Search

The Program Evaluation and Review Technique (PERT) has been developed and implemented as a technique for planning and evaluating the progress of research and development programs Probability theory is used in PERT to calculate a measure of the uncertainty in meeting scheduled dates These computed probabilities can then be used by managers to aid then judgment in evaluating the validity of schedules for accomplishing program objectives This paper investigates the effect of subdividing activities in PERT flow diagrams on computed probabilities for meeting scheduled dates It is shown that the mechanics of activity subdivision can increase or decrease the computed probabilities, depending upon whether the scheduled date for an event is later or earlier in time than the expected date Management personnel who are using PERT probabilities to aid their judgment in evaluating schedules should be aware of the potential biases inherent in the computation of probabilities. [ABSTRACT FROM AUTHOR]

This paper comprises an extension of Model I of a recent paper of GLUSS, Opns Res 7, 468ñ477 (1969), and its purpose is to dictate strategies that minimize the expected cost (in time) of locating a fault in a complex system of equipment. These strategies are specialized for use with automatic testing equipment. The model assumes that the complex system consists of N modules containing n(1), , n(N) elements respectively, that the cost of examining the modules are t1, tN respectively, and that the costs of examining the elements within the rth module are tr1 , , trn(r). It is further assumed that module tests are performed to find which module is faulty before element tests are performed, and that there exist probabilities at each stage that errors of two kinds can be made (1) that the test fails to detect an actual fault in the module or item tested, (2) that the test finds a fault that does not exist. The estimation of the probabilities of faults lying in respective modules or elements is performed in a different way from that in Gluss' paper they are computed from element reliability data by manipulation of their γ parameters, where γ is the element failure rate. Furthermore, consideration is given to fault symptoms that are supplied by weighting the probabilities according to the symptom information. Because of the anticipated difficulty in obtaining the necessary parameter estimates, the analysis may be most useful for its illumination of the influence the several required estimates have on the optimum search routine. [ABSTRACT FROM AUTHOR]

Calculation of the probability of kill from firing a salvo of ammunition rounds often is a very complicated and time-consuming problem Thus there is a tendency to sacrifice realism to obtain greater mathematical simplicity There are, however, some worthwhile salvo-firing problems where realism and mathematical simplicity are both attainable Let the salvo size, the lethality properties of the ammunition, the shape and vulnerability properties of the target, and the probability properties of the salvo-aiming system against this target all be arbitrary but given Consider determination of the maximum salvo kill-probability value obtainable through use of the optimum hit-location probability distribution for the ammunition rounds Evaluation of this maximum salvo kill probability, and of the corresponding optimum round-hit-location probability distribution, is the problem analyzed in this paper Although this problem is solved for a very general class of situations, the computations required are not extensive The results derived also can be used to indicate the effects on salvo kill probability of salvo size, target shape and vulnerability distribution, target maneuver, fire-control-system properties, etc [ABSTRACT FROM AUTHOR]

This paper considers the decision maker whose evaluation and consequent choice of actions is accomplished through the use of the expected utility hypothesis. In cases where the utility function is increasing with upper and lower bounds then the utility function can be characterized by a distribution function, and we can take advantage of the various properties of such functions as well as existing results with respect to such functions. Using these properties and results we can determine the certainty equivalents as a function of the parameters of the distribution function (utility function) and the parameters of the probability distribution on the uncertain payoff. The following cases are considered: (1) Gaussian distribution function and Gaussian probability distribution, (2) Exponential distribution function and exponential distribution and (3) Exponential distribution function and Gaussian probability distribution. [ABSTRACT FROM AUTHOR]

This paper is concerned with characterizing decision rules for the sequential E-model of chance-constrained programming. A key feature of our characterization will be a detailed discussion of various interpretations of the probability operator in the chance constraints. Specifically we define two new classes of decision rules by exhibiting those sets of constraints which locally support the corresponding probability requirements. The question of how the probabilistic constraints for future periods are affected by previous decisions and realizations of the random variables is considered in detail. Since we are primarily concerned with the feasibility of decision rules, we deal mainly with the constraints of the model. The procedure for selecting the optimum rule from among a particular class of feasible rules depends on the objective function and is briefly discussed in the final section along with some implications concerning the form of the optimum rule. The application of our proposed rules to a two-period example previously appearing in the literature concludes the paper. [ABSTRACT FROM AUTHOR]

