Logics in realization and modeling from data

We consider the modelling and analysis of public transportation networks, such as railway or subway networks, governed by a timetable. Specifically, we study a (max,+)-linear model of a generic transportation network and thereby give a self-contained introduction to the key ideas underlying the (max,+) algebra. We elaborate on the algebraic structure implied by the (max,+)-model to formulate (and solve) the control problem in the deterministic as well as in the stochastic case. The control problem is here whether a train should wait on a connecting train which is delayed. Our objective is then to minimise the propagation of the delay through the network while maintaining as many connections as possible. With respect to the deterministic control problem, we present some recent ideas concerning the use of (max,+)-techniques for analysing the propagation of delays. Moreover, we show how one can use the (max,+)-algebra to drastically reduce the search space for the deterministic control problem. For the stochastic control problem, we consider a parameterised version of the control problem, that is, we describe the control policy by means of a real-valued parameter, say \theta . Finding the optimal control is then turned into an optimisation problem with respect to \theta . We address the problem by incorporating an estimator of the derivative of the expected performance with respect to \theta into a stochastic approximation algorithm.

Cost functors and optimal realization

The determination of an optimal drug administration policy is formulated as a stochastic optimal control problem for a linear system subjected to additive white noise. The maximum allowable dosage is explicitly considered as a continuous state constraint to the control problem. The stochastic problem is transformed into a deterministic problem and subsequently solved by the technique of control parametrization. Examples are presented to highlight the interesting features of the optimal policy.

: Given a mean-square continuous stochastic vector process y with stationary increments and a rational spectral density Phi such that Phi(infinity) is finite an nonsingular, consider the problem of finding all minimal Gauss-Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal Stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations) require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman-Bucy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algorithm is presented which generates families of external realizations defined on the same probability space and totally ordered with respect to state covariances. (Author)

We describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this covariance extension equation (CEE) is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem, answering a conjecture due to Georgiou (1987) in the affirmative. We also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the CEE, yielding some insights into the structure of solutions to the minimal partial stochastic realization problem. By combining this parameterization with some of the classical approaches in partial realization theory, we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations. As a corollary to these results, we note that, in sharp contrast with the deterministic case, there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length.

Construction projects are characterized by their dynamic nature and operational details. This paper presents a hybrid simulation methodology; designed to model construction projects. The methodology utilizes Discrete Event Simulation (DES) and System Dynamics (SD). DES has been widely used in modeling construction operations; however, it lacks the ability to model the global aspects of operations being modeled and the cause-effect relations of simulation variables. SD is utilized to circumvent these limitations. Both simulation methods provide valuable decision support but none is individually capable of capturing the holistic nature of the operation being modeled. The developed methodology integrates DES and SD to utilize their respective advantages in simulating construction operations. The developed methodology encompasses five stages: 1) identification of model objectives, 2) decision criteria to assist in selecting simulation methodology, 3) building simulation model and identification of interface variables, 4) computation framework and 5) implementation and testing. The paper describes the essential features of the developed methodology and its computational framework and focuses primarily on the modeling aspects of SD. A case study project is analyzed to demonstrate the use of the developed methodology and to highlight its capabilities.

We review a class of online planning algorithms for deterministic and stochastic optimal control problems, modeled as Markov decision processes. At each discrete time step, these algorithms maximize the predicted value of planning policies from the current state, and apply the first action of the best policy found. An overall receding-horizon algorithm results, which can also be seen as a type of model-predictive control. The space of planning policies is explored optimistically, focusing on areas with largest upper bounds on the value - or upper confidence bounds, in the stochastic case. The resulting optimistic planning framework integrates several types of optimism previously used in planning, optimization, and reinforcement learning, in order to obtain several intuitive algorithms with good performance guarantees. We describe in detail three recent such algorithms, outline the theoretical guarantees on their performance, and illustrate their behavior in a numerical example.

It has been suggested by many supply chain practitioners that in certain cases inventory can have a stimulating effect on the demand. In mathematical terms this amounts to the demand being a function of the inventory level alone. In this work we propose a logistic growth model for the inventory dependent demand rate and solve first the continuous time deterministic optimal control problem of maximising the present value of the total net profit over an infinite horizon. It is shown that under a strict condition there is a unique optimal stock level which the inventory planner should maintain in order to satisfy demand. The stochastic version of the optimal control problem is considered next. A bang-bang type of optimal control problem is formulated and the associated Hamilton-Jacobi-Bellman equation is solved. The inventory level that signifies a switch in the ordering strategy is worked out in the stochastic case.

Generalized spectral decomposition for stochastic nonlinear problems

In this paper we consider a stochastic bilinear fractional max-min problem with separate linear constraints. In the case when all the coefficients of the objective function are simple randomized, two ways for solving this problem will be employed: the minimum-risk approach and Kataoka's model. We prove that, under some positivity conditions, these stochastic problems are equivalent with certain deterministic bilinear-fractional min-max (or max-min) problems. For solving these deterministic optimization problems a parametrical procedure will be presented.

