Supply chain coordination with defective items and quantity discount

This study develops an integrated inventory system involving defective items and quantity discount for optimal pricing and ordering strategies. The model analysed in this study is one in which the buyer orders a quantity, the vendor produces more than buyer's order quantity in order to reduce set-up cost, and then he/she offers an all-units quantity discount to the buyer. Our objective is to determine the optimal order quantity, retail price, mark-up rate, and the number of shipments per production run from the vendor to the buyer, so that the entire supply chain joint total profit incurred has a maximum value. Furthermore, an algorithm of finding the optimal solution is developed. Numerical examples are provided to illustrate the theoretical results.     

Trade credit financing in the vendor–buyer inventory system with ordering cost reduction,transportation cost and backorder price discount when the received quantity is uncertain

Previous ordering cost reduction vendor–buyer inventory models with backorder price discount usually assumed that the buyer must pay to the vendor for the ordered items as soon as the items are received, the received quantity is same as the ordered quantity and the transportation cost is independent of the shipment lot-size. In practice, however, the vendor is willing to offer the buyer a certain credit period without interest to promote market competition as well as the buyer's quantity received may not match with the ordered quantity due to unavailability of the raw material, worker's strike, human errors in counting, transcribing, etc. Furthermore, the discounts are offered for the transportation cost of large ordered quantities. This paper derives a single-vendor single-buyer supply chain model for the ordering cost reduction inventory system with backorder price discount, taking into consideration the effect of transportation cost discount and the condition of permissible delay in payments include the case where the buyer's received quantity does not necessarily match the quantity requisitioned. We take the transportation cost as a function of the shipment lot-size and it is taken to be in an all-unit-discount cost format. Thus we incorporate transportation cost explicitly into the model and develop optimal solution procedures for solving the proposed inventory problem. Numerical example and sensitivity analyses are given to demonstrate the applications and performance of the proposed methodology.     

Supply chain coordination model with controllable lead time and service level constraint

We consider the coordination issue in a decentralized supply chain composed of a vendor and a buyer in this paper. The vendor offers a single product to the buyer who is faced with service level constraint. In addition, lead time can be reduced by added crashing cost. We analyze two supply chain inventory models. The first one is developed under decentralized mode based on Stackelberg model, the other one is developed under centralized mode of the integrated supply chain. The solution procedures are also provided to get the optimal solutions of these two models. Finally, a price discount mechanism is proposed to induce both the vendor and the buyer to accept the centralized model. The feasibility and efficiency of the proposed models are manifested by numerical examples and some managerial implications are highlighted.     

Supply chain coordination using time-based temporary price discounts

In this paper, the issue of supply chain coordination (SCC) in a buyer–seller supply chain (SC) with an order size constraint is investigated. The buyer keeps safety stock to cope with lead time demand uncertainties from customers’ side. Unsatisfied demand will be lost. Therefore the whole SC sales volume depends on the service level provided by the buyer. By proposing a time-based temporary price discount in each replenishment cycle, the seller intends to convince the buyer to optimize its safety stock globally. Maximum and minimum discounts, which are acceptable for both parties, are determined and an appropriate discount schedule is derived. A set of numerical experiments are conducted to show performance of the proposed model. The results show that the safety stock coordination is profitable; the proposed model is capable of coordinating supply chain. In addition, the model can share extra benefits between SC members fairly.     

Buyer–vendor coordination for fixed lifetime product with quantity discount under finite production rate

Buyer–vendor coordination has been widely addressed; however, the fixed lifetime of the product is seldom considered. In this paper, we study the coordination of an integrated production-inventory system with quantity discount for a fixed lifetime product under finite production rate and deterministic demand. We first derive the buyer’s ordering policy and the vendor’s production batch size in decentralised and centralised systems. We then compare the two systems and show the non-coordination of the ordering policies and the production batch sizes. To improve the supply chain efficiency, we propose quantity discount contract and prove that the contract can coordinate the buyer–vendor supply chain. Finally, we present analytically tractable solutions and give a numerical example to illustrate the benefits of the proposed quantity discount strategy.     