This paper establishes sufficient conditions for decision rules to be optimal for two n-period P-models of chance-constrained programming. The models considered are the triangular model with total probability constraints and the block triangular model with conditional probability constraints. It is shown that, for both these models, a sufficient condition for optimality is that the decision rule be the optimal piecewise linear rule. Proofs of theorems are based on results contained in earlier papers by the authors, on n-period E-models of chance-constrained programming. [ABSTRACT FROM AUTHOR]

This paper reports the results of six experiments and analyses performed to explore the applicability of the non-constant-sum case of the theories of von Neumann-Morgenstern, and others, to the actual behavior of people playing games or involved in bargaining situations. The paper suggests directions in which the theory of games might be modified and extended to improve its applicability and usefulness. A "split-the-difference principle" is suggested to augment the usual theory, so as to specify the exact amount of payments to be made in an ordinary two-person bargaining situation such as the sale of a used car. The application of this principle seems satisfactory in the experiments. One experiment suggests that, in a sequence of trials in the same game situation, people tend to start near an equilibrium point and then try to find a better equilibrium, if there is one. The experiments show examples of non-optimal behavior of the bargainers when the judgment necessary to estimate the relevant payoff is obscure. A fair division of five parcels of objects among five players when each player attaches different values to the parcels is outlined and computed, and the effect of coalitions is discussed. [ABSTRACT FROM AUTHOR]

An introduction to the papers published in the 1972 issue of the "Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability," Vol. VI, is presented.

Intuitively, a risk averter diversifies between two investments if there is some sort of negative interdependence. In [3], Samuelson gives the example of buying shares in a coal company and an ice company. It is of interest to characterize this concept of negative interdependence more sharply. The most commonly used characterization of negative interdependence of two random variables X and Y is that the Pearson linear correlation coefficient be negative. An equivalent characterization is, of course, that Cov(X,Y) <0, where Cov(X,Y) = E(XY)-EX·EY. A reader of Samuelson's discussion on page 7 of [3] might be led to the conclusion that negative linear correlation between two investments is sufficient for diversification by a risk-averse investor (i.e., strictly concave increasing utility function). In fact, negative correlation is sufficient only in some very special situations, such as x and Y having a bivariate normal distribution or the investor having a quadratic utility function. Examples will be presented later to demonstrate that negative correlation is neither necessary nor sufficient for diversification. Another characterization of negative interdependence is given by Lehmann [1, p. 198]. This characterization does lead to a sufficient condition for diversification (Theorem 5) and is related to that of Theorem IV in [3]. A weaker characterization which is also sufficient for diversification is provided in Theorem 4. In the context of this discussion, the most appropriate definition of negative interdependence would be: X and Y are negatively interdependent if diversification by all risk-averse investors is optimal. This definition is made operational by its characterization in terms of the joint distribution function of x and Y in Theorem 3. In this paper, terms increasing, decreasing, and concave, without the adjective strict, should be taken in their weak sense; i.e., increasing means nondecreasing. [ABSTRACT FROM AUTHOR]

While the activity of marketing research can be fruitfully viewed within a statistical decision theoretic model, relatively little is known concerning the descriptive aspects of how people-managers or consumers-revise probabilities in the light of new information. This paper reports the results of a behavioral study in probability revision and the implications of these findings for the operational use of decision theoretic concepts in prescriptive and descriptive choice-making models [ABSTRACT FROM AUTHOR]

Someone has said that maternity is a matter of fact, while paternity is, at best, a matter of opinion. Concerning the field to which this issue of the Journal is devoted, however, there is little difference of opinion — Harry Markowitz is almost universally considered to be the father of portfolio analysis. The essence of the theory appeared in his 1952 article, although the full exposition was not available until several years later. [ABSTRACT FROM AUTHOR]

