Selecting preferred solutions in the minimax approach to dynamic programming problems under flexible constraints

Dynamic Programming is a powerful approach to the optimization of sequential or multistage decision processes, e.g., in planning or in system control. In this paper, we consider both theoretical and algorithmic issues in sequential decision processes under flexible constraints. Such processes must attain a given goal within some tolerance. Tolerances or preferences also apply to the values the decision variables may take or on the action chosen at each step. Such problems boil down to maximin optimization. Unfortunately, this approach suffers from the so-called “drowning effect” (lack of discrimination) and the optimality principle of dynamic programming is not always verified. In this context, we introduce a general framework for refined minimax optimization procedures in order to compare and select preferred alternatives. This framework encompasses already introduced methods such as LexiMin and DiscriMin, but it allows their extension to the comparison of vectors of unequal lengths. We show that these refined comparisons restore compatibility with the optimality principle, and that classical algorithms can be adapted to compute such preferred solutions, by exploiting existing results on idempotent semirings.     

Temperature-frequency reduction of the dynamic functions of black-filled rubbers in the nonlinear region

The possibility of using temperature-frequency reduction of the dynamic functions of black-filled rubbers in the nonlinear region is demonstrated and an additional applicability criterion — similar slopes of the isotherms of the amplitude dependences of the dynamic functions — is formulated. In the case of heavily filled rubbers it may be necessary to determine the constants of the Williams-Landel-Ferry equation and the reference temperature T_S from the experimental data — they may not coincide with the universal values. Master curves of the dynamic functions are obtained for a series of shock-absorbing rubbers at a shear strain amplitude =0.01.A. V. Topchiev Institute of Petrochemical Synthesis, Academy of Sciences of the USSR, Moscow. Scientific-Research Institute of the Rubber Industry, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 751–754, July–August, 1973.     

The sugar production factory—A dynamic decision task

This article describes the Sugar Production Factor and its structural equivalent, the Personal Interaction task. These are two simple, individual dynamic decision-making tasks in which subjects make interdependent decisions to reach a goal, and receive feedback on the outcome of their efforts along the way. An important result from human learning experiments using these two tasks and their variants is that subjects reliably improve their ability to reach the goal over a moderate number of training trials (40–90) but do not show consistent improvement in other measures of task knowledge. These other measures focus on subjects' ability to accurately predict the task environment's response to their actions and subjects' ability to produce useful heuristics. This pattern of results runs counter to the idea that decision makers' performance in dynamic decision tasks depends critically on the predictive accuracy their internal models of the task environment. Variants of both tasks have been used to manipulate this pattern of results and explore more deeply the nature of the internal models that subjects form of the task environment. These variants are discussed in the context of other relevant findings in the dynamic decision making literature.     

Maximizing Queueing Network Utility Subject to Stability: Greedy Primal-Dual Algorithm

We study a model of controlled queueing network, which operates and makes control decisions in discrete time. An underlying random network mode determines the set of available controls in each time slot. Each control decision “produces” a certain vector of “commodities”; it also has associated “traditional” queueing control effect, i.e., it determines traffic (customer) arrival rates, service rates at the nodes, and random routing of processed customers among the nodes. The problem is to find a dynamic control strategy which maximizes a concave utility function H(X), where X is the average value of commodity vector, subject to the constraint that network queues remain stable.We introduce a dynamic control algorithm, which we call Greedy Primal-Dual (GPD) algorithm, and prove its asymptotic optimality. We show that our network model and GPD algorithm accommodate a wide range of applications. As one example, we consider the problem of congestion control of networks where both traffic sources and network processing nodes may be randomly time-varying and interdependent. We also discuss a variety of resource allocation problems in wireless networks, which in particular involve average power consumption constraints and/or optimization, as well as traffic rate constraints.     

On the game of googol

In the classical secretary problem the decision maker can only observe the relative ranks of the items presented. Recently, Ferguson — building on ideas of Stewart — showed that, in a game theoretic sense, there is no advantage if the actual values of the random variables underlying the relative ranks can be observed (game of googol). We extend this to the case where the number of items is unknown with a known upper bound. Corollary 3 extends one of the main results in [HK] toall randomized stopping times. We also include a modified, somewhat more formal argument for Ferguson's result.The first author was supported in part by NSF Grant DMS-89-01267.     

