Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period

A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the trade credit scenario such as permissible delay in payments. In real life business, we observe that the demand is a function of both the selling price and credit period rather than the constant demand. Incorporating this demand function to the retailer of a supply chain, we develop an EPQ – based model for perishable items under two-echelon trade financing. The purpose of this paper is to maximize the profit by determining the optimal selling price, credit period and replenishment time. It is shown that the model developed by Jaggi et al. [Jaggi, J. K., Goyal, S. K., & Goel, S. K., 2008. Retailer’s optimal replenishment decisions with creditlinked demand under permissible delay in payments. European Journal of Operational Research, 190, 130–135] can be treated as a special case of this paper. Finally, through numerical examples, sensitivity analysis shows the influence of key model parameters.     

A complete proof on the solution procedure for non-instantaneous deteriorating items with permissible delay in payment

Ouyang et al. [Ouyang, L. Y., Wu, K. S., & Yang C. T. (2006). A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments. Computers & Industrial Engineering, 51, 637–651] present an inventory model for non-instantaneous deteriorating items with permissible delay in payments. They develop some theorems to characterize the optimal solutions and provide an easy-to-use method to find the optimal replenishment cycle time. Although, their inventory models are correct and interesting, processes of arguments to derive those theorems and the easy-to-use method to search for the optimal replenishment cycle time are not complete. So, the main purpose of this paper is to overcome those shortcomings and present complete proofs for Ouyang et al. (2006).     

Optimal ordering policies for deteriorating items using a discounted cash-flow analysis when a trade credit is linked to order quantity

In today’s competitive market, in order to obtain a competition advantage, the supplier often offers the purchaser a longer permissible delay in payments or a price discount if the order quantity is greater than or equal to a predetermined quantity. As a result, in this paper, we establish an inventory model for the purchaser in which the supplier provides different trade credits. We then solve the inventory problem by using a discounted cash-flow (DCF) approach, characterize the optimal solution, and obtain some theoretical results to find the optimal order quantity and the optimal replenishment time. Finally, we provide several numerical examples to illustrate the results.     

An inventory model for deteriorating items with partial backlogging and permissible delay in payments

The paper deals with an inventory model with a varying rate of deterioration and partial backlogging rate under the condition of permissible delay in payments. The existing literature on the subject generally deal with situations where the payment of an order is made on the receipt of items by the inventory system and shortages are either completely backlogged or fully lost. In this paper, a varying deterioration rate of time and the condition of permissible delay in payments used in conjunction with the economic order quantity model are the focus of discussion. In addition, the shortages are neither completely backlogged nor completely lost assuming the backlogging rate to be inversely proportional to the waiting time for the next replenishment. Numerical examples are presented to illustrate the model.     

基于订货阈值的供应商延期支付策略设计

针对完全信息下以供应商为核心企业的二级供应链库存系统,将全单位延期支付与部分延期支付两种手段相结合,并在考虑延期支付期限与订货量相关的条件下,设计了基于供应商视角和订货阈值的延期支付策略,从而得出此策略能使系统利润达到帕累托最优.最后,通过数值算例对相关结论进行了验证和灵敏度分析.     

Optimal inventory replenishment policy for the EPQ model under trade credit derived without derivatives

Within the economic production quantity (EPQ) framework, the main purpose of this article is to deal with Chung and Huang's model (K.J. Chung, and Y.F. Huang,“The optimal cycle time for EPQ inventory model under permissible delay in payments,” Int. J. Prod. Econ., 84, pp. 307–318, 2003) and extend Chung and Huang's model (2003) by considering the unit selling price higher than the unit purchasing cost using the algebraic method to determine the optimal inventory replenishment policy for the retailer under trade credit. This article provides this algebraic approach which could be used easily to introduce the basic inventory theories to younger students who lack the knowledge of calculus. In addition, we develop an easy-to-use procedure to find the optimal inventory replenishment policy for the retailer in the extended model developed in this article. Finally, numerical examples are given to illustrate the result obtained in our extended model.     