Lee Smith's paper is interesting as much for the questions he raises as for the answer he proposes. Those of us who have tried to elicit subjective probability distributions in the context of corporate systems can sympathize with his position that no "objective" way has yet been proposed. It is no wonder that the procedures which he cites as precursors to his proposal are ad hoc and, shall we say, "flexible." The main issue, however, is whether the procedure advocated in his paper overcomes the limitations of previous devices. Related questions concern: (a) the compatability of probabilistic thinking with the culture of line managemerit; and (b) the accuracy of the elicited judgments, using the "best" of methods. [ABSTRACT FROM AUTHOR]

This paper studies a control policy for an M/G/1 nonpreemptive-priority queuing system with removable server and two priority classes. This policy turns off the server when the system is empty and turns him on when a given linear combination of the numbers of class 1 and class 2 customers in the system exceeds a certain value. Expressions for the long-run average numbers of class 1 and class 2 customers in the system are derived. [ABSTRACT FROM AUTHOR]

This paper considers the static flowshop-scheduling problem with the objective of minimizing, as a cost function, the mean job-completion time. Within the more general framework of combinatorial optimization problems, it defines a heuristic search technique--an approach that has been successful in the past in obtaining near-optimal solutions for problems that could not be solved exactly, either for lack of theory or because of exorbitant computational requirements. The paper presents a two-phase algorithm: The first phase searches among schedules with identical processing orders on all machines; the second refines the schedule by allowing passing. Results of computer study are presented for a large ensemble of pseudorandom problems, and for two particular problems previously cited in the literature. The method is shown to provide solutions that are exceptionally low in cost, and superior to those provided by sampling techniques in the cases for which comparison is possible. Computation time is also discussed and is given in machine-independent terms. [ABSTRACT FROM AUTHOR]

This paper concerns developing methods for approximating a chance-constrained set when any information concerning the random variables must be derived from actual samples. Such a situation has not been presented in the literature. When existing chance-constrained programming techniques are used, it is not possible to relate the accuracy of sample-based assumptions to actual constraint satisfaction. The methods presented here employ the concept of a distribution-free tolerance region to construct various sets whose elements have the common property of satisfying the chance constraint with a preassigned level of confidence. The sample size required to meet the desired confidence is readily available in tabular or graphical form. [ABSTRACT FROM AUTHOR]

This paper discusses the M/M/1 queue embedded in a Markovian environment (or, subject to extraneous phase changes). It studies the busy period, equilibrium conditions, and emptiness probabilities by using analytic matric functions, and makes comparisons with the work of NEUTS and of NAOR and YECHIALI. [ABSTRACT FROM AUTHOR]

This paper investigates the stochastic behavior before a total system failure for a standby redundant system with repair maintenance. It derives the expected number of visits to state k before a total system failure and other measures explicitly by considering 'taboo' probabilities for an imbedded Markov chain generated by the system. The transition probabilities and the first-passage distributions are also discussed. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

This paper shows that analytical techniques can be applied in the analysis of a wide class of combat problems by specializing the inputs of a general functional model. The generating functions of the losses, and the corresponding expected values and variances, are derived and expressed as integrals with respect to a supplementary time variable. [ABSTRACT FROM AUTHOR]

This article reviews the formulation of the constrained capital budgeting problem. The article compares the merits of integer programming and continuous linear programming as methods of solving this problem. The paper discusses capital constraints and the question of an appropriate objective function. Additional constraints are presented and problems arising because of the binary nature of capital projects are discussed. Some attempts have been made to include probability considerations in mathematical programming approaches to capital budgeting.

This article analyzes the relationship between stochastic dominance (SD) and mean-semivariance (ES) criteria. This paper contains two main results. First, there are good conceptual bases for anticipating greater consistency between ES and second SD (SSD) rules than between mean-variance (EV) and SSD rules. In particular, with one possible and relatively minor exception (identical mean and semivariance, the efficient set of portfolios derived from the ES rule with semivariance calculated around a fixed point is a subset of the SSD efficient set.