An approach to a consistent qualification of decision procedures in probabilistic information processing

Pattern recognition seems to be a rather unique field of interwoven logical inference and decision theory applications. The existence of hundreds of theoretical and real pattern recognition devices forms an ideal basis for research on the structures of various approaches and their comparison. The task of pattern recognition is to select a hypotheses out of a set (e.g.: figures 0, …, 9) on the basis of given data (e.g. the black and white points of a digitized picture). There exists an ideal classifier to solve this problem as the theorem of Bayes provides a logically perfect connection between the input data and the result. But as the so called Bayes-machine proves completely unpractical for real purposesit is “approximated” by more or less complex “real” decision procedures.Thus the theorem of Bayes provides a starting point for the application of statistical considerations and information theory to the analysis of the structures of real decision procedures. The results allow a rather consistent and simple comparison of most decision procedures and provide a tool to estimate the performance of a given procedure in a given environment. The results apply not only to pattern recognition but also to many other fields such as imminence analysis and medical diagnosis.     

Dynamic properties of certain unvulcanized rubbers in the high-elastic and fluid states

The dynamic properties of a series of unvulcanized rubbers (cis-polybutadienes, Na-polybutadiene, cis-polyisoprene, butyl rubber, polyisobutylene, and ethylene-propylene copolymer) have been investigated in a low-frequency dynamic testing machine and a frequency rheometer on the frequency range from 5 · 10^–2 to 4.5 · 10³ Hz at temperatures of 25 and 60°C. At these temperatures the mechanical loss factor is the most sensitive criterion of transitions from one physical state of the polymer to another. For all the specimens investigated on the experimental range of angular frequencies the modulus of the complex dynamic viscosity and the effective viscosity determined under static conditions coincide, assuming the equivalence of angular frequencies and shear rates.A. V. Topchiev Institute of Petrochemical Synthesis, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 698–703, July–August, 1969.     

Consistency of Posterior Mixtures in the Gaussian Family on a Hilbert Space and Its Applications

Majumdar (1994, J. Multivariate Anal.48 87-105) compounds (in the sense of Robbins, 1951, "Proceedings, Second Berkeley Sympos. Math. Statist. Probab.," pp. 131-148, Univ. of California Press, Berkeley) the estimation problem in the mean-parameter family of Gaussian distributions on a real separable infinite dimensional Hilbert space. The question of asymptotic optimality of compound estimators that are Bayes versus a hyperprior mixture of i.i.d. priors on the compound parameter is reduced there, under a compactness restriction on the parameter space, to the question of consistency, in an extended sense, of a certain posterior mixture for the empirical mixture. For mixing hyperpriors with full topological support, that consistency result is obtained in this paper. A corollary of the consistency result is applied to obtain asymptotically optimal decision rules in the empirical Bayes problem involving the mean-parameter Gaussian family and a sufficiently smooth risk function.     

Study of the process of unstable flow of elastomers in open and closed channels and its connection with mechanical characteristics of the material

The unstable flow process has been studied for examples of a series of unvulcanized rubbers of different structures by a complex of methods — capillary, tribometric, and dynamic methods. A connection has been found between the critical parameters which characterize the onset of this process and the dynamic mechanical characteristics. The possibility of modeling the unstable flow process in open channels is shown.A. V. Topchiev Institute for Petrochemical Synthesis, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 530–536, May–June, 1970.     

Allowance for inertia of test piece in dynamic measurements of the rheological characteristics of polymers

The determination of the dynamic shear modulus and dynamic viscosity of polymer materials on apparatus of the Fitzgerald-Ferry type is discussed. Limiting values of the test piece height for which it is permissible to neglect wave propagation of the oscillations are determined. Agreement of the values of the rheological constants determined with and without allowance for inertia is observed at ratios of test piece height to wavelength not exceeding 0.02–0.05. Above this limit the discrepancy rapidly increases and reaches values excluding the use of the formulas recommended in [1–4]. Relations are presented from which the rheological constants should be determined with allowance for the inertia of the test piece.Mekhanika Polimerov, Vol. 2, No. 1, pp. 130–138, 1966     

A new method of measuring the viscoelastic properties of polymer systems in fluid and highly elastic states using forced vibrations in the ultralow-frequency region

A method is considered for measuring the dynamic mechanical characteristics of polyer systems using forced vibrations in the ultralow-frequency region down to 10^–6 Hz. The essence of this method lies in the fact that during the measurement process the motion of the polymer sample is controlled with a set amplitude and frequency, and the dynamic characteristics of the polymer are determined from the controlling mechanical stress. The method is illustrated using the results of measurements of the components of the complex modulus for polybutadiene over a wide range of temperatures and frequencies without making use of superimposition.The Central Constructional Bureau for Prototype Instrument Construction, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 344–347, March–April, 1976.     

