On the Global Convergence of Stochastic Fictitious Play

We establish global convergence results for stochastic fictitious play for four classes of games: games with an interior ESS, zero sum games, potential games, and supermodular games. We do so by appealing to techniques from stochastic approximation theory, which relate the limit behavior of a stochastic process to the limit behavior of a differential equation defined by the expected motion of the process. The key result in our analysis of supermodular games is that the relevant differential equation defines a strongly monotone dynamical system. Our analyses of the other cases combine Lyapunov function arguments with a discrete choice theory result: that the choice probabilities generated by any additive random utility model can be derived from a deterministic model based on payoff perturbations that depend nonlinearly on the vector of choice probabilities.     

Flexible Servers in Understaffed Tandem Lines

We study the dynamic assignment of cross‐trained servers to stations in understaffed lines with finite buffers. Our objective is to maximize the production rate. We identify optimal server assignment policies for systems with three stations, two servers, different flexibility structures, and either deterministic service times and arbitrary buffers or exponential service times and small buffers. We use these policies to develop server assignment heuristics for Markovian systems with larger buffer sizes that appear to yield near‐optimal throughput. In the deterministic setting, we prove that the best possible production rate with full server flexibility and infinite buffers can be attained with partial flexibility and zero buffers, and we identify the critical skills required to achieve this goal. We then present numerical results showing that these critical skills, employed with an effective server assignment policy, also yield near‐optimal throughput in the Markovian setting, even for small buffer sizes. Thus, our results suggest that partial flexibility is sufficient for near‐optimal performance, and that flexibility structures that are effective for deterministic and infinite‐buffered systems are also likely to perform well for finite‐buffered stochastic systems.     

Optimal and Heuristic Models for Lot Splitting in a Flow Shop

This paper investigates optimal lot-splitting policies in a multiprocess flow shop environment with the objective of minimizing either mean flow time or makespan. Using a quadratic programming approach to the mean flow time problem, we determine the optimal way of splitting a job into smaller sublots under various setup times to run time ratios, number of machines in the flow shop, and number of allowed sublots. Our results come from a deterministic flow shop environment, but also provide insights into the repetitive lots scheme using equal lot splits for job shop scheduling in a stochastic environment. We indicate those conditions in which managers should implement the repetitive lots scheme and where other lot-splitting schemes should work better.     

Testing for Stochastic Monotonicity

We propose a test of the hypothesis of stochastic monotonicity. This hypothesis is of interest in many applications in economics. Our test is based on the supremum of a rescaled U‐statistic. We show that its asymptotic distribution is Gumbel. The proof is difficult because the approximating Gaussian stochastic process contains both a stationary and a nonstationary part, and so we have to extend existing results that only apply to either one or the other case. We also propose a refinement to the asymptotic approximation that we show works much better in finite samples. We apply our test to the study of intergenerational income mobility.     

Assessing the impact of uncertainty and the level of aggregation in case mix planning

In this article, we present a framework for evaluating the impact of uncertainty and the use of different aggregation levels in case mix planning on the quality of strategic decisions regarding the case mix of a hospital. In particular, we analyze the effect of modeling (i) demand, (ii) resource use, and (iii) resource availability as stochastic input parameters on the performance of case mix planning models. In addition, the consequences of taking the weekly structure with inactive days without surgeries into account are assessed (iv). The purpose of this paper is to provide a guideline for the decision-maker planning the case mix on the consideration of stochastic aspects and different aggregation levels. We formulate a mixed integer programming model for case mix planning along with different stochastic and deterministic extensions. The value of the different extensions is analyzed using a factorial design. The resulting stochastic models are solved using sample average approximation. Simulation is used to evaluate the strategies derived by the different models using real-world data from a large German hospital. We find that highly aggregated basic case mix planning models can overestimate the objective value by up to 10% and potentially lead to biased results. Refining the problem decreased the gap between projected case mix planning results and simulated results considerably and led to improved solutions.     

Lot Splitting in Stochastic Flow Shop and Job Shop Environments*

In recent years many firms have been implementing small lot size production. Lot splitting breaks large orders into smaller transfer lots and offers the ability to move parts more quickly through the production process. This paper extends the deterministic studies by investigating various lot splitting policies in both stochastic job shop and stochastic flow shop settings using performance measures of mean flow time and the standard deviation of flow time. Using a computer simulation experiment, we found that in stochastic dynamic job shops, the number of lot splits is more important than the exact form of splitting. However, when optimal job sizes are determined for each scenario, we found a few circumstances where the implementation of a small initial split, called a “flag,” can provide measurable improvement in flow time performance. Interestingly, the vast majority of previous research indicates that methods other than equal lot splitting typically improves makespan performance. The earlier research, however, has been set in the static, deterministic flow shop environment. Thus, our results are of practical interest since they show that the specific method of lot splitting is important in only a small set of realistic environments while the choice of an appropriate number of splits is typically more important.     