In hypothesis testing, the level of significance, α, often performs a screening as well as an evaluating function. Results which are not "significant" at some preset value of α are simply not reported. Unfortunately, this value of α is usually arbitrarily set at .05 or .01 and bears no relation to the needs of potential users of the study. This paper suggests an approach to making the selection of α more responsive to these potential applications. [ABSTRACT FROM AUTHOR]

This paper considers the use of antithetic variates to reduce the variance of estimates obtained in the simulation of a GI/G/1 queue. Two experimental configurations are considered: in the first, 2n observations are taken in a single run; in the second, n observations are taken in each of two runs. If the sequences of uniform random variables that generate the realizations of the queuing system in the two runs are antithetic, we show that the variance of estimates of the mean and distribution of stationary waiting time and number in the queue is less in the second configuration than in the first. We also obtain sufficient conditions for the covariance of functions of a vector of uniform random variables to be nonnegative. Experimental results are given for M/M/1 queuing simulations to illustrate the magnitude of the variance reduction. [ABSTRACT FROM AUTHOR]

This paper presents an improved stochastic model for the behavior of the daily census and associated waiting variables in general-acute hospitals. The development and form of the distributions derived follow the general approach of a previous paper by Shonick [J. Amer. Statist. Assoc. 65, 1474–1500 (1970)], but greater flexibility is provided for the health-services planner by increasing the number of decision choices available for achieving given outcomes. This increased flexibility is attained via a generalization of the queuing discipline, permitting admissions of elective patients to be suspended when the number of occupied beds reaches a predetermined level (which may be less than the total bed complement). For any emergency-elective mix of the demand for hospitalization, this model permits the computation of many measures of operating efficiency, including expected overfill rate, percentage occupancy, waiting-list length, and loss of emergency patients. A variant of the model permits computation of the expected number of patient days that will be served in ‘nonapproved’ facilities when emergencies arriving during ‘full’ periods are accommodated therein, rather than turned away. [ABSTRACT FROM AUTHOR]

This paper describes and discusses some search problems, starting with the case of no false contacts, to provide a foundation for developing a search theory in the presence of false contacts. Important properties of the search plan are defined and illustrated by examples. After a general description of false-contact generators and alternative actions that may be used when contacts are made, the problem is formulated for the case in which false contacts are generated by real stationary objects that are investigated when contacted. It is shown that the formulation and solution of the standard optimization functionals are contingent on the number of false targets found, in general, and possibly on the locations and times of contact as well. The optimization functional is difficult to write and more difficult to solve. The formulation is made for the expected-time functional when the number of false targets is bounded. Finally, the paper outlines a procedure for ending the solution and illustrates it with an example. [ABSTRACT FROM AUTHOR]

This paper examines Markovian decision processes in which the transition probabilities corresponding to alternative decisions are not known with certainty. The processes are assumed to be finite-state, discrete-time, and stationary. The rewards are time discounted. Both a game-theoretic and the Bayesian formulation are considered. In the game-theoretic formulation, variants of a policy-iteration algorithm are provided for both the max-min and the max-max cases. An implicit enumeration algorithm is discussed for the Bayesian formulation where upper and lower bounds on the total expected discounted return are provided by the max-max and max-rain optimal policies. Finally, the paper discusses asymptotically Bayes-optimal policies. [ABSTRACT FROM AUTHOR]

This paper considers an n-phase generalization of the typical M/M/1 queuing model, where the queuing-type birth-and-death process is defined on a continuous-time n-state Markov chain. It shows that many models analyzed in the literature can be considered special cases of this framework. The paper focuses on the steady-state regime, and observes that, in general, closed-form results for the limiting probabilities are circuit to obtain, if at all possible. Hence, numerical methods should be employed. For an interesting special case, explicit results are obtained that are analagous to the classical solutions for the simple M/M/1 queue. [ABSTRACT FROM AUTHOR]

This paper considers the problem of optimal search for a stationary target when the detection capability of the search sensor, as characterized by its sweep width, is fixed but not known in advance. Most of the results are confined to the assumption of a gamma prior sweep-width distribution and to the cases of normal and uniform prior target-location distributions. Explicit formulas are obtained for the optimal search plans, the probability of detection versus time, the expected time to detection, and the posterior marginal distributions for target location and sweep width. In the case of a normal prior target-location distribution, expected time to detection is compared for the optimal plan and search plans based on the assumption of a known value for sweep width; this includes a sensitivity analysis treating actual sweep width as a parameter. For a target of value, the search plan that maximizes expected net return is determined. This plan requires specification of a rule for terminating an unsuccessful search. Explicit results are given assuming a nonincreasing target value and a nondecreasing search-cost rate. [ABSTRACT FROM AUTHOR]

