An Inventory Model for Perishable Products with Stock-Dependent Demand and Trade Credit under Inflation

1 Graduate School of Information Management and Security, Korea University, Anam-dong 5-ga, Seongbuk-gu, Seoul 136-713, Republic of Korea 2 School of Industrial Management Engineering and Graduate School of Management of Technology, Korea University, Anam-dong 5-ga, Seongbuk-gu, Seoul 136-713, Republic of Korea 3 Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, BC, Canada V6T 1Z2


Introduction
After the global economic crisis, developing countries (and even some developed countries) have suffered from large scale inflation.Meanwhile, the inflation in food market is especially severe.For example, according to the BBC Online, the global grain price increased 10% in July 2012 because of the hot and dry weather in the USA and Eastern Europe.Since the demand for food is quite rigid, the inflation increases the poverty level in developing countries.Purchase with large amount may reduce the cost and bring about more sales, but it may increase the market risk and spoilage due to the characteristic of perishable products.In traditional EOQ model, the payment time does not affect the profit and replenishment policy.If the inflation is considered, the order quantity and payment time can influence both the supplier and retailer's decisions.The supplier may offer a credit period to promote the market, and the retailer may order more since the funding pressure is less.Therefore, we should consider all these factors in order to make a reasonable replenishment policy under inflation.This research tries to determine the optimal order quantity for the inventory management of perishable products under inflation when the supplier offers a credit period.
Traditional inventory models assume that the demand rate is independent of the inventory level.However, the observations in supermarkets show that a large pile of goods induce consumers to buy more.This occurs because larger stockpiles receive more visibility; especially for some perishable food (e.g., vegetables, fruits, and bread), high inventory may also suggest that they are fresh and popular.Silver and Peterson [1] observed that the consumption rate could be proportional to the stock level displayed.Later, Baker and Urban [2] considered a more practical assumption that the demand rate is a polynomial function of instantaneous stock level.Sana and Chaudhuri [3] considered a deterministic EOQ model with delays in payments and price discount offers, incorporating stock-dependent demand and other demand patterns.Since the perishable products may deteriorate over time, Mandal and Phaujdar [4] presented an inventory model for perishable Mathematical Problems in Engineering products with a constant deterioration rate, assuming that the demand rate is a linear function of instantaneous stock level.Some of the relevant works for inventory management of deteriorating and stock-dependent demand items are by Sana [5], Padmanabhan and Vrat [6], Mandal and Maiti [7], Dye and Ouyang [8], Chang et al. [9], and others.
Moreover, inflation and time value of money are ignored because they may not influence the inventory policy significantly.However, after global financial crisis, many countries have suffered from large scale inflation.Therefore, inflation and time value of money should be considered when the inventory policy is made.The pioneer researcher in this area was Buzacott [10], who developed an EOQ model under inflation subject to different types of pricing policies.Vrat and Padmanabhan [11] considered an inventory model with initial-stock-dependent demand under a constant inflation rate.Bose et al. [12] presented an inventory model for deteriorating items with time-dependent demand and shortages under inflation and time discounting.Recently, Sarkar et al. [13] developed an economic manufacturing quantity model for an imperfect production system with inflation and time value of money to determine the optimal production reliability and production rate that maximize the profit.Sana [14] proposed a control policy for a production system under inflation, assuming that the demand is dependent on the stock and the sales team's promotional effort.Some related works can be found in Chung and Lin [15], Hou [16], Jaggi et al. [17], Roy et al. [18], and others.When the inflation is considered, the credit period may be employed in replacement of price discounts or financial service and also affect the replenishment policy.Therefore, some suppliers are willing to offer a credit period to promote the market competition.Chang [19] developed an inventory model for deteriorating items under inflation and time discounting, assuming that the supplier offers a trade credit if the retailer's order size is larger than a certain level.
Although there are many exiting research works on the inventory management of perishable products, few papers have discussed the inventory management for perishable products with stock-dependent demand under inflation and time discounting.This paper deals with this problem, and it provides an optimal solution to minimize the retailer's present value of cost.

