Integrated Vendor Buyer Research Articles

Minimax distribution-free procedure for integrated inventory model with backorder price discount and controllable lead time

The integrated vendor–buyer inventory model has been an object of study for a long time, but little is done about the effect of reducing lead time on the integrated inventory models with backorder price discount and defective items. The aim is to minimise the total related cost in the continuous review model by considering the order quantity, reorder point, lead-time and backorder price discount as decision variables. Moreover, we assume that the lead-time demand distribution is unknown but the mean and variance of lead-time demand are known. We apply the minimax distribution-free procedure to find the optimal solution. In the integrated system we discussed here, it assumes that, when the buyer places an order, the vendor produces m-times the order quantity in each production cycle, to reduce the set-up cost. In addition, the buyer offers a backorder price discount to the customers to motivate them for backorders. The arrival lot may contain some defective items, so its treatment is also incorporated in this paper. We develop an effective iterative procedure for finding the optimal policy and give a numerical example to illustrate the solution procedure.

Optimizing inventory and sales decisions in a two-stage supply chain with imperfect production and backorders

This paper develops a model of an integrated vendor–buyer supply chain with imperfect production and shortages. We assume that market demand is sensitive to the buyer’s selling price and thus study combined operations and pricing decisions in the supply chain. We first derive the expected profit per unit of time using the well-known renewal-reward theorem, and then maximize profit for the cases of independent and joint optimization. Numerical examples and a sensitivity analysis are provided to illustrate the proposed models. The results indicate that coordination and backordering improve the total expected profit of the system, and that both measures become more important for the supply chain as the price sensitivity of demand increases. Furthermore, the coordinated supply chain often prefers to perform inspection at the vendor, who is more familiar with the product and its deficiencies than the buyer in many cases.

Joint determination of the lot size and number of shipments for a family of integrated vendor–buyer systems considering defective products

In this paper we present two solution procedures to obtain the solution for a family of economic production quantity inventory models for an integrated vendor–buyer system considering that production system generates defective products, in which the number of shipments must be a discrete value and that the lot size can take continuous (case 1) or integer (case 2) values. These solution procedures provide a more realistic approach for solving a family of inventory models. Previous works only derive the optimal lot size considering the number of shipments as a given value or derive both the lot size and the number of shipments considering both decision variables as continuous. The number of shipments takes integer values in real life and thus it must be considered as a discrete variable. This paper revisits and solves previous inventory models considering the variables (lot size and number of shipments) according to their real-life nature.

Integrated vendor–buyer supply chain model with vendor’s setup cost reduction

This paper deals with an integrated vendor–buyer supply chain model. Two models are constructed based on the probability distribution of the lead time demand. The lead time demand follows a normal distribution in the first model. In the second model, we consider the distribution free approach for the lead time demand. For the second model, only mean and standard deviation are known. The aim of our model is to reduce the total system cost by considering the setup cost reduction of the vendor. Finally, some numerical examples are presented to illustrate the models.

An operational policy for an integrated inventory system under consignment stock policy with controllable lead time and buyers’ space limitation

This paper aims to enable the decision maker of an integrated vendor–buyer system, under Consignment Stock (CS) policy, to make the optimal/sub-optimal production/replenishment decisions where the buyer places a space limitation to the vendor and the lead-time is controllable with an extra investment. Within any production cycle, the vendor produces at a finite rate and ships the outputs to the buyer with a number of equal-sized lots. With a long-term consignment stock agreement, the vendor takes responsibility to maintain a certain inventory level in the buyer's warehouse. Some of the shipments are delayed so that the buyer's inventory does not go beyond the capacity limitation. The buyer compensates the vendor after the complete consumption of the products. The holding cost consists of a storage component and a financial component. Two constraint four-variable non-linear integer optimization models are established wherein the buyer space limitation is considered. Because the developed models are mathematically very difficult to solve, three doubly hybrid meta-heuristic algorithms are employed to solve the models. The computational results show that one of these three algorithms works very well both in the sense of the success rate and the mean CPU time. The analysis of the computational example also reveals the quantitative effects the buyer space limitation may have to the annual joint total expected cost (JTEC) of the integrated system.

An integrated vendor–buyer inventory model with imperfect items and planned back orders

In this paper, we develop a mathematical model to determine an integrated vendor–buyer inventory policy, where the vendor’s production process is imperfect and produces a certain number of defective items with a known probability density function. The vendor prepares for the repeating flow of orders of size \( {Q_{\mathrm{P}}}=nQ \) from the buyer by producing items in batches of size QP and planning to have each batch delivered to the buyers in n deliveries, each with a lot of Q units. Once the buyer receives the items, a 100 % screening process is conducted. We assume the screening process and demand take place simultaneously. We also assume that shortages are allowed and are completely back ordered. The objective is to minimize the total joint annual costs incurred by the vendor and the buyer. The expected annual integrated total cost is derived and a solution procedure is provided to find the optimal solution. Numerical examples show that the integrated model gives an impressive cost reduction in comparison to an independent decision by the buyer. The results also show that even though there is a cost associated with each back order, it is profitable for the company to have planned back orders if customers are willing to wait for the next delivery when a shortage occurs.