This paper presents several fault-testing models. Our system is composed of n components that either work or fail. The systems treated are the k-out-of-n type (k/n). This system works if and only if k or more of its components work, making the series (n/n) and parallel (1/n) systems special cases. Components can be individually tested end tests give perfect information. Each test has a time to complete it and testing procedures are judged by the expected cumulative testing time they require. In the first model, the components are assumed to function or fail independently of each other. A feasible test procedure must determine (as) if the system is working or has failed. This is solved for the series and parallel systems and, under a possibly restrictive hypothesis, for the general k/n system. In another model, the components are assumed to be nominally independent conditioned on the system being failed. Here a feasible test procedure must locate all failed components. This problem is solved for the general k/n system. [ABSTRACT FROM AUTHOR]

This paper formulates a stochastically constrained equipment replacement model. Its aim is to determine a sequence of replacement dates such that the total ‘current account’ cost of all future costs and capital expenditures over an infinite time horizon for the n initial incumbent machines is minimized subject to the constraints that, at each point in time, the expected number of machines in each of several ‘utility classes’ has a prescribed value. The theoretical model is then solved under a set of simplifying assumptions and applied to the problem of ship replacement. [ABSTRACT FROM AUTHOR]

This paper deals with a search model for allocating effort when there are many target objects of differing sizes present in a search area. All target objects are initially present in the search area and remain at least until discovered; no new objects are generated during the search process. Targets are not moveable. The search process is characterized by the quantity of search effort. The model describes the distribution of the number and sizes of target objects present. The probability of discovery is assumed proportional to object size. The paper gives many results, including the probability distribution of the number and sizes of discovered objects, and the prior and posterior distributions of the number of objects remaining undiscovered. Then it gives the method of maximum likelihood estimation of the model parameters, and solves the sequential decision problem of when to terminate the search process. [ABSTRACT FROM AUTHOR]

In this model the income process of a firm is assumed to be a homogeneous Markov process (discrete or continuous). One is interested in an optimal dividend strategy, i.e., a strategy that maximizes the expected sum of the discounted dividend payments. This paper derives general results in the case where the income process is skip free to the fight; in particular, it establishes the factorization formula and shows it to be closely connected with the probability of ruin. The theory is illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

An M/G/1 batched-service queuing model is described for a waiting room of size m. When the waiting room is filled, the arrival rate changes from λ to λp; i.e., 100 (1-p) per cent of all arrivals balk if the waiting room is full. Since p may be zero, the model includes finite queues as special cases. The special case with m=0 has been handled by TAkÁCS. However, in this paper the steady-state probability formulas for length of queue given by Takáes are corrected, and the results extended for m&gE1. [ABSTRACT FROM AUTHOR]

This paper analyzes a single-server queuing system in which service is alternated between two queues. Each queue is assumed to have an independent Poisson input and an independent general service-time distribution. The alternating priority rule is followed. Independent general distributions are assumed for the intervals required to switch service from one queue to the other. The following aspects of the steady-state solution are presented for each queue: average delay of an entering customer, average length, average durations of the busy period, and average rate at which service is offered to a queue. The model has been applied to a generalized data-communications system in which transmission between two data stations is possible in only one direction at a time. [ABSTRACT FROM AUTHOR]

A finite discrete nonstationary Markov chain is completely characterized (after the initial probability distribution has taken effect) by its time sequence of transition probability matrices Pi. The ith causative matrix Ci is defined as the product Pi-1 (if it exists) times Pi+1 Thus, the causative matrices are analogous to derivatives in calculus as an indication of rate of change. The eigenvalues and eigenvectors of a constant causative matrix C have been found useful in their connection with the tendency of the chain to be convergent or divergent. Results for two-state chains are presented in some detail. A comprehensive bibliography of papers on non-stationary chains is included. [ABSTRACT FROM AUTHOR]

This paper considers a multiserver system with preemptive priorities and derives recurrence relations for the moments of the lifetime distribution. [ABSTRACT FROM AUTHOR]

