Designing a two-echelon distribution network under demand uncertainty

This paper proposes a comprehensive methodology for the stochastic multi-period two-echelon distribution network design problem (2E-DDP) where product flows to ship-to-points are directed from an upper layer of primary warehouses to distribution platforms (DPs) before being transported to the ship-to-points. A temporal hierarchy characterizes the design level dealing with DP location and capacity decisions, as well as the operational level involving transportation decisions as origin-destination flows. These design decisions must be calibrated to minimize the expected distribution cost associated with the two-echelon transportation schema on this network under stochastic demand. We consider a multi-period planning horizon where demand varies dynamically from one planning period to the next. Thus, the design of the two-echelon distribution network under uncertain customer demand gives rise to a complex multi-stage decisional problem. Given the strategic structure of the problem, we introduce alternative modeling approaches based on two-stage stochastic programming with recourse. We solve the resulting models using a Benders decomposition approach. The size of the scenario set is tuned using the sample average approximation (SAA) approach. Then, a scenario-based evaluation procedure is introduced to post-evaluate the design solutions obtained. We conduct extensive computational experiments based on several types of instances to validate the proposed models and assess the efficiency of the solution approaches. The evaluation of the quality of the stochastic solution underlines the impact of uncertainty in the two-echelon distribution network design problem (2E-DDP).     

A static-dynamic strategy for spare part inventory systems with nonstationary stochastic demand

Inventory control is especially difficult when demand is stochastic and nonstationary. We consider a spare part inventory control problem with multiple-period replenishment lead time, and describe a static-dynamic strategy for the problem. By solving a static-dynamic uncertainty model, the strategy first makes decisions on the replenishment periods and order-up-to-levels over the planning horizon, but implements only the decisions of the first period. It then uses the rolling horizon approach in the next period when the inventory status is revised, and the multi-period problem is updated as better forecasts become available. In light of structural property of the developed static-dynamic uncertainty model, the optimal solution to the model can be obtained without much computational effort and thus the strategy can be easily implemented. Computational experiments and result of a case study verify the efficacy of the proposed strategy.     

Competitive Multi-period Pricing for Perishable Products: A Robust Optimization Approach

We study a multi-period oligopolistic market for a single perishable product with fixed inventory. Our goal is to address the competitive aspect of the problem together with demand uncertainty using ideas from robust optimization and variational inequalities. The demand function for each seller has some associated uncertainty and we assume that the sellers would like to adopt a policy that is robust to adverse uncertain circumstances. We believe this is the first paper that uses robust optimization for dynamic pricing under competition. In particular, starting with a given fixed inventory, each seller competes over a multi-period time horizon in the market by setting prices and protection levels for each period at the beginning of the time horizon. Any unsold inventory at the end of the horizon is worthless. The sellers do not have the option of periodically reviewing and replenishing their inventory. We study non-cooperative Nash equilibrium policies for sellers under such a model. This kind of a setup can be used to model pricing of air fares, hotel reservations, bandwidth in communication networks, etc. In this paper we demonstrate our results through some numerical examples.     

Production planning via scenario modelling

Several Linear Programming (LP) and Mixed Integer Programming (MIP) models for the production and capacity planning problems with uncertainty in demand are proposed. In contrast to traditional mathematical programming approaches, we use scenarios to characterize the uncertainty in demand. Solutions are obtained for each scenario and then these individual scenario solutions are aggregated to yield a nonanticipative or implementable policy. Such an approach makes it possible to model nonstationarity in demand as well as a variety of recourse decision types. Two scenario-based models for formalizing implementable policies are presented. The first model is a LP model for multi-product, multi-period, single-level production planning to determine the production volume and product inventory for each period, such that the expected cost of holding inventory and lost demand is minimized. The second model is a MIP model for multi-product, multi-period, single-level production planning to help in sourcing decisions for raw materials supply. Although these formulations lead to very large scale mathematical programming problems, our computational experience with LP models for real-life instances is very encouraging.     

On justification of use of stochastic functions for multimodal optimization models

A model of a complicated function under uncertainty is constructed axiomatically, formalizing suppositions on rationality of information on a considered function.     

