An inventory model for increasing demand under two levels of trade credit linked to order quantity

During the growth stage of a product life cycle especially for high-tech products, the demand function increases with time. In this paper, we extend the constant demand to a linear non-decreasing demand function of time and incorporate a permissible delay in payment under two levels of trade credit into the model. The supplier offers a permissible delay linked to order quantity, and the retailer also provides a downstream trade credit period to its customers. The objective is to find the optimal replenishment cycle that minimizes the retailer’s annual total relevant cost per unit time. The condition for an optimal solution to the generalized model is presented and some fundamental theoretical results are established. Finally, numerical examples to illustrate the proposed model are provided. Sensitivity analysis is performed and some relevant managerial insights are obtained.     

An incentive-compatible solution for trade credit term incorporating default risk

This paper studies the optimal trade credit term decision in an extended economic ordering quantity (EOQ) framework that incorporates a default risk component. A principal-agent bilevel programming model with costs minimization objectives is set up to derive the incentive-compatible credit term. The supplier determines the credit term as the leader in the first level programming, by balancing her/his financing capacity with the retailer’s default risk, order behavior and cost shifting. At the second level, the retailer makes decisions on ordering and payment time by reacting on the term offered by the supplier. A first order condition solution procedure is derived for the bilevel programming when credit term is confined within the practically feasible interval. Two key results are obtained – the condition to derive incentive-compatible credit term, and an equation system to derive threshold default risk criterion filtering retailers suitable for credit granting. Numerical experiments show that the capital cost of the supplier is the most important factor determining the credit term. Default risk acts like a filtering criterion for selecting retailers suitable for credit granting. Empirical evidence supporting our theoretical considerations is obtained by estimating three panel econometric models, using a dataset from China’s listed companies.     

The optimal pricing and ordering policy for an integrated inventory model when trade credit linked to order quantity

In traditional inventory models, it is implicitly assumed that the buyer must pay for the purchased items as soon as they have been received. However, in many practical situations, the vendor is willing to provide the buyer with a permissible delay period when the buyer’s order quantity exceeds a given threshold. Therefore, to incorporate the concept of vendor–buyer integration and order-size-dependent trade credit, we present a stylized model to determine the optimal strategy for an integrated vendor–buyer inventory system under the condition of trade credit linked to the order quantity, where the demand rate is considered to be a decreasing function of the retail price. By analyzing the total channel profit function, we developed some useful results to characterize the optimal solution and provide an iterative algorithm to find the retail price, buyer’s order quantity, and the numbers of shipment per production run from the vendor to the buyer. Numerical examples and sensitivity analysis are given to illustrate the theoretical results, and some managerial insights are also obtained.     

Augmenting the lot sizing order quantity when demand is probabilistic

In this paper we consider a single item, discrete time, lot sizing situation where demand is random and its parameters (e.g., mean and standard deviation) can change with time. For the appealing criterion of minimizing expected total relevant costs per unit time until the moment of the next replenishment we develop two heuristic ways of selecting an appropriate augmentation quantity beyond the expected total demand through to the planned (deterministic) time of the next replenishment. The results of a set of numerical experiments show that augmentation is important, particularly when orders occur frequently (i.e., the fixed cost of a replenishment is low relative to the costs of carrying one period of demand in stock) and the coefficient of variability of demand is relatively low, but also under other specified circumstances. The heuristic procedures are also shown to perform very favourably against a hindsight, baseline (s, S) policy, especially for larger levels of non-stationarity.     

Optimal ordering policies for perishable multi-item under stock-dependent demand and two-level trade credit

In this paper, a multi-item inventory model for perishable items is developed, where the demand rates of the items are stock dependent, two-level trade credit is adopted and the restriction of inventory capacity is also considered. The major objective is to determine the optimal cycle time and order quantities such that the total profit is maximized. The existence and uniqueness of the optimal cycle is discussed by Lagrange approach, and line search algorithms are developed to find the optimal solution of the model. Furthermore, numerical examples are given to illustrate the methods. The sensitivity of the solution to changes in the values of different parameters is also discussed.     

