Goyal's Model Research Articles

Observations on 'a joint economic-lot-size model for purchaser and vendor'

One of the supply chain coordination methods that has been central to the literature is the joint economic lot-sizing (JELS) problems. The JELS methods have shown to reduce the total cost of a supply chain system. This paper revisits the work of Goyal (1988), one of the earliest papers on JELS, where the vendor produces an integer multiplier of equal size shipment lots for a buyer and provides three observations on that. Subsequently, two new models have been developed based on the second and third observations. This paper provided numerical examples and sensitivity analyses to illustrate the similarities and differences among all models. The supply chain performance improves with alternate models for a wider range of attributes, while Goyal's model provides better results for the specific example.

Priority-based model for network partition problems for mobile ad hoc networks

In mobile ad hoc networks, it is collection of nodes used for communication among the nodes. Due to the node mobility, the network may be divided into a number of parts. This paper proposes a priority-based model for mobile ad hoc networks, which assigns the weights to all nodes. Here, the high-priority nodes must maintain connections against those of low-priority nodes. The proposed model has been simulated with the help of 100 mobile nodes in the network using Network Simulator (NS-2.34). The proposed model was simulated and tested under the various environments. It was compared with Goyal model and Hahner model. The simulation results have shown that the proposed model has reached the top position in terms of the network size, percentage of path reconstruction and average delay.

The optimal ordering time interval under trade credit financing

This paper discusses the economic order quantity (EOQ) under conditions of permissible delay in payments. In 1985, Goyal assumes that the unit selling price and the unit purchasing price are equal. This view is debatable sometimes. The main purpose of this paper is to modify Goyal's model to allow theunit selling price and the unit purchasing price not necessarily be equal to reflect the real-life situations. A modified inventory model is developed. Furthermore, three theorems are developed to efficiently determine the optimal cycle time and the optimal order quantity. Goyal's model is considered as a special case. Numerical examples are given to illustrate these theorems

Retailer's optimal ordering policy under limited warehouse space and payments delay

This paper is to modify and extend Goyal's model to allow the unit selling price and the unit purchasing price not necessarily be equal and the different payment rule will be adopted to reflect the real-life situations. We also extend the assumption that the retailer's storage space is limited, and the modified inventory replenishment model is developed so that the annual total cost is minimized. Five theorems are developed to efficiently determine the optimal cycle time and the optimal order quantity. These theorems can help the retailer accurately and quickly to determine the optimal replenishment policies. Numerical examples are given to illustrate all results obtained in this paper

Retailer's optimal ordering policies with trade credit financing

In this article, we extended Goyal's model to develop an Economic Order Quantity (EOQ) model in which the supplier offers the retailer the permissible delay period M, and the retailer in turn provides the trade credit period N (with N ≤ M) to his/her customers. In addition, we assume that (1) the retailer's selling price per unit is necessarily higher than its unit cost, and (2) the interest rate charged by a supplier or a bank is not necessarily higher than the retailer's investment return rate. We then establish an appropriate EOQ model with trade credit financing, and provide an easy-to-use closed-form solution to the problem. Furthermore, we find it is possible that a well-established buyer may order a lower quantity and take the benefit of the permissible delay more frequently, which contradicts to the result by the previous researchers. Finally, we perform some sensitivity analyses to illustrate the theoretical results and obtain some managerial results.

Global optimal policy for vendor–buyer integrated inventory system within just in time environment

Traditionally, inventory problems for the vendor and the buyer are treated separately. In modern enterprises, however, the integration of vendor---buyer inventory system is an important issue. This co-operative approach to inventory management contributes to the success of supply chain management by minimizing the joint inventory cost. The joint inventory cost and the response time can further be reduced when the buyer orders and the vendor replenishes the required items just in time (JIT) for their consumption. The inclusion of the JIT concept in this model contributes significantly to a joint inventory cost reduction. A numerical example and sensitivity analysis are carried out. The derived results show an impressive cost reduction when compared with Goyal's model.