Deterministic inventory model for deteriorating items with trade credit financing and capacity constraints

This work presents an inventory model for optimizing the replenishment cycle time for a single deteriorating item under a permissible delay in payments and constraints on warehouse capacity (owned warehouse capacity, with excess inventory stored in rental warehouses). Rented warehouses are assumed to charge higher unit holding costs than owned warehouses. Furthermore, item deterioration rates are assumed to differ between warehouses. This study has two main purposes: First, the mathematical models of the inventory system are established under the above conditions. Second, this study demonstrates that the optimal solution not only exists but is unique, and two theorems are devised for determining the optimal replenishment cycle time. Finally, numerical examples are presented to illustrate the resulting theorems.     

Two ant-colony algorithms for minimizing total flowtime in permutation flowshops

The problem of scheduling in flowshops with the objective of minimizing total flowtime is studied. For solving the problem two ant-colony algorithms are proposed and analyzed. The first algorithm refers to some extent to ideas by Stuetzle [Stuetzle, T. (1998). An ant approach for the flow shop problem. In: Proceedings of the sixth European Congress on intelligent techniques and soft computing (EUFIT '98) (Vol. 3) (pp. 1560–1564). Aachen: Verlag Mainz] and Merkle and Middendorf [Merkle, D., & Middendorf, M. (2000). An ant algorithm with a new pheromone evaluation rule for total tardiness problems. In: Proceedings of the EvoWorkshops 2000, lecture notes in computer science 1803 (pp. 287–296). Berlin: Springer]. The second algorithm is newly developed. The proposed ant-colony algorithms have been applied to 90 benchmark problems taken from Taillard [Taillard, E. (1993). Benchmarks for basic scheduling problems. European Journal of Operational Research, 64, 278–285]. A comparison of the solutions yielded by the ant-colony algorithms with the best heuristic solutions known for the benchmark problems up to now, as published in extensive studies by Liu and Reeves [Liu, J., & Reeves, C.R. (2001). Constructive and composite heuristic solutions to the P//ΣC_i scheduling problem. European Journal of Operational Research, 132, 439–452, and Rajendran and Ziegler [Rajendran, C., & Ziegler, H. (2004). Ant-colony algorithms for permutation flowshop scheduling to minimize makespan/total flowtime of jobs. European Journal of Operational Research, 155, 426–438], shows that the presented ant-colony algorithms are better, on an average, than the heuristics analyzed by Liu and Reeves and Rajendran and Ziegler.     

Supply chain models for deteriorating products with ramp type demand rate under permissible delay in payments

In this paper an order level inventory model for deteriorating items with general ramp type demand rate under conditions of permissible delay in payments is proposed. In this model shortages are allowed and partially backlogged. The backlogging rate is variable and dependent on the waiting time for the next replenishment. Its study requires exploring the feasible ordering relations between the time parameters appeared, which leads to three models. For each model the optimal replenishment policy is determined. The sufficient conditions of the existence and uniqueness of the optimal solutions are also provided. Suitably selected numerical examples highlight the obtained results. Sensitivity analysis of the optimal solution with respect to major parameters of the system has been carried out and the implications are discussed.     

A comprehensive note on “Lot‐sizing decisions for deteriorating items with two warehouses under an order‐size‐dependent trade credit”

To reduce inventory and increase sales, the supplier frequently offers the retailer a permissible delay in payments if the retailer orders more than or equal to a predetermined quantity. In 2012, Liao et al. proposed an economic order quantity model for a retailer with two warehouses when the supplier offers a permissible delay linked to order quantity. In this paper, we attempt to overcome some shortcomings of their mathematical model. Then, we apply some existing theoretical results in fractional convex programs to prove that the annual total variable cost is pseudoconvex. Hence, the optimal solution exists uniquely, which simplifies the search for the global minimum solution to a local minimum solution. Finally, we run a couple of numerical examples to illustrate the problem and compare the optimal solutions between theirs and ours.     