Optimal diffusion of a new technology when both demand and supply are nonstatic

The investment problem of a monopolized sector selling an innovated product is explored. Learning by doing is supposed to occur on the supply side, while learning by using is introduced to explain demand growth. Pontryagin's maximum principle is applied to the resulting optimal control problem, which includes supply capacity and cumulative output as state variables. The optimal investment policy turns out to be of a very simple form: all profit is retained and invested until capacity achieves its optimal size. In spite of this, the new technology price displays a variety of time patterns that heavily depend on the actual demand and cost conditions, as one would expect in the real world.The authors express their gratitude to Professor Sergio Rinaldi for helpful comments. This work was partially supported by Centro Teoria dei Sistemi, CNR, Milano, Italy.     

An uncooperative order model for items with trade credit,inventory-dependent demand and limited displayed-shelf space

This paper considers a two-echelon supply chain where a supplier sells a single product through a retailer, who faces an inventory-dependent demand. The supplier hopes to incentive the retailer to order more items by offering trade credit. The retailer places the ordered items on the display shelf (DS) with limited space and stocks the remaining items (if any) that exceed the shelf capacity in his/her backroom/warehouse (BW). From the supplier’s perspective, we focus mainly on under which conditions the supplier should offer trade credit and how he/she should design such trade credit policy and corresponding ordering policy to obtain much more benefits. From the retailer’s perspective, we discuss whether the retailer needs BW and exactly how many items need to be stocked in BW when the supplier offers trade credit. We formulate a “supplier-Stackelberg” game model, from which we obtain the conditions under which the presented simple trade credit policy not only increases the overall chain profit but also each member’s profit. We also show that the trade credit policy is always more beneficial to the retailer than to the supplier if it is offered.     

Deterministic economic order quantity models with partial backlogging when demand and cost are fluctuating with time

Journal of the Operational Research Society - In today's time-based competition, the unit cost of a high-tech product declines significantly over its short life cycle while its demand...     

Comments on: A comparative analysis for determining optimal price and order quantity when a sale increases demand

Although the net present value (NPV) criterion is theoretically the correct approach to developing optimal inventory policies, in the classical EOQ case, the average profit criterion generates solutions that are practically identical to those resulting from the NPV criterion. Nevertheless, a recent paper suggests that, when the demand for a product is price-elastic and a wholesaler offers a one-time-only price discount, use of the average profit criterion may obtain policies that are drastically suboptimal compared to the policies obtained by using the NPV criterion. We show that this suggestion is based on inaccurate models and inconsistent comparisons. Although in cases of large one-time-only discounts, there may be significant differences in the policies and consequences resulting from the two criteria, such large discounts are unrealistic. Furthermore, the larger the discount, the less practicable are the optimal order quantities based on either one of these criteria. Thus, in most real-life situations, the use of the average profit criterion does not result in serious suboptimization. In these situations, what may be important is not whether a retailer uses the NPV criterion or the average profit criterion, but whether the retailer can and does implement the optimal decisions resulting from the use of either criterion.     

Optimal replenishment policies for deteriorating items with stock sensitive demand under two-level trade credit and limited capacity

Recently, Min et al. [18] established an inventory model for deteriorating items under stock-dependent demand and two-level trade credit and obtained the optimal replenishment policy. Their analysis imposed a terminal condition of zero ending-inventory. However, with a stock-dependent demand, it may be desirable to order large quantities, resulting in stock remaining at the end of the cycle, due to the potential profits resulting from the increased demand. As a result, to make the theory more applicable in practice, we extend their model to allow for: (1) an ending-inventory to be nonzero, (2) a maximum inventory ceiling to reflect the facts that too much stock leaves a negative impression on the buyer and the amount of shelf/display space is limited.     