Model Formulation
This section introduces the assumptions, notations, and mathematical formulation used in our perishable product inventory model.

Assumptions.
The following assumptions are used throughout the paper.
(1) The supplier sells one single item to the retailer in quantity.(2) The items are replenished when the stock level becomes zero.(3) The supplier provides a credit period, which is dependent on the order quantity.
(4) The deteriorating rate is constant.The deteriorating items cannot be repaired and the salvage value is zero.(5) The demand rate () at the retailer's end is dependent on the instantaneous inventory level (), which means () =  + (),  ≥ 0, 1 ≥  ≥ 0. (6) The lead time is zero and shortages are not allowed.(7) The planning horizon of the inventory system is finite.
The number of cycles must be integer in the planning horizon.

Mathematical
By integrating both sides of (1) with respect to , With the boundary condition () = 0, the solution of the integral equation is With (0) = , we get On the above assumptions, there are two scenarios to arise: Scenario A,  ≥ ; Scenario B,  < .

Scenario A:
When  ≥ .Since the credit period  is longer than the replenishment cycle length , the retailer can sell all the items before the end of credit period, as shown in Figure 1.Therefore, there is no interest charged by the supplier.The elements of the retailer's cost are as follows: ordering cost, purchasing cost, and holding cost.Since the replenishment is made at the beginning of each cycle, the present value of the ordering cost during the first cycle is .
The purchasing cost is paid at the end of credit period ; the present value of the purchasing cost during the first cycle is Because the holding cost occurs all over the replenishment cycle, the present value of the holding cost during the first cycle is Therefore, the net present value of the cost during the first cycle is As shown in Figure 2, there are  cycles in the planning horizon.Therefore, the present value of the total cost over the planning horizon  is Proposition 1.When there exists a unique  * at which Proof.Let   1 () and   1 () represent the first and the second derivatives of  1 () with respect to , respectively.Taking the first derivative of  1 () with respect to , Let ) > 0, and they are decreasing on .

Scenario B: When 𝑀 < 𝑇
Case 1. Letting   ∫  0 () −  ≥ , which means at time , the revenue earned is more than the purchasing cost, then the revenue is enough to pay the purchasing cost.There is no interest charged by the supplier, although the credit period  is shorter than the replenishment cycle length , as shown in Figure 3.
The objective function is the same with that under Scenario A: Case 2. Let   ∫  0 () −  < , but there is budget.That means, although the revenue earned by time  is less than the purchasing cost, there is budget to pay the short purchasing cost at time .Therefore, there is still no interest charged by the supplier.
The objective function is the same with that under Scenario A: Case 3. Let   ∫  0 () −  < , and there is no budget.As a result, all the revenue earned by time  is used to pay the purchasing cost and the supplier charges interest rate   from  to  for the unpaid balance.The interest and the remaining payments should be made at the end of the replenishment cycle .Therefore, there are four elements in the retailer's cost: ordering cost, holding cost, the purchasing cost paid at time , and the interest and the remaining payments made at the end of replenishment cycle .Two of the elements are different from Scenario A: the purchasing cost paid at time  and the interest and the remaining payments made at the end of the replenishment cycle .The present value of the purchasing cost paid at time  during the first cycle is equal to the present value of the revenue earned by time : The present value of the remaining payments and interest paid at the end of the replenishment cycle during the first cycle is The net present value of the cost during the first cycle is The present value of the total cost over the planning horizon  is Proposition 2. When there exists a unique  * at which  2 ()/| = * = 0 for (, +∞), then  2 () is mini- Proof.Let   2 () and   2 () represent the first and the second derivatives of  2 () with respect to , respectively.Taking the first derivative of  2 () with respect to , Letting Taking the first derivative of  2 () with respect to , we get Therefore, we can get ) ≤ 0 and lim  → +∞  2 () = +∞, we can get that there is a unique solution  * , at which  2 ( * ) = 0, and  2 () is minimized from the Intermediate Value Theorem ( [20]).Therefore, if  2 () =   2 ()(  − 1) −  2 () ≤ 0,  * is the optimal solution.Otherwise,  2 () is increasing on (, +∞), and  =  is the optimal solution.