Single-Stage and Serial Production Line Systems with Unreliable Machines and General Probability Distributions

This paper presents stochastic models for analyzing production to stock and production on demand systems with arbitrary demand, production, machine breakdown and repair processes, and a finite buffer. Expressions are developed for the transition probabilities among the different states, the proportion of transitions that take the process to a particular state, the probabilities that the process is at a particular state, and the expected state residence times. Also, expressions are developed for some quantities of interest, like average inventory, percentage of lost sales, and expected waiting time. Applications are discussed for various demand, production, machine breakdown, and repair processes and numerical examples are presented. Guidelines are given for utilizing the models in evaluating the economic performance of production systems. Using an approximation procedure, the models are used for modeling multi-stage serial production to stock and production on demand lines. Simulation results indicated that the quality of the approximation is very good. The multi-stage models are useful in practical situations, because they allow the use of different probability distributions for the various processes (production, machine failure, etc.) of each stage.     

Stochasticity in Physiologically Based Kinetics Models: Implications for Cancer Risk Assessment

In case of low-dose exposure to a substance, its concentration in cells is likely to be stochastic. Assessing the consequences of this stochasticity in toxicological risk assessment requires the coupling of macroscopic dynamics models describing whole-body kinetics with microscopic tools designed to simulate stochasticity. In this article, we propose an approach to approximate stochastic cell concentration of butadiene in the cells of diverse organs. We adapted the dynamics equations of a physiologically based pharmacokinetic (PBPK) model and used a stochastic simulator for the system of equations that we derived. We then coupled kinetics simulations with a deterministic hockey stick model of carcinogenicity. Stochasticity induced substantial modifications relative to dose-response curve, compared with the deterministic situation. In particular, there was nonlinearity in the response and the stochastic apparent threshold was lower than the deterministic one. The approach that we developed could easily be extended to other biological studies to assess the influence of stochasticity at macroscopic scale for compound dynamics at the cell level.     

Extending the Stochastic Two-Stage Model of Carcinogenesis to Include Self-Regulation of the Nonmalignant Cell Population

One of the challenges of introducing greater biological realism into stochastic models of cancer induction is to find a way to represent the homeostatic control of the normal cell population over its own size without complicating the analysis too much to obtain useful results. Current two-stage models of carcinogenesis typically ignore homeostatic control. Instead, a deterministic growth path is specified for the population of "normal" cells, while the population of "initiated" cells is assumed to grow randomly according to a birth-death process with random immigrations from the normal population. This paper introduces a simple model of homeostatically controlled cell division for mature tissues, in which the size of the nonmalignant population remains essentially constant over time. Growth of the nonmalignant cell population (normal and initiated cells) is restricted by allowing cells to divide only to fill the "openings" left by cells that die or differentiate, thus maintaining the constant size of the nonmalignant cell population. The fundamental technical insight from this model is that random walks, rather than birth-and-death processes, are the appropriate stochastic processes for describing the kinetics of the initiated cell population. Qualitative and analytic results are presented, drawn from the mathematical theories of random walks and diffusion processes, that describe the probability of spontaneous extinction and the size distribution of surviving initiated populations when the death/differentiation rates of normal and initiated cells are known. The constraint that the nonmalignant population size must remain approximately constant leads to much simpler analytic formulas and approximations, flowing directly from random walk theory, than in previous birth-death models.(ABSTRACT TRUNCATED AT 250 WORDS)     

Optimal Pricing,Ordering, and Return Policies for Consumer Goods

Our research addresses a firm that sells a product to consumers who are sensitive to both price and return policy. The operational decisions of interest are the selling price, return policy, and quantity of new product to purchase. We model a single selling season that is split into two periods where the boundary between periods is delineated by the opportunity to recover product returns and resell them. That is, returns in the first period can be recovered and sold in the second period. Returns also arise in the second period, but these may only be salvaged. We first analyze both deterministic and stochastic models, finding that the deterministic results largely carry over to the stochastic case. In addition, our results indicate that the model is quite insensitive to errors in the estimates of the parameter values, except for purchase cost and parameters related to demand. Finally, we perform an analysis on the value of various investments to improve financial performance. Results indicate that investments to reduce the recovery cost of returns or reduce returns uncertainty are minimal, while investments to increase recovery speed, reduce market uncertainty, and reduce the return rate can be quite valuable.     

