A spatiotemporal Data Envelopment Analysis (S-T DEA) approach: the need to assess evolving units

We consider a two-stage supply chain with one supplier and one manufacturer. The manufacturer faces a Poisson demand process where the arrival rate depends on the selling price, the announced delivery time, and the delivery reliability defined as the probability of satisfying the announced delivery time. Such a demand model generalizes the works in the literature by simultaneously considering the above three demand sensitivity factors. The main purpose of this paper is to study the equilibrium decisions in the supply chain with an all-unit quantity discount contract. We consider four scenarios regarding whether the leadtime standard, the delivery reliability standard, and the manufacturer’s capacity are endogenous, and whether the manufacturer’s production cost is its private information. We find that an all-unit quantity discount scheme can coordinate the supply chain for most cases. Managerial insights are observed regarding the impact of the three demand sensitivity factors. For example, the breakpoint in an optimal quantity discount contract always increases with the delivery reliability sensitivity under an exogenous delivery reliability, but may decrease under an endogenous delivery reliability; with asymmetric information, a higher variance of the manufacturer’s unit production costs leads to a lower unit wholesale price for the low-cost manufacturer.     

基于多种运输渠道的经济批量问题的多项式时间算法

为了集中管理, 降低成本, 增强竞争优势, 供应商通常只负责生产,而把产品的配送业务外包给某个配送中心, 由配送中心按照零售商的需求决定发货运输的方式和时间.这样供应商, 配送中心和零售商就构成了二级供应链系统.本文研究的是运输方式为不同参数的全单位数量折扣结构时, 二级供应链系统的经济批量问题.分析了最优解的性质,并对此问题的一种特殊情形给出一个多项式时间算法.     

具有非线性采购成本库存控制问题的研究现状与挑战

库存管理是基于运筹学而发展起来的一门学科,并成为近几十年来运筹学和管理科学重要的研究领域之一。在库存系统中,采购成本是必不可少的成本之一,主要包含产品成本、运输成本、装卸成本等。现实中,采购成本依赖于采购量,且往往是采购量的非线性函数。介绍了几类常见的采购成本函数：依赖于采购量的固定成本、增量折扣、全量折扣、车载容量折扣和凸采购成本等。基于周期盘点库存模型和连续盘点库存模型,综述了带有这些非线性采购成本函数的库存模型研究进展。虽然经过了几十年的研究,但很多带有非线性采购成本的库存模型的最优采购策略因为其复杂性至今未能被完整刻画。通过综述来简单讨论该类问题的挑战和机会。     

Polynomial cases of the economic lot sizing problem with cost discounts

In this paper we study the economic lot sizing problem with cost discounts. In the economic lot sizing problem a facility faces known demands over a discrete finite horizon. At each period, the ordering cost function and the holding cost function are given and they can be different from period to period. There are no constraints on the quantity ordered in each period and backlogging is not allowed. The objective is to decide when and how much to order so as to minimize the total ordering and holding costs over the finite horizon without any shortages. We study two different cost discount functions. The modified all-unit discount cost function alternates increasing and flat sections, starting with a flat section that indicates a minimum charge for small quantities. While in general the economic lot sizing problem with modified all-unit discount cost function is known to be NP-hard, we assume that the cost functions do not vary from period to period and identify a polynomial case. Then we study the incremental discount cost function which is an increasing piecewise linear function with no flat sections. The efficiency of the solution algorithms follows from properties of the optimal solution. We computationally test the polynomial algorithms against the use of CPLEX.     

