Multistage Set Research Articles

A multi-stage multi-criterion group decision-making method for emergency management based on alternative chain and trust radius of experts

Multi-criterion group decision-making (MCGDM) is common when a group of experts collectively evaluate feasible alternatives regarding a set of criteria. With the increasing complexity of decision-making activities, there may be intricate interrelationships between criteria. In addition, many MCGDM problems have a dynamic characteristic when alternatives are presented to experts sequentially. In such a multi-stage setting, previous experts have influence on later experts. To overcome these challenges in MCGDM, this study first introduces two concepts, namely, trust radius, to model the social influence of experts, and alternative chain, to represent multiple alternatives that can be coordinated at adjacent stages, and then proposes a multi-stage MCGDM method based on the alternative chain and trust radius of experts. In this method, the weights of experts are determined according to the degree that an expert is influenced by others and the degree that an expert has influence on others. The weights of criteria are determined by the Decision Making Trial and Evaluation Laboratory (DEMATEL) method. This study obtains the optimal alternative chain from a 0–1 optimization model based on comprehensive scores of alternatives at each stage and compatible relationships between adjacent stages. The applicability of the proposed method is demonstrated by an illustrative example regarding emergency management of dammed lakes after natural disasters. Comparative analyses show advantages of this method.

Effective Scenarios in Multistage Distributionally Robust Optimization with a Focus on Total Variation Distance

We study multistage distributionally robust optimization (DRO) to hedge against ambiguity in quantifying the underlying uncertainty of a problem. Recognizing that not all the realizations and scenario paths might have an “effect” on the optimal value, we investigate the question of how to define and identify critical scenarios for nested multistage DRO problems. Our analysis extends the work of Rahimian, Bayraksan, and Homem-de-Mello [Math. Program., 173 (2019), pp. 393--430], which was in the context of a static/two-stage setting, to the multistage setting. To this end, we define the notions of effectiveness of scenario paths and the conditional effectiveness of realizations along a scenario path for a general class of multistage DRO problems. We then propose easy-to-check conditions to identify the effectiveness of scenario paths in the multistage setting when the distributional ambiguity is modeled via the total variation distance. Numerical results show that these notions provide useful insight on the underlying uncertainty of the problem.

A serial inventory system with lead-time-dependent backordering: A reduced-state approximation

We study a serial inventory system where the external customers may have a maximum time that they would be willing to wait for delivery in cases of stock-out and the demand would be lost if the remaining delivery lead time of the next available item is longer. This lead-time-dependent backordering behavior subsumes the models of partial backordering regardless of the wait that a customer would experience. In the inventory literature, this behavior has only been analyzed in single-location settings. We study this behavior in a multi-stage setting. We consider continuous review policies at all stages facing external Poisson demands. Using the method of supplementary variables, we define the stochastic process representing the inventory system and obtain the expressions for the operating characteristics of the inventory system. Based on the solution structures for the special cases, we propose an approximate solution which rests on replacing the state-dependent purchasing decision of the customer with an averaged-out purchase probability computed using only the age of the oldest item. An extensive numerical study indicates that the proposed approximation performs very well. Our numerical study provides additional insights about the sensitivity and allocation of stock levels across stages.

Multistage knapsack

Many systems have to be maintained while the underlying constraints, costs and/or profits change over time. Although the state of a system may evolve during time, a non-negligible transition cost is incurred for transitioning from one state to another. In order to model such situations, we look at a recently introduced multistage model where the input is a sequence of instances (one for each time step), and the goal is to find a sequence of solutions (one for each time step) that are both (i) near optimal for each time step and (ii) as stable as possible. We propose a PTAS for the Multistage Knapsack problem. This is the first approximation scheme for a combinatorial optimization problem in the considered multistage setting, and its existence contrasts with the inapproximability results for other combinatorial optimization problems that are even polynomial-time solvable in the static case.

Markovian Decentralized Ensemble Control for Demand Response

With the advancement in smart grid and smart energy devices, demand response becomes one of the most economic and feasible solutions to ease the load stress of the power grids during peak hours. In this work, we propose a fully decentralized ensemble control framework with consensus for demand response (DR) events and compatible control methods based on random policies. We show that under the consensus that is tailored to DR, our proposed decentralized control method yields the same optimality as the centralized control method in both myopic and multistage settings.

Multistage planning of electric transit charging facilities under build-operate-transfer model

Careful advanced planning of charging facilities is required to support the successful functioning of the electric transit system. However, transit agencies generally lack sufficient experiences and professional skills to plan and build charging systems. A great way for them is to cooperate with other companies that provide professional charging services. This cooperation is often executed by a build-operate-transfer (BOT) model such that the charging service provider builds and plans the charging system, operates it for several years and then transfers the ownership to the transit agency. This research aims to investigate charging facilities planning for electric transit systems under the BOT model. As the transit electrification is conducted in a gradual manner, we consider the planning under a multistage setting. The problem is modeled as a mixed-integer program. We apply the model into a real-world transit system in a metropolis of China and some meaningful managerial insights are derived. • Online-to-offline mobile charging system for electric vehicles. • Scenario-sampling-based online policy for operating the mobile chargers. • Simulation-based optimization framework for planning the system.

