Matthew J. Sobel

Forecasting the Effects of Global Warming on Biodiversity

ABSTRACT The demand for accurate forecasting of the effects of global warming on biodiversity is growing, but current methods for forecasting have limitations. In this article, we compare and discuss the different uses of four forecasting methods: (1) models that consider species individually, (2) niche-theory models that group species by habitat (more specifically, by environmental conditions under which a species can persist or does persist), (3) general circulation models and coupled ocean–atmosphere–biosphere models, and (4) species–area curve models that consider all species or large aggregates of species. After outlining the different uses and limitations of these methods, we make eight primary suggestions for improving forecasts. We find that greater use of the fossil record and of modern genetic studies would improve forecasting methods. We note a Quaternary conundrum: While current empirical and theoretical ecological results suggest that many species could be at risk from global warming, during the recent ice ages surprisingly few species became extinct. The potential resolution of this conundrum gives insights into the requirements for more accurate and reliable forecasting. Our eight suggestions also point to constructive synergies in the solution to the different problems.

Inventory control with an exponential utility criterion

A base-stock policy is shown to be optimal when a dynamic version of the “news vendor” model is optimized with respect to an exponential utility criterion.

The variance of discounted Markov decision processes

Formulae are presented for the variance and higher moments of the present value of single-stage rewards in a finite Markov decision process. Similar formulae are exhibited for a semi-Markov decision process. There is a short discussion of the obstacles to using the variance formula in algorithms to maximize the mean minus a multiple of the standard deviation.

Dynamic Oligopoly with Inventories

This paper develops a dynamic model of oligopoly and discusses the existence and characteristics of optimal policies for firms in such a model. The firms are assumed to face a random demand so they hold inventories which fluctuate from one period to the next. This necessitates a dynamic model rather than a static one. Our extension of the equilibrium concept to the oligopoly model is founded on recent generalizations of Shapleys stochastic game. We show the existence of equilibrium price-quantity strategies for the firms and also (i) an equilibrium strategy may be found by solving an appropriate static game and (ii) the quantity part of the strategy is often a constant (time invariant).

Discounted MDP's: distribution functions and exponential utility maximization

The present value of the rewards associated with a discrete-time Markov process has a probability distribution which depends on the initial state. The first part of the paper applies fixed point theory to a system of equations for the distribution functions of the present value. The second part of the paper expands the model to a Markov decision process (MDP) and considers the maximization of the expected utility of the present value when the utility function is exponential.

Inventory Policies for Systems with Stochastic and Deterministic Demand

We consider a periodic review inventory system with demand arriving simultaneously from a deterministic source and a random source. The deterministic demand has to be satisfied immediately and the stochastic demand can be backordered. Assuming that the stochastic demand is never backlogged if there is stock in the system, we prove that a modified ( s, S) policy is optimal under general conditions if there is a setup cost. If there is a smoothing cost instead of the setup cost, we observe that the problem corresponds to a standard model with one source of demand.

New Product Innovation with Multiple Features and Technology Constraints

We model a firms decisions about product innovation, focusing on the extent to which features should be improved or changed in the succession of models that comprise a life cycle. We show that the structure of the internal and external environment in which a firm operates suggests when to innovate to the technology frontier. The criterion is maximization of the expected present value of profits during the life cycle. Computational studies complement the theoretical results and lead to insights about when to bundle innovations across features. The formalization was influenced by extensive interviews with managers in a high-technology firm that dominates its industry.

Manufacturing lead-time rules: Customer retention versus tardiness costs

Inaccurate production backlog information is a major cause of late deliveries, which can result in penalty fees and loss of reputation. We identify conditions when it is particularly worthwhile to improve an information system to provide good lead-time information. We first analyze a sequential decision process model of lead-time decisions at a firm which manufactures standard products to order, and has complete backlog information. There are Poisson arrivals, stochastic processing times, customers may balk in response to quoted delivery dates, and revenues are offset by tardiness penalties. We characterize an optimal policy and show how to accelerate computations. The second part of the paper is a computational comparison of this optimum (with full backlog information) with a lead-time quotation rule that is optimal with statistical shop-status information. This reveals when the partial-information method does well and when it is worth implementing measures to improve information transfer between operations and sales.

Optimal Operation of Queues

The early growth of queueing theory was motivated by the design of telephone and other service systems. As the literature expanded, much of its growth consisted of mathematical theory suggested by the literature itself rather than by congested service systems. This self-sustaining theoretical growth has been coupled recently with a renewed pragmatic motivation. The result has been an increase in the volume of prescriptive research in queueing theory. Most of these recent contributions begin by embellishing a standard queueing model with a cost structure. Although they go on to address the question of optimal design or of optimal operation, one might imagine that including a cost structure is merely hanging bells and jangles on a standard model. Although that conjecture may be valid, the analysis of normative queueing models often differs fundamentally from the derivation of properties of descriptive models and it provides new insights into how congestion should be managed.

Fill Rates of Single-Stage and Multistage Supply Systems

A supply systems fill rate is the fraction of demand that is met from on-hand inventory. This paper presents formulas for the fill rate of periodic review supply systems that use base-stock-level policies. The first part of the paper contains fill-rate formulas for a single-stage system and general distributions of demand. When demand is normally distributed, an exact expression uses only the standard normal distribution and density functions, and a good approximation uses only the standard normal distribution function. The second part of the paper derives the probability distribution of the finished goods inventory level for serial systems with buffer inventories between stages. This distribution leads to fill-rate formulas and the conclusion that shorter supply chains have higher fill rates.