多供应商多零售商下经济批量问题的多项式时间算法研究

基于多供应商和多零售商构成的经济批量问题,通过构建优化模型,分析了订购费用为全部单位数量折扣和增加数量折扣两种情形模型最优解的相关性质。将这些性质应用到动态规划算法设计中,对订购费用为全部单位数量折扣时的一种特殊情形及增加数量折扣的一般情形分别设计了求解问题最优解的多项式时间算法,并用算例说明了算法的执行过程和有效性。     

满足多断点折扣费用函数的经济批量问题的多项式时间算法

在进货费用为全单位数量折扣函数的基础上,建立了一类有限时期内的经济批量问题.通过分析最优解的性质,设计了一个计算复杂性为O(T3+mT2)的动态规划算法,其中m为全单位数量折扣费用中的断点数,T为时期数.最后的算例进一步说明了该算法的有效性.     

Optimal solutions for the economic lot-sizing problem with multiple suppliers and cost structures

This paper considers a multi-supplier economic lot-sizing problem in which the retailer replenishes his inventory from several suppliers. Each supplier is characterized by one of three types of order cost structures: incremental quantity discount cost structure, multiple set-ups cost structure and all-unit quantity discount cost structure. The problem is challenging due to the mix of different cost structures. For all cases of the problem where each supplier is characterized by one of the first two cost structures, some optimality properties are proposed and optimal algorithms based on dynamic programming are designed. For the case where all suppliers are characterized by all-unit quantity discount cost structures, it is hard to design a polynomial time algorithm by the analyzed optimal properties. However, it is proved that one of its special cases can be solved in polynomial time.     

The single-item newsboy problem with dual performance measures and quantity discounts

This paper considers models for the single-item newsboy problem with quantity discount and the following dual performance measure: “maximize the expected profit subject to a constraint that the probability of achieving a target profit level is no less than a predetermined risk level.” We also consider two types of quantity discount: all-unit and incremental. For our models with zero shortage cost, a closed-form solution for determining the optimal order quantity is derived. However, models with positive shortage cost can only be solved numerically.     

碳排放交易机制下的供应链运作策略及协调研究

碳排放权交易是控制和减少碳排放的有效工具,同时也使得企业的运作成本增加,供应链的决策管理更复杂。考虑由零售商和制造商组成的二阶段供应链,研究不同碳排放权交易政策(供应链成员内部碳排放权交易、外部市场碳排放权交易)下基于数量折扣契约的供应链协调问题,并与无碳排放约束的情形进行比较。研究结果表明:在不同碳排放权交易政策下,数量折扣契约能够实现供应链的协调;无碳排放约束的供应链最优订货量大于外部碳排放权交易的供应链最优订货量,成员内部碳排放交易下的供应链最优订货量和供应链期望利润随着碳排放配额的增加最终等于无碳排放约束下的供应链最优订货量和供应链期望利润;外部市场碳排放交易政策下供应链的期望利润随着碳排放配额的增加而增加。     

Price discount based on early order commitment in a single manufacturer–multiple retailer supply chain

Early order commitment (EOC) is a strategy for supply chain coordination, wherein the retailer commits to purchasing from a manufacturer a fixed order quantity a few periods in advance of the regular delivery lead time. In this paper, we formulate and analyze the EOC strategy for a decentralized, two-level supply chain consisting of a single manufacturer and multiple retailers, who face external demands that follow an autocorrelated AR(1) process over time. We characterize the special structure of the optimal solutions for the retailers’ EOC periods to minimize the total supply chain cost and discuss the impact of demand parameters and cost parameters. We then develop and compare three solution approaches to solving the optimal solution. Using this optimal cost as the benchmark, we investigate the effectiveness of using the wholesale price-discount scheme for the manufacturer to coordinate this decentralized system. We give numerical examples to show the benefits of EOC to the whole supply chain, examine the efficiency of the discount scheme in general situation, and provide the special conditions when the full coordination is achieved.     