This paper deals with two problems of asymptotic aggregation of an ergodie Markov chain. The first is concerned with the determination of an aggregated Markov chain, the asymptotic probabilities of which are just equal to conditional asymptotic probabilities defined on the original Markov chain. The second one relates to the determination of an aggregated chain, the asymptotic probabilities of which are equal to the asymptotic probabilities of the elements of a given state partition. Both problems are approached from a topological point of view and solutions are given in terms of signal-flow graph techniques. Comments about the main computational features of the method are also included. [ABSTRACT FROM AUTHOR]

One formulation of the machine sequencing problem is that of finding a minimaximal path in a disjunctive graph. This paper describes an implicit enumeration procedure that solves the problem by generating a sequence of circuit-free graphs and solving a slightly amended critical-path problem for each graph in the sequence. Each new term of the sequence is generated from an earlier one by complementing one disjunctive arc. The search tree is drastically cut down by the fact that the only disjunctive arcs that have to be considered for being complemented are those on a critical path. An evaluation of these candidates is used to direct the search at each stage, The procedure can start with any feasible schedule (like the one actually used in production, or generated by some heuristics), and gradually improve it. Thus one can possibly stop short of the optimum, with a reasonably 'good' feasible schedule. Storage requirements are limited to the data pertinent to the current node of the search tree. [ABSTRACT FROM AUTHOR]

This paper studies a particular random tour (continuous random walk): A particle moves in two dimensions at constant speed by choosing successive travel directions that are independent and uniformly distributed between 0 and 2π, with the lengths of the steps between direction changes being independent, exponentially distributed, random variables. An analytic expression for the probability density of the particle's position after a time t is derived, and an application is made to a military situation where the 'particle' is a target trying to escape being destroyed by an unseen enemy. The bulk of the paper is devoted to deriving the density function. [ABSTRACT FROM AUTHOR]

In this paper, exact, recursive, and asymptotic formulas' are derived for a queuing system in which access to the servers takes place at regular intervals. Steady-state results, already known, are extended and utilized to construct delay distributions that are adaptable to computer work. [ABSTRACT FROM AUTHOR]

This paper studies the virtual waiting time for the M/G/1 queue under this priority rule: within each generation, customers are served in the order of shortest (or longest) service times. It also considers the limiting behavior of the virtual waiting time, and compares the asymptotic means with those of the first-come, first-served discipline. [ABSTRACT FROM AUTHOR]

This paper gives an explicit formula for the waiting-time distribution in a single-server system with general holding times subject to a Poisson input from any number of priority classes. In the general case, where the holdingtime distributions of the demands in each priority case are different, the Laplace-Stieltjes transform and the moments of the distribution are found. The queue discipline is last-come first-served in each priority class. The cases of preemptive resume and nonpreemptive priorities are considered. It is shown that the variance of the waiting-time distribution (whatever the holding-time distribution) is greater than the corresponding variance when the queue discipline is first-come first-served. [ABSTRACT FROM AUTHOR]

This paper considers the mathematical properties of chance constrained programming problems where the restriction is on the joint probability of a multivariate random event. One model that is considered arises when the right-handside constants of the linear constraints are random. Another model treated here occurs when the coefficients of the linear programming variables are described by a multinormal distribution. It is shown that under certain restrictions both situations can be viewed as a deterministic nonlinear programming problem. Since most computational methods for solving nonlinear programming models require the constraints be concave, this paper explores whether the resultant problem meets the concavity assumption. For many probability laws of practical importance, the constraint in the first type of model is shown to violate concavity, However, a simple logarithmic transformation does produce a concave restriction for an important class of problems. The paper also surveys the generalized linear programming method for solving such problems when the logarithmic transformation is justified. For the second type model, the constraint is demonstrated to be nonconcave. [ABSTRACT FROM AUTHOR]

Consider a single-server queuing process with Poisson input and general service time distribution. Define the remaining busy period x(t) as the time from a given instant t until the server becomes idle for the first time. Let ξ(t) be the number of customers in the system at time t. Let x1(t) be the remaining service time for the customer being served at time t. In this paper we shall study the stochastic law of the remaining service time x1O) given that ξ(0)=i and ξ(t)-r. The result will then be applied to the problem of finding the joint distribution of the remaining busy period x (t) and the number of customers served during x (t) given that ξ(0) = i and ξ(t)=r. [ABSTRACT FROM AUTHOR]

