Piecewise linear approximations for the static–dynamic uncertainty strategy in stochastic lot-sizing

In this paper, we develop a unified mixed integer linear modelling approach to compute near-optimal policy parameters for the non-stationary stochastic lot sizing problem under static–dynamic uncertainty strategy. The proposed approach applies to settings in which unmet demand is backordered or lost; and it can accommodate variants of the problem for which the quality of service is captured by means of backorder penalty costs, non-stockout probabilities, or fill rate constraints. This approach has a number of advantages with respect to existing methods in the literature: it enables seamless modelling of different variants of the stochastic lot sizing problem, some of which have been previously tackled via ad hoc solution methods and some others that have not yet been addressed in the literature; and it produces an accurate estimation of the expected total cost, expressed in terms of upper and lower bounds based on piecewise linearisation of the first order loss function. We illustrate the effectiveness and flexibility of the proposed approach by means of a computational study.     

Two-stage stochastic mixed-integer linear programming: The conditional scenario approach

In this paper we consider the two-stage stochastic mixed-integer linear programming problem with recourse, which we call the RP problem. A common way to approximate the RP problem, which is usually formulated in terms of scenarios, is to formulate the so-called Expected Value (EV) problem, which only considers the expectation of the random parameters of the RP problem. In this paper we introduce the Conditional Scenario (CS) problem which represents a midpoint between the RP and the EV problems regarding computational tractability and ability to deal with uncertainty. In the theoretical section we have analyzed some useful bounds related to the RP, EV and CS problems. In the numerical example here presented, the CS problem has outperformed both the EV problem in terms of solution quality, and the RP problem with the same number of scenarios as in the CS problem, in terms of solution time.     

OPTIMAL SAFETY-STOCK INVESTMENT THROUGH SUBJECTIVE EVALUATION OF STOCKOUT COSTS

This paper solves the problem of setting safety stocks by combining a subjective evaluation of the stockout-cost function with the usual holding-cost function. The technique given in this paper for estimating a decision maker's (DM's) disvalue function for stockouts is robust in terms of the shapes permissible for this function and allows for uncertainty on the part of the DM in expressing his/her trade-offs. The estimation technique could be applied to situations other than the safety-stock problem. We provide an optimization routine for the safety-stock problem which is designed to operate in conjunction with the estimation technique. Safety-stock levels are arrived at iteratively.     

Bayesian Forecasting via Deterministic Model

Rational decision making requires that the total uncertainty about a variate of interest (a predictand) be quantified in terms of a probability distribution, conditional on all available information and knowledge. Suppose the state-of-knowledge is embodied in a deterministic model, which is imperfect and outputs only an estimate of the predictand. Fundamentals are presented of two Bayesian methods for producing a probabilistic forecast via any deterministic model. The Bayesian Processor of Forecast (BPF) quantifies the total uncertainty in terms of a posterior distribution, conditional on model output. The Bayesian Forecasting System (BFS) decomposes the total uncertainty into input uncertainty and model uncertainty, which are characterized independently and then integrated into a predictive distribution. The BFS is compared with Monte Carlo simulation and ensemble forecasting technique, none of which can alone produce a probabilistic forecast that quantifies the total uncertainty, but each can serve as a component of the BFS.     

Lognormal Approximations of Fault Tree Uncertainty Distributions

Fault trees are used in reliability modeling to create logical models of fault combinations that can lead to undesirable events. The output of a fault tree analysis (the top event probability) is expressed in terms of the failure probabilities of basic events that are input to the model. Typically, the basic event probabilities are not known exactly, but are modeled as probability distributions: therefore, the top event probability is also represented as an uncertainty distribution. Monte Carlo methods are generally used for evaluating the uncertainty distribution, but such calculations are computationally intensive and do not readily reveal the dominant contributors to the uncertainty. In this article, a closed‐form approximation for the fault tree top event uncertainty distribution is developed, which is applicable when the uncertainties in the basic events of the model are lognormally distributed. The results of the approximate method are compared with results from two sampling‐based methods: namely, the Monte Carlo method and the Wilks method based on order statistics. It is shown that the closed‐form expression can provide a reasonable approximation to results obtained by Monte Carlo sampling, without incurring the computational expense. The Wilks method is found to be a useful means of providing an upper bound for the percentiles of the uncertainty distribution while being computationally inexpensive compared with full Monte Carlo sampling. The lognormal approximation method and Wilks’s method appear attractive, practical alternatives for the evaluation of uncertainty in the output of fault trees and similar multilinear models.     

