Jamal et al. [2000. Optimal payment time for a retailer under permitted delay of payment by the wholesaler. International Journal of Production Economics 66, 59–66] study a wholesaler–retailer supply chain where the retailer is given a permissible credit period to pay back the dues without paying any interest to the wholesaler. The problem is modeled as a cost minimization problem with two decision variables: the payment time P of the retailer and the length of the inventory cycle T. While the model represents, in general, an interesting problem, we show in this note that there are a number of nontrivial flaws contained in the development of their model and therefore the solution they derived is not correct. We give the correct model and derive the corresponding optimal solution.
The problem studied by Jamal et al. (2000) can be described as follows: A retailer orders a product from a wholesaler, which is perishable at a deterioration rate θ . Assume that the order quantity is Q , and the demand for the product has a constant rate D . Then, at the beginning t=0t=0, the inventory level is Q , and at t=Tt=T the inventory level reduces to zero, where T is called the length of the inventory cycle. It is further assumed that there is a permitted credit period [0,M][0,M], during which no interest is incurred for the retailer. After that, the retailer must pay for any unpaid balance at an interest rate Ip . The retailer can also use the interest earned in the period [0,M][0,M] (with a known interest rate Ie) to pay off the wholesaler.
When is the best time P for the retailer to complete the payment to the wholesaler, so that his overall cost is minimized? The main purpose of Jamal, Sarker and Wang's paper is to address this question. This is an interesting problem, which models a significant situation in management of wholesaler–retailer supply chains with perishable products. An optimal solution for the problem may provide not only a desirable strategy for the retailer, but also a deeper understanding on other important issues such as development of wholesale–retailer contracts in such a situation. This line of research has received some attention in the literature. For example, problems with permissible delay in payments of retailers have been considered by Aggarwal and Jaggi (1995), Chung et al. (2005), Goyal (1985), Arcelus et al. (2002), Hwang and Shinn (1997), and Sarker and Pan (1994), Kim et al. (1997) have investigated the problem of developing an optimal credit policy to maximize the wholesaler's profit.
Jamal et al.'s (2000) paper first argues that the total cost for the retailer is represented by a function TVC(P,T)=TC(P,T)×TTVC(P,T)=TC(P,T)×T, and then suggests an iterative search approach to compute the optimal solution (P,T)(P,T) that minimizes TC(P,T)TC(P,T). Unfortunately, there are a number of nontrivial technical flaws contained in their development. In the following sections, we will show that:
(i)
their claim (lines 6–12, left column of p. 63) that the total cost function TC (P ,T ) is convex in P and T is not correct, and TC (P ,T ) is actually a saddle function with respect to P and T . Consequently, the payment time P they obtained is in fact the maximum point of TC (P ,T*T*) instead of the minimum point;
(ii)
they overlooked the possibility that the interest cost to be paid by the retailer may become negative, which implies a pathological scenario that the wholesaler pays interest to the retailer; and
(iii)
the deteriorated value of the perishable product is missed in the development of the interest cost function PT, and therefore the solution derived based on this function is not valid.
We will correct these mistakes and derive the optimal solution. Numerical results will also be reported.
