of parameters , ,   , and  varies.Table 1 presents the observed results with various parameters , ,   , and .
The following inferences can be made based on Table 1.
(1) The retailer's optimal order quantity  * and the optimal average profit Π * increase as the value of  increases.It implies that when the market demand is more sensitive to the inventory level, the retailer will increase his/her order quantity to make more profit under permissible delay in payments.
(2) The optimal order quantity  * and the maximum average profit Π * decrease as the value of  increases.
(3) When   increases, the optimal order quantity  * and the optimal cycle time  * are increasing but the optimal average profit Π * is decreasing.
(4) The retailer's order quantity  * is increasing as  increases.This verifies the fact that the retailer could indeed increase the sales quantity by adopting the trade credit policy provided by his/her supplier.

Conclusions
In this paper, we develop an EOQ model for deteriorating items under permissible delay in payments.The primary difference of this paper as compared to previous related studies is the following four aspects: (1) the demand rate of items is dependent on the retailer's current stock level, (2) many items deteriorate continuously such as fruits and vegetables, (3) the retailer who purchases the items enjoys a fixed credit period offered by the supplier if the order quantity is greater than or equal to the predetermined quantity, and (4) we here use maximizing profit as the objective to find the optimal replenishment policies, which could better describe the essence of the inventory system than a cost-minimization model.In addition, we have discussed in detail the conditions of the existence and uniqueness of the optimal solutions to the model.Three easy-to-use theorems are developed to find the optimal ordering policies for the considered problem, and these theoretical results are illustrated by numerical example.By studying the effects of , ,   , and  on the optimal order quantity  * and the optimal average profit Π * , some managerial insights are derived.
The presented model can be further extended to some more practical situations.For example, we could allow for shortages, quantity discounts, time value of money and inflation, price-sensitive and stock-dependent demand, and so forth.From ( 11) and ( 13), we get From ( 26) and ( 24), we have

E. Proof of Theorems 4 and 5
Before the proof of Theorems 4 and 5, according to the anterior analysis in Section 4, we can get the following results.

Subcase 2 . 1 (
≤ ).Since the cycle time  is shorter than the credit period , from time 0 to  the retailer sells the goods and continues to accumulate the sales revenue to earn interest   ∫  0 ∫  0 () , and from time  to  the retailer can use the sales revenue generated in [0, ] to earn interest   ∫  0 ()( − ).Therefore, the interest earned from time 0 to  per cycle ) If the trade credit period  is offered, the retailer would settle the account at  =  and pay for the interest charges on items in stock with rate   over the interval [, ] as  ≥ , whereas the retailer settles the account at  =  and does not need to pay any interest charge of items in stock during the whole cycle as  ≤ .

Table 1 :
The impact of change of , ,   , and  on the optimal solutions.