A production-transportation problem with stochastic demand and concave production costs

Well known extensions of the classical transportation problem are obtained by including fixed costs for the production of goods at the supply points (facility location) and/or by introducing stochastic demand, modeled by convex nonlinear costs, at the demand points (the stochastic transportation problem, [STP]). However, the simultaneous use of concave and convex costs is not very well treated in the literature. Economies of scale often yield concave cost functions other than fixed charges, so in this paper we consider a problem with general concave costs at the supply points, as well as convex costs at the demand points. The objective function can then be represented as the difference of two convex functions, and is therefore called a d.c. function. We propose a solution method which reduces the problem to a d.c. optimization problem in a much smaller space, then solves the latter by a branch and bound procedure in which bounding is based on solving subproblems of the form of [STP]. We prove convergence of the method and report computational tests that indicate that quite large problems can be solved efficiently. Problems up to the size of 100 supply points and 500 demand points are solved. Received October 11, 1993 / Revised version received July 31, 1995 Published online November 24, 1998     

Optimizing the economic lot size of a three-stage supply chain with backordering derived without derivatives

Multi-stage supply chain management integration provides a key to successful international business operations. This is because the integrated approach improves the global system performance and cost efficiency. The integrated production inventory models using differential calculus to solve the multi-variable problems are prevalent in operational research. This paper extends the integrated vendor–buyer inventory problem by Yang and Wee [Yang, P.C., Wee, H.M., 2002. The economic lot size of the integrated vendor–buyer system derived without derivatives. Optimal Control Applications and Methods 23, 163–169] to solve the multi-variable problems in the supply chain, and simplifies the solution procedure using a simple algebraic method. As a result, the solution procedure may be easily understood and applied so as to derive the optimal solution.     

Managerial and economic impacts of reducing delivery variance in the supply chain

This paper addresses the managerial and economic impacts of improving delivery performance in a serial supply chain when delivery performance is evaluated with respect to a delivery window. Building on contemporary management theories that advocate variance reduction as the critical step in improving the overall performance of a system, we model the variance of delivery time to the final customer as a function of the investment to reduce delivery variance and the costs associated with untimely delivery (expected earliness and lateness). A logarithmic investment function is used and the model solution involves the minimization of a convex–concave total cost function. A numerical example is provided to illustrate the model and the solution procedure. The model presented provides guidelines for determining the optimal level of financial investment for reducing delivery variance. The managerial implications as well as the economic aspects of delivery variance reduction in supply chain management are discussed.     

Single-supplier/multiple-buyer supply chain coordination: Incorporating buyers’ expectations under vertical information sharing

We address the coordination problem in a single-supplier/multiple-buyer supply chain. The supplier wishes to coordinate the supply chain by offering quantity discounts. To obtain their complete cost information, the supplier exchanges his own cost parameters with buyers leading to vertical information sharing. The supplier thinks that the buyers, as they have access to supplier’s setup and holding cost information, may demand a portion of the anticipated coordination savings based on the partial information they hold about the cost structure of the entire supply chain. We model each buyer’s expectations based on her limited view of the entire supply chain which consists of herself and the supplier only. These expectations are then incorporated into the modeling of the supply chain, which results in a generalization of the traditional Stackelberg type models. We discuss alternative efficiency sharing mechanisms, and propose methods to design the associated discount schemes that take buyers’ expectations into account. In designing the discount schemes, we consider both price discriminatory and non-price discriminatory approaches. The study adds to the existing body of work by incorporating buyers’ expectations into a constrained Stackelberg structure, and by achieving coordination without forcing buyers to explicitly comply with the supplier’s replenishment period in choosing their order quantities. The numerical analysis of the coordination efficiency and allocation of the net savings of the proposed discount schemes shows that the supplier is still able to coordinate the supply chain with high efficiency levels, and retain a significant portion of the net savings.     

Environmentally conscious optimization of supply chain networks

The optimization of supply chain structures considering both economic and environmental performances is nowadays an important research topic. However, enterprises are commonly faced with the competing issues of reduced cost, improved customer service and increased environmental factors as a multi-faceted trade-off problem when designing supply chains. Hence, this paper proposes an environmentally conscious optimization model of a supply chain network with a broader and more comprehensive objective function that considers not just the transportation costs, but also the costs for the amount of greenhouse gas emissions, fuel consumption, transportation times, noise and road roughness. The paper sheds light on the trade-offs between various parameters such as vehicle speed, fuel, time, emissions, noise and their total cost, and offers managerial insights on economies of environmentally conscious supply chain optimization. An integer non-linear programming model is developed to help decision makers find the optimal solution under mentioned considerations. The proposed model is validated through the solution of an example, where its applicability to supply chain problems is demonstrated for managerial insights.     

