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

Abstract

We consider an inventory model for perishable products with stock-dependent demand under inflation. It is assumed that the supplier offers a credit period to the retailer, and the length of credit period is dependent on the order quantity. The retailer does not need to pay the purchasing cost until the end of credit period. If the revenue earned by the end of credit period is enough to pay the purchasing cost or there is budget, the balance is settled and the supplier does not charge any interest. Otherwise, the supplier charges interest for unpaid balance after credit period, and the interest and the remaining payments are made at the end of the replenishment cycle. The objective is to minimize the retailer’s (net) present value of cost. We show that there is an optimal cycle length to minimize the present value of cost; furthermore, a solution procedure is given to find the optimal solution. Numerical experiments are provided to illustrate the proposed model.

1. 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 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.

2. Model Formulation

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

2.1. 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 𝐷(𝑡) at the retailer’s end is dependent on the instantaneous inventory level 𝑞(𝑡), which means 𝐷(𝑡)=𝑎+𝑏𝑞(𝑡), 𝑎≥0, 1≥𝑏≥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.

2.2. Mathematical Model

The replenishment cycle starts with the initial inventory level 𝑄 and ends with zero stock. Since the inventory is depleted by the effect of both stock-dependent consumption and deterioration, we can describe the retailer’s inventory level 𝑞(𝑡) by the following differential equation: 𝑑𝑞(𝑡)(𝑡)(𝑡𝑑𝑡=−𝑎+𝑏𝑞+𝜃𝑞),0≤𝑡≤𝑇. (1) By integrating both sides of (1) with respect to 𝑡, ^𝑇₀−1(𝑡)(𝑡)(𝑡)𝑎+𝑏𝑞+𝜃𝑞𝑑𝑞=^𝑇₀𝑑𝑡. (2) With the boundary condition 𝑞(𝑇)=0, the solution of the integral equation is 𝑞(𝑡)=𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)(𝑇−𝑡)}−1. (3) With 𝑞(0)=𝑄, we get 𝑎𝑄=𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1. (4) On the above assumptions, there are two scenarios to arise: Scenario A, 𝑀≥𝑇; Scenario B, 𝑀<𝑇.

2.2.1. Scenario A: When 𝑀≥𝑇

Since the credit period 𝑀 is longer than the replenishment cycle length 𝑇, 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.

Figure 1 

The retailer’s inventory level of the first cycle when 𝑀≥𝑇.

Since the replenishment is made at the beginning of each cycle, the present value of the ordering cost during the first cycle is 𝐴.

The purchasing cost is paid at the end of credit period 𝑀; the present value of the purchasing cost during the first cycle is 𝐶𝑄𝑒^−𝑟𝑀=𝐶𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1𝑒^−𝑟𝑀. (5)

Because the holding cost occurs all over the replenishment cycle, the present value of the holding cost during the first cycle is ℎ^𝑇₀𝑞(𝑡)𝑒^−𝑟𝑡𝑑𝑡=𝑎ℎ()𝑟𝑒𝑏+𝜃^−𝑟𝑇+−1𝑎ℎ()𝑒𝑏+𝜃)(𝑟+𝑏+𝜃^{(𝑏+𝜃)𝑇}−𝑒^−𝑟𝑇. (6) Therefore, the net present value of the cost during the first cycle is 𝑃𝑉₁(𝑇)=𝐴+𝐶𝑄𝑒^−𝑟𝑀+ℎ^𝑇₀𝑞(𝑡)𝑒^−𝑟𝑡𝑑𝑡=𝐴+𝐶𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1𝑒^−𝑟𝑀+𝑎ℎ𝑒𝑏+𝜃^−𝑟𝑇−1𝑟+𝑒^{(𝑏+𝜃)𝑇}−𝑒^−𝑟𝑇.𝑟+𝑏+𝜃 (7)