Recoverable robust spanning tree problem under interval uncertainty representations

This paper deals with the recoverable robust spanning tree problem under interval uncertainty representations. A strongly polynomial time, combinatorial algorithm for the recoverable spanning tree problem is first constructed. This problem generalizes the incremental spanning tree problem, previously discussed in literature. The algorithm built is then applied to solve the recoverable robust spanning tree problem, under the traditional interval uncertainty representation, in polynomial time. Moreover, the algorithm allows to obtain several approximation results for the recoverable robust spanning tree problem under the Bertsimas and Sim interval uncertainty representation and the interval uncertainty representation with a budget constraint.     

Sensitivity Analysis, Monte Carlo Risk Analysis, and Bayesian Uncertainty Assessment

Standard statistical methods understate the uncertainty one should attach to effect estimates obtained from observational data. Among the methods used to address this problem are sensitivity analysis, Monte Carlo risk analysis (MCRA), and Bayesian uncertainty assessment. Estimates from MCRAs have been presented as if they were valid frequentist or Bayesian results, but examples show that they need not be either in actual applications. It is concluded that both sensitivity analyses and MCRA should begin with the same type of prior specification effort as Bayesian analysis.     

Testing and Characterizing Properties of Nonadditive Measures Through Violations of the Sure‐Thing Principle

In expected utility theory, risk attitudes are modeled entirely in terms of utility. In the rank‐dependent theories, a new dimension is added: chance attitude, modeled in terms of nonadditive measures or nonlinear probability transformations that are independent of utility. Most empirical studies of chance attitude assume probabilities given and adopt parametric fitting for estimating the probability transformation. Only a few qualitative conditions have been proposed or tested as yet, usually quasi‐concavity or quasi‐convexity in the case of given probabilities. This paper presents a general method of studying qualitative properties of chance attitude such as optimism, pessimism, and the “inverse‐S shape” pattern, both for risk and for uncertainty. These qualitative properties can be characterized by permitting appropriate, relatively simple, violations of the sure‐thing principle. In particular, this paper solves a hitherto open problem: the preference axiomatization of convex (“pessimistic” or “uncertainty averse”) nonadditive measures under uncertainty. The axioms of this paper preserve the central feature of rank‐dependent theories, i.e. the separation of chance attitude and utility.     

Development of a Preliminary Framework for Informing the Risk Analysis and Risk Management of Nanoparticles

Decisions are often made even when there is uncertainty about the possible outcomes. However, methods for making decisions with uncertainty in the problem framework are scarce. Presently, safety assessment for a product containing engineered nano-scale particles is a very poorly structured problem. Many fields of study may inform the safety assessment of such particles (e.g., ultrafines, aerosols, debris from medical devices), but engineered nano-scale particles may present such unique properties that extrapolating from other types of studies may introduce, and not resolve, uncertainty. Some screening-level health effects studies conducted specifically on engineered nano-scale materials have been published and many more are underway. However, it is clear that the extent of research needed to fully and confidently understand the potential for health or environmental risk from engineered nano-scale particles may take years or even decades to complete. In spite of the great uncertainty, there is existing research and experience among researchers that can help to provide a taxonomy of particle properties, perhaps indicating a relative likelihood of risk, in order to prioritize nanoparticle risk research. To help structure this problem, a framework was developed from expert interviews of nanotechnology researchers. The analysis organizes the information as a system based on the risk assessment framework, in order to support the decision about safety. In the long term, this framework is designed to incorporate research results as they are generated, and therefore serve as a tool for estimating the potential for human health and environmental risk.     