An integrated vendor–buyer inventory model with backorder price discount and effective investment to reduce ordering cost

The single-vendor single-buyer integrated production inventory system has been an object of study for a long time, but little is known about the effect of investing in reducing ordering cost on the integrated inventory models with backorder price discount and variable lead time. The purpose of this article is to investigate in the continuous review model with backorder price discount and variable lead time to effectively increase investment and to reduce the joint expected annual total cost. The integrated strategy discussed here is one in which the buyer orders a quantity, then the vendor produces n times order quantity in each production cycle, in order to reduce setup cost. In addition, the buyer offers backorder price discounts to the customers that may motivate the customers’ desire for backorders, and buyer ordering cost can be reduced through effective investment. An integrated inventory model is established to find the optimal solutions of order quantity, ordering cost, backorder price discount, lead time, and the number of shipments from the vendor to the buyer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, numerical examples are used to demonstrate the benefits of the model.     

Quantity discount model to increase vendor's profits and decrease buyer's costs

A generalized optimal quantity discount is developed which minimizes the buyer's costs and maximizes the vendor's profit. Through the dual variable, a relationship is established between vendor and buyer models, and related pricing schemes are developed.     

Supply chain coordination with two production modes and random demand depending on advertising expenditure and selling price

This article discusses production and order as well as advertising coordination issues in a single-manufacturer single-buyer supply chain, where the manufacturer sells a newsvendor-type product through the buyer who faces a random demand depending on advertising expenditure and selling price. The buyer has two ordering opportunities: the one happens before the beginning of the season, and the other takes place at the end of the season. The ordered items are produced by the manufacturer in two production modes for different requirements. The first production mode is relatively cheap but requires a long lead-time, whereas the second is expensive but offers quick response. Under such a setting, the centralised and decentralised decision models are developed, respectively, and the closed form solution to each model is provided as well. Moreover, we point out that the traditional revenue-sharing contract fails to coordinate the supply chain. We thus propose an improved revenue-sharing contract that requests the manufacturer not only shares the buyer's revenue but also bears a portion of the buyer's operating costs. Such a contract can achieve perfect coordination of the supply chain and arbitrarily allocate its profit between two parties.     

Two level supply chain coordination with delay in payments for fixed lifetime products

In this paper, a single-vendor, single-buyer supply chain system for fixed lifetime products is considered in the settings of both decentralized and centralized models. In the decentralized model, the vendor is the decision-maker of the supply chain. In particular, we study the coordination between the vendor and the buyer that allows the buyer to delay his payment in compensation for altering his order size. This policy has been studied in the literature for the products with unlimited lifetime. In this paper, we focus on the products with fixed and limited lifetime which is common in practice. To evaluate the efficiency of the proposed delay in payments policy, a centralized decision-making problem is modeled, where there is a common decision-maker for both the vendor and the buyer. We derive analytically tractable solutions to the proposed models. Furthermore, we prove that the decentralized model can achieve the same minimal cost as the centralized model when the vendor and the buyer’s costs of capital are equal. A detailed numerical example is presented to illustrate the benefit of the proposed delay in payments policy.     

Cost allocation model for optimizing supply chain inventory with controllable lead time

The coordination issue of a decentralized supply chain composed of a vendor and a buyer is considered in this paper. The vendor offers a single product to the buyer and the lead time can be controllable with adding crashing cost. Two supply chain inventory models with controllable lead time under different decision modes are considered, one is proposed under decentralized model based on Stackelberg model, the other is proposed under centralized model of the integrated supply chain. The solution procedures are also suggested to get the optimal solutions of these two models. In addition, an asymmetric Nash bargaining model based on satisfaction level is also developed to get the best cost allocation ratio between the vendor and the buyer by taking their individual rationalities into consideration. The results of numerical example show that shortening lead time reasonably can reduce inventory cost and the cost allocation model based on satisfaction level developed in this paper is effective.     