Multistage Open-Shop-Problem: Schedules for N Tasks on K Machines with Different Technical Characteristics for an Arbitrary order of Tasks

The statements and mathematical models of the open — shop-problem are considered: performing N tasks on K machines with an arbitrary order of processing each task on all machines. Unlike most publications on solving this problem, the task posed is solved in a multi-stage formulation under the conditions of a sequential chain of sections and workshops of the enterprise, as well as in the presence of restrictions on the start and end times of tasks and the permissible operating times of machines. In addition, for scheduling systems of enterprises, it is extremely important to consider this task in a multi-stage setting, i.e. in a sequential chain of plots and workshops. The properties of admissible and optimal schedules, as well as algorithms of exact and approximate methods for solving these problems by successive optimization algorithms are investigated. At the early stages of solving the problem, the fact of incompatibility of the system of constraints of the task and the definition of a subset of tasks or resources, the time range of which should be expanded, is established. The described algorithms for solving the problem are illustrated by numerical examples.

The Importance of Modeling Data Missingness in Algorithmic Fairness: A Causal Perspective

Training datasets for machine learning often have some form of missingness. For example, to learn a model for deciding whom to give a loan, the available training data includes individuals who were given a loan in the past, but not those who were not. This missingness, if ignored, nullifies any fairness guarantee of the training procedure when the model is deployed. Using causal graphs, we characterize the missingness mechanisms in different real-world scenarios. We show conditions under which various distributions, used in popular fairness algorithms, can or can not be recovered from the training data. Our theoretical results imply that many of these algorithms can not guarantee fairness in practice. Modeling missingness also helps to identify correct design principles for fair algorithms. For example, in multi-stage settings where decisions are made in multiple screening rounds, we use our framework to derive the minimal distributions required to design a fair algorithm. Our proposed algorithm also decentralizes the decision-making process and still achieves similar performance to the optimal algorithm that requires centralization and non-recoverable distributions.

Multistage graph problems on a global budget

Time-evolving or temporal graphs gain more and more popularity when exploring complex networks. In this context, the multistage view on computational problems is among the most natural frameworks. Roughly speaking, herein one studies the different (time) layers of a temporal graph (effectively meaning that the edge set may change over time, but the vertex set remains unchanged), and one searches for a solution of a given graph problem for each layer. The twist in the multistage setting is that the solutions found must not differ too much between subsequent layers. We relax on this already established notion by introducing a global instead of the local budget view studied so far. More specifically, we allow for few disruptive changes between subsequent layers but request that overall, that is, summing over all layers, the degree of change is moderate. Studying several classical graph problems (both NP-hard and polynomial-time solvable ones) from a parameterized complexity angle, we encounter both fixed-parameter tractability and parameterized hardness results. Surprisingly, we find that sometimes the global multistage versions of NP-hard problems such as Vertex Cover turn out to be computationally more tractable than the ones of polynomial-time solvable problems such as Matching. In addition to time complexity, we also analyze the space efficiency of our algorithms.

On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling

We consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation. We study a sample average approximation (SAA) approach that uses nested sampling, in which at each stage, a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next-stage constraints. This approach provides an approximate solution to the multistage problem. To establish the consistency of the SAA approach, we first consider a two-stage problem and show that in the second-stage problem, given a scenario, the optimal values and solutions of the SAA converge to those of the true problem with probability one when the sample sizes go to infinity. These convergence results do not hold uniformly over all possible scenarios for the second stage problem. We are nevertheless able to prove that the optimal values and solutions of the SAA converge to the true ones with probability one when the sample sizes at both stages increase to infinity. We also prove exponential convergence of the probability of a large deviation for the optimal value of the SAA, the true value of an optimal solution of the SAA, and the probability that any optimal solution to the SAA is an optimal solution of the true problem. All of these results can be extended to a multistage setting and we explain how to do it. Our framework and SAA results cover a large variety of resource allocation problems for which at each stage after the first one, new information becomes available and the allocation can be readjusted, under constraints that involve expectations estimated by Monte Carlo. As an illustration, we apply this SAA method to a staffing problem in a call center, in which the goal is to optimize the numbers of agents of each type under some constraints on the quality of service (QoS). The staffing allocation has to be decided under an uncertain arrival rate with a prior distribution in the first stage, and can be adjusted at some additional cost when better information on the arrival rate becomes available in later stages.