As shown in Figure 2, there are 𝑚 cycles in the planning horizon. Therefore, the present value of the total cost over the planning horizon 𝐿 is 𝑃𝑉𝐻₁(𝑇)=^𝑚−1_𝑛=0𝑃𝑉₁(𝑇)𝑒^{−𝑟𝑛𝑇}=1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇𝑃𝑉₁(𝑇)=1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇𝐴+𝐶𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1𝑒^−𝑟𝑀+𝑎ℎ()𝑟𝑒𝑏+𝜃^−𝑟𝑇+−1𝑎ℎ()𝑒𝑏+𝜃)(𝑟+𝑏+𝜃^{(𝑏+𝜃)𝑇}−𝑒^−𝑟𝑇. (8)

Figure 2 

The retailer’s inventory cycles in the planning horizon.

Proposition 1. When there exists a unique 𝑇^∗ at which 𝑑𝑃𝑉𝐻₁(𝑇)/𝑑𝑇|_𝑇=𝑇^∗=0 for (0,𝑀], then 𝑃𝑉𝐻₁(𝑇) is minimized at 𝑇=𝑇^∗if 𝑃𝑉^′₁(𝑀)(𝑒^𝑟𝑀−1)−𝑟𝑃𝑉₁(𝑀)≥0. Otherwise, 𝑇=𝑀 is the optimal solution.

Proof. Let 𝑃𝑉^′₁(𝑇) and 𝑃𝑉^′′₁(𝑇) represent the first and the second derivatives of 𝑃𝑉₁(𝑇) with respect to 𝑇, respectively. Taking the first derivative of 𝑃𝑉𝐻₁(𝑇) with respect to 𝑇, 𝑑𝑃𝑉𝐻₁(𝑇)=𝑒𝑑𝑇^−𝑟𝑇1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇2𝑃𝑉^′₁(𝑇)𝑒^𝑟𝑇−1−𝑟𝑃𝑉₁(𝑇). (9) Let 𝑓₁(𝑇)=𝑃𝑉^′₁(𝑇)(𝑒^𝑟𝑇−1)−𝑟𝑃𝑉₁(𝑇); then 𝑑𝑃𝑉𝐻₁(𝑇)/𝑑𝑇=(𝑒^−𝑟𝑇(1−𝑒^−𝑟𝐿)/(1−𝑒^−𝑟𝑇)²)𝑓₁(𝑇). 𝑒^−𝑟𝑇(1−𝑒^−𝑟𝐿)/(1−𝑒^−𝑟𝑇)²=(1−𝑒^−𝑟𝐿)(1/(𝑒^𝑟𝑇+𝑒^−𝑟𝑇−2))>0, and they are decreasing on 𝑇.
Taking the first derivative of 𝑓₁(𝑇) with respect to 𝑇, we get 𝑓^′₁(𝑇)=𝑃𝑉^′′₁(𝑇)𝑒^𝑟𝑇−1+𝑟𝑃𝑉^′₁(𝑇)𝑒^𝑟𝑇−1. (10)
In 𝑓^′₁(𝑇), 𝑃𝑉^′₁(𝑇)=𝐶𝑎𝑒^{(𝑏+𝜃)𝑇−𝑟𝑀}−𝑎ℎ()𝑒𝑏+𝜃^−𝑟𝑇+𝑎ℎ()×)𝑒𝑏+𝜃)(𝑟+𝑏+𝜃(𝑏+𝜃^{(𝑏+𝜃)𝑇}+𝑟𝑒^−𝑟𝑇𝑃𝑉^′′₁(𝑇)=𝐶𝑎(𝑏+𝜃)𝑒^{(𝑏+𝜃)𝑇−𝑟𝑀}+𝑎ℎ𝑟()𝑒𝑏+𝜃^−𝑟𝑇+𝑎ℎ()×)𝑏+𝜃)(𝑟+𝑏+𝜃(𝑏+𝜃²𝑒^{(𝑏+𝜃)𝑇}−𝑟²𝑒^−𝑟𝑇. (11)
Obviously, 𝑃𝑉^′₁(𝑇) and 𝑃𝑉^′′₁(𝑇) are both more than zero. Therefore, 𝑓^′₁(𝑇)>0; 𝑓₁(𝑇) is increasing on 𝑇. From 𝑓₁(0)=−𝑟𝐴 and lim_𝑇→∞𝑓₁(𝑇)=+∞, the Intermediate Value Theorem (Thomas and Finney [20]) implies that there exists a unique solution 𝑇^∗, which makes 𝑓₁(𝑇^∗)=0. Hence, 𝑑𝑃𝑉𝐻₁(𝑇)=𝑒𝑑𝑇^−𝑟𝑇1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇2𝑓₁(𝑇)<0if0<𝑇<𝑇^∗=0if𝑇=𝑇^∗>0if𝑇>𝑇^∗. (12)
Therefore, if 𝑓₁(𝑀)=𝑃𝑉^′₁(𝑀)(𝑒^𝑟𝑀−1)−𝑟𝑃𝑉₁(𝑀)≥0, 𝑇^∗ is the optimal solution. Otherwise, 𝑃𝑉𝐻₁(𝑇) is decreasing on (0,𝑀] and 𝑇=𝑀 is the optimal solution.

