Optimization and Dynamics with Their Applications

Iterative methods, play an important role in computational sciences. In this chapter, we present new semilocal and local convergence results for the Newton-Kantorovich method. These new results extend the applicability of the Newton-Kantorovich method on approximate zeros by improving the convergence domain and ratio given in earlier studies.

It is demonstrated how the concept of the Limit-Nash bargaining solution as defined in Forgó and Szidarovszky (Eur J Oper Res 147:108–116, 2003) can be carried over to the non-symmetric case. It is studied how externally given weights of the players and the relative magnitude of penalties for not being able to come to an agreement influence the solution.

Even after many years one kind picture still floating in my eyes: I see professor Szidarovszky, facing the blackboard, sponge and chalk at the upheld left and right hands, and writing and cleaning the lines simultaneously he put on the table, to fill up our heads with numerical methods. This style expresses his very dynamic and efficient research work at the same time, and hopefully this paper indicates that his efforts have not been useless as numerical methods are very intensively used in order to characterize the nature of lot sizing problems in JIT environment. In JIT environment the jidoka principle empowers employees to signal quality problems, and these result in frequent stoppages. This way we consider the output of the assembly line random variable that follows Beta distribution, but with low beta values. For specific beta values we derive explicit forms of the expected values of the inventory related and the annual total costs as function of alpha, the other parameter of the Beta distribution. But increasing alpha expresses increasing process quality. We found that increasing process quality decreases the expected annual cost, and the explicit forms give the saved cost volumes. Two simulation analyses are conducted to reveal the development of the variance of annual costs. The estimations of the variance of the minimum total annual costs indicate that with process improvement the variance of the minimum of the annual total costs will decrease.

In a public goods game, while cooperators need to make contributions, defectors can take a free ride after the realization of the public good. The Nash equilibrium in this game is simply zero contribution from all the players. A conventional approach to encourage cooperation and achieve the public good is using rewards to compensate the difference between the cooperators’ and the defectors’ payoffs. However, the public good may not be realized due to the uncertainty in the game, and the conventional way could underestimate the required rewards to achieve the public good. On the other hand, public good did realize in human history when people cannot survive from a natural disaster, such as a big flood, without cooperating to build a solid embankment, and most of them are willing to choose cooperation without rewards. The realization of the public good leads to the reduction of the defection cost and the contribution, which has a potential to encourage the players’ cooperation. Then the conventional method may overestimate the necessary rewards to realize the public good. In this paper, a public goods game is employed to model the interaction among boundedly rational players with the rewards for large-scale cooperation, and a behavioral game-theoretic approach is developed to describe their decision making processes with the consideration of the risk and impact of public good in the game. It turns out that the conventional rewards to achieve the large-scale cooperation can be reduced for a favorable group of the players.

In the Cournot duopoly game with unimodal piecewise-linear reaction functions (tent maps) proposed by Rand (J Math Econ, 5:173–184, 1978) to show the occurrence of robust chaotic dynamics, a maximum production constraint is imposed in order to explore its effects on the long run dynamics. The presence of such constraint causes the replacement of chaotic dynamics with asymptotic periodic behaviour, characterized by fast convergence to superstable cycles. The creation of new periodic patters, as well as the possible coexistence of several stable cycles, each with its own basin of attraction, are described in terms of border collision bifurcations, a kind of global bifurcation recently introduced in the literature on non-smooth dynamical systems. These bifurcations, caused by the presence of maximum production constraint, give rise to quite particular bifurcation structures. Hence the duopoly model with constraints proposed in this paper can be seen as a simple exemplary case for the exploration of the properties of piecewise smooth dynamical systems.

In an industry of competing firms the market price function is not completely known, however based on repeated price information the firms are able to continuously adjust their beliefs. Under simplifying conditions, it is shown that these beliefs converge to the true price function as time goes to infinity, that is, successful learning can be achieved. The same result holds if there is continuously distributed delay in the price information, if the weighting function is exponential, however in the case of fixed delays stability may be lost if the delay is sufficiently large.

An agent based simulation model is presented to investigate the long-term behavior of firms in an industrial district. The firms are interconnected with each other through input-output relations, product markets, labor, and innovation spillover. The prices of the products depend on the supply-demand balance of the market as well as on the innovation levels of the firms. Dynamic strategies of the firms are examined and conditions for successful industrial cluster formation are developed.

A common problem in systems and control theory is to provide an approximation to non-linear systems. We provide a novel approach as a general solution to this problem originally conceived by Gamkrelidze. We consider and solve a general approximation problem which provides the fundamentals for various switching-type systems thus encompassing a wide range of systems theory problems.

A dynamic model of a love affair between two people is examined under different conditions. First the two-dimensional model is analyzed without time delays in the interaction of the lovers. Conditions are derived for the existence of a unique as well as for multiple steady states. The nonzero steady states are always stable and the stability of the zero steady state depends on model parameters. Then a delay is assumed in the mutual-reaction process called the Gaining-affection process. Similarly to the no-delay case, the nonzero steady states are always stable. The zero steady state is either always stable or always unstable or it is stable for small delays and at a certain threshold stability is lost in which case the steady state bifurcates to a limit cycle. When delay is introduced to the self-reaction process called the Losing-memory process, then the asymptotic behavior of the steady state becomes more complex. The stability of the nonzero steady state is lost at a certain value of the delay and bifurcates to a limit cycle, while the stability of the zero steady state depends on model parameters and there is the possibility of multiple stability switches with stability losses and regains. All stability conditions and stability switches are derived analytically, which are also verified and illustrated by using computer simulation.

