Production and payment policies for an imperfect manufacturing system with discount cash flows analysis in fuzzy random environments

ABSTRACT

This paper considers an imperfect manufacturing system with credit policies in fuzzy random environments. The supplier simultaneously offers the retailer either a permissible delay in payments or a cash discount and retailer in turn provides its customer a permissible delay period. We used an alternate approach – discount cash flow analysis to establish an inventory problem. It is assumed that the elapsed time until the machine shifts from ‘in-control’ state to ‘out-of-control’ state is characterized as a fuzzy random variable. As a function of this parameter, the profit function is also a random fuzzy variable. Based on the credibility measure of fuzzy event, the model with fuzzy random elapsed time can be transformed into a crisp model . We establish several theoretical results to obtain the solution that provides the largest present value of all future cash flows. Finally, numerical example is given to illustrate the results and obtain some managerial insights.     

两货栈及延期支付和现金折扣情形下变质产品的最优订货策略

已有的两货栈库存模型通常不考虑将延期支付和现金折扣相结合的情形,但实际上,供应商在给予销售商延期支付政策的同时,也会实施现金折扣策略以激励销售商尽快付款,加快资金周转,减少坏账损失。为此,本文建立了延期支付和现金折扣情形下变质产品的两货栈库存模型,并对模型的最优解进行理论分析,给出了最优解的求解步骤。最后通过数值算例对模型的可行性进行了验证,并分析了模型参数变化对最优订货策略和最优付款时间的影响。     

两货栈及延期支付和现金折扣情形下变质产品的最优订货策略

已有的两货栈库存模型通常不考虑将延期支付和现金折扣相结合的情形,但实际上,供应商在给予销售商延期支付政策的同时,也会实施现金折扣策略以激励销售商尽快付款,加快资金周转,减少坏账损失。为此,本文建立了延期支付和现金折扣情形下变质产品的两货栈库存模型,并对模型的最优解进行理论分析,给出了最优解的求解步骤。最后通过数值算例对模型的可行性进行了验证,并分析了模型参数变化对最优订货策略和最优付款时间的影响。     

Cost minimization and the stochastic discount factor

It is shown that a cost function subject to internal costs of adjustment induces a stochastic discount factor (pricing kernel) that is a function of random output, input and output prices, existing capital stock, and investment. The only assumption on firm preferences is that they are increasing in current period consumption and future stochastic consumption. This ensures that the firm will always act to minimize current period cost of providing future consumption, and it is the first-order conditions for this cost minimization problem that generate the stochastic discount factor, which itself can be interpreted as the marginal variable cost of varying stochastic output. A cost-based pricing kernel is estimated using annual time-series data on macroeconomic variables and returns data for the S&P 500 and commercial paper.     

Comonotonic approximations to quantiles of life annuity conditional expected present value

In large portfolios, the risk borne by annuity providers (insurance companies or pension funds) is basically driven by the randomness in the future mortality rates. To fix the ideas, we adopt here the standard Lee-Carter framework, where the future forces of mortality are decomposed in a log-bilinear way. This paper aims to provide accurate approximations for the quantiles of the conditional expected present value of the payments to the annuity provider, given the future path of the Lee-Carter time index. Mortality is stochastic while the discount factors are derived from a zero-coupon yield curve and are assumed to be deterministic. Numerical illustrations based on Belgian mortality (general population and insurance market statistics) show that the accuracy of the approximations proposed in this paper is remarkable, with relative difference less than 1% for most probability levels.     

Optimal tax depreciation with uncertain future cash-flows

Since taxable income consists of cash-flows reduced with depreciation charges, the choice of the depreciation method affects taxable income in future periods. A manager can therefore try to minimize the present value of future tax payments by choosing a particular depreciation method among those that are accepted by the tax authorities. We focus here on the choice between the two most commonly used methods in practice, i.e. the straight line depreciation method (SDM) and an accelerated depreciation method (ADM), such as the double declining balance (DDB) method and the sum of the years–digits (SYD) method. We show how the optimal choice depends on the discount factor, the degree of uncertainty in future cash-flows, and the structure of the tax system.     

一种随机需求下的供应链激励机制

研究了在随机需求条件下，供应链中用价格折扣策略协调供需双方利益的问题。考虑随机性需求有可能造成分销商的库存积压，本提出了生产商给予分销商的积压商品价格折扣的策略，分析了其激励机制，给出了最优价格折扣的模型和算法。最后用数值方法验证了这种价格折扣策略能够给生产商和分销商带来利益改善，而且需求波动越大，该策略的协调效果越好。     

The optimal order and payment policies for deteriorating items in discount cash flows analysis under the alternatives of conditionally permissible delay in payments and cash discount

This study investigates a deteriorating inventory problem in which the supplier simultaneously offers the retailer either a conditionally permissible delay in payments or a cash discount. In the case of a conditionally permissible delay, if the retailer orders more than a predetermined quantity, then he/she has a grace period to make the full payment. Otherwise, he/she must pay the payment for goods of certain proportion first while receiving the goods and has a grace period to pay off the rest. As to a cash discount, if the retailer pays for the entire amount of the order within a certain short period, then he/she will receive a cash discount from the supplier. In additions, from a financial standpoint, all cash outflows related to the inventory control that occur at different points of time have different values. Hence, it is necessary to take account of the factor of time value of monetary when drafting the replenishment policy. In a word, this paper uses an alternate approach-discount cash flow (DCF) analysis to establish an inventory problem for deteriorating items in which the supplier provides the retailer either a conditionally permissible delay or a cash discount. We then study the necessary and sufficient conditions for finding the optimal solution. Furthermore, we establish several theoretical results to obtain the solution that provides the smallest present value of all future cash flows. Finally, several numerical examples are given to illustrate the results and obtain some managerial insights.     