2.2.2. Scenario B: When 𝑀<𝑇

Case 1. Letting 𝑃𝑒^𝑟𝑀∫^𝑀₀𝐷(𝑡)𝑒^−𝑟𝑡𝑑𝑡≥𝐶𝑄, which means at time 𝑀, 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 𝑀 is shorter than the replenishment cycle length 𝑇, as shown in Figure 3.

Figure 3 

The retailer’s inventory level of the first cycle when 𝑀<𝑇.

The objective function is the same with that under Scenario A: 𝑃𝑉𝐻₁(𝑇)=^𝑚−1_𝑛=0𝑃𝑉₁(𝑇)𝑒^{−𝑟𝑛𝑇}=1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇𝑃𝑉₁(𝑇)=1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇𝐴+𝐶𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1𝑒^−𝑟𝑀+𝑎ℎ()𝑟𝑒𝑏+𝜃^−𝑟𝑇+−1𝑎ℎ()𝑒𝑏+𝜃)(𝑟+𝑏+𝜃^{(𝑏+𝜃)𝑇}−𝑒^−𝑟𝑇. (13)

Case 2. Let 𝑃𝑒^𝑟𝑀∫^𝑀₀𝐷(𝑡)𝑒^−𝑟𝑡𝑑𝑡<𝐶𝑄, but there is budget. That means, although the revenue earned by time 𝑀 is less than the purchasing cost, there is budget to pay the short purchasing cost at time 𝑀. Therefore, there is still no interest charged by the supplier.
The objective function is the same with that under Scenario A: 𝑃𝑉𝐻₁(𝑇)=^𝑚−1_𝑛=0𝑃𝑉₁(𝑇)𝑒^{−𝑟𝑛𝑇}=1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇𝑃𝑉₁(𝑇)=1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇𝐴+𝐶𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1𝑒^−𝑟𝑀+𝑎ℎ()𝑟𝑒𝑏+𝜃^−𝑟𝑇+−1𝑎ℎ()𝑒𝑏+𝜃)(𝑟+𝑏+𝜃^{(𝑏+𝜃)𝑇}−𝑒^−𝑟𝑇. (14)

Case 3. Let 𝑃𝑒^𝑟𝑀∫^𝑀₀𝐷(𝑡)𝑒^−𝑟𝑡𝑑𝑡<𝐶𝑄, and there is no budget. As a result, all the revenue earned by time 𝑀 is used to pay the purchasing cost and the supplier charges interest rate 𝐼_𝑐 from 𝑀 to 𝑇 for the unpaid balance. The interest and the remaining payments should be made at the end of the replenishment cycle 𝑇. Therefore, there are four elements in the retailer’s cost: ordering cost, holding cost, the purchasing cost paid at time 𝑀, and the interest and the remaining payments made at the end of replenishment cycle 𝑇. Two of the elements are different from Scenario A: the purchasing cost paid at time 𝑀 and the interest and the remaining payments made at the end of the replenishment cycle 𝑇.

