EOQ under Date-terms Supplier Credit

This paper examines EOQ under date-terms supplier credit, making explicit the separate effects on inventory policy of the two components of carrying cost-namely, financing cost and other variable holding costs. When a distinction between these types of holding costs is made, EOQ can no longer be expressed as a simple formula. Rather, optimal order quantity must be determined by search over a well-defined range of order quantities which encompasses the classical EOQ. The conclusion currently contained in the literature that the optimal order quantity under date terms is always given by an integer multiple of monthly demands no longer applies. In particular, a unique feature of date-terms credit is the possible existence of multiple EOQs.     

Using the complete squares method to analyze a lot size model when the quantity backordered and the quantity received are both uncertain

Several researchers have recently derived formulae for economic order quantities (EOQs) with some variants without reference to the use of derivatives, neither for first-order necessary conditions nor for second-order sufficient conditions. In addition, this algebraic derivation immediately produces an individual formula for evaluating the minimum expected annual cost. The purpose of this paper is threefold. First, this study extends the previous result to the EOQ formula, taking into account the scenario where the quantity backordered and the quantity received are both uncertain. Second, the complete squares method can readily derive global optimal expressions from a non-convex objective function in an algebraic manner. Third, the explicit identification of some analytic cases can be obtained: it is not as easy to do this using decomposition by projection. A numerical example has been solved to illustrate the solution procedure. Finally, some special cases can be deduced from the EOQ model under study, and concluding remarks are drawn.     

基于CRE的供应链渠道协作利润优化研究

为了实现供应链合作伙伴的双赢和多赢,在由一个制造商和两个批发商组成的供应链中,以制定最优共同补货周期策略为核心,制造商作为盟主指定共同补货周期和折扣率,批发商作为成员企业按共同补货同期的整数倍进行补货.制造商采用一致价格折扣的方式弥补批发商成本的增加.考虑到需求的可变性和是否实行共同补货周期策略,建立了制造商、批发商和整个供应链系统的利润优化模型并进行了模拟.结果表明共同补货周期策略可以实现多方收益的帕累托改进,供应链三方协作追求整体收益,可以实现"共赢".     

传统分销系统的经销商的库存管理

目前我国快速消费品行业中的中小企业的分销系统大都是传统的分销系统,在它之中经销商是处于核心地位.本文通过对经销商的库存进行重点分析,以了解产生经销商跨区冲货的原因及如何有效控制经销商合理库存的方法.作者认为分销系统是一个典型的系统工程.     

Credit risk estimation using payment history data: a comparative study of Turkish retail stores

Our paper presents a comparative study applying logistic regression and multiple criteria decision analysis tools to the operations of wholesalers to assess the credit risk of their retailers using payment history data and to cluster the risky customers by ranking their risk levels. Our sample comprises approximately 6,000 retailer customers and 600.000 transactions of one of the major wholesalers of Turkey. Our findings emphasize the importance of using payment history and some non-financial factors data for predicting the creditworthiness of a firm.     

Three-level inventory deployment for a luxury watch company facing various perturbations

A well-known Swiss watch brand, active in the top-end luxury market, is facing a complex inventory deployment problem where watches of different models (more than 100 different models) must be dispatched first to wholesalers to finally reach the shops where consumers come in. Along the way, different perturbations are expected at three levels (production plan, demand, and dispatching), and accurate reactions must be taken to fit to these uncertainties. Solution methods are proposed to solve realistic instances. Results show the relevance of the methods and the robustness of the solutions.     

Coordination in decentralized assembly systems with uncertain component yields

The literature on assembly systems with random component yields has focused on centralized systems, where a single decision maker chooses all components’ production quantities and incurs all the costs. We consider a decentralized setting where the component suppliers choose their production quantities based solely on their own cost/reward structure, and the assembly firm makes ordering decisions based on its own cost/reward structure. When the suppliers control their inputs but the outputs exhibit random yields, coordination in such systems becomes quite complex. In such situations, incentive alignment control mechanisms are needed so that the suppliers will choose production quantities as in the centralized system case. One such mechanism is to penalize the supplier with the worse delivery performance. We analyze the conditions under which system coordination is achieved while respecting participation constraints. Further, we determine the optimal component ordering policy for the assembly firm and derive the optimal coordinating penalties.     

Statistical inference for DEA estimators of directional distances

In productivity and efficiency analysis, the technical efficiency of a production unit is measured through its distance to the efficient frontier of the production set. The most familiar non-parametric methods use Farrell–Debreu, Shephard, or hyperbolic radial measures. These approaches require that inputs and outputs be non-negative, which can be problematic when using financial data. Recently, Chambers et al. (1998) have introduced directional distance functions which can be viewed as additive (rather than multiplicative) measures efficiency. Directional distance functions are not restricted to non-negative input and output quantities; in addition, the traditional input and output-oriented measures are nested as special cases of directional distance functions. Consequently, directional distances provide greater flexibility. However, until now, only free disposal hull (FDH) estimators of directional distances (and their conditional and robust extensions) have known statistical properties (Simar and Vanhems, 2012). This paper develops the statistical properties of directional d estimators, which are especially useful when the production set is assumed convex. We first establish that the directional Data Envelopment Analysis (DEA) estimators share the known properties of the traditional radial DEA estimators. We then use these properties to develop consistent bootstrap procedures for statistical inference about directional distance, estimation of confidence intervals, and bias correction. The methods are illustrated in some empirical examples.     