Evaluating Risk Communications: Completing and Correcting Mental Models of Hazardous Processes, Part II

We propose a decision-analytic framework, called the mental models approach , for evaluating the impact of risk communications. It employs multiple evaluation methods, including think-aloud protocol analysis, problem solving, and a true-false test that allows respondents to express uncertainty about their answers. The approach is illustrated in empirical comparisons of three brochures about indoor radon.     

Optimal design of supply chain networks under uncertain transient demand variations

We consider a detailed mathematical formulation for the problem of designing supply chain networks comprising multiproduct production facilities with shared production resources, warehouses, distribution centers and customer zones and operating under time varying demand uncertainty. Uncertainty is captured in terms of a number of likely scenarios possible to materialize during the lifetime of the network. The problem is formulated as a mixed-integer linear programming problem and solved to global optimality using standard branch-and-bound techniques. A case study concerned with the establishment of Europe-wide supply chain is used to illustrate the applicability and efficiency of the proposed approach. The results obtained provide a good indication of the value of having a model that takes into account the complex interactions that exist in such networks and the effect of inventory levels to the design and operation.     

Potential Uncertainty Reduction in Model‐Averaged Benchmark Dose Estimates Informed by an Additional Dose Study

A methodology is presented for assessing the information value of an additional dosage experiment in existing bioassay studies. The analysis demonstrates the potential reduction in the uncertainty of toxicity metrics derived from expanded studies, providing insights for future studies. Bayesian methods are used to fit alternative dose‐response models using Markov chain Monte Carlo (MCMC) simulation for parameter estimation and Bayesian model averaging (BMA) is used to compare and combine the alternative models. BMA predictions for benchmark dose (BMD) are developed, with uncertainty in these predictions used to derive the lower bound BMDL. The MCMC and BMA results provide a basis for a subsequent Monte Carlo analysis that backcasts the dosage where an additional test group would have been most beneficial in reducing the uncertainty in the BMD prediction, along with the magnitude of the expected uncertainty reduction. Uncertainty reductions are measured in terms of reduced interval widths of predicted BMD values and increases in BMDL values that occur as a result of this reduced uncertainty. The methodology is illustrated using two existing data sets for TCDD carcinogenicity, fitted with two alternative dose‐response models (logistic and quantal‐linear). The example shows that an additional dose at a relatively high value would have been most effective for reducing the uncertainty in BMA BMD estimates, with predicted reductions in the widths of uncertainty intervals of approximately 30%, and expected increases in BMDL values of 5–10%. The results demonstrate that dose selection for studies that subsequently inform dose‐response models can benefit from consideration of how these models will be fit, combined, and interpreted.     

Strategic thinking—Dealing with uncertainty

Frameworks for strategic planning provide questions, but the answers depend on executives' strategic thinking. Three methods of strategic thinking are delineated. Assessment involves classification and modelling of the situation; problem identification and analysis involve goal formulation and strategy design; and synthesis involves inferences about assumptions, dialectical analysis and reframing. The methods provide executives with different ways of dealing with uncertainty. Either problem analysis of synthesis is used when objectives and the appropriate corrective action are certain. Problem identification is used when there is ambiguity about strategic direction, and assessment is a style adapted to ambiguity about both objectives and corrective action.     

Robust Predictions in Games With Incomplete Information

We analyze games of incomplete information and offer equilibrium predictions that are valid for, and in this sense robust to, all possible private information structures that the agents may have. The set of outcomes that can arise in equilibrium for some information structure is equal to the set of Bayes correlated equilibria. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action–state distribution. We derive exact bounds on how prior knowledge about the private information refines the set of equilibrium predictions. We consider information sharing among firms under demand uncertainty and find new optimal information policies via the Bayes correlated equilibria. We also reverse the perspective and investigate the identification problem under concerns for robustness to private information. The presence of private information leads to set rather than point identification of the structural parameters of the game.     