An EPQ model for deteriorating items with inventory-level-dependent demand and permissible delay in payments

This article develops an inventory model for exponentially deteriorating items under conditions of permissible delay in payments. Unlike the existing related models, we assume that the items are replenished at a finite rate and the demand rate of the items is dependent on the current inventory level. The objective is to determine the optimal replenishment policies in order to maximise the system's average profit per unit of time. A simple method is shown for finding the optimal solution of the model based on the derived properties of the objective function. In addition, we deduce some previously published results as the special cases of the model. Finally, numerical examples are used to illustrate the proposed model. Some managerial insights are also inferred from the sensitive analysis of model parameters.     

A multi-product EOQ model with pricing consideration—T. C.E. Cheng''s momdel revisited

We present two major revisions/corrections regarding a recent paper by T.C.E. Cheng [Computers ind. Engng 18, 529–534 (1990)]. Firstly, we note that a critical assumption of equal replenishment cycle length for all products is stated, but not incorporated into the mathematical formulation in Cheng. In this paper, we re-formulate the problem with equal replenishment cycle length incorporated and derive the corresponding Kuhn-Tucker optimality conditions. Next, under linear demand assumption, we show that the closed-form solutions provided by Cheng may result in non-optimal solutions. The reason is that Cheng failed to derive conditions under which closed-form solutions may be optimal. In this paper, by employing trigonometric methods {see e.g. Chap. 2 of Griffiths [Introduction of the Theory of Equations. Wiley, New York (1945).] or Appendix of Porteus [Mgmt Sci. 31, 998–1010 (1985)]}, we derive an optimal closed-form solution that is unqiue and obtain conditions under which the optimal closed-form solution is valid.     

A finite time horizon inventory model with deterioration and time-value of money under the conditions of permissible delay in payments

A varying deterioration rate, time-value of money and the condition of permissible delay in payments used in conjunction with the EOQ model are the focus of discussion. The replenishment number and fraction of each cycle in which there is no shortage are both determined to minimize the present value of inventory cost over a finite planning horizon. Two special cases and numerical examples are presented to illustrate the model.     

A production lot size inventory model for deteriorating items

In this paper, an attempt has been made to develop a production lot size model which incorporates an unfilled-order backlog for an inventory system with exponential decaying items. Approximate expressions are obtained for the optimum production lot size, the production cycle time and the total cycle time. The theory is illustrated by considering a numerical example of this class. Also it is shown that earlier models developed by Ghare and Schrader [3] and Misra [5] can be obtained as particular cases by choosing appropriate values for the various parameters of the model.     

An accurate and reliable solution procedure for the EOQ model with deteriorating items under supplier credits linked to the ordering quantity

The inventory problem consists of two parts: (1) the modelling and (2) the solution procedure. The modelling can provide insight to solve the inventory problem and the solution procedure involves the implementation of the inventory model. Basically, the modelling and the solution procedure to the inventory problem are equally important. Chang et al.’s inventory model [Chang, C. T., Ouyang, L. Y., & Teng, J. T (2003). An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. Applied Mathematical Modelling, 27, 983–996] is correct and interesting. However, they ignore the explorations of the functional behaviors of the annual total relevant cost to locate the optimal solutions such that proofs of their solution procedures are not perfect from the viewpoint of logic. The main purposes of this paper are to provide accurate and reliable solution procedures to improve [Chang, C. T., Ouyang, L. Y., & Teng, J. T (2003). An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. Applied Mathematical Modelling, 27, 983–996].     

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-steady friction in transient polytropic flow

An accurate method of integrating quasi-steady friction in transient polytropic flows is introduced and is justified both physically and analytically. When applied to subsonic flows, it is found to be especially informative at higher Mach numbers where existing methods of integration can lead to serious errors in mass conservation. The method is used herein to assess the accuracy of widely used finite difference representations of quasi-steady friction, attention focusing principally on solutions obtained using the method of characteristics.