基于风险-均值分析的供应商选择与订单分配集成优化模型与求解

针对不确定环境下具有不同供应合约的供应商选择与订单分配问题,本文构建了基于风险-均值分析的模糊两阶段多周期集成优化模型。与传统的该问题研究并未充分考虑供应商选择与订单分配两阶段决策的交互影响不同,在该模型中,第一阶段供应商选择的评价目标依赖于后期实际运营中的订单分配决策;并考虑未来需求和实际运营成本的不确定性,引入在险价值和期望值两种决策准则对供应商选择方案的绩效进行评价。提出了该模型的分析求解方法,在险价值得以精确评估,期望值被控制在确定的误差范围内,并可以达到足够的精度要求。     

Multi-period planning and uncertainty issues in cellular manufacturing: A review and future directions

In this paper we review research that has been done to address cellular manufacturing under conditions of multi-period planning horizons, with demand and resource uncertainties. Most traditional cell formation procedures ignore any changes in demand over time caused by product redesign and uncertainties due to volume variation, part mix variation, and resource unreliability. However in today’s business environment, product life cycles are short, and demand volumes and product mix can vary frequently. Thus cell design needs to address these issues. It is only recently that researchers have been modelling uncertainty and multi-period issues. In this paper we conduct a comprehensive review of the work that addresses these issues. We present mathematical programming formulations as well as a taxonomy of existing models. Finally we suggest some directions for future research.     

因子结构下的新型多期投资组合选择模型

首次将因子模型与多期均值-方差模型相结合,建立了单因子结构下的新型多期均值-方差模型,并且利用二次规划方法得到了此模型的解析最优投资策略.为使这一结论具有普适性,又将它推广到了多因子模型情形,并且得到了与单因子模型下类似的结论.最后,通过数值算例验证了本文的理论结果.新模型确定最优投资策略的方法,是一个科学且可操作性强的方法.     

A stochastic programming model for the optimal issuance of government bonds

Sovereign states issue fixed and floating securities to fund their public debt. The value of such portfolios strongly depends on the fluctuations of the term structure of interest rates. This is a typical example of planning under uncertainty, where decisions have to be taken on the base of the key stochastic economic factors underneath the model.     

“酒店+OTA”双渠道供应链超订策略研究

为解决市场需求不确定环境下,酒店和在线旅行网站(Online Travel Agency,OTA)合作时的能力超订量与在线房间预留量的决策问题,建立了基于佣金合作模式的数学模型,给出了实现酒店整体期望收益最大化的在线房间预留量与能力超订量。借助数值分析,进一步研究了佣金率与需求不确定性对最优决策的影响。结果表明,当佣金率与门店需求不确定性较小时,酒店采取双渠道策略并且实施超订;当佣金率与门店需求不确定性很大时,酒店采取门店单渠道策略但不实施超订。另外,在线房间预留量随着佣金率、门店需求不确定性的增大而减小。     

Robust non-zero-sum stochastic differential reinsurance game

This paper considers the non-zero-sum stochastic differential game problem between two ambiguity-averse insurers (AAIs) who encounter model uncertainty and seek the optimal reinsurance decision under relative performance concerns. Each AAI manages her own risks by purchasing reinsurance with the objective of maximizing the expected utility of her relative terminal surplus with respect to that of her counterparty. The two AAIs’ decisions influence each other through the insurers’ relative performance concerns and the correlation between their surplus processes. We establish a general framework of Nash equilibrium for the associated non-zero-sum game with model uncertainty. For the representative case of exponential utilities, we solve the equilibrium strategies explicitly. Numerical studies are conducted to draw economic interpretations.     

可再生能源系统的不确定性优化模型及求解

结合新提出的满意度方法和混合整数规划方法,给出了多态不确定性环境下可再生能源规划模型.该模型综合描述了多地区、多时期、多品种、多部门之间的可再生能源管理系统的复杂性、系统性、动态性和不确定性.最后的实例求解结果说明该模型能很好地反映能源安全性、系统可靠性与系统成本之间的关系,并能给出不同系统违反水平下的能源配置方式及增容计划,为决策者提供决策参考.     

Determination of the crop mix of a rubber and oil palm plantation—A programming approach

This paper sets out a multi-period multi-objective linear programming model developed to capture as accurate as possible, the complex relationship between factors of production and the constraints operating on these factors of production for a perennial crop plantation. The primary objective is to provide a tool to determine the optimal crop mix under conditions of price uncertainty. Information necessary for the firm's capital budgeting can be simultaneously derived. Uncertainty is accomodated in a linear risk constraint in the model and the level of uncertainty is parametrised to obtain the efficiency frontier.To illustrate the methodology developed, the paper analyses investment alternatives in a Malaysian plantation. Sensitivity analyses were performed on the usage of factors of production as well as rate of export duty the firm has to pay. Such analyses are useful for intertemporal production planning under conditions of uncertainty.     

The risk-averse newsvendor problem with random capacity

We study the effect of capacity uncertainty on the inventory decisions of a risk-averse newsvendor. We consider two well-known risk criteria, namely Value-at-Risk (VaR) included as a constraint and Conditional Value-at-Risk (CVaR). For the risk-neutral newsvendor, we find that the optimal order quantity is not affected by the capacity uncertainty. However, this result does not hold for the risk-averse newsvendor problem. Specifically, we find that capacity uncertainty decreases the order quantity under the CVaR criterion. Under the VaR constraint, capacity uncertainty leads to an order decrease for low confidence levels, but to an order increase for high confidence levels. This implies that the risk criterion should be carefully selected as it has an important effect on inventory decisions. This is shown for the newsvendor problem, but is also likely to hold for other inventory control problems that future research can address.     