M-estimation in Multistage Sampling Procedures

Multi-stage (designed) procedures, obtained by splitting the sampling budget suitably across stages, and designing the sampling at a particular stage based on information about the parameter obtained from previous stages, are often advantageous from the perspective of precise inference. We develop a generic framework for M-estimation in a multistage setting and apply empirical process techniques to develop limit theorems that describe the large sample behavior of the resulting M-estimates. Applications to change-point estimation, inverse isotonic regression, classification, mode estimation and cusp estimation are provided: it is typically seen that the multistage procedure accentuates the efficiency of the M-estimates by accelerating the rate of convergence, relative to one-stage procedures. The step-by-step process induces dependence across stages and complicates the analysis in such problems, which we address through careful conditioning arguments.

Privatization neutrality with quality and subsidies

This study introduces a subsidy policy on product quality in a quality-then-price game to remedy the quality distortion under a mixed oligopoly (one public firm and one private firm) framework. We show that the multi-stage setting for firms is crucial for the validity of privatization neutrality. Since firms have different objectives, their asymmetric strategic consideration on price will spill over to the quality competition if there exists price differentiation in equilibrium under partial privatization. This spillover effect results in lower social welfare levels than the first-best outcome, and the neutrality of privatization in White (Economics Letters 53:189–195) no longer holds in our multi-stage model. Specifically, the optimal privatization policy is either fully public or completely private, where the social welfare attains the first-best outcome.

Operational results on the fully automatic LHC collimator alignment

The Large Hadron Collider has a complex collimation system installed to protect its sensitive equipment from normal and abnormal beam losses. The collimators are set around the beam following a multistage transverse setting hierarchy. The insertion position of each collimator is established using beam-based alignment techniques to determine the local beam position and rms beam size at each collimator. During previous years, collimator alignments were performed semiautomatically, with collimation experts present to oversee and control the alignment. During run II, a new fully automatic alignment tool was developed and used for collimator alignments throughout 2018. This paper discusses the improvements on the alignment software to automate it using machine learning, whilst focusing on the operational results obtained when testing the new software in the LHC. The alignment tests were conducted with both proton and ion beams, and angular alignments were performed with proton beams. This upgraded software successfully decreased the alignment time by a factor of 3 and made the results more reproducible, which is particularly important when performing angular alignments.

Modelling HSLA steel product quality under multi-stage manufacturing set up using multi-block partial least square regression

In order to have an understanding about the quality of steel product, involving multiple stages of manufacturing process, adequate assessment about the input-output relationship is necessary to ensure high-quality product. In this study, a high-strength low-alloy (HSLA) steel product quality, comprising of two stages of manufacturing, is modelled using partial least square regression (PLSR) and multi-block PLSR (MBPLSR) approaches. The alloy chemistry and rolling parameters are considered here as input variables along with strength and ductility of the finished steel as responses. Hotelling's T2 statistic is used for diagnosis of faults in batches of heat along with developing fault detection system through significant input variables. Both the modelling approaches are found to be useful for this purpose. However, the MBPLSR-based modelling approach is preferable since it helps to locate the source of the problem quicker at appropriate stages of operation with the help of input variables.

Adaptive and Dynamic Team Formation for Strategic and Tactical Planning

Past work in security games has mainly focused on the problem static resource allocation; how to optimally deploy a given fixed team of resources. My research aims to address the challenge of integrating operational planning into security games, where resources are heterogeneous and the defender is tasked with optimizing over both the investment into these resources, as well as their deployment in the field. This allows the defender to design more adaptive strategies, reason about the efficiency of their use of these resources as well as their effectiveness in their deployment. This thesis explores the challenges in integrating these two optimization problems in both the single stage and multi-stage setting and provides a formal model of this problem, which we refer to as the Simultaneous Optimization of Resource Teams and Tactics (SORT) as a new fundamental research problem in security games that combines strategic and tactical decision making. The main contributions of this work are solution methods to the SORT problem under various settings as well as exploring various types of tradeoffs that can arise in these settings. These include managing budget for investment in resources as well as capacity constraints on use of resources. My work addresses scenarios when the tactical decision problem (optimal deployment) is difficult, and thus evaluating the performance of any given team is difficult. Additionally, I address domains where we are tasked with making repeated strategic level decision and where, due to changing domain features, fluctuations in time dependent processes or the realization of uncertain parameters in the problem, it becomes necessary to re-evaluate and adapt to new information.