An Integrated Vendor-Buyer Cooperative Inventory Model for Items with Imperfect Quality and Shortage Backordering

We develop a model to determine an integrated vendor-buyer inventory policy for items with imperfect quality and planned backorders. The production process is imperfect and produces a certain number of defective items with a known probability density function. The vendor delivers the items to the buyer in small lots of equally sized shipments. Upon receipt of the items, the buyer will conduct a 100% inspection. Since each lot contains a variable number of defective items, shortages may occur at the buyer. We assume that shortages are permitted and are completely backordered. The objective is to minimize the total joint annual costs incurred by the vendor and the buyer. The expected total annual integrated cost is derived and a solution procedure is provided to find the optimal solution. Numerical examples show that the integrated model gives an impressive cost reduction in comparison to an independent decision by the buyer.

An easy method to derive the integrated vendor–buyer production–inventory model with backordering using cost-difference rate comparison approach

In growing competitive markets, close cooperative strategies among the vendors and the buyers are essential to cut the joint inventory cost and to have less response time to the supply chain players. Recently, Yang et al. (2007) [2] have presented an integrated two-stage production–inventory model and used the classical differential calculus method to derive the optimal lot size for the vendor and the buyer, and the number of deliveries in supply chain. Moreover, numerous researchers adopted other algebraic methods to derive the optimal solution for inventory models. However, in these papers, the cost comparisons method and arithmetic–geometric mean inequality do not focus on explicitly developing the mathematical expressions for two-stage inventory system with the backorders. In this study, we extend the inventory model proposed by Yang et al. (2007) [2], and consider a three-variable inventory problem and present an alternative approach to determine the optimal replenishment policy for the integrated vendor–buyer inventory system with backordering utilizing the cost-difference rate comparison approach.

Integrated vendor–buyer cooperative inventory model with controllable lead time, ordering cost reduction, and service-level constraint

In this study, an integrated vendor–buyer inventory model is proposed. The vendor produces a product in a batch environment and delivers it to a buyer facing normally distributed demand. In the current study, several assumptions are used that the combination of these assumptions has not been studied in previous papers. Here is the list of these assumptions: service-level constraint instead of shortage cost is used; ordering cost reduction is allowed, and decreasing lead time by a cost dependent on two factors including shortened lead time and ordering quantity is considered. An algorithm is developed to obtain the optimal solution of the proposed model. A numerical example is included to illustrate the results of the model. Also, the effects of assumptions used in this paper are evaluated and discussed.

An integrated vendor–buyer cooperative inventory model in an imperfect production process with shortage backordering

In this paper, we develop an integrated inventory model for vendor–buyer coordination under an imperfect production process. The proportion of defective items in each production lot is assumed to be stochastic and follows a known probability density function. The vendor inspects the items while they are being produced and delivers good-quality items to the buyer in small lots over multiple shipments. We assume that shortages are permitted and are completely backordered. The expected annual integrated total cost is derived and a solution procedure is provided to find the optimal solution. Numerical examples show that the integrated model gives an impressive cost reduction in comparison to an independent decision by the buyer. The results also show that if back orders are permitted, the expected integrated annual cost is less than that when back orders are not permitted. Therefore, if customers are willing to wait for the next shipment when a shortage occurs, it is profitable for the company to allow back orders although the delay incurs a penalty cost.

Integrated vendor buyer cooperative models for deteriorating inventory with quantity discount and price sensitive stock dependent demand

In the proposed problem, an optimal pricing and replenishment policy are discussed for a single – vendor and single – buyer. The demand is considered to be linear increasing function of stock displayed and linear decreasing function of retail price. The units in inventory are subject to constant rate of deterioration. It is observed that the joint decision is more beneficial to the vender as compared to the buyer. The quantity discount pricing is incorporated to attract the buyer to bid for the join decision. A negotiation factor is used to share the profits between vendor and buyer. Sensitivity analysis with numerical example suggests that the integrated total profit is very sensitive to the stock – dependent parameter, price-sensitive parameter and deterioration rate. It is seen that the total joint profit increases, therefore it is advantageous to opt for the joint decision with quantity discount price

An Integrated Vendor-Buyer Model with Uncertain Lead Time, Life Time under Inflation and Variable Holding Cost

This study considers the problem of a vendor which supplies an item to the buyer with imprecise partial backlogging rate of unsatisfied demand and non instantaneous deterioration rate considering variable holding cost, the effect of inflation and time value of money. The supplier’s lead time is a stochastic function of his managing cost. The extra costs incurred by the retailer due to the uncertain lead time in terms of shortage costs or lost sales costs should be owed by the supplier. A numerical example is cited to illustrate the results and its significant features. Finally, to study the effect of changes of demand parameters, deterioration, inflation and managing cost on supplier and the retailer’s profit, a sensitivity analysis is presented numerically.