Optimal Ordering Policy for Deteriorating Items with Partial Backlogging under Permissible Delay in Payments

In 1985, Goyal developed an Economic order quantity (EOQ) model under conditions of permissible delay in payments. Jamal et al. then generalized Goyal's model for deteriorating items with completely backlogging. However, they only ran several simulations to indicate that the total relevant cost may be convex. Recently, Teng amended Goyal's model by considering the difference between unit price and unit cost, and provided an alternative conclusion that it makes economic sense for some retailers to order less quantity and take the benefits of the permissible delay more frequently. However, he did not consider deteriorating items and partial backlogging. In this paper, we establish a general EOQ model for deteriorating items when the supplier offers a permissible delay in payments. For generality, our model allows not only the partial backlogging rate to be related to the waiting time but also the unit selling price to be larger than the unit purchase cost. Consequently, the proposed model includes numerous previous models as special cases. In addition, we mathematically prove that the total relevant cost is strictly pseudo-convex so that the optimal solution exists and is unique. Finally, our computational results reveal six managerial phenomena.

RETAILER'S INVENTORY POLICY UNDER SUPPLIER'S PARTIAL TRADE CREDIT POLICY

This paper discusses the economic order quantity ( EOQ ) under partial trade credit. In 1985, Goyal assumed that: (i) The unit selling price and the unit purchasing price were equal. (ii) The supplier would offer the retailer full trade credit under condition of delay payments. The main purpose of this paper wants to modify Goyal's model to presume that the unit selling price and the unit purchasing price are not necessarily equal. Furthermore, in this paper, we assume that the supplier would offer the retailer partial trade credit not full trade credit. That is, the retailer must make a partial payment to the supplier when the order is received. Then, the retailer must pay off the remaining balance at the end of the permissible delay period. Under these conditions, we model the retailer's inventory system as a cost minimization problem to determine the retailer's optimal inventory cycle time and optimal order quantity. One theorem is developed to efficiently determine the optimal inventory policy for the retailer. We deduce some previously published results of other researchers as special cases. Finally, numerical examples are given to illustrate the theorem obtained in this paper.

The optimal and heuristic solutions of a distribution network

Competitive environment of global networking and customization call for cost reduction in arborescent (tree-like) inventory system. One of the keys to cost reduction is the integration of the supply chain. Much practical research has been done in the area of integrating the inventory system in recent years. Integration involves the joint decision making among the producer, the distributors and the retailers in the supply chain. By revising Goyal's model, an optimal solution is derived in this paper. A heuristic solution model is also developed for a producer–distributors–retailers inventory system using the principle of strategic partnership. Numerical results indicate the heuristic solution is a good estimation of the optimal solution.

Optimal retailer's replenishment policy for the EPQ model under the supplier's trade credit policy

This paper investigates the optimal replenishment policy under conditions of permissible delay in payments within an economic production quantity (EPQ) framework. In 1985, Goyal assumed that: 1. The unit selling price and the unit purchasing price are equal. 2. The replenishment rate is infinite. 3. At the end of the credit period, the account is settled. The retailer starts paying for higher interest charges on the items in stock and returns money of the remaining balance immediately when the items are sold. The main purpose of this paper is to modify Goyal's model to presume that the unit selling price and the unit purchasing price are not necessarily equal to reflect the real-life situations, and assumes the replenishment rate is finite. Furthermore, this paper will propose that at the end of the credit period, the retailer will make a partial payment on total purchasing cost and pay off the remaining balance by loan from the bank. One theorem is obtained to explore the convexity of the total annual relevant cost function. Three theorems are developed to determine the optimal cycle time and the optimal order quantity. Numerical examples are given to illustrate these theorems.