Densities of Ruin-Related Quantities in the Cramér-Lundberg Model with Pareto Claims

In this paper, we consider the classical yet widely applicable Cramér-Lundberg risk model with Pareto distributed claim sizes. Building on the previously known expression for the ruin probability we derive distributions of different ruin-related quantities. The results rely on the theory of scale functions and are intended to illustrate the simplicity and effectiveness of the theory. A particular emphasis is put on the tail behavior of the distributions of ruin-related quantities and their tail index value is established. Numerical illustrations are provided to show the influence of the claim sizes distribution tail index on the tails of the ruin-related quantities distribution.     

Some quantities related to monotonicity and bounded variation of functions

The aim of this paper is to introduce and study some quantities connected with monotonicity and bounded variation of real functions. These quantities will also be considered for bounded subsets of spaces of bounded or continuous functions on a compact interval, equipped with the standard supremum norm. Several properties of such quantities are established, and possible applications are briefly sketched. Moreover, some relation with the theory of measures of noncompactness is discussed as well.     

The rise of complex phenomena in Cournot duopoly games due to demand functions without inflection points

In this paper, we propose Cournot duopoly games where quantity-setting firms use non-linear demand functions that have no inflection points. Two different kinds of repeated games are introduced based on rationality process of firms and Puu’s incomplete approach. First, a model of two rational firms that are in competition and produce homogenous commodities is introduced. The equilibrium points of this model are obtained and their dynamical characteristics such as stability, bifurcation and chaos are investigated. By using rationality process firms do not need to solve any optimization problem but they adjust their production based on estimation of the marginal profit. Using Puu’s incomplete information approach a new model is introduced. As in the first model, the equilibrium points are obtained and their dynamical characteristics are investigated. By using Puu’s approach firms only need to know their profits and the quantities produced in the past two times. We compare the properties of the two models under the two approaches. The paper extends and generalizes the results of other authors that consider similar processes.     

Nonlinear dynamics in a heterogeneous duopoly game with adjusting players and diseconomies of scale

A repeated, discrete time, heterogeneous Cournot duopoly game with bounded rational and adaptive players adjusting the quantities of production is subject of investigation. Linear inverse demand function and quadratic cost functions reflecting decreasing returns to scale are assumed. The game is modeled with a system of two difference equations. Evolution of outputs over time is obtained by iteration of a two dimensional nonlinear map. Existing equilibria and their stability are analyzed. In face of diseconomies of scale, bounded rational and adaptive duopolists are shown to experience a decrease in the latitude of their output adjustment decisions with respect to the market stability compared to constant returns to scale and ceteris paribus. Chaotic dynamics is confirmed to depend mainly on the adjustment behavior of the bounded rational player, who if overshoots leaves the adaptive player with limited opportunities to stabilize the market again, hence industries facing diseconomies of scale are found to be less stable than those with constant marginal costs. Complexity of the dynamical system is examined by means of numerical simulations, where the paper extends the results of other authors who considered analogous games assuming linear cost functions. Intermittent transition to chaos and attractor merging crisis are shown among others.     

On the total operating costs up to default in a renewal risk model

The paper proposes a new approach to study a general class of ruin-related quantities in the context of a renewal risk model. While the classical approaches in Sparre Andersen models have their own merits, the approach presented in this paper has its advantages from the following perspectives. (1) The underlying surplus process has the flexibility to reflect a broad range of scenarios for surplus growth including dividend policies and interest returns. (2) The solution method provides a general framework to unify a great variety of existing ruin-related quantities such as Gerber–Shiu functions and the expected present value of dividends paid up to ruin, and facilitates derivations of new ruin-related quantities such as the expected present value of total claim costs up to ruin, etc. In the end, many specific examples are explored to demonstrate its application in renewal risk models.     

Eliciting density ratio classes

The probability distributions of uncertain quantities needed for predictive modelling and decision support are frequently elicited from subject matter experts. However, experts are often uncertain about quantifying their beliefs using precise probability distributions. Therefore, it seems natural to describe their uncertain beliefs using sets of probability distributions. There are various possible structures, or classes, for defining set membership of continuous random variables. The Density Ratio Class has desirable properties, but there is no established procedure for eliciting this class. Thus, we propose a method for constructing Density Ratio Classes that builds on conventional quantile or probability elicitation, but allows the expert to state intervals for these quantities. Parametric shape functions, ideally also suggested by the expert, are then used to bound the nonparametric set of shapes of densities that belong to the class and are compatible with the stated intervals. This leads to a natural metric for the size of the class based on the ratio of the total areas under upper and lower bounding shape functions. This ratio will be determined by the characteristics of the shape functions, the scatter of the elicited values, and the explicit expert imprecision, as characterized by the width of the stated intervals. We provide some examples, both didactic and real, and conclude with recommendations for the further development and application of the Density Ratio Class.     