The new method is exact in the special case of steady flows. Of the approximate methods considered, the popular approximation V_t¦V_{t − Δt}¦Δt is found to perform well. A modification introduced by Karney and McInnis (Journal of Hydology Engineering, ACSE, 1992, 118(7), 1014–1030 [1]) improves the correlation for downstream characteristic lines, but has the opposite effect for upstream characteristics. A simple cubic scheme, loosely related to a more complex one used successfully by Murray (Proceedings of the International Conference on Unsteady Flow and Fluid Transients, 29 September–1 October. Balkema, Amsterdam, 1992, pp. 143–157 [2]), illustrates difficulties in higher-order schemes.

The new method is not exact in transient flows, but it gives good accuracy in slowly varying flows and it is plausible in rapidly varying flows. Using it as a benchmark, the approximation V_t¦V_{t − Δt}¦Δt is again shown to perform well. All of the schemes considered are unreliable when the flow is about to become choked within the region of integration     

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.     

A supplier selection and order allocation problem with stochastic demands

We consider a system comprising a retailer and a set of candidate suppliers that operates within a finite planning horizon of multiple periods. The retailer replenishes its inventory from the suppliers and satisfies stochastic customer demands. At the beginning of each period, the retailer makes decisions on the replenishment quantity, supplier selection and order allocation among the selected suppliers. An optimisation problem is formulated to minimise the total expected system cost, which includes an outer level stochastic dynamic program for the optimal replenishment quantity and an inner level integer program for supplier selection and order allocation with a given replenishment quantity. For the inner level subproblem, we develop a polynomial algorithm to obtain optimal decisions. For the outer level subproblem, we propose an efficient heuristic for the system with integer-valued inventory, based on the structural properties of the system with real-valued inventory. We investigate the efficiency of the proposed solution approach, as well as the impact of parameters on the optimal replenishment decision with numerical experiments.     

On the validity of the arithmetic–geometric mean method to locate the optimal solution in a supply chain system

Cardenas-Barron [Cardenas-Barron, L.E. (2010) ‘A Simple Method to Compute Economic order Quantities: Some Observations’, Applied Mathematical Modelling, 34, 1684–1688] indicates that there are several functions in which the arithmetic–geometric mean method (AGM) does not give the minimum. This article presents another situation to reveal that the AGM inequality to locate the optimal solution may be invalid for Teng, Chen, and Goyal [Teng, J.T., Chen, J., and Goyal S.K. (2009), ‘A Comprehensive Note on: An Inventory Model under Two Levels of Trade Credit and Limited Storage Space Derived without Derivatives’, Applied Mathematical Modelling, 33, 4388–4396], Teng and Goyal [Teng, J.T., and Goyal S.K. (2009), ‘Comment on ‘Optimal Inventory Replenishment Policy for the EPQ Model under Trade Credit Derived without Derivatives’, International Journal of Systems Science, 40, 1095–1098] and Hsieh, Chang, Weng, and Dye [Hsieh, T.P., Chang, H.J., Weng, M.W., and Dye, C.Y. (2008), ‘A Simple Approach to an Integrated Single-vendor Single-buyer Inventory System with Shortage’, Production Planning and Control, 19, 601–604]. So, the main purpose of this article is to adopt the calculus approach not only to overcome shortcomings of the arithmetic–geometric mean method of Teng et al. (2009), Teng and Goyal (2009 Teng, JT, Chen, J and Goyal, SK. 2009. A Comprehensive Note on: An Inventory Model under Two Levels of Trade Credit and Limited Storage Space Derived without Derivatives. Applied Mathematical Modelling, 33: 4388–4396. [Crossref], [Web of Science ®] , [Google Scholar]) and Hsieh et al. (2008), but also to develop the complete solution procedures for them.