Centralization versus decentralization: Risk pooling,risk diversification,and supply chain disruptions

We investigate optimal system design in a multi-location system in which supply is subject to disruptions. We examine the expected costs and cost variances of the system in both a centralized and a decentralized inventory system. We show that, when demand is deterministic and supply may be disrupted, using a decentralized inventory design reduces cost variance through the risk diversification effect, and therefore a decentralized inventory system is optimal. This is in contrast to the classical result that when supply is deterministic and demand is stochastic, centralization is optimal due to the risk-pooling effect. When both supply may be disrupted and demand is stochastic, we demonstrate that a risk-averse firm should typically choose a decentralized inventory system design.     

Cash Flow Modelling and Forecasting in Capital Budgeting Under Uncertainty*

The subject of this paper is modelling and forecasting of cash flows generated by a capital investment. The paper proposes the application of autoregressive, moving average, and mixed autoregressive moving-average processes to capital budgeting. In addition, models for deterministic, stochastic and seasonal trends are considered. For each class of cash flows, analytical expressions are developed for the mean and variance of a project's net present value (NPV). Also considered are several equilibrium pricing models. For two of them, the forecasting methodology developed here is integrated into the pricing equation. Although the overall emphasis of the paper is on modelling and forecasting of cash flows, the implications for NPV pricing and risk analysis are also investigated. Several examples are used to illustrate the impact of particular cash flow models on the price and risk of a project.     

Approximating soft-capacitated facility location problem with uncertainty

In this paper we devise the stochastic and robust approaches to study the soft-capacitated facility location problem with uncertainty. We first present a new stochastic soft-capacitated model called The 2-Stage Soft Capacitated Facility Location Problem and solve it via an approximation algorithm by reducing it to linear-cost version of 2-stage facility location problem and dynamic facility location problem. We then present a novel robust model of soft-capacitated facility location, The Robust Soft Capacitated Facility Location Problem. To solve it, we improve the approximation algorithm proposed by Byrka et al. (LP-rounding algorithms for facility-location problems. CoRR, 2010a) for RFTFL and then treat it similarly as in the stochastic case. The improvement results in an approximation factor of \(\alpha + 4\) for the robust fault-tolerant facility location problem, which is best so far.     

Generalized Method of Integrated Moments for High‐Frequency Data

We propose a semiparametric two‐step inference procedure for a finite‐dimensional parameter based on moment conditions constructed from high‐frequency data. The population moment conditions take the form of temporally integrated functionals of state‐variable processes that include the latent stochastic volatility process of an asset. In the first step, we nonparametrically recover the volatility path from high‐frequency asset returns. The nonparametric volatility estimator is then used to form sample moment functions in the second‐step GMM estimation, which requires the correction of a high‐order nonlinearity bias from the first step. We show that the proposed estimator is consistent and asymptotically mixed Gaussian and propose a consistent estimator for the conditional asymptotic variance. We also construct a Bierens‐type consistent specification test. These infill asymptotic results are based on a novel empirical‐process‐type theory for general integrated functionals of noisy semimartingale processes.     

Point‐of‐Use Hybrid Inventory Policy for Hospitals

Modern point‐of‐use technology at hospitals has enabled new replenishment policies for medical supplies. One of these new policies, which we call the hybrid policy, is currently in use at a large U.S. Midwest hospital. The hybrid policy combines a low‐cost periodic replenishment epoch with a high‐cost continuous replenishment option to avoid costly stockouts. We study this new hybrid policy under deterministic and stochastic demand. We develop a parameter search engine using simulation to optimize the long‐run average cost per unit time and, via a computational study, we provide insights on the benefits (reduction in cost, inventory, and number of replenishments) that hospitals may obtain by using the hybrid policy instead of the commonly used periodic policies. We also use the optimal hybrid policy parameters from the deterministic analysis to propose approximate expressions for the stochastic hybrid policy parameters that can be easily used by hospital management.      