Multiproduct multiple-buyer single-vendor supply chain problem with stochastic demand, variable lead-time, and multi-chance constraint

In this paper, a multi-product multi-chance constraint joint single-vendor multi-buyers inventory problem is considered in which the demand follows a uniform distribution, the lead-time is assumed to vary linearly with respect to the lot size, and the shortage in combination of backorder and lost-sale is assumed. Furthermore, the orders are placed in multiple of packets, there is a limited space available for the vendor, there are chance constraints on the vendor service rate to supply the products, and there is a limited budget for each buyer to purchase the products. While the elements of the buyers’ cost function are holding, shortage, order and transportation costs, the set up and holding costs are assumed for the vendor. The goal is to determine the re-order point and the order quantity of each product for each buyer such that the chain total cost is minimized. We show the model of this problem to be a mixed integer nonlinear programming type and in order to solve it a particle swarm optimization (PSO) approach is used. To justify the results of the proposed PSO algorithm, a genetic algorithm (GA) is applied as well to solve the problem. Then, the quality of the results and the CPU times of reaching the solution are compared through three numerical examples that are given to demonstrate the applicability of the proposed methodology in real world inventory control problems. The comparison results show the PSO approach has better performances than the GA method.     

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.     

Two-echelon supply chain inventory model with controllable lead time and service level constraint

This paper considers a two-echelon supply chain inventory problem consisting of a single-vendor and a single-buyer. In the system under study, a vendor produces a product in a batch production environment and supplies it to a buyer facing a stochastic demand, which is assumed to be normally distributed. Also, buyer’s lead time is controllable which can be shortened at an added cost and all shortages are backordered. A model has been formulated for an integrated vendor–buyer problem to jointly determine the optimal order quantity, lead time and the number of shipments from the vendor to the buyer during a production cycle while minimizing the total expected cost of the vendor–buyer integrated system. It is often difficult to estimate the shortage cost in inventory systems. Therefore, instead of having a shortage cost term in the objective function, a service level constraint (SLC) is included in the model that requires a certain proportion of demands to be met in each cycle. An efficient procedure has been suggested to find the bounds on number of shipments and then, an algorithm is developed to obtain the optimal solution of the proposed model. A numerical example is included to illustrate the algorithmic procedure and the effects of key parameters are studied to analyze the behavior of the model. Finally, the savings of buyer and vendor are investigated from implementation of joint optimization model over the model in which they minimize their own cost independently.     

Integrated vendor--buyer inventory system with sublot sampling inspection policy and controllable lead time

This article investigates the impact of inspection policy and lead time reduction on an integrated vendor--buyer inventory system. We assume that an arriving order contains some defective items. The buyer adopts a sublot sampled inspection policy to inspect selected items. The number of defective items in the sublot sampling is a random variable. The buyer's lead time is assumed reducible by adding crash cost. Two integrated inventory models with backorders and lost sales are derived. We first assume that the lead time demand follows a normal distribution, and then relax the assumption about the lead time demand distribution function and apply the minimax distribution-free procedure to solve the problem. Consequently, the order quantity, reorder point, lead time and the number of shipments per lot from the vendor to the buyer are decision variables. Iterative procedures are developed to obtain the optimal strategy.     

Nash equilibrium solution in a vendor–buyer supply chain model with permissible delay in payments

In practice, vendors (or sellers) often offer their buyers a fixed credit period to settle the account. The benefits of trade credit are not only to attract new buyers but also to avoid lasting price competition. On the other hand, the policy of granting a permissible delay adds not only an additional cost but also an additional dimension of default risk to vendors. In this paper, we will incorporate the fact that granting a permissible delay has a positive impact on demand but negative impacts on both costs and default risks to establish vendor–buyer supply chain models. Then we will derive the necessary and sufficient conditions to obtain the optimal solution for both the vendor and the buyer under non-cooperative Nash equilibrium. Finally, we will use two numerical examples to show that (1) granting a permissible delay may significantly improve profits for both the vendor and the buyer, and (2) the sensitivity analysis on the optimal solution with respect to each parameter.     