Economic ordering policies for items under permissible delay in payments

This paper discusses the economic order quantity (EOQ) under conditions of permissible delay in payments. In 1985, Goyal assumed that: (1) The unit selling price and the unit purchasing price are equal. (2) At the end of the credit period, the account is settled. The retailer starts paying for higher interest charges on the items in stock and returns money of the remaining balance immediately when the items are sold. The main purpose of this paper wants to modify Goyal's model to presume that the unit selling price and the unit purchasing price are not necessarily equal to reflect the real-life situations. Furthermore, this paper will propose that at the end of credit period, the retailer will make a partial payment on total purchasing cost and pay off the remaining balance by loan from the bank. Comparing with Goyal's model, we also show that the optimal cycle times in this paper are not longer than those of Goyal's model. Three theorems are developed to determine the optimal cycle time and the optimal order quantity. Numerical examples are given to illustrate these theorems.

Optimal strategy in vendor-buyer alliances with quantity discount

This study develops an optimal pricing and replenishment policy of an integrated vendor-buyer inventory system with a finite vendor's replenishment rate. A quantity discount pricing strategy is derived to entice the buyer to order larger lot sizes in order to achieve the positive result of collaboration. A negotiation factor is incorporated to balance the cost saving between the vendor and the buyer. By revising Goyal's model, a significant cost reduction inventory model is derived. A numerical example and sensitivity analysis are carried out to illustrate the theory.

A reply to “Note on: An optimal batch size for a production system operating under periodic delivery policy”

Goyal and Cardenas-Barron (2002), have presented a comparative study of solutions, in a note in the current issue, of Goyal's model [Oper. Res. Q. 28 (1977) 539] and Sarker and Khan's model [Comput. Ind. Engng 37 (1999) 711] on joint batch sizing problems. In this reply, we would like to point out that these two models are not directly comparable. The differences of these two models are discussed and the validity of Sarker and Khan's model is demonstrated.

On the economic order quantity under conditions of permissible delay in payments

In this note, I amend Goyal's model by considering the difference between unit price and unit cost. I then establish an easy analytical closed-form solution to the problem. The theoretical results obtained here reveal the following two managerial phenomena. (1) In certain cases, the economic replenishment interval and order quantity decreases under the permissible delay in payments, which contradicts to Goyal's conclusion. It makes economic sense for some customers to order less quantity (or shorten the replenishment time interval) and to take the benefits of the permissible delay more frequently. (2) If a supplier wants to reduce his/her large level of inventory, then he/she should charge an excessive interest rate on his/her customer's outstanding amount after the credit term expires. Consequently, his/her customers will order to buy more quantity than the classical economic order quantity. As a matter of fact, these two managerial phenomena have been demonstrated in the decision making of using credit cards. For example, most credit card companies provide card holders 25 days of grace period, and charge 18–20% interest on the amount past due (ie, the second phenomenon). However, for a well-established credit card holder, he/she will take the benefit of 25 days of grace period constantly, but will not spend over his/her limit and face an excessive finance charge (ie, the first phenomenon).

The economic lot size of the integrated vendor‐buyer inventory system derived without derivatives

AbstractIn previous modellings of the integrated vendor–buyer system, the buyer's economic order quantity and the vendor's optimal number of deliveries are derived by setting the first derivatives to zero and solving the simultaneous equations. The Hessian matrix of second derivatives is used to prove the convexity of the objective function. This procedure can be difficult for students who lack the background of differential calculus. This study develops algebraically the optimal policy of the integrated vendor–buyer inventory system without using differential calculus. A significant cost reduction is also achieved when Goyal's model is modified. Copyright © 2002 John Wiley & Sons, Ltd.

Note on: An optimal batch size for a production system operating under periodic delivery policy

Goyal and Cardenas-Barron (Int. J. Prod. Econ. 72 (2001) 99) have presented a comparative study of Goyal's model (Oper. Res. Q. 28–3 (1977) 539) and Jamal and Sarker's model (Int. J. Prod. Econ. 32 (1993) 255) on joint batch sizing problems using a numerical example. In this note, we like to point out that these two models are not directly comparable. The differences of these two models are discussed and a possible proof for correctness of Jamal and Sarker's model is provided.