Mixture cure models in credit scoring: If and when borrowers default

Mixture cure models were originally proposed in medical statistics to model long-term survival of cancer patients in terms of two distinct subpopulations - those that are cured of the event of interest and will never relapse, along with those that are uncured and are susceptible to the event. In the present paper, we introduce mixture cure models to the area of credit scoring, where, similarly to the medical setting, a large proportion of the dataset may not experience the event of interest during the loan term, i.e. default. We estimate a mixture cure model predicting (time to) default on a UK personal loan portfolio, and compare its performance to the Cox proportional hazards method and standard logistic regression. Results for credit scoring at an account level and prediction of the number of defaults at a portfolio level are presented; model performance is evaluated through cross validation on discrimination and calibration measures. Discrimination performance for all three approaches was found to be high and competitive. Calibration performance for the survival approaches was found to be superior to logistic regression for intermediate time intervals and useful for fixed 12 month time horizon estimates, reinforcing the flexibility of survival analysis as both a risk ranking tool and for providing robust estimates of probability of default over time. Furthermore, the mixture cure model’s ability to distinguish between two subpopulations can offer additional insights by estimating the parameters that determine susceptibility to default in addition to parameters that influence time to default of a borrower.     

An inventory model for non-instantaneous deteriorating items with quadratic demand rate and shortages under trade credit policy

noindent In this paper, we propose an appropriate inventory model for non-instantaneous deteriorating items over quadratic demand rate with permissible delay in payments and time dependent deterioration rate. In this model, the completely backlogged shortages are allowed. In several existing results, the authors discussed that the deterioration rate is constant in each cycle. However, the deterioration rate of items are not constant in real world applications. Motivated by this fact, we consider that the items are deteriorated with respect to time. To minimize the total relevant inventory cost, we prove some useful theorems to illustrate the optimal solutions by finding an optimal cycle time with the necessary and enough conditions for the existence and uniqueness of the optimal solutions. Finally, we discuss the numerical instance and sensitivity of the proposed model.     

Optimal portfolio and consumption selection with default risk

We investigate an optimal portfolio and consumption choice problem with a defaultable security. Under the goal of maximizing the expected discounted utility of the average past consumption, a dynamic programming principle is applied to derive a pair of second-order parabolic Hamilton-Jacobi-Bellman (HJB) equations with gradient constraints. We explore these HJB equations by a viscosity solution approach and characterize the post-default and pre-default value functions as a unique pair of constrained viscosity solutions to the HJB equations.     

Modeling credit portfolio derivatives,including both a default and a prepayment feature

Apart from heteronomy exit events such as, for example credit default or death, several financial agreements allow policy holders to voluntarily terminate the contract. Examples include callable mortgages or life insurance contracts. For the contractual counterpart, the result is a cash‐flow uncertainty called prepayment risk. Despite the high relevance of this implicit option, only few portfolio models consider both a default and a cancellability feature. On a portfolio level, this is especially critical because empirical observations of the mortgage market suggest that prepayment risk is an important determinant for the pricing of mortgage‐backed securities. Furthermore, defaults and prepayments tend to occur in clusters, and there is evidence for a negative association between the two risks. This paper presents a realistic and tractable portfolio model that takes into account these observations. Technically, we rely on an Archimedean dependence structure. A suitable parameterization allows to fit the likelihood of default and prepayment clusters separately and accounts for the postulated negative interdependence. Moreover, this structure turns out to be tractable enough for real‐time evaluation of portfolio derivatives. As an application, the pricing of loan credit default swaps, an example of a portfolio derivative that includes a cancellability feature, is discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal consumption and investment with insurer default risk

We solve the optimal consumption and investment problem in an incomplete market, where borrowing constraints and insurer default risk are considered jointly. We derive in closed-form the optimal consumption and investment strategies. We find two main results by quantitative analysis. As insurer default risk increases, the proportion of wealth invested in stocks could increase when wealth is small, and decrease when wealth is large. As risk aversion increases, the voluntary annuity demand could increase when insurer default risk is low, and decrease when this risk is high.     

Inventory management under date-terms supplier trade credit with stochastic demand and leadtime

Executives and academics alike are expressing increasing interest in supplier trade credit. We consider a form of credit known as ‘date-terms’ where the credit period extends to a specified date in the month following the invoice. We extend published research to consider both demand and supply uncertainty, employing a gamma distribution to model demand during the leadtime and reorder period. Taking an applied perspective, we evaluate four heuristics against an optimal solution for the case where pragmatic restrictions are placed on the reorder period. We evaluate how the reorder period and the performance of heuristics are affected by various environmental parameters (based on industrial data) and comment on the ramifications of this form of trade credit.     