Solution Procedure
In this section, we develop two algorithms to find the optimal solution under the condition of whether there is budget to pay the purchasing cost at the end of the credit period.
Case 1.If there is budget, the interest will never be charged by the supplier.
Algorithm A. We have the following steps.
Step 1. Input all the initial data, and set the optimal cycle length, the optimal present value of the total cost, and the number of cycles to be  opt = 0, () * = +∞, and  * = 0.
Step 3. If  1 () ≤ () * , update  opt , () * , and  * , and go to Step 2. Otherwise, the current  opt , () * , and  * are the optimal solutions.Case 2. There is no budget.Therefore, when the revenue earned by the end of credit period is not enough to pay the purchasing cost, the supplier charges interest for the unpaid balance.
Algorithm B. We have the following steps.

Numerical Example
This section presents two cases where the results are illustrated.The following parameters are used in the first case. = 200,  = 0.3,  = 0.1,  = 1000, ℎ = 1,  = 2, and   = 0.2.We assume the supplier offers a credit period of  ∈ [0.05, 0.15, 0.25] when the retailer orders more than 300 items per time.There is no budget.
From Table 1, one can see that when the credit period is short, the retailer prefers to order less to decrease the interest charged by the supplier.When the credit period is so long enough that the retailer could earn enough revenue to pay the purchasing cost, the order quantity increases significantly.Therefore, the credit period is a good promotion means to attract more orders.
Then we consider a special case where the planning horizon is infinite.The following parameters are used in this case:  = 200,  = 0.3,  = 0.1,  = 1000, ℎ = 1,  = +∞, and   = 0.2.We assume the supplier offers a credit period of  ∈ [0.05, 0.15, 0.25] when the retailer orders more than 200 items per time.There is budget to pay the purchasing cost at the end of the credit period.
Table 2 shows that, as the discount rate increases, the retailer chooses to shorten the replenishment cycle and accelerate the fund flow.That explains why people prefer short term and low risk investment when inflation is significant.The credit period could also increase the order quantity when the retailer has budget, but the influence is small.The retailer just regards the trade credit as a discount.Therefore, the credit period policy is much more attractive for small retailers or the ones who are in financial distress.

Conclusions
In this paper, an inventory model for perishable products with stock-dependent demand and credit period under inflation and time discounting has been proposed.The credit period is dependent on the purchasing quantity.If the purchasing cost is totally paid at the end of the credit period, the supplier does not charge any interest.Otherwise, the supplier charges interest for unpaid balance after credit period.All remaining payments should be made at the end of each cycle.From the results we can see that, as inflation rate goes up, the cycle length and order quantity decrease.The longer credit period offered by the supplier encourages the retailer to buy more, especially for these small retailers.The inflation could restrain the consumption for the perishable products with stock-dependent demand, and offering a trade credit is a good promotion for the supplier to enlarge the market under inflation.The results show that additional cost savings may be obtained by adjusting the order quantity with consideration of the inflation and time value of money.Therefore, this research proposes a better replenishment policy than the basic EOQ model in terms of the total cost when inflation and time value of money variation are considered.
The proposed model may be extended in several directions.First, we may further incorporate the pricing strategy into the analysis.Second, shortage is allowed and the unsatisfied demand could be lost, totally backordered, or partially backordered.Third, the deterministic demand may be changed to a stochastic demand.

Figure 1 :Figure 2 :
Figure 1: The retailer's inventory level of the first cycle when  ≥ .

Figure 3 :
Figure 3: The retailer's inventory level of the first cycle when  < .

Table 1 :
Effect of  and  on decisions without budget.

Table 2 :
Effect of  and  on decisions with budget and infinite .