The present value of the purchasing cost paid at time 𝑀 during the first cycle is equal to the present value of the revenue earned by time 𝑀: 𝑃^𝑀₀𝐷(𝑡)𝑒^−𝑟𝑡𝑑𝑡=𝑃𝑎𝑏(𝑏+𝜃)𝑟1−𝑒^−𝑟𝑀+𝑎𝑏()𝑒𝑏+𝜃)(𝑏+𝜃+𝑟^{(𝑏+𝜃)𝑇}1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}. (15)

The present value of the remaining payments and interest paid at the end of the replenishment cycle during the first cycle is 𝐶𝑄−𝑃𝑒^𝑟𝑀𝑀₀𝐷(𝑡)𝑒^−𝑟𝑡𝑑𝑡1+𝐼_𝑐(𝑇−𝑀)𝑒^−𝑟𝑇=𝐶𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1−𝑃𝑒^𝑟𝑀×𝑎𝑏(𝑏+𝜃)𝑟1−𝑒^−𝑟𝑀+𝑎𝑏(𝑏+𝜃)(𝑏+𝜃+𝑟)×𝑒^{(𝑏+𝜃)𝑇}1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}×1+𝐼_𝑐(𝑇−𝑀)𝑒^−𝑟𝑇. (16)

The net present value of the cost during the first cycle is 𝑃𝑉₂(𝑇)=𝐴+ℎ^𝑇₀𝑞(𝑡)𝑒^−𝑟𝑡+𝑑𝑡𝐶𝑄−𝑃𝑒^𝑟𝑀𝑀₀𝐷(𝑡)𝑒^−𝑟𝑡×𝑑𝑡1+𝐼_𝑐(𝑇−𝑀)𝑒^−𝑟𝑇+𝑃^𝑀₀𝐷(𝑡)𝑒^−𝑟𝑡𝑑𝑡=𝐴+𝑎ℎ()𝑟𝑒𝑏+𝜃^−𝑟𝑇+−1𝑎ℎ()𝑒𝑏+𝜃)(𝑏+𝜃+𝑟^{(𝑏+𝜃)𝑇}−𝑒^−𝑟𝑇+𝐶𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1−𝑃𝑒^𝑟𝑀×𝑎𝑏(𝑏+𝜃)𝑟1−𝑒^−𝑟𝑀+𝑎𝑏()𝑒𝑏+𝜃)(𝑏+𝜃+𝑟^{(𝑏+𝜃)𝑇}×1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}×1+𝐼_𝑐(𝑇−𝑀)𝑒^−𝑟𝑇+𝑃𝑎𝑏(𝑏+𝜃)𝑟1−𝑒^−𝑟𝑀+𝑎𝑏()𝑒𝑏+𝜃)(𝑏+𝜃+𝑟^{(𝑏+𝜃)𝑇}1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}. (17)

The present value of the total cost over the planning horizon 𝐿 is 𝑃𝑉𝐻₂(𝑇)=^𝑚−1_𝑛=0𝑃𝑉₂(𝑇)𝑒^{−𝑟𝑛𝑇}=1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇𝑃𝑉₂(𝑇)=1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇𝐴+ℎ^𝑇₀𝑞(𝑡)𝑒^−𝑟𝑡+𝑑𝑡𝐶𝑄−𝑃𝑒^𝑟𝑀𝑀₀𝐷(𝑡)𝑒^−𝑟𝑡×𝐼𝑑𝑡_𝑐(𝑇−𝑀)+1×𝑒^−𝑟𝑇+𝑃^𝑀₀𝐷(𝑡)𝑒^−𝑟𝑡=𝑑𝑡1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇𝐴+𝑎ℎ()𝑟𝑒𝑏+𝜃^−𝑟𝑇+−1𝑎ℎ()𝑒𝑏+𝜃)(𝑟+𝑏+𝜃^{(𝑏+𝜃)𝑇}−𝑒^−𝑟𝑇+𝐶𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1−𝑃𝑒^𝑟𝑀×𝑎𝑏(𝑏+𝜃)𝑟1−𝑒^−𝑟𝑀+𝑎𝑏()𝑒𝑏+𝜃)(𝑏+𝜃+𝑟^{(𝑏+𝜃)𝑇}×1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}×1+𝐼_𝑐(𝑇−𝑀)𝑒^−𝑟𝑇+𝑃𝑎𝑏(𝑏+𝜃)𝑟1−𝑒^−𝑟𝑀+𝑎𝑏()𝑒𝑏+𝜃)(𝑏+𝜃+𝑟^{(𝑏+𝜃)𝑇}×1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}. (18)

