We consider an inventory model for perishable products with stockdependent 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.
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 [
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 [
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 stockdependent 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.
This section introduces the assumptions, notations, and mathematical formulation used in our perishable product inventory model.
The following assumptions are used throughout the paper.
The supplier sells one single item to the retailer in quantity.
The items are replenished when the stock level becomes zero.
The supplier provides a credit period, which is dependent on the order quantity.
The deteriorating rate is constant. The deteriorating items cannot be repaired and the salvage value is zero.
The demand rate
The lead time is zero and shortages are not allowed.
The planning horizon of the inventory system is finite. The number of cycles must be integer in the planning horizon.
The replenishment cycle starts with the initial inventory level
Since the credit period
The retailer’s inventory level of the first cycle when
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
Because the holding cost occurs all over the replenishment cycle, the present value of the holding cost during the first cycle is
As shown in Figure
The retailer’s inventory cycles in the planning horizon.
When there exists a unique
Let
Taking the first derivative of
In
Obviously,
Therefore, if
Letting
The retailer’s inventory level of the first cycle when
The objective function is the same with that under Scenario A:
Let
The objective function is the same with that under Scenario A:
Let
The present value of the purchasing cost paid at 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
When there exists a unique
Let
Letting
Taking the first derivative of
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.
If there is budget, the interest will never be charged by the supplier.
We have the following steps.
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.
We have the following steps.
This section presents two cases where the results are illustrated. The following parameters are used in the first case.
From Table
Effect of




Present value of total cost 

0.1  0.05  2/9  232.40 

0.15  2/7  302.68 
 
0.25  2/7  302.68 
 
 
0.13  0.05  2/9  232.40 

0.15  2/7  302.68 
 
0.25  2/7  302.68 
 
 
0.16  0.05  2/9  232.40 

0.15  2/7  302.68 
 
0.25  2/7  302.68 

Then we consider a special case where the planning horizon is infinite. The following parameters are used in this case:
Table
Effect of




Present value of total cost 

0.1  0.05  0.2492  262.04 

0.15  0.2502  263.15 
 
0.25  0.2512  264.26 
 
 
0.13  0.05  0.2433  255.50 

0.15  0.2446  256.91 
 
0.25  0.2458  258.33 
 
 
0.16  0.05  0.2378  249.43 

0.15  0.2393  251.14 
 
0.25  0.2409  252.85 

In this paper, an inventory model for perishable products with stockdependent 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 stockdependent 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.
Selling price per unit
Inventory level at time
Initial inventory level
Purchasing cost per unit, with
Replenishment cycle length (decision variable)
Holding cost per unit
Ordering cost
Planning horizon
Discount rate (i.e., opportunity cost) per unit time, which is related to the time value of money and inflation rate
Deteriorating rate
Credit period,
The interest charged per $ per unit time by the supplier when
This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF2011013D00137). This work was also supported by NSFC (no. 71371061); Coordination and Disruption Coping for Customer Oriented Coopetition Supply Chain Networks.