In most inventory models one of the costs considered is shortage cost. However, this cost is often very hard to measure. For the models considered in this paper, instead of introducing a shortage cost, we define a stock-out penalty function and impose a constraint on it. For example, a constraint on the probability of stock-out can be viewed this way. We exhibit the general, randomized, optimal policy and give graphical and numerical methods of calculating it. We also discuss how our stock-out penalty function is related to shortage cost. [ABSTRACT FROM AUTHOR]

Traditionally the probability of killing a point target is analyzed by means of employing a lethal circle (in two dimensions) or a lethal sphere (in three dimensions). The concept of a lethal circle or sphere is based on the assumption of a `kill-no kill' situation in which a target anywhere within the circle (sphere) is killed and a target anywhere outside of the lethal region is not killed. Another aspect of the `kill-no kill' approach is that it combines all lethal effects (heat, shock, x-rays, etc.) in one parameter-the radius R of the lethal circle (sphere). In many problems of weapon systems effectiveness this approach is extremely useful, but in other areas the attendant assumptions are unrealistic and the purpose of this paper is to present a useful alternative methodology. All of the formulas derived are amenable to desk-calculator computation. [ABSTRACT FROM AUTHOR]

One of the most important concepts in value theory or utility theory is the notion of independence among variables or additivity of values. Its importance stems from numerous multiple-criteria procedures used for rating people, products, and other things. Most of these rating procedures rely on the notion of independence (often implicitly) for their validity. However, a satisfactory definition of independence (additivity), based on multidimensional consequences and hypothetical gambles composed of such consequences, has not appeared. This paper therefore presents a definition of independence for cases where the set of consequences X is a product set X1×X2× … ×X each element in X being an ordered n-tuple (x1, The definition is stated in terms of indifference between special pairs of gambles formed from X. It is then shown that if the condition of the definition holds, the utility of each (x1, x2, …, xn) in X can be written in the additive form ϕ(x1, x2, …, xn)=ϕ1(x1)+ϕ2(x2)+…, ϕn(xn), where ϕi is a real-valued function defined on the set Xi i=1, 2, …, n. The development is free of any specific assumptions about ϕ(e.g., continuity, differentiability) except that it be a von Neumann-Morgenstern utility function, and places no restrictions on the natures of the Xi. [ABSTRACT FROM AUTHOR]

A commuter living outside a circular city arrives at a point on the circumference and is equally likely to wish to travel to every point of the interior. In the present paper we give the probability distribution for trip length, and the probability distribution for traffic weight at each point, under the assumption of six different routing systems. [ABSTRACT FROM AUTHOR]

This paper deals with tradeoffs and major interactions among the different activities in Jogistics systems. Specifically, the problem of 'interchange- ability' between maintenance and supply, embodied in the question of 'subassembly vs. piece-part replacement,' is considered. The approach presented consists of a novel method of systems representation: A basic model, which is fundamentally a multistage circular queuing system, is developed for each of the units (assemblies, subassemblies, etc.) that constitute an equipment system. These elementary systems are then connected together with the pattern of linkage made to correspond to the technological breakdown of the over-all system. An interesting result of this approach is that the important measure of effectiveness for any unit is not only its unavailability (probability of stockout), but also the duration of unavailability. Iterative solutions in the case of linear cost functions, a total cost constraint, and the usual exponential assumptions are also presented. [ABSTRACT FROM AUTHOR]

In this paper, treating time as a discrete variable, we study the transient behavior of a bulk-service queuing system in which the number of customers arriving within a fixed time interval follows a binomial probability distribution. The service times are assumed to be identically distributed and statistically independent. The busy period and queue length distributions have been obtained in terms of probability generating functions. [ABSTRACT FROM AUTHOR]

In this paper the behaviour of a complex system consisting of n components of constant failure rates λ[SUB1], λ[SUB2] , λ[SUBn] and repair rates μ[SUB1], μ[SUB2] , , μ[SUBn] is examined The probability that the system is operating at time t is obtained Also the probability that it is in outage stage i at time t has been determined At the end, a particular case where μ[SUB1]'s are equal is considered. [ABSTRACT FROM AUTHOR]