Proposition 2. When there exists a unique 𝑇^∗ at which 𝑑𝑃𝑉𝐻₂(𝑇)/𝑑𝑇|_𝑇=𝑇∗=0 for (𝑀,+∞), then 𝑃𝑉𝐻₂(𝑇) is minimized at 𝑇=𝑇^∗ if 𝑃𝑉^′₂(𝑀)(𝑒^𝑟𝑀−1)−𝑟𝑃𝑉₂(𝑀)≤0. Otherwise, 𝑇=𝑀 is the optimal solution.

Proof. Let 𝑃𝑉^′₂(𝑇) and 𝑃𝑉^′′₂(𝑇) represent the first and the second derivatives of 𝑃𝑉₂(𝑇) with respect to 𝑇, respectively. Taking the first derivative of 𝑃𝑉𝐻₂(𝑇) with respect to 𝑇, 𝑑𝑃𝑉𝐻₂(𝑇)=𝑒𝑑𝑇^−𝑟𝑇1−𝑒^−𝑟𝐿1−𝑒^−𝑟𝑇2×𝑃𝑉^′₂(𝑇)𝑒^𝑟𝑇−1−𝑟𝑃𝑉₂(𝑇). (19)
Letting 𝑓₂(𝑇)=𝑃𝑉^′₂(𝑇)(𝑒^𝑟𝑇−1)−𝑟𝑃𝑉₂(𝑇), then 𝑑𝑃𝑉𝐻₂(𝑇)/𝑑𝑇=(𝑒^−𝑟𝑇(1−𝑒^−𝑟𝐿)/(1−𝑒^−𝑟𝑇)²)𝑓₂(𝑇).
Taking the first derivative of 𝑓₂(𝑇) with respect to 𝑇, we get 𝑓^′₂(𝑇)=𝑃𝑉^′′₂(𝑇)(𝑒^𝑟𝑇−1)+𝑟𝑃𝑉^′₂(𝑇)(𝑒^𝑟𝑇−1). In 𝑓^′₂(𝑇), 𝑃𝑉^′₂(𝑇)=−𝑎ℎ()𝑒𝑏+𝜃^−𝑟𝑇+𝑎ℎ()×)𝑒𝑏+𝜃)(𝑟+𝑏+𝜃(𝑏+𝜃^{𝑏𝑇+𝜃𝑇}+𝑟𝑒^−𝑟𝑇+𝐶𝑎𝑒^{(𝑏+𝜃)𝑇}−𝑎𝑏𝑏+𝜃+𝑟1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}×𝐼_𝑐(𝑇−𝑀)+1𝑒^−𝑟𝑇+𝐶𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1−𝑃𝑒^𝑟𝑀×𝑎𝑏(𝑏+𝜃)𝑟1−𝑒^−𝑟𝑀+𝑎𝑏()𝑒𝑏+𝜃)(𝑏+𝜃+𝑟^{(𝑏+𝜃)𝑇}×1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}×𝐼_𝑐−𝑟−𝐼_𝑐𝑟(𝑇−𝑀)𝑒^−𝑟𝑇+𝑃𝑎𝑏𝑒𝑏+𝜃+𝑟^{(𝑏+𝜃)𝑇}1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}>0𝑃𝑉^′′₂(𝑇)=𝑎ℎ𝑟()𝑒𝑏+𝜃^−𝑟𝑇+𝑎ℎ()×𝑏+𝜃)(𝑟+𝑏+𝜃𝑏+𝜃)²𝑒^{(𝑏+𝜃)𝑇}−𝑟²𝑒^−𝑟𝑇+(𝑏+𝜃)𝐶𝑎𝑒^{(𝑏+𝜃)𝑇}−𝑎𝑏(𝑏+𝜃)×𝑏+𝜃+𝑟1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}𝑒^{(𝑏+𝜃)𝑇}×1+𝐼_𝑐(𝑇−𝑀)𝑒^−𝑟𝑇+2𝐶𝑎𝑒^{(𝑏+𝜃)𝑇}−𝑎𝑏×𝑏+𝜃+𝑟1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}𝑒^{(𝑏+𝜃)𝑇}×𝐼_𝑐−𝑟−𝐼_𝑐𝑟(𝑇−𝑀)𝑒^−𝑟𝑇+𝐶𝑎𝑒𝑏+𝜃^{(𝑏+𝜃)𝑇}−1−𝑃𝑒^𝑟𝑀×𝑎𝑏(𝑏+𝜃)𝑟1−𝑒^−𝑟𝑀+𝑎𝑏()𝑒𝑏+𝜃)(𝑏+𝜃+𝑟^{(𝑏+𝜃)𝑇}×1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}×𝑟²−2𝐼_𝑐𝑟+𝐼_𝑐𝑟²(𝑇−𝑀)𝑒^−𝑟𝑇+𝑃𝑎𝑏(𝑏+𝜃)𝑒𝑏+𝜃+𝑟^{(𝑏+𝜃)𝑇}1−𝑒^{−(𝑏+𝜃+𝑟)𝑀}>0. (20) Therefore, we can get 𝑓^′₂(𝑇)=𝑃𝑉^′′₂(𝑇)(𝑒^𝑟𝑇−1)+𝑟𝑃𝑉^′₂(𝑇)(𝑒^𝑟𝑇−1)>0. From 𝑓₂(𝑀)=𝑃𝑉^′₂(𝑀)(𝑒^𝑟𝑀−1)−𝑟𝑃𝑉₂(𝑀)≤0 and lim_𝑇→+∞𝑓₂(𝑇)=+∞, we can get that there is a unique solution 𝑇^∗, at which 𝑓₂(𝑇^∗)=0, and 𝑃𝑉𝐻₂(𝑇) is minimized from the Intermediate Value Theorem ([20]). Therefore, if 𝑓₂(𝑀)=𝑃𝑉^′₂(𝑀)(𝑒^𝑟𝑀−1)−𝑟𝑃𝑉₂(𝑀)≤0, 𝑇^∗ is the optimal solution. Otherwise, 𝑃𝑉𝐻₂(𝑇) is increasing on (𝑀,+∞), and 𝑇=𝑀 is the optimal solution.