Optimal pricing contracts and level of information asymmetry in a supply chain

This study considers a stylized supply chain model consisting of a dominant supplier and a buyer, in which the latter possesses superior knowledge of his private cost information. The supplier's imperfect knowledge about the buyer's cost is denoted by a subjective distribution. By assuming that the distribution is uniform, we first derive the explicit expressions of the optimal equilibrium outcomes of two contract formats offered by the supplier, the simple price‐only contract and sophisticated menu of contracts, respectively. Based on the optimal results, we continue to investigate how the supplier's expected profit varies with the level of information asymmetry, which is measured by the variance of the supplier's subjective distribution. Our findings indicate that the supplier's expected profits initially decrease and then increase as the supplier's knowledge of the buyer's cost becomes increasingly uncertain. That is, if the supplier could choose the level of information asymmetry, she would prefer a symmetric case (with the variance being zero) or a totally asymmetric case (with the variance being as high as possible). Numerical experiments demonstrate that our result still holds when we take into account such distributions as a truncated gamma and normal, and a fixed support of the distribution function instead of a moving support.     

Supply chain coordination for a deteriorating item with stock and price‐dependent demand under revenue sharing contract

This paper deals with an integrated single‐manufacturer single‐retailer supply chain model for a single item. The market demand is assumed to be dependent on both the on‐hand stock and price, and the manufacturer and the retailer are in an agreement of lot‐for‐lot policy. The proposed model is developed under the contract that the retailer offers the manufacturer a percentage of revenue (s)he generates by selling a lot. We determine optimal policies for both the centralized and decentralized coordination systems. A comparison of these policies is made with a numerical example. Sensitivity analysis is performed to examine the stability of the solution.     

Pricing,manufacturing and inventory policies for raw material in a three-level supply chain

We studied a decentralised three-layer supply chain including a supplier, a producer and some retailers. All the retailers order their demands to the producer and the producer order his demands to the supplier. We assumed that the demand is price sensitive and shortage is not permitted. The goal of the paper is to optimise the total cost of the supply chain network by coordinating decision-making policy using Stackelberg–Nash equilibrium. The decision variables of our model are the supplier's price, the producer's price and the number of shipments received by the supplier and producer, respectively. To illustrate the applicability of the proposed model numerical examples are presented.     

Vendor–buyer inventory models with trade credit financing under both non-cooperative and integrated environments

Most researchers studied vendor–buyer supply chain inventory policies only from the perspective of an integrated model, which provides us the best cooperative solution. However, in reality, not many vendors and buyers are wholly integrated. Hence, it is necessary to study the optimal policies not only under an integrated environment but also under a non-cooperative environment. In this article, we develop a supply chain vendor–buyer inventory model with trade credit financing linked to order quantity. We then study the optimal policies for both the vendor and the buyer under a non-cooperative environment first, and then under a cooperative integrated situation. Further, we provide some numerical examples to illustrate the theoretical results, compare the differences between these two distinct solutions, and obtain some managerial insights. For example, in a cooperative environment, to reduce the total cost for both parties, the vendor should either provide a simple permissible delay without order quantity restriction or offer a long permissible delay linked order quantity. By contrast, in a non-cooperative environment, the vendor should provide a short permissible delay to reduce its total cost.     

Quasi-optimal integrated production, inventory and maintenance policies for a single-vendor single-buyer system with imperfect production process

In this paper we deal with the integrated supply chain management problem in the context of a single vendor-single buyer system for which the production unit is assumed to randomly shift from an in-control to an out-of-control state. At the end of each production cycle, a corrective or preventive maintenance action is performed, depending on the state of the production unit, and a new setup is carried out. Two different integrated production, shipment and maintenance strategies are proposed to satisfy the buyer’s demand at minimum total cost. The first one suggests that the buyer orders batches of size nQ and the vendor produces nQ and makes equal shipments of size Q. The second policy proposes that to satisfy the same ordered quantity, the vendor produces separately smaller batches of size Q, n times. The total integrated average cost per time unit corresponding to each strategy is considered as the performance criterion allowing choosing the best policy for any given situation.