Adaptation and approximate strategies for solving the lot-sizing and scheduling problem under multistage demand uncertainty

This work addresses the lot-sizing and scheduling problem under multistage demand uncertainty. A flexible production system is considered, with the possibility to adjust the size and the schedule of lots in every time period based on a rolling-horizon planning scheme. Computationally intractable multistage stochastic programming models are often employed on this problem. An adaptation strategy to the multistage setting for two-stage programming and robust optimization models is proposed. We also present an approximate heuristic strategy to address the problem more efficiently, relying on multistage stochastic programming and adjustable robust optimization. In order to evaluate each strategy and model proposed, a Monte Carlo simulation experiment under a rolling-horizon scheme is performed. Results show that the strategies are promising in solving large-scale problems: the approximate strategy based on adjustable robust optimization has, on average, 6.72% better performance and is 7.9 times faster than the deterministic model.

Efficiency Measurement of Multistage Processes: Context Dependent Numbers of Stages

An important area of research involving the benchmarking methodology data envelopment analysis (DEA), concerns the modeling of multistage situations. In the usual multistage settings, it is generally assumed that all decision-making units (DMUs) have the same number and configuration of stages. However, in many real-world examples, this assumption does not hold. Consider, for example, a supply chain setting where for some DMUs, products are shipped directly from a supplier to a retailer (single-stage), while for other DMUs, products can be transshipped through distribution centers (two or more stages). In the current paper, we investigate an efficiency measurement situation where the DMUs exhibit a mix of single and two-stage setups. The particular case examined involves a set of high technology firms that can be thought of as falling into two groups; those firms where the output of interest is the annual revenue generated, and those that not only generate revenue, but as well invest a portion of that revenue in R&D. Firms in the first group can be viewed as being single-stage DMUs while those in the other group are of the two-stage type. The modeling complication here is that the set of DMUs do not explicitly form a homogeneous set of units. We develop a DEA-style model aimed at measuring efficiency in the presence of such nonhomogeneous two-group structures.

Robust optimization with time-dependent uncertainty in radiation therapy

ABSTRACTIn the recent past, robust optimization methods have been developed and successfully applied to a variety of single-stage problems. More recently, some of these approaches have been extended to multi-stage settings with fixed uncertainties. However, in many real-world applications, uncertainties evolve over time, rendering the robust solutions suboptimal. This issue is particularly prevalent in medical decision making, where a patient’s condition can change during the course of the treatment. In the context of radiation therapy, changes in cell oxygenation directly affect the response to radiation. To address such uncertain changes, we provide a general robust optimization framework that incorporates time-dependent uncertainty sets in a tractable fashion. Temporal changes reside within a cone, whose projection at each step yields the current uncertainty set. We develop conic robust two-stage linear problems and provide their robust counterparts for uncertain constraint parameters, covering the range of radiation therapy problems. For a clinical prostate cancer case, the time-dependent robust approach improves the tumor control throughout the treatment, as opposed to current methods that lose efficacy at some stage. We show that this advantage does not bear additional risks compared to current clinical methods. For intermediate diagnostics, we provide the optimal observation timing that maximizes the value of information. While these findings are relevant to clinical settings, they are also general and can be applied to a broad range of applications; e.g., in maintenance scheduling.

More powerful permutation test based on multistage ranked set sampling

ABSTRACTMany researches have used ranked set sampling (RSS) method instead of simple random sampling (SRS) to improve power of some nonparametric tests. In this study, the two-sample permutation test within multistage ranked set sampling (MSRSS) is proposed and investigated. The power of this test is compared with the SRS permutation test for some symmetric and asymmetric distributions through Monte Carlo simulations. It has been found that this test is more powerful than the SRS permutation test; its power increased by set size and/or number of cycles and/or number of stages. Symmetric distributions power increased better than asymmetric distributions power.

Topology Design Games and Dynamics in Adversarial Environments

We study the problem of network topology design within a set of policy-compliant topologies as a game between a designer and an adversary. At any time instant, the designer aims to operate the network in an optimal topology within the set of policy compliant topologies with respect to a desired network property. Simultaneously, the adversary counters the designer trying to force operation in a suboptimal topology. Specifically, if the designer and the attacker choose the same link in the current topology to defend/grow and attack, respectively, then the latter is thwarted. However, if the defender does not correctly guess where the attacker is going to attack, and, hence, acts elsewhere, the topology reverts to the best policy-compliant configuration after a successful attack. We show the existence of various mixed strategy equilibria in this game and systematically study its structural properties. We study the effect of parameters, such as probability of a successful attack, and characterize the steady state behavior of the underlying Markov chain. While the intuitive adversarial strategy here is to attack the most important links, the Nash equilibrium strategy is for the designer to defend the most crucial links and for the adversary to focus attack on the lesser crucial links. We validate these properties through two use cases with example sets of network topologies. Next, we consider a multi-stage framework where the designer is not only interested in the instantaneous network property costs but a discounted sum of costs over many time instances. We establish structural properties of the equilibrium strategies in the multi-stage setting, and also demonstrate that applying algorithms based on the Q-Learning and Rollout methods can result in significant benefits for the designer compared with strategies resulting from a one-shot based game.