Multi-period supplier selection under price uncertainty

We consider a problem faced by a procurement manager who needs to purchase a large volume of multiple items over multiple periods from multiple suppliers that provide base prices and discounts. Discounts are contingent on meeting various conditions on total volume or spend, and some are tied to future realizations of random events that can be mutually verified. We formulate a scenario-based multi-stage stochastic optimization model that allows us to consider random events such as a drop in price because of the most favoured customer clauses, a price change in the spot market or a new discount offer. We propose certainty-equivalent heuristics and evaluate the regret of using them. We use our model for three bidding events of a large manufacturing company. The results show that considering most favored customer clauses in supplier offers may create substantial savings that may surpass the savings from regular discount offers.     

Markov Decision Processes on Borel Spaces with Total Cost and Random Horizon

This paper deals with Markov Decision Processes (MDPs) on Borel spaces with possibly unbounded costs. The criterion to be optimized is the expected total cost with a random horizon of infinite support. In this paper, it is observed that this performance criterion is equivalent to the expected total discounted cost with an infinite horizon and a varying-time discount factor. Then, the optimal value function and the optimal policy are characterized through some suitable versions of the Dynamic Programming Equation. Moreover, it is proved that the optimal value function of the optimal control problem with a random horizon can be bounded from above by the optimal value function of a discounted optimal control problem with a fixed discount factor. In this case, the discount factor is defined in an adequate way by the parameters introduced for the study of the optimal control problem with a random horizon. To illustrate the theory developed, a version of the Linear-Quadratic model with a random horizon and a Logarithm Consumption-Investment model are presented.     

Four Payment Models for the Multi-Mode Resource Constrained Project Scheduling Problem with Discounted Cash Flows

In this paper, the multi-mode resource constrained project scheduling problem with discounted cash flows is considered. The objective is the maximization of the net present value of all cash flows. Time value of money is taken into consideration, and cash in- and out-flows are associated with activities and/or events. The resources can be of renewable, nonrenewable, and doubly constrained resource types. Four payment models are considered: lump sum payment at the terminal event, payments at prespecified event nodes, payments at prespecified time points and progress payments. For finding solutions to problems proposed, a genetic algorithm (GA) approach is employed, which uses a special crossover operator that can exploit the multi-component nature of the problem. The models are investigated at the hand of an example problem. Sensitivity analyses are performed over the mark up and the discount rate. A set of 93 problems from literature are solved under the four different payment models and resource type combinations with the GA approach employed resulting in satisfactory computation times. The GA approach is compared with a domain specific heuristic for the lump sum payment case with renewable resources and is shown to outperform it.     

带观察时的跳服从Erlang(n)分布的对偶模型的红利贴现问题

研究了跳服从Erlang(n)分布,随机观察时服从指数分布的对偶风险模型.假设在边值策略下红利分发只在观察时发生,建立了红利期望贴现函数V(u;b)的微积分方程组.给出了当收益额服从PH(m)分布时V(u;b)的解析解.探讨了当收益额服从指数分布时V(u;b)的具体求解方法.     

An EOQ model for deteriorating items under trade credits

In the classical inventory economic order quantity (or EOQ) model, it was assumed that the supplier is paid for the items immediately after the items are received. However, in practices, the supplier may simultaneously offer the customer: (1) a permissible delay in payments to attract new customers and increase sales, and (2) a cash discount to motivate faster payment and reduce credit expenses. In this paper, we provide the optimal policy for the customer to obtain its minimum cost when the supplier offers not only a permissible delay but also a cash discount. We first establish a proper model, and then characterize the optimal solution and provide an easy-to-use algorithm to find the optimal order quantity and replenishment time. Furthermore, we also compare the optimal order quantity under supplier credits to the classical economic order quantity. Finally, several numerical examples are given to illustrate the theoretical results.     

Optimal three-level presort loading of commercial bulk mail in the postal service industry

This study focuses on the presort loading of commercial bulk mail. Here, presort is the process by which a mailer prepares mail such that it is sorted to at least the finest extent required by the postal service provider for a claimed (discounted) price. We formulated this presort loading problem (PLP) as a special case of transportation problem with minimum quantity commitment (MQC) constraints. In addition, we developed a polynomial time optimal solution algorithm for the PLP and performed computational experiments on randomly generated problem instances under various discount structures. Results of the computational experiments show that mailers can potentially reduce their costs by sending mail less frequently, using small-sized mail trays; however, the discount structure does not affect the main results. There is some evidence that smaller cost reductions on mailing fees occur as the variation in the discount rate increases; however, the effects of discount structure are nominal compared with the gains from changing the mailing frequency. Mailing fee savings are more heavily influenced by the discount structure when MQC constraints become tighter.     