碳交易规制下信息不对称供应链的减排和低碳推广博弈研究

论文在碳交易规制下,研究单一制造商和双零售商组成的供应链减排与低碳推广决策以及零售商对制造商的成本信息分享问题,分析了零售商对制造商分享成本信息的条件,及不同情形下的制造商减排和零售商低碳推广策略。研究发现:零售商的最优低碳推广水平只与自身相关参数有关,零售商只有在自身低碳推广效率足够高时才会与制造商分享信息;制造商最优减排量随消费者低碳意识、碳交易价格、零售商低碳推广效率及其不确定性增大而提高,随零售商之间竞争程度提高而降低;无论零售商是否对制造商分享成本信息,供应链成员的最优利润随零售商之间竞争程度提高而增加,随消费者低碳意识、碳交易价格提高而提高;零售商低碳推广效率的不确定性越大则制造商的最优利润越低,零售商的最优利润越高;仅一家零售商分享成本信息时,制造商无法通过转移支付使另一家零售商与其分享信息。     

Supply chain optimization of petroleum organization under uncertainty in market demands and prices

A multi-period stochastic planning model has been developed and implemented for a supply chain network of a petroleum organization operating in an oil producing country under uncertain market conditions. The proposed supply chain network consists of all activities related to crude oil production, processing and distribution. Uncertainties were introduced in market demands and prices. A deterministic optimization model was first developed and tested. The impact of uncertainty on the supply chain was studied by performing a sensitivity analysis in which ±20% deviations were introduced in market demands and prices of different commodities. A stochastic formulation was then proposed, which is based on the two-stage problem with finite number of realizations. The proposed stochastic programming approach proved to be quite effective in developing resilient production plans in light of high degree of uncertainty in market conditions. The anticipated production plans have a considerably lower expected value of perfect information (EVPI). The main conclusion of this study is that for an oil producing country with oil processing capabilities, the impact of economic uncertainties may be tolerated by an appropriate balance between crude exports and processing capacities.     

基于再制造的多周期闭环供应链网络均衡模型

针对单周期模型在刻画废旧产品回收和再制造过程中的局限性,以多周期的再制造闭环供应链网络为研究对象,利用均衡理论和变分不等式的方法构建了基于再制造的多周期闭环供应链网络均衡模型,得到了制造商和零售商及整个供应链网络的均衡条件及其经济解释,并分析了均衡解的唯一性条件。针对所建立的模型,利用Euler算法进行了算例分析,说明了潜在市场份额对供应链成员决策的影响。     

Computational strategies for non-convex multistage MINLP models with decision-dependent uncertainty and gradual uncertainty resolution

In many planning problems under uncertainty the uncertainties are decision-dependent and resolve gradually depending on the decisions made. In this paper, we address a generic non-convex MINLP model for such planning problems where the uncertain parameters are assumed to follow discrete distributions and the decisions are made on a discrete time horizon. In order to account for the decision-dependent uncertainties and gradual uncertainty resolution, we propose a multistage stochastic programming model in which the non-anticipativity constraints in the model are not prespecified but change as a function of the decisions made. Furthermore, planning problems consist of several scenario subproblems where each subproblem is modeled as a nonconvex mixed-integer nonlinear program. We propose a solution strategy that combines global optimization and outer-approximation in order to optimize the planning decisions. We apply this generic problem structure and the proposed solution algorithm to several planning problems to illustrate the efficiency of the proposed method with respect to the method that uses only global optimization.     

The impact of customer returns on pricing and order decisions

In this paper, we address the simultaneous determination of price and inventory replenishment when customers return product to the firm. We examine cases when the quantity of returned product is a function of both the quantity sold and the price, in single and multi-period problems, with and without uncertainty in demand.     

Bayesian decision-making in inventory modelling

The objective of this paper is to advocate the use of Bayesianmethods in tackling decision problems with limited past data.It is assumed that a Bayesian approach is least likely to besuccessful when there is no information on which to base a meaningfulprior. Here we use a limiting, invariant, form of the conjugateprior distribution to represent this ignorance. The resultsof decisions based on Bayesian methods with this ‘non-informative’prior are compared with those which result from deriving a pointestimate for the unknown parameter. The particular context consideredhere is that of a single-period inventory model with compoundPoisson demand made up of a known demand size distribution butan unknown demand rate. The demand rate is assumed to be highenough for a normal approximation to the compound Poisson distributionto be used, in which case it is possible to analyse the behaviourdirectly. An extension to the multi-period model with zero leadtime is considered briefly. The results lend support to theuse of Bayesian methods, with or without a meaningful prior,for which the analysis and computation are no more complex thanthose required by standard methods.