A Combined Monte Carlo and Possibilistic Approach to Uncertainty Propagation in Event Tree Analysis

In risk analysis, the treatment of the epistemic uncertainty associated to the probability of occurrence of an event is fundamental. Traditionally, probabilistic distributions have been used to characterize the epistemic uncertainty due to imprecise knowledge of the parameters in risk models. On the other hand, it has been argued that in certain instances such uncertainty may be best accounted for by fuzzy or possibilistic distributions. This seems the case in particular for parameters for which the information available is scarce and of qualitative nature. In practice, it is to be expected that a risk model contains some parameters affected by uncertainties that may be best represented by probability distributions and some other parameters that may be more properly described in terms of fuzzy or possibilistic distributions. In this article, a hybrid method that jointly propagates probabilistic and possibilistic uncertainties is considered and compared with pure probabilistic and pure fuzzy methods for uncertainty propagation. The analyses are carried out on a case study concerning the uncertainties in the probabilities of occurrence of accident sequences in an event tree analysis of a nuclear power plant.     

Distributionally robust maximum probability shortest path problem

In this study, we discuss and develop a distributionally robust joint chance-constrained optimization model and apply it for the shortest path problem under resource uncertainty. In sch a case, robust chance constraints are approximated by constraints that can be reformulated using convex programming. Since the issue we are discussing here is of the multi-resource type, the resource related to cost is deterministic; however, we consider a robust set for other resources where covariance and mean are known. Thus, the chance-constrained problem can be expressed in terms of a cone constraint. In addition, since our problem is joint chance-constrained optimization, we can use Bonferroni approximation to divide the problem into L separate problems in order to build convex approximations of distributionally robust joint chance constraints. Finally, numerical results are presented to illustrate the rigidity of the bounds and the value of the distributionally robust approach.

     

考虑长期运力合同的班轮收益管理运输路径优化模型

基于收益管理的方法,文章对随机需求环境下班轮运力分配和路径优化问题进行了定量研究。首先针对海运收益管理的特征,建立了考虑长期运力合同、空箱调运的轮运力分配和路径选择随机规划模型,然后应用稳健优化方法对此模型进行求解。最后,通过数值仿真得到了优化的舱位分配方案,比较发现稳健优化模型取得了较确定性规划模型更好的收益,显示了模型和方法对于集装箱海运企业的收益管理问题具有应用价值。     

考虑空箱调运的集装箱海运收益管理能力分配优化模型

文章基于收益管理的思想对面临不确定需求的海运集装箱能力分配问题进行了定量研究。首先描述了集装箱海运收益管理问题与航空客运收益管理的差异,接着建立了考虑和不考虑空箱调运的海运集装箱能力分配随机规划模型,并应用稳健优化方法对模型进行求解,最后通过数值仿真,说明了模型和求解方法对于海运集装箱运输企业的收益管理问题具有重要的应用价值。     

On multi-criteria chance-constrained capacitated single-source discrete facility location problems

This work aims at investigating multi-criteria modeling frameworks for discrete stochastic facility location problems with single sourcing. We assume that demand is stochastic and also that a service level is imposed. This situation is modeled using a set of probabilistic constraints. We also consider a minimum throughput at the facilities to justify opening them. We investigate two paradigms in terms of multi-criteria optimization: vectorial optimization and goal programming. Additionally, we discuss the joint use of objective functions that are relevant in the context of some humanitarian logistics problems. We apply the general modeling frameworks proposed to the so-called stochastic shelter site location problem. This is a problem emerging in the context of preventive disaster management. We test the models proposed using two real benchmark data sets. The results show that considering uncertainty and multiple objectives in the type of facility location problems investigated leads to solutions that may better support decision making.