Optimal reinsurance with regulatory initial capital and default risk

In a reinsurance contract, a reinsurer promises to pay the part of the loss faced by an insurer in exchange for receiving a reinsurance premium from the insurer. However, the reinsurer may fail to pay the promised amount when the promised amount exceeds the reinsurer’s solvency. As a seller of a reinsurance contract, the initial capital or reserve of a reinsurer should meet some regulatory requirements. We assume that the initial capital or reserve of a reinsurer is regulated by the value-at-risk (VaR) of its promised indemnity. When the promised indemnity exceeds the total of the reinsurer’s initial capital and the reinsurance premium, the reinsurer may fail to pay the promised amount or default may occur. In the presence of the regulatory initial capital and the counterparty default risk, we investigate optimal reinsurance designs from an insurer’s point of view and derive optimal reinsurance strategies that maximize the expected utility of an insurer’s terminal wealth or minimize the VaR of an insurer’s total retained risk. It turns out that optimal reinsurance strategies in the presence of the regulatory initial capital and the counterparty default risk are different both from optimal reinsurance strategies in the absence of the counterparty default risk and from optimal reinsurance strategies in the presence of the counterparty default risk but without the regulatory initial capital.     

Analysis of a fuzzy economic order quantity model for deteriorating items under retailer partial trade credit financing in a supply chain

This paper investigates the economic order quantity (EOQ) — based inventory model for a retailer under two levels of trade credit to reflect the supply chain management situation in the fuzzy sense. It is assumed that the retailer maintains a powerful position and can obtain the full trade credit offered by the supplier yet the retailer just offers a partial trade credit to customers. The demand rate, holding cost, ordering cost, purchasing cost and selling price are taken as fuzzy numbers. Under these conditions, the retailer can obtain the most benefits. Study also investigates the retailer’s inventory policy for deteriorating items in a supply chain management situation as a cost minimization problem in the fuzzy sense. The annual total variable cost for the retailer in fuzzy sense is defuzzified using Graded Mean Integration Representation method. Then the present study shows that the defuzzified annual total variable cost for the retailer is convex, that is, a unique solution exists. Mathematical theorems and algorithms are developed to efficiently determine the optimal inventory policy for the retailer. Numerical examples are given to illustrate the theorems and the algorithms. Finally, the results in this paper generalize some already published results in the crisp sense.     

Estimation of rating classes and default probabilities in credit risk models with dependencies

Let Y = m(X) + ε be a regression model with a dichotomous output Y and a one‐step regression function m . In the literature, estimators for the three parameters of m , that is, the breakpoint θ and the levels a and b , are proposed for independent and identically distributed (i.i.d.) observations. We show that these standard estimators also work in a non‐i.i.d. framework, that is, that they are strongly consistent under mild conditions. For that purpose, we use a linear one‐factor model for the input X and a Bernoulli mixture model for the output Y . The estimators for the split point and the risk levels are applied to a problem arising in credit rating systems. In particular, we divide the range of individuals' creditworthiness into two groups. The first group has a higher probability of default and the second group has a lower one. We also stress connections between the standard estimator for the cutoff θ and concepts prevalent in credit risk modeling, for example, receiver operating characteristic. Copyright © 2014 John Wiley & Sons, Ltd.     

An economic production quantity inventory model involving fuzzy demand rate and fuzzy deterioration rate

Generally, in deriving the solution of economic production quantity (EPQ) inventory model, we consider the demand rate and deterioration rate as constant quantity. But in case of real life problems, the demand rate and deterioration rate are not actually constant but slightly disturbed from their original crisp value. The motivation of this paper is to consider a more realistic EPQ inventory model with finite production rate, fuzzy demand rate and fuzzy deterioration rate. The effect of the loss in production quantity due to faulty/old machine have also been taken into consideration. The methodology to obtain the optimum value of the fuzzy total cost is derived and a numerical example is used to illustrate the computation procedure. A sensitivity analysis is also carried out to get the sensitiveness of the tolarance of different input parameters.