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 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.     

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.     

An inventory model involving back-order price discount when the amount received is uncertain

This paper presents and analyses the continuous review inventory model with order quantity, safety factor, back-order price discount, ordering cost and lead time as decision variables. Our work is based on the paper of Huang (2010 Huang, S.-P. (2010). Using simple and efficient algorithm involving ordering cost reduction and backorder price discount on inventory system under variable lead time. Information Technology Journal, 9(4), 804–810.[Crossref] , [Google Scholar]). We extend the model to incorporate the situation when the amount received is uncertain. The lead time demand is assumed follows a normal distribution. A solution procedure is developed to find the optimal solution. A numerical example is given to illustrate the model. A sensitivity analysis is also included to describe the effects of changes in the model parameters on the expected annual cost.     

An integrated production–distribution inventory system involving probabilistic defective and errors in quality inspection under variable setup cost

This paper presents a probabilistic defective vendor–buyer integrated inventory model with the consideration of quality inspection errors at the buyer's end and setup cost as function of capital investment. An integrated inventory model is established to find the optimal solutions of lot size, setup cost, and the total number of shipments from the vendor to the buyer in one production run, so that the joint expected total cost incurred has the minimum value. We consider three types of continuous probabilistic defective function to find the associated cost of the system. The expected total cost function is derived for each of these three distributions, its convexity is proved via differential calculus. An efficient iterative algorithm is designed to obtain the optimal solution of the model. The computational effort and time are small for the proposed algorithm and it is simple to implement. Numerical examples and sensitivity analysis are used to demonstrate the application and the performance of the proposed methodology. The computational results indicate that if we make decisions with the capital investment in reducing setup cost, it will help to lower the system cost, and we obtain a significant amount of savings to increase the competitive edge in business.     

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.     

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 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.     

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.     

A periodic review inventory model involving fuzzy expected demand short and fuzzy backorder rate

The purpose of this paper is to extend [Ouyang, L. Y., Chuang, B. R. (2001). A periodic review inventory-control system with variable lead time. International Journal of Information and Management Sciences, 12, 1–13] periodic review inventory model with variable lead time by considering the fuzziness of expected demand shortage and backorder rate. We fuzzify the expected shortage quantity at the end of cycle and the backorder (or lost sales) rate, and then obtain the fuzzy total expected annual cost. Using the signed distance method to defuzzify, we derive the estimate of total expected annual cost in the fuzzy sense. For the proposed model, we provide a solution procedure to find the optimal review period and optimal lead time in the fuzzy sense so that the total expected annual cost in the fuzzy sense has a minimum value. Furthermore, a numerical example is provided and the results of fuzzy and crisp models are compared.     

Quality improvement and backorder price discount under controllable lead time in an inventory model

The proposed study investigates a continuous review inventory model with order quantity, reorder point, backorder price discount, process quality, and lead time as decision variables. An investment function is used to improve the process quality. Two models are developed based on the probability distribution of lead time demand. The lead time demand follows a normal distribution in the first model and in the second model it does not follow any specific distribution but mean and standard deviation are known. We prove two lemmas to obtain optimal solutions for the normal distribution model and distribution free model. Finally, some numerical examples are given to illustrate the model.     

Investing in lead-time variability reduction in a collaborative vendor–buyer supply chain model with stochastic lead time

This study deals with investing in lead-time variability reduction problems for the integrated vendor–buyer supply chain system with partial backlogging under stochastic lead time. We consider that lead time variability can be reduced through further investment; more specifically, a logarithmic investment function is used that allows investment to be made to reduce lead-time variability. By using the proposed supply chain model, considerable savings can be achieved to increase the competitive edge. The objective is to derive the optimal production/ordering strategy, and the best investment policy to minimize joint total cost. A computer code using the software, Mathematica, is developed to derive the optimal solution. Furthermore, we discuss the sensitivity of the optimal solution together with the changes of the values of the parameters associated with the model for decision-making. Various numerical examples are given to illustrate the results.     

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.     

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.     

可控前置时间的连续盘点联合库存策略研究

鉴于控制前置时间对精益生产系统的重要性,在考虑买方与卖方合作的同时,扩展Goyal生产批量交货的假设,假设需求服从正态分布,以订购数量、运送次数与前置时间为决策变量,建立前置时间可控制的联合库存模型以确定适当的库存水平,使得库存总成本最小化,且可以通过协商在买卖双方之间进行节省成本的分配。进行了数值范例,并将联合库存模型与Banerjee模型、Goyal模型进行了比较。     

Joint replenishment policy with backordering and special sale

In this paper, an inventory control model with a joint replenishment policy and a temporary discount is developed. We assume that shortage is allowed and buyer uses an economic order quantity inventory control model. Different cases based on ordering policies for the first joint replenishment if the special order is not taken, and coincidence of a special period length with a positive or negative inventory level of the last regular period length, are investigated. Furthermore, several theorems are proved through which closed-form solutions are obtained. At the end, two numerical examples illustrate the different situations that the buyer may face and sensitivity analyses for both examples are reported.     

Optimal transfer,ordering and payment policies for joint supplier–buyer inventory model with price-sensitive trapezoidal demand and net credit

This study aims at formulating an integrated supplier–buyer inventory model when market demand is variable price-sensitive trapezoidal and the supplier offers a choice between discount in unit price and permissible delay period for settling the accounts due against the purchases made. This type of trade credit is termed as ‘net credit’. In this policy, if the buyer pays within offered time M₁, then the buyer is entitled for a cash discount; otherwise the full account must be settled by the time M₂; where M₂ > M₁ ? 0. The goal is to determine the optimal selling price, procurement quantity, number of transfers from the supplier to the buyer and payment time to maximise the joint profit per unit time. An algorithm is worked out to obtain the optimal solution. A numerical example is given to validate the proposed model. The managerial insights based on sensitivity analysis are deduced.     

Combined optimal price and optimal inventory replenishment policies when a sale results in increase in demand

When a supplier reduces the price of a product temporarily a buyer might place a large order and offer a sale on these units to its customers. In most cases a price discount results in an increase in demand. In this paper we relax the constant demand assumption made in most studies of inventory systems with price changes. We analyze the options available to a buyer and develop profit functions for different combinations of sales period and replenishment time and present optimal ordering policies. The paper also presents a procedure to include any relationship between price and demand to determine the combined optimal price and optimal order quantity.     

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.     

Channel coordination with price discount mechanism under price‐sensitive market demand

This paper considers a two‐echelon supply chain with single vendor single buyer for trading a single product. The buyer's demand has been assumed to be price‐sensitive. We have assumed three‐level inspection at the vendor's end in order to maintain good quality of the items delivered to the buyer. We have developed models for both decentralized and centralized scenarios and have determined the optimal solution using the basic concepts of analytic geometry and algebra. In addition, we have proposed a price discount mechanism, where the vendor has provided discounts on the purchase cost to the buyer against the buyer's (increased) order quantity. Finally, in order to illustrate and validate the proposed model, a numerical example and sensitivity analysis is carried out, which has provided some important managerial implications. It has been observed from the numerical study that the proposed price discount policy coordinates the supply chain and has improved the profitability of the supply chain and its members.