基于D-S理论的延期支付与存货质押业务决策模型

由于市场的不确定性,延期支付货款和存货质押融资中存在诸多不确定因素,使得资金需求企业难以做出有效的选择。针对此问题,从资金需求企业利润最大化的角度出发,首先分别给出了延期支付和存货质押两种业务下的企业利润模型,进而基于D-S理论建立了两种业务的决策模型。该模型以存货质押和延期支付业务下企业利润差额为目标函数,根据证据推理,以信任函数和似然函数构造出利润差额的上下界概率分布,并据此给出两种业务的决策依据。算例分析表明,对资金需求企业来说,存货质押融资业务并不一定优于延期支付,在特定的市场环境下,需要利用本文提出的模型进行定量计算,从而做出最优的选择。     

Optimal dividend policy when risk reserves follow a jump–diffusion process with a completely monotone jump density under Markov-regime switching

The paper studies optimal dividend distribution for an insurance company whose risk reserves in the absence of dividends follow a Markov-modulated jump–diffusion process with a completely monotone jump density where jump densities and parameters including discount rate are modulated by a finite-state irreducible Markov chain. The major goal is to maximize the expected cumulative discounted dividend payments until ruin time when risk reserve is less than or equal to zero for the first time. I extend the results of Jiang (2015) for a Markov-modulated jump–diffusion process from exponential jump densities to completely monotone jump densities by proving that it is also optimal to take a modulated barrier strategy at some positive regime-dependent levels and that value function as the fixed point of a contraction is explicitly characterized.     

For what trading strategies is the tax payment stream of infinite variation?

For an Itô asset price process and under quite mild structural assumptions, we show that the accumulated payments of a linear tax on trading gains are of infinite variation if the quadratic covariation of the trading strategy and the asset price is negative. By contrast, if the strategy is a smooth function of the asset price and some finite variation processes with positive partial derivative with respect to the price variable, then accumulated tax payments are of finite variation. An interesting example are constant proportion portfolio insurance (CPPI) strategies which we extend to models with capital gains taxes. The associated tax payment stream is of finite variation if the tax-adjusted constant multiple of the cushion which is invested in the risky asset is bigger or equal to one. Otherwise, it is of infinite variation.     

Online product returns risk assessment and management

Commonly viewed as a cost center from an operations perspective, product returns have the potential to strongly influence operating margins and business profitability, thus constituting a risk for online retailers. This work addresses the problem of how to assess and manage product returns costs using a risk analysis methodology. Online product returns are seen as a random phenomenon that fluctuates in severity over time, threatening the profitability of the online store. Therefore, the starting point is that this risk can be modeled as a future random stream of payments. Given one or many future time periods, we aim to assess and manage this risk by answering two important questions: (1) Pricing—or what dollar amount factored on top of the current price of goods sold online would cover the cost of product returns, and (2) Reserving—or how much capital does an online retailer need to reserve at the beginning of each period to cover the cost of online product returns. We develop our analysis for one period (a month) by a closed formula model, and for multi-period (a year) by a dynamic simulation model. Risk measurements are executed in both cases to answer the two main questions above. We exemplify this methodology using an anonymized archival database of actual purchase and return history provided by a large size US women’s apparel online retailer.     

延期支付条件下零售商的最优售价和订货策略

在供应商给予零售商延期支付和现金折扣的优惠政策下,进一步假设产品的年需求量依赖于零售商产品售价的基础上,建立了由一个零售商和一个供应商所构成的库存决策模型,扩展了经典的经济生产批量(EPQ)模型。通过模型的分析求解,可以得出零售商在上述情况下的最优订货周期、最优售价及最优付款时间的简单判定方法。最后,通过算例,验证了模型的可行性,得出了与实际相符的结论。     

Comments on Taylor's see-saw approach to claims reserving

Taylor (1981) introduces the See-Saw (SS) model for claims reserving in order to make allowance for speed of finalization. The model is applied to live data given in Taylor (1981) and there is prima facie evidence to suggest that it does well, especially in the light of comparisons of actual versus expected payments for each payment year.The purpose of the present paper is to demonstrate that from the point of view of operational forecasting, which is the object of the claims reserving exercise, the fitting of the linear SS can be improved upon. Moreover, we employ the SS as a vehicle for indicating the kind of validation tests that ought to be carried out once the parameters of a proposed model have been estimated from the data. Essentially, we indicate how the properties of the residuals may be used for diagnostic checking of the model.Many researchers involved in the claims reserving area are of the view that the data are extremely noisy especially if the model put forward only explains a small proportion of the total variation. Why not test whether this is the case? It turns out that the particular linear SS used by Taylor does not explain the signal accurately so that the balance of the variation is not entirely due to noise.