Distribution of Aggregate Utility Using Stochastic Elements of Additive Multiattribute Utility Models

Conventionally, elements of a multiattribute utility model characterizing a decision maker's preferences, such as attribute weights and attribute utilities, are treated as deterministic, which may be unrealistic because assessment of such elements can be imprecise and erroneous, or differ among a group of individuals. Moreover, attempting to make precise assessments can be time consuming and cognitively demanding. We propose to treat such elements as stochastic variables to account for inconsistency and imprecision in such assessments. Under these assumptions, we develop procedures for computing the probability distribution of aggregate utility for an additive multiattribute utility function (MAUF), based on the Edgeworth expansion. When the distributions of aggregate utility for all alternatives in a decision problem are known, stochastic dominance can then be invoked to filter inferior alternatives. We show that, under certain mild conditions, the aggregate utility distribution approaches normality as the number of attributes increases. Thus, only a few terms from the Edgeworth expansion with a standard normal density as the base function will be sufficient for approximating an aggregate utility distribution in practice. Moreover, the more symmetric the attribute utility distributions, the fewer the attributes to achieve normality. The Edgeworth expansion thus can provide a basis for a computationally viable approach for representing an aggregate utility distribution with imprecisely specified attribute weights and utilities assessments (or differing weights and utilities across individuals). Practical guidelines for using the Edgeworth approximation are given. The proposed methodology is illustrated using a vendor selection problem.     

Stabilized‐Cycle Strategy for Capacitated Lot Sizing with Multiple Products: Fill‐Rate Constraints in Rolling Schedules

In practice, deterministic, multi‐period lot‐sizing models are implemented in rolling schedules since this allows the revision of decisions beyond the frozen horizon. Thus, rolling schedules are able to take realizations and updated forecasts of uncertain data (e.g., customer demands) into account. Furthermore, it is common to hold safety stocks to ensure given service levels (e.g., fill rate). As we will show, this approach, implemented in rolling schedules, often results in increased setup and holding costs while (over‐)accomplishing given fill rates. A well‐known alternative to deterministic planning models are stochastic, static, multi‐period planning models used in the static uncertainty strategy, which results in stable plans. However, these models have a lack of flexibility to react to the realization of uncertain data. As a result, actual costs may differ widely from planned costs, and downside deviations of actual fill rates from those given are very high. We propose a new strategy, namely the stabilized cycle. This combines and expands upon ideas from the literature for minimizing setup and holding costs in rolling schedules, while controlling actual product‐specific fill rates for a finite reporting period. A computational study with a multi‐item capacitated medium‐term production planning model has been executed in rolling schedules. On the one hand, it demonstrates that the stabilized‐cycle strategy yields a good compromise between costs and downside deviations. Furthermore, the stabilized‐cycle strategy weakly dominates the order‐based strategy for both constant and seasonal demands.     

Statistical Inference for Stochastic Dominance and for the Measurement of Poverty and Inequality

We derive the asymptotic sampling distribution of various estimators frequently used to order distributions in terms of poverty, welfare, and inequality. This includes estimators of most of the poverty indices currently in use, as well as estimators of the curves used to infer stochastic dominance of any order. These curves can be used to determine whether poverty, inequality, or social welfare is greater in one distribution than in another for general classes of indices and for ranges of possible poverty lines. We also derive the sampling distribution of the maximal poverty lines up to which we may confidently assert that poverty is greater in one distribution than in another. The sampling distribution of convenient dual estimators for the measurement of poverty is also established. The statistical results are established for deterministic or stochastic poverty lines as well as for paired or independent samples of incomes. Our results are briefly illustrated using data for four countries drawn from the Luxembourg Income Study data bases.     

Genericity and Markovian Behavior in Stochastic Games

This paper examines Markov perfect equilibria of general, finite state stochastic games. Our main result is that the number of such equilibria is finite for a set of stochastic game payoffs with full Lebesgue measure. We further discuss extensions to lower dimensional stochastic games like the alternating move game.     

A shelf-space optimization model when demand is stochastic and space-elastic

The more customer demand is impulse-driven, the more it is space-dependent and the more it is subject to variation. We investigate the corresponding problem of retail shelf-space planning when demand is stochastic and sensitive to the number and position of facings. We develop a model to maximize a retailer׳s profit by selecting the number of facings and their shelf position under the assumption of limited space. The model is particularly applicable to promotional or temporary products.We develop the first optimization model and solution approach that takes stochastic demand into account, since the current literature applies deterministic models for shelf-space planning. By the means of an innovative modeling approach for the case with space- and positioning effects and the conversion of our problem into a mixed-integer problem, we obtain optimal results within very short run times for large-scale instances relevant in practice. Furthermore, we develop a solution approach to account for cross-space elasticity, and solve it using an own heuristic, which efficiently yields near-optimal results. We demonstrate that correctly considering space elasticity and demand variation is essential. The corresponding impacts on profits and solution structures become even more significant when space elasticity and stochastic demand interact, resulting in up to 5% higher profits and up to 80% differences in solution structures, if both effects are correctly accounted for. We develop an efficient modeling approach, compare the model results with approaches applied in practice and derive rules-of-thumb for planners.