3. 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 𝑇_opt=0, 𝑃𝑉𝐻(𝑇)^∗=+∞, and 𝑚^∗=0.
Step 2. Set 𝑚=𝑚+1. Let 𝑇=𝐻/𝑚 and get 𝑄 from 𝑄=(𝑎/(𝑏+𝜃))[𝑒^{(𝑏+𝜃)𝑇}−1]. Find the corresponding 𝑀_𝑖 from 𝑄. Then, we can get 𝑃𝑉𝐻₁(𝑇).
Step 3. If 𝑃𝑉𝐻₁(𝑇)≤𝑃𝑉𝐻(𝑇)^∗, update 𝑇_opt, 𝑃𝑉𝐻(𝑇)^∗, and 𝑚^∗, and go to Step 2. Otherwise, the current 𝑇_opt, 𝑃𝑉𝐻(𝑇)^∗, and 𝑚^∗ 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.
Step 1. Input all the initial data, and set the optimal cycle length, the optimal present value of the total cost, the optimal 𝑃𝑉𝐻₂(𝑇)^∗, and the number of cycles to be 𝑇_opt=0, 𝑃𝑉𝐻(𝑇)^∗=+∞, 𝑃𝑉𝐻₂(𝑇)^∗=+∞, and 𝑚^∗=0.
Step 2. Set 𝑚=𝑚+1. Let 𝑇=𝐻/𝑚 and get Q from 𝑄=(𝑎/(𝑏+𝜃))[𝑒^{(𝑏+𝜃)𝑇}−1]. Find the corresponding 𝑀_𝑖 from Q. If 𝑇≥𝑀_𝑖 and 𝑃𝑒^𝑟𝑀∫^𝑀₀𝐷(𝑡)𝑒^−𝑟𝑡𝑑𝑡<𝐶𝑄, 𝑃𝑉𝐻₂(𝑇) is the optimal solution. Otherwise, 𝑃𝑉𝐻₁(𝑇) is the optimal solution. If we get 𝑃𝑉𝐻₁(𝑇), go to Step  3. Otherwise, go to Step  4.
Step 3. If 𝑃𝑉𝐻₁(𝑇)≤𝑃𝑉𝐻(𝑇)^∗, update 𝑇_opt, 𝑃𝑉𝐻(𝑇)^∗, and 𝑚^∗, and then go to Step  2. Otherwise, the current 𝑇_opt, 𝑃𝑉𝐻(𝑇)^∗, and 𝑚^∗ are the optimal solutions.
Step 4. If 𝑃𝑉𝐻₂(𝑇)≤𝑃𝑉𝐻(𝑇)^∗, update 𝑇_opt, 𝑃𝑉𝐻(𝑇)^∗, 𝑃𝑉𝐻₂(𝑇)^∗, and 𝑚^∗, and then go to Step  2. If 𝑃𝑉𝐻(𝑇)^∗<𝑃𝑉𝐻₂(𝑇)≤𝑃𝑉𝐻₂(𝑇)^∗, update 𝑃𝑉𝐻₂(𝑇)^∗ and go to Step  2. Otherwise, the current 𝑇_opt, 𝑃𝑉𝐻(𝑇)^∗, and 𝑚^∗ are the optimal solutions.