Collisions between baseballs, softballs and bats are complex and therefore their models are complex. One purpose of this paper is to show how complex these collisions can be, while still being modeled using only Newton’s principles and the conservation laws of physics. This paper presents models for the speed and spin of balls and bats. These models and equations for bat-ball collisions are intended for use by high school and college physics students, engineering students and most importantly students of the science of baseball. Unlike in previous papers, these models use only simple Newtonian principles to explain simple collision configurations.

Energy generation from wind turbines shows an increasing trend for the last two decades. As the amount of wind generation increases, wind turbine (WT) operators face challenges with finding alternative disposal options for WTs over their useful life. Wind farm operator (decision makers) can benefit from a well-designed reverse logistics network to determine the best disposal alternative for WT end-of-life use (EOL). This chapter is an example of the recovery of valuable material that can be recycled/recovered or remanufactured at the end of WTs useful life by designing an effective reverse logistics network. Here, a mixed integer linear programming (MILP) model is proposed to determine a long-term strategy for WT EOL. The objective in this model is to minimize the transportation and operating cost as well as finding the best locations for recycling and remanufacturing centers. The results of this study show that due to the high operating cost at remanufacturing centers, sending most WTs to them is costlier than sending them to recycling centers. In addition, it was found that transportation cost depends on the amount of flow that has been sent to the recycling or remanufacturing center.

We address a maintenance outsourcing problem where the owner of a piece of critical equipment plans on outsourcing preventive and failure replacement services to a service agent. The owner (i.e., customer) and the agent negotiate on the maintenance policy and spare part ordering strategy in the service contract. We first provide the classical Nash bargaining solution to the problem and analytically determine the optimal threat values the decision makers can use in negotiation. We then extend the model and show how the decision makers can increase their profits through a price discount scheme, which requires the total profit to be achieved at the maximum level. The total maximum profit is analytically determined, and the effects of the price discount scheme and threats on the individual and total profits are illustrated through a numerical study.

A model for the planification of agricultural production is proposed in the Alto Rio Lerma Irrigation District (ARLID) located in the state of Guanajuato in Mexico. The ARLID have a limited water supply from ground and surface sources, as well as area restrictions. In addition, producer faces the problem of high price uncertainty, which affect seriously the amount of expected profit in each season. Therefore, farmers need to distribute their available land between crops in a fuzzy environment. A multiobjective linear programming model in a fuzzy environment (MLFM) is proposed to approach the problem described above. Ten price scenarios are considered according to the records of the last 10 years, these prices were given in the same scale base 2009. In the results all available area, 112,000 ha, was used. Each price scenario generate one objective to be maximize, some price scenarios generate high profits, while others low profits. The results obtained in the MLFM produce the best expected benefit, 4,820 million of pesos, when prices behave as random variables. For Winter season land is distributed mainly between red tomato and wheat, while in the Spring season between corn and wheat. Sorghum was the only second crop to be sown. The model applied in this particular problem of agricultural planification, show the best land use distribution when market fluctuations are expected.

Time-to-failure distributions with mound-shaped failure rates are examined, and the existence of optimal preventive replacement policies is studied. Sufficient and necessary conditions are derived for the existence and the uniqueness of the optimal solutions. The cases of lognormal, log-logistic, log-gamma and log-Weibull variables are discussed in detail. Examples of lognormal cases are provided for illustration.

Increasing water demand is threatening many hydro-environmental systems, particularly lakes in arid regions. The goal of this research is to assess restoration plans for a drying saline lake, Urmia Lake, in Iran. In order to include interactions of different lake sub-systems, effectiveness of the plans, as a challenging question for decision makers, is studied by a system dynamics model. The plans that are studied and modeled are increasing irrigation efficiency, decreasing irrigated land area, cloud seeding, inter-basin water transfer projects, and using refined wastewater. Here, it is found that increasing irrigation efficiency by 4% annually and controlling irrigated lands would have around 60% effect in revitalizing the lake to its ecological level, among those considered restoration plans. By linking potential policies to their outcomes, this modeling effort is a step toward supporting the consensus to restore the lake.

Over-pumping of groundwater resources is a serious problem world-wide. In addition to depleting this valuable water supply resource, hydraulically connected wetlands and surface water bodies are often impacted and even destroyed by over-pumping. Effectively managing groundwater resources in a way that satisfy human needs while preserving natural resources is a daunting problem that will only worsen with growing populations and climate change. What further complicates management of these systems is that even when pumping rates of wells are held fairly constant, their hydraulic effects are often highly transient due to variable weather and hydrologic conditions. Despite this, transient conditions are rarely if ever accounted for by management models due to the difficulties in separating pumping effects from natural factors like weather. To address this shortcoming, a conceptual real-time decision support system for managing complex groundwater/surface water systems affected by variable weather, hydrologic, and pumping conditions over space and time is presented in this chapter. For the hypothetical but realistic groundwater/surface water system presented here, the decision support system, based upon previous work by Coppola et al. (2003a, b, 2005a, b, c, 2007, 2014) consists of real-time data streams combined with artificial neural network (ANN) prediction models and formal optimization. Time variable response coefficients derived from ANN prediction models are used by an optimization model to maximize total groundwater pumping in a multi-layered aquifer system while protecting against aquifer over-draft, streamflow depletion, and dewatering of riparian areas. Optimization is performed for different management constraint sets for both the wet and dry seasons, resulting in significantly different groundwater pumping extraction solutions. Stochastic optimization is also performed for different precipitation forecast events to address corresponding uncertainty associated with weather-dependent irrigation pumping demand. This data-driven support system can continuously adapt in real-time to existing and forecasted hydrological and weather conditions, as well as water demand, providing superior management solutions.