The present value of resources with large discount rates

This paper describes a method to detect limit cycles for optimal control problems in the plain. The procedure includes two steps. First, the solution paths are analytically studied for large discount rates. Second, we demonstrate by means of computer simulations how the dynamics found can be traced back to small discount rates. We apply this method to two specific examples from resource management: a taxation problem and an exploited system of predator-prey interaction which show that the limit cycles may grow as the discount rates decrease. The principle that small discount rates are more conservative than large ones is therefore questionable. The relation of our results to theorems in optimal growth theory is also discussed. This paper is part of a research project on “Cyclical Resource Management” financially supported by the German Science Foundation (DFG).     

二供应商经济批量问题的多项式时间算法

为了从采购费用结构不同的供应商中找到最佳补货策略,考虑一个零售商从两个供应商补货的二供应商经济批量问题.零售商在两个供应商处的采购费用结构分别为复合安装费用和全单位数量折扣费用结构.通过对问题结构性质的分析论证,将问题的可行解转化为一个有向网络,降低问题求解的计算复杂性.综合动态规划和Dijkstra最短路算法证明了该问题是多项式时间可解的.     

An analytical method for cost analysis in multi-stage supply chains: A stochastic network model approach

This paper studies the cost distribution characteristics in multi-stage supply chain networks. Based on the graphical evaluation and review technique, we propose a novel stochastic network mathematical model for cost distribution analysis in multi-stage supply chain networks. Further, to investigate the effects of cost components, including the procurement costs, inventory costs, shortage costs, production costs and transportation costs of supply chain members, on the total supply chain operation cost, we propose the concept of cost sensitivity and provide corresponding algorithms based on the proposed stochastic network model. Then the model is extended to analyze the cost performance of supply chain robustness under different order compensation ability scenarios and the corresponding algorithms are developed. Simulation experiment shows the effectiveness and flexibility of the proposed model, and also promotes a better understanding of the model approach and its managerial implications in cost management of supply chains.     

An interactive method for inventory control with fuzzy lead-time and dynamic demand

Normally, the real-world inventory control problems are imprecisely defined and human interventions are often required to solve these decision-making problems. In this paper, a realistic inventory model with imprecise demand, lead-time and inventory costs have been formulated and an inventory policy is proposed to minimize the cost using man–machine interaction. Here, demand increases with time at a decreasing rate. The imprecise parameters of lead-time, inventory costs and demand are expressed through linear/non-linear membership functions. These are represented by different types of membership functions, linear or quadratic, depending upon the prevailing supply condition and marketing environment. The imprecise parameters are first transformed into corresponding interval numbers and then following the interval mathematics, the objective function for average cost is changed into respective multi-objective functions. These functions are minimized and solved for a Pareto-optimum solution by interactive fuzzy decision-making procedure. This process leads to man–machine interaction for optimum and appropriate decision acceptable to the decision maker’s firm. The model is illustrated numerically and the results are presented in tabular forms.     

Optimal ordering and pricing policy with supplier quantity discounts and price-dependent stochastic demand

We study a joint ordering and pricing problem for a retailer whose supplier provides all-unit quantity discount for the product. Both generalized disjunctive programming model and mixed integer nonlinear programming model are presented to formulate the problem. Some properties of the problem are analysed, based on which a solution algorithm is developed. Two numerical examples are presented to illustrate the problem, which are solved by our solution algorithm. Managerial analysis indicates that supplier quantity discount has much influence on the ordering and pricing policy of the retailer and more profit can be obtained when the supplier provides quantity discount.