This paper offers a personal perspective on the development of decision theory and related subjects during the past half century. It first reviews six milestones in the foundations of decision analysis that are associated with Frank P. Ramsey, John von Neumann and Oskar Morgenstern, Leonard J. Savage, Maurice Allais and Ward Edwards, West Churchman and Russell Ackoff, and Kenneth Arrow. It then gives a personal account of further developments over the past 30 years in linear utility theory, subjective probability and ambiguity, nonlinear preference and utility, stochastic dominance and inequality analysis, multiattribute utility theory, and the theory of social choice. The paper could be viewed as a supplement to my extensive review of utility theory in Management Science some 20 years ago. However, it makes no claim of completeness since its main aim is to recall important debts and describe what it has been like to be a part of the development of decision analysis in the past few decades.

We introduce a general framework that guides the management scientist's formulation of deterministic models of production processes. Using the framework, we reformulate the constraints of familiar linear programming-based planning models to specifically treat components of production lead time, thereby realizing a more accurate representation of the production process. In addition, the reformulation accommodates noninteger values for lead times as well as unequal-length planning periods. Manufacturing Resources Planning (MRP) and the Critical Path Method (CPM) are recast in terms of the framework, revealing opportunities for model generalization and extension, and their relationship to linear programming models.

This paper presents an alternative procedure for computing the steady state probability vector of an M/M/1 queue with randomly varying arrival and service rates. By exploiting the structure of the infinitesimal generator of the underlying continuous-time Markov chain, the approach represents an efficient adaptation of the state reduction method introduced by Grassmann for solving problems involving M/M/1 queues under a random environment. We compare computational requirements of the proposed approach with the method of Neuts and block elimination under different rush-hour congestion patterns while keeping the overall traffic intensity constant as well as under different traffic intensities. We demonstrate that the proposed method requires minimal computing time to reach convergence and moreover the time requirement does not change much when traffic intensity increases.

Analyzing machine investment opportunities from a systems perspective requires an understanding of interdependencies between machine investment and utilization decisions. Machine investment decisions involve determining the number of machines to purchase and their types. Machine utilization decisions involve determining part allocations and production cycles. Interdependencies between these decisions result in trade-offs between investment and operating costs. This paper develops an analytical approach for studying machine investment opportunities in a broad systems context. It begins with a discussion of the interdependencies between investment and utilization decisions. The focus of this research and a review of existing literature related to machine investment decisions are then discussed. Next a mathematical model that quantifies these interdependencies is formulated. The paper ends with a simple example that demonstrates the usefulness of the model and highlights the importance of using a systems approach. The main contribution of this work is the simultaneous consideration of trade-offs between machine investment and utilization decisions.

We consider a multi-priority, N-server, Poisson arrival, nonpreemptive queue, motivated by police applications. The number of servers requested by an arrival has a known priority dependent probability distribution. All servers requested by a customer must start service simultaneously; the servers' service times are independent and exponentially distributed with parameter μ, independent of priority, server identity or system state. In order to save available servers for higher priority customers, arriving customers of each lower priority are deliberately queued whenever the number of servers busy equals or exceeds a given priority-dependent cutoff number. Whenever all higher priority queues are empty, the longest waiting priority i customer will enter service the instant there is a service completion from a state having precisely N i − k + 1 servers busy, where k is the number of servers requested by the customer and N i is the server cutoff number for priority i. The queueing discipline is in a sense HOL by priorities. We derive the priority i waiting time distribution (in transform domain) and other system statistics. Illustrative computational results are given.

An influence diagram is a network representation of probabilistic inference and decision analysis models. The nodes correspond to variables that can be either constants, uncertain quantities, decisions, or objectives. The arcs reveal probabilistic dependence of the uncertain quantities and information available at the time of the decisions. The influence diagram focuses attention on relationships among the variables. As a result, it is increasingly popular for eliciting and communicating the structure of a decision or probabilistic model. This paper develops the framework for assessment and analysis of linear-quadratic-Gaussian models within the influence diagram representation. The “Gaussian influence diagram” exploits conditional independence in a model to simplify elicitation of parameters for the multivariate normal distribution. It is straightforward to assess and maintain a positive (semi-)definite covariance matrix. Problems of inference and decision making can be analyzed using simple transformations to the assessed model, and these procedures have attractive numerical properties. Algorithms are also provided to translate between the Gaussian influence diagram and covariance matrix representations for the normal distribution.

Stratified sampling is perhaps the most natural of the variance reduction techniques. However its use is often frustrated by the high dimensionality of the sample space. This paper investigates the difficulty and suggests a basic sampling scheme for use in such problems. The accuracy of estimators when this method of sampling is used is examined in detail. A way of implementing the scheme in practice is suggested which makes use of shadow response variables (variables which have similar properties to control variables). This reduces the dimensionality of the sample space to a tractable size. Two detailed examples are given for which a 10% to 90% reduction in variance is obtained compared with crude Monte Carlo.