4. Numerical Example

This section presents two cases where the results are illustrated. The following parameters are used in the first case. 𝑎=200, 𝑏=0.3, 𝜃=0.1, 𝐴=1000, ℎ=1, 𝐿=2, and 𝐼_𝑐=0.2. We assume the supplier offers a credit period of 𝑀∈[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.

Table 1 

Effect of 𝑟 and 𝑀 on decisions without budget.

Then we consider a special case where the planning horizon is infinite. The following parameters are used in this case: 𝑎=200, 𝑏=0.3, 𝜃=0.1, 𝐴=1000, ℎ=1, 𝐿=+∞, and 𝐼_𝑐=0.2. We assume the supplier offers a credit period of 𝑀∈[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.

Table 2 

Effect of 𝑟 and 𝑀 on decisions with budget and infinite 𝐿.

5. 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.

Notations

𝑃:	Selling price per unit
𝑞(𝑡):	Inventory level at time 𝑡
𝑄:	Initial inventory level (𝑞(0)=𝑄)
𝐶:	Purchasing cost per unit, with 𝑃>𝐶
𝑇:	Replenishment cycle length (decision variable)
𝐻:	Holding cost per unit
𝐴:	Ordering cost
𝐿:	Planning horizon
𝑅:	Discount rate (i.e., opportunity cost) per unit time, which is related to the time value of money and inflation rate
𝜃:	Deteriorating rate
𝑀:	Credit period, 𝑀={𝑀0,𝑞0≤𝑄<𝑞1;𝑀1,𝑞1≤𝑄<𝑞2;…;𝑀𝑘,𝑞𝑘≤𝑄}
𝐼𝑐:	The interest charged per $ per unit time by the supplier when 𝑇>𝑀, with 𝐼𝑐>𝑒𝑟.

Acknowledgments

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2011-013-D00137). This work was also supported by NSFC (no. 71371061); Coordination and Disruption Coping for Customer Oriented Coopetition Supply Chain Networks.