A scheduling policy for adjusting economic lot quantities to a feasible solution

A heuristic scheduling policy is introduced for a multi-item, single-machine production facility. The scheduling policy uses the presumed optimal order quantities derived from solving an Economic Lot Size Problem and checks that the quantities obtain a feasible production schedule according to current inventory levels and expected demand rates. If not, the scheduling policy modifies the order quantities to achieve a possible solution without shortages. The scheduling policy is inspired by modification of the similar heuristic Dynamic Cycle Lengths Policy by Leachman and Gascon from 1988, 1991. The main characteristics of this scheduling policy are successive batches of the same item are treated explicitly, due to that it is quite possible that one item be manufactured several times before one other item is manufactured once more; the batches are ordered in increasing run-out time; if the existing situation creates stock-outs with ordinary order quantities, then the order quantities are decreased with a common scaling factor to try to prevent inventory shortages; in case the decrease of the order quantities changes expected run-out times, the batches are reordered after new run-out times; no filling up to an explicit inventory level is done, the filling up is done by the desirable order quantity; to prevent possible excess inventory the policy suggests time periods where no production should be performed. The scheduling policy contains no economical evaluation; this is supposed to be done when the order quantities are calculated, the policy prevents shortages and excess inventory. A numerical example illustrates the suggested scheduling policy. Finally, it is discussed as to how the policy can also take into account stochastic behaviour of the demand rates and compensate the schedule by applying appropriate safety times.     

On some measures of noncompactness in the space of continuous functions

We consider some quantities in the space of functions continuous on a bounded interval, which are related to monotonicity of functions. Based on those quantities we construct a few measures of noncompactness in the mentioned function space. Several properties of those measures are established; among others it is shown that they are regular or “partly” regular measures and equivalent to the Hausdorff measure of noncompactness.     

Leonhard Euler’s use and understanding of mathematical transcendence

Leonhard Euler primarily applied the term “transcendental” to quantities which could be variable or determined. Analyzing Euler’s use and understanding of mathematical transcendence as applied to operations, functions, progressions, and determined quantities as well as the eighteenth century practice of definition allows the author to evaluate claims that Euler provided the first modern definition of a transcendental number. The author argues that Euler’s informal and pragmatic use of mathematical transcendence highlights the general nature of eighteenth century mathematics and proposes an alternate perspective on the issue at hand: transcendental numbers inherited their transcendental classification from functions.     

On per-iteration complexity of high order Chebyshev methods for sparse functions with banded Hessians

Some algorithms for unconstrained and differentiable optimization problems involve the evaluation of quantities related to high order derivatives. The cost of these evaluations depends widely on the technique used to obtain the derivatives and on some characteristics of the objective function: its size, structure and complexity. Functions with banded Hessian are a special case that we study in this paper. Because of their partial separability, the cost of obtaining their high order derivatives, subtly computed by the technique of automatic differentiation, makes High order Chebyshev methods more interesting for banded systems than for dense functions. These methods have an attractive efficiency as we can improve their convergence order without increasing significantly their algorithmic costs. This paper provides an analysis of the per-iteration complexities of High order Chebyshev methods applied to sparse functions with banded Hessians. The main result can be summarized as: the per-iteration complexity of a High order Chebyshev method is of order of the objective function’s. This theoretical analysis is verified by numerical illustrations.     

Environmental considerations on the optimal product mix

Several types of regulations limit the amount of different emissions that a firm may create from its production processes. Depending on the emission, these regulations could include threshold values, penalties and taxes, and/or emission allowances that can be traded. However, many firms try to comply with these regulations without a systematic plan, often leading not only to emission violations and high penalties, but also to high costs. In this paper, we present two mathematical models that can be used by firms to determine their optimal product mix and production quantities in the presence of several different types of environmental constraints, in addition to typical production constraints. Both models are comprehensive and incorporate several diverse production and environmental issues. The first model, which assumes that each product has just one operating procedure, is a linear program while the second model, which assumes that the firm has the option of producing each product using more than one operating procedure, is a mixed integer linear program. The solutions of both models identify the products that the firm should produce along with their production quantities. These models can be used by firms to quickly analyze several “what if” scenarios such as the impact of changes in emission threshold values, emission taxes, trading allowances, and trading transaction costs.     

On comonotonic functions of uncertain variables

Uncertain variable is used to model quantities in uncertainty. This paper considers comonotonic functions of an uncertain variable, and gives their uncertainty distributions. Besides, it proves the linearity of expected value operator on comonotonic functions of an uncertain variable.