Although a smoothly running supply chain is ideal, the reality is to deal with imperfectness in transportations. This paper tries to propose a mathematical model for a supply chain under the effect of unexpected disruptions in transport. Supplier offers the retailer a trade credit period

Much of the logistician’s planning and control effort is directed toward running an efficient operation under normal conditions. At the same time, global trades such as Wal-Mart, Home Depot, and Dollar General are facing the extraordinary circumstances (such as earth quake, mishandling in transport, shipping damage, and misplacing products) that may result a risk in delivery from a supplier to a retailer. The supply disruptions take the form of high-impact and low-probability contingencies which can threaten decision makers of a supply chain. Mathematical modeling helps decision makers to evaluate optimal ordering policies against an incredibly complex and dynamic set of risks and constraints.

In the classical logistics models, it was assumed that the retailers and their customers must pay for the items as soon as the items are received. However, in practices, the supplier/retailer would allow a specified credit period (say 30 days) to their retailers/customers for payment without penalty to stimulate the demand of the consumable products. This credit term in financial management is denoted as “net 30.” Teng [

This paper investigates a supply chain model in which the supplier is willing to provide the retailer a full trade credit period for payments and the retailer offers the full trade credit to his/her customer. This is called two-echelon (or two-level) trade credit financing. In practice, this two-level trade credit financing at a retailer is more matched to real-life situations in a supply chain. Companies, like TATA and Toyato, can delay the full amount of purchasing cost until the end of the delay period offered by his suppliers. But these companies only offer partial delay payment to his dealership on the permissible credit period.

This paper attempts to develop a mathematical model for the supply chain with two-level trade credits and probabilistic considerations of supply disruptions. To manage the risk in delivery, the retailer arranges some alternatives to rework those defective items which involve defective costs. Here, the retailer offers trade credits to his customers and also receives full trade credit from the supplier. The retailer replenishes his inventory noninstantaneously and faces probabilistic risks due to supply disruptions. According to risk management in operations research, two situations such as (a) risk neutral and (b) risk averse are considered. The solution procedures are described for the retailer in both cases.

The rest of this paper is organized as follows. The literature review is presented in Section

During the past few years, many researchers have studied inventory models for permissible delay in payments. Goyal [

For the literature related to supply disruptions, Gulyani [

TC(

This paper considers the following assumptions.

The present model is confined to single supplier, single retailer, and multiple customers.

The inventory system deals with only one type of item.

Shortages are not allowed.

Demand rate (

Lead time is zero.

Time period is infinite.

The supplier offers the full trade credit

The retailer also offers the trade credit

Elapsed time (

If the products are defective due to contingency in delivery, the retailers need to find supply sources to recover these defective products. It accounts for the defective cost

The objective is to minimize the annual total cost incurred at the retailer:

Annual ordering cost =

Excluding interest charges, the annual stock holding cost is

Interest earned by the retailer.

Consider the following:

Consider the following:

There is no interest earned for the retailer since retailer’s credit period is prior to customer’s credit period.

Interest payable by the retailer.

Consider the following:

There is no interest payable for the retailer.

Consider the following:

Annual defective cost due to disruption in supply.

If the number of defective items in each replenishment cycle, as in Tsao [

The annual defective cost is

Using the approximation

Therefore, the total cost incurred at the retailer TC(

Here, two situations, namely, (a) risk neutral and (b) risk averse are considered. To minimize the annual total cost TC(

The first-order and second-order derivatives of TC_{1}(

The first-order and second-order derivatives of

The first-order and second-order derivatives of

If

If

In this section, a solution procedure is given to find optimal replenishment policy by limiting the expected number of defective items (up to

Kuhn-Tucker conditions are used to solve the constrained optimization as in (

In this case, the following are the Kuhn-Tucker conditions:

In this case, the following are the Kuhn-Tucker conditions:

In order to illustrate the model, we consider the following examples.

Let

For the risk-averse solution, we limit the expected number of defective items at most

The previous examples illustrate that when the number of defective items is limited to 2, the risk-neutral solution is better than the risk-averse solution; if number of defective items is greater than or equal to 3, then the risk-averse solution is better than the risk-neutral one.

The illustrated model is very useful for low-probability high-consequence contingency event. In this paper, an EPQ-based model in which retailer offers trade credit to his customer and fixes the bound for defectiveness due to contingency such as shipping damages, misplacing products, earthquake, and hurricane is developed. Theorem

For the future research, this paper can be extended by considering perishable items or seasonal products.