On the correctness of a linear closure of set of recognition algorithms of the decision tree type

A recognition operator of the decision tree type is proposed in this paper that is used for the canonical representation of a recognition algorithm, and a theorem is proved about the correctness of the linear closure of a class of incorrect recognition algorithms based on the construction of decision trees.Translated from Dinamicheskie Sistemy, No. 5, pp. 91–94, 1986.     

Solvable classes of discrete dynamic programming

As finite state models to represent a discrete optimization problem given in the form of an r-ddp (recursive discrete decision process), three subclasses of r-msdp (recursive monotone sequential decision process) are introduced in this paper. They all have a feature that the functional equations of dynamic programming hold and there exists an algorithm (in the sense of the theory of computation) to obtain the set of optimal policies. (In this sense, we may call them solvable classes of discrete dynamic programming.) Besides the algorithms for obtaining optimal policies, two types of representations are extensively studied for each class of r-msdp's. Other related decision problems are also discussed. It turns out that some of them are solvable while the rest of them are unsolvable.     

Dynamic Plots in Virtual Negotiations

Advanced information and communications technology provides the basis for continuous monitoring of, and rapid data exchange about, crucial operations. Of special interest are those conflict situations where organizations continuously readjust mutually affecting decisions, considering the competitors' choices, but without communicating verbally. An example refers to trucking companies who base their decisions, which trucks to assign to different routes, on the competitors' assignments. To support the decision process for these types of virtual negotiation settings, a new dynamic plot approach is proposed. Dynamic plots can be used to visualize the decision topology of all parties and the impact of making a decision on all parties. Of special interest in this paper are dynamic plots with an individual stability equilibrium, where competitors do not revise their decisions unless a change in the market occurs. Dynamic plots for 2 × 2 conflict situations are discussed first, followed by a discussion of 3 × 3 conflict situations. The paper closes with a discussion of a computer implementation and empirical evidence of virtual negotiations with dynamic plots. The results of these virtual negotiations suggest that dynamic plots stimulate virtual negotiations and support efficiency, equity, and system optimum for these types of real-time negotiations.     

Uniqueness of solution in the nonlinear theory of dynamic viscoelasticity

The assertion of uniqueness of solution in the dynamic theory of physically nonlinear viscoelasticity is proved.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 1120–1122, November–December, 1971.     

Dynamic Game Problems of Approach for Fractional-Order Equations

We propose a general method for the solution of game problems of approach for dynamic systems with Volterra evolution. This method is based on the method of decision functions and uses the apparatus of the theory of set-valued mappings. Game problems for systems with Riemann–Liouville fractional derivatives and regularized Dzhrbashyan–Nersesyan derivatives (fractal games) are studied in more detail on the basis of matrix Mittag-Leffler functions introduced in this paper.     

Validation of a Structural Optimization Algorithm Transforming Dynamic Loads into Equivalent Static Loads

Generally, structural optimization is carried out based on external static loads. However, all forces have dynamic characteristics in the real world. Mathematical optimization with dynamic loads is almost impossible in a large-scale problem. Therefore, in engineering practice, dynamic loads are often transformed into static loads via dynamic factors, design codes, and so on. Recently, a systematic transformation of dynamic loads into equivalent static loads has been proposed in Refs. 1–3. Equivalent static loads are made to generate at each time step the same displacement field as the one generated by the dynamic loads. In this research, it is verified that the solution obtained via the algorithm of Refs. 1–3 satisfies the Karush–Kuhn–Tucker necessary conditions. Application of the algorithm is discussed.     

Determination of the elastic and damping properties of the human femur in bending

The results of an experimental investigation of the deflection of the human femur subjected to a bending moment are presented. The existence of two principal bending planes is established. The equivalent flexural rigidity of the dry bone is calculated for both static and dynamic loading. It is found that the human femur has a nonlinear elastic characteristic. The stiffness and damping properties obtained in static and dynamic testing are compared.Paper presented at the First All-Union Conference on Engineering and Medical Biomechanics, Riga, October, 1975.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 642–646, July–August, 1975.     

Robustness and information levels in case-based multiple criteria sorting

Case-based preference elicitation methods for multiple criteria sorting problems have the advantage of posing rather small cognitive demands on a decision maker, but they may lead to ambiguous results when preference parameters are not uniquely determined. We use a simulation approach to determine the extent of this problem and to study the impact of additional case information on the quality of results. Our experiments compare two decision analysis tools, case-based distance sorting and the simple additive weighting method, in terms of the effects of additional case information on sorting performance, depending on problem dimension – number of groups, number of criteria, etc. Our results confirm the expected benefit of additional case information on the precision of estimates of the decision maker’s preferences. Problem dimension, however, has some unexpected effects.