The study of system dynamics starts from model construction and simulation to understand and solve dynamical complicated problems. Traditional approaches of modeling process depend on experts’ experiences and the trial-and-error pro- cedure, so it is difficult to guarantee a useful model. Because a system dynam- ics model is equivalent to a specially-designed artificial neural network, both of which operate under the same numerical propagation constraints, we use the ar- tificial neural network training algorithms and take advantage of historical data to assist system dynamics model construction. Experimental studies show that this approach is feasible.

This paper studies the reliable uncapacitated facility location problem in which facilities are subject to uncertain disruptions. A two-stage distributionally robust model is formulated, which optimizes the facility location decisions so as to minimize the fixed facility location cost and the expected transportation cost of serving customers under the worst-case disruption distribution. The model is formulated in a general form, where the uncertain joint distribution of disruptions is partially characterized and is allowed to have any prespecified dependency structure. This model extends several related models in the literature, including the stochastic one with explicitly given disruption distribution and the robust one with moment information on disruptions. An efficient cutting plane algorithm is proposed to solve this model, where the separation problem is solved respectively by a polynomial-time algorithm in the stochastic case and by a column generation approach in the robust case. Extensive numerical study shows that the proposed cutting plane algorithm not only outperforms the best-known algorithm in the literature for the stochastic problem under independent disruptions but also efficiently solves the robust problem under correlated disruptions. The practical performance of the robust models is verified in a simulation based on historical typhoon data in China. The numerical results further indicate that the robust model with even a small amount of information on disruption correlation can mitigate the conservativeness and improve the location decision significantly. Summary of Contribution: In this paper, we study the reliable uncapacitated facility location problem under uncertain facility disruptions. The problem is formulated as a two-stage distributionally robust model, which generalizes several related models in the literature, including the stochastic one with explicitly given disruption distribution and the robust one with moment information on disruptions. To solve this generalized model, we propose a cutting plane algorithm, where the separation problem is solved respectively by a polynomial-time algorithm in the stochastic case and by a column generation approach in the robust case. The efficiency and effectiveness of the proposed algorithm are validated through extensive numerical experiments. We also conduct a data-driven simulation based on historical typhoon data in China to verify the practical performance of the proposed robust model. The numerical results further reveal insights into the value of information on disruption correlation in improving the robust location decisions.

When studying deterministic and stochastic population models, the actual problems are the formalization of processes, taking into account new effects caused by the interaction of species, and the development of computer research methods. Computer research methods make it possible to analyze the trajectories of multidimensional population systems. We consider the “two competitors - two migration areas” model, which takes into account intraspecific and interspecific competition in two populations, as well as bidirectional migration of both populations. For this model, we take into account the variability of the reproduction rates of species. A formalized description of the four-dimensional model “two competitors - two migration areas” and its modifications is proposed. Using the implementation of the evolutionary algorithm, a set of parameters is obtained that ensure the coexistence of populations under conditions of competition between two species in the main area, taking into account the migration of these species. Taking into account the obtained set of parameters, a positive stationary state is found. Two-dimensional and three-dimensional projections of phase portraits are constructed. Stochastization of the model “two competitors - two migration areas” is carried out based on the method of self-consistent one-step models constructing. The Fokker-Planck equations are used to describe the structure of the model. A transition to a four-dimensional stochastic differential equation in the Langevin form is performed. To carry out numerical experiments, a specialized software package is used to construct and study stochastic models, and a computer program based on differential evolution is developed. Algorithms for generating trajectories of the Wiener process and multipoint distributions and modifications of the Runge-Kutta method are used. In the deterministic and stochastic cases, the dynamics of the trajectories of populationmigration systems is studied. A comparative analysis of deterministic and stochastic models is carried out. The results can be used in modeling of different classes of dynamic systems.

In the body of systems thinking, system dynamics (SD) is considered to be a powerful tool. However, when dealing with a social entity such as a smallholder group, SD is considered to be a researcher-centrist methodology that may not accurately reflect perspectives and social power sensitivities of all stakeholders. This paper reports on research in which, in an effort to enhance SD, these deficiencies are addressed through complementary application of soft systems methodology, which has strength in acknowledging multiple perspectives, and critical systems heuristics, which can explicitly address power asymmetry. A series of interviews and workshops was undertaken to identify the problematic situation of smallholder beef farming in Java. To describe its linkages, this problematic situation was then translated into a causal loop diagram from which the systems archetypes were identified. The nature of each archetype is described, and the implications for identification of the possible system leverage points are discussed. Copyright © 2014 John Wiley & Sons, Ltd.

We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of 1- 1/e ¿ 0.632, a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the 1 - 1/e bound. Our main result is a 0.67-approximation online algorithm for stochastic bipartite matching, breaking this 1 - ¿ barrier. Furthermore, we show that no online algorithm can produce a 1 - ¿ approximation for an arbitrarily small e for this problem. Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-)matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution. At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application.