Note on: An integrated vendor–buyer cooperative inventory model for items with imperfect quality

Huang [Huang, C., 2002. An integrated vendor–buyer cooperative inventory model for items with imperfect quality. Production Planning and Control, 13 (4), 355–361] proposes an integrated inventory model which allows a vendor and a buyer to minimise their expected integrated total cost function when the vendor's production process is not perfect. This note identifies two errors in the paper and provides a reformulation of the model.

Stochastic integrated vendor–buyer model with unstable lead time and setup cost

This paper presents a new vendor-buyer system where there are different objectives for both sides. The proposed method of this paper is different from the other previously published works since it considers different objectives for both sides. In this paper, the vendor’s emphasis is on the crashing of the setup cost, which not only helps him compete in the market but also provides better services to his customers; and the buyer’s aim is to reduce the lead time, which not only facilitates the buyer to fulfill the customers’ demand on time but also enables him to earn a good reputation in the market or vice versa. In the light of the above stated facts, an integrated vendor-buyer stochastic inventory model is also developed. The propsed model considers two cases for demand during lead time: Case (i) Complete demand information, Case (ii) Partial demand information. The proposed model jointly optimizes the buyer’s ordered quantity and lead time along with vendor’s setup cost and the number of shipments. The results are demonstrated with the help of numerical examples.

The economic lot size of the integrated vendor–buyer inventory system derived without derivatives: A simple derivation

In this paper, we first complement an inappropriate mathematical error on the total cost in two previous published research papers in inventory field. Then an arithmetic–geometric inequality method is proposed not only to simplify the algebraic method of completing prefect squares, but also to complement their shortcomings. In addition, we also provide a closed-form optimal solution to the integrated vendor–buyer inventory system without using complex derivatives. Finally, the proposed method seems to be easy-to-understand and simple-to-apply by individuals who are not familiar with differential calculus.

An optimal integrated vendor–buyer inventory policy under conditions of order-processing time reduction and permissible delay in payments

This study deals with the order-processing cost reduction and permissible delay in payments problem in the single-vendor single-buyer integrated inventory model. We consider that the order-processing cost can be reduced at an extra crashing cost, which varies with the reduction in the order-processing time length . In addition, the buyer is allowed a fixed time period before they settle the account with the vendor. The objective of this study is to minimize the annual integrated total cost by optimizing simultaneously the delivery interval, the number of deliveries per order and the investment cost in order-processing time. An integrated total cost function is derived, and an algorithm procedure is proposed for determining the optimal decision variables. Finally, numerical examples are provided to illustrate the algorithm procedure.

An integrated vendor–buyer inventory model with order-processing cost reduction and permissible delay in payments

Trade credit plays an important role in financing for many businesses and industries. For the buyers, purchased inventory can be considered to be financed in whole or in part with permissible delay in payments during the purchasing process. On the other hand, both the vendor and buyer take part in order-processing cost reduction by applying information technologies, such as EDI (electronic data interchange). The order-processing cost can also be reduced by adding certain capital investments, and this will affect the lot size decisions. This article develops an integrated inventory model to determine the optimal inventory policy under conditions of order-processing cost reduction and permissible delay in payments, and shows that the total annual variable cost function possesses some kinds of convexities. A solution procedure is provided to determine the optimal order policy. Finally, a numerical example is presented to illustrate the solution procedure.

An integrated vendor–buyer model with stock-dependent demand

We develop an integrated vendor–buyer model for a two-stage supply chain. The vendor manufactures the product and delivers it in a number of equal-sized batches to the buyer. The items delivered are presented to the end customers in a display area. Demand is assumed to be positively dependent on the amount of items displayed. The objective is to maximize total supply chain profit. The numerical analysis shows that buyer–vendor coordination is more profitable in situations when demand is more stock dependent. It also shows that the effect of double marginalization provides a link between the non-coordinated and the coordinated case.

“A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives”: A comment

Wee and Chung [Wee, H.M., Chung, C.T., 2007. A note on the economic lot size of the integrated vendor–buyer inventory system derived without derivatives. European Journal of Operational Research 177, 1289–1293] use the complete squares method to locate the optimal solution of the integrated system’s total cost TC ( Q , B , n ). However, their algebra method has shortcomings such that it may be invalid. The main purpose of this paper is to overcome those shortcomings and present complete proofs for Wee and Chung (2007).

Coordination between vendor and buyer considering trade credit and items of imperfect quality

Vendor–buyer models for trade credit issues in the existing literature always consider the items produced as perfect. However, a number of items in a lot may be of imperfect quality. This paper extends Chen and Kang's [2007. Integrated vendor–buyer cooperative models with variant permissible delay in payments. European Journal of Operational Research 183 (2), 658–673] study to develop integrated vendor–buyer models that consider a permissible delay in payment and imperfect quality to determine the optimal solutions of the buyer's order quantity and the frequency for each vendor's production run. To establish a cooperative relationship between the vendor and buyer for a win–win situation, a simple solution algorithm is presented to determine the cost saving of each side when the warranty cost per unit is increased. A numerical example is used to illustrate the results for the proposed models.