Convex duality for finite-fuel problems in singular stochastic control

Upon introducing a finite-fuel constraint in a stochastic control system, the convex duality formulation can be set up to represent the original singular control problem as a minimization problem over the space of vector measures at each level of available fuel. This minimization problem is imbedded tightly into a related weak problem, which is actually a mathematical programming problem over a convex,w*-compact space of vector-valued Radon measures. Then, through the Fenchel duality principle, the dual for the finite-fuel control problems is to seek the maximum of smooth subsolutions to a dynamic programming variational inequality. The approach is basically in the spirit of Fleming and Vermes, and the results of this paper extend those of Vinter and Lewis in deterministic control problems to the finite-fuel problems in singular stochastic control. Meanwhile, we also obtain the characterization of the value function as a solution to the dynamic programming variational inequality in the sense of the Schwartz distribution.The author is much indebted to Professor Wendell H. Fleming for his constant support and many helpful discussions during the preparation of this paper.     

A novel variable step size fractional order incremental conductance algorithm to maximize power tracking of fuel cells

The maximum power tracking of Proton Exchange Membrane Fuel Cells (PEMFC) is important for the optimization of fuel cell system design. It is necessary to operate a fuel cell at maximum power to ensure full efficiency. This study presents a novel Fractional Order Incremental Conductance Algorithm (FOINC) with variable step size control which can be used for maximum power point tracking in the design of fuel cells. The method has high maximum power point tracking speed and good steady-state response, and does not require extra sensing elements for different fuel cell equipment. When compared to the traditional Incremental Conductance (INC) and Perturbation and Observation (P&O) methods, the system simulation results show the method to be feasible and effective.     

Optimal assignment of NOP due-dates and sequencing in a single machine shop

We consider the problem of optimal assignment of NOP due-dates ton jobs and sequencing them on a single machine to minimize a penalty function depending on the values of assigned constant waiting allowance and maximum job tardiness. It is shown that the earliest due date (EDD) order is an optimal sequence. For finding optimal constant waiting allowance, we reduce the problem to a multiple objective piecewise linear programming with single variable. An efficient algorithm for optimal solution of the problem is given.     

Routing vehicles to minimize fuel consumption

We consider a generalization of the capacitated vehicle routing problem known as the cumulative vehicle routing problem in the literature. Cumulative VRPs are known to be a simple model for fuel consumption in VRPs. We examine four variants of the problem, and give constant factor approximation algorithms. Our results are based on a well-known heuristic of partitioning the traveling salesman tours and the use of the averaging argument.     

Path problems in generalized stars, complete graphs, and brick wall graphs

Path problems such as the maximum edge-disjoint paths problem, the path coloring problem, and the maximum path coloring problem are relevant for resource allocation in communication networks, in particular all-optical networks. In this paper, it is shown that maximum path coloring can be solved optimally in polynomial time for bidirected generalized stars, even in the weighted case. Furthermore, the maximum edge-disjoint paths problem is proved NP-hard for complete graphs (undirected or bidirected), a constant-factor off-line approximation algorithm is presented for the weighted case, and an on-line algorithm with constant competitive ratio is given for the unweighted case. Finally, an open problem concerning the existence of routings that simultaneously minimize the maximum load and the number of colors is solved: an example for a graph and a set of requests is given such that any routing that minimizes the maximum load requires strictly more colors for path coloring than a routing that minimizes the number of colors.     

Problems of maximizing the compliance of curvilinear rods

A number of problems of finding the shape of a thin curvilinear rod (the support element of an artificial lens) of constant cross-section and specified length with its ends at specified points and under specified loading conditions with maximum compliance for characteristic types of end restraint and loading are considered. It is shown that the boundary-value problem arising for the non-linear Euler equation may have a set (possibly denumerable) of solutions, one of which gives the absolute maximum compliance, and the others the local maxima. The problem is analysed in detail, analytical solutions are obtained and the corresponding shapes are constructed in a number of important cases.     

Ultracentrifuge Sedimentation during the Period of Constant Acceleration

Initially, the angular velocity of an ultracentrifuge increasesat a uniform rate until a given maximum is reached, after whichit is maintained at a constant value. The concentration distributionduring the accelerating period is obtained as the solution ofa one-dimensional approximation to the diffusion equation, analogousto that previously used (Fujita & MacCosham, 1959) for thecase of constant angular velocity.     

Singular Arcs in the Optimal Evasion Against a Proportional Navigation Vehicle

The two-dimensional optimal evasion problem against a proportional navigation pursuer is analyzed using a nonlinear model. The velocities of both players have constant modulus, but change in direction. The problem is to determine the time-minimum trajectory (disengagement) or time-maximum trajectory (evasion) of the evader while moving from the assigned initial conditions to the final conditions. A maximum principle procedure allows one to reduce the optimal control problem to the phase portrait analysis of a system of two differential equations. The qualitative features of the optimal process are determined.     

Light reflection is nonlinear optimization

In this paper, we show that the near field reflector problem is a nonlinear optimization problem. From the corresponding functional and constraint function, we derive the Monge–Ampère type equation for such a problem.     

Optimal abort landing trajectories in the presence of windshear

This paper is concerned with optimal flight trajectories in the presence of windshear. The abort landing problem is considered with reference to flight in a vertical plane. It is assumed that, upon sensing that the airplane is in a windshear, the pilot increases the power setting at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant. Hence, the only control is the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative.The performance index being minimized is the peak value of the altitude drop. The resulting optimization problem is a minimax problem or Chebyshev problem of optimal control, which can be converted into a Bolza problem through suitable transformations. The Bolza problem is then solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems. Numerical results are obtained for several combinations of windshear intensities, initial altitudes, and power setting rates.For strong-to-severe windshears, the following conclusions are reached: (i) the optimal trajectory includes three branches: a descending flight branch, followed by a nearly horizontal flight branch, followed by an ascending flight branch after the aircraft has passed through the shear region; (ii) along an optimal trajectory, the point of minimum velocity is reached at about the time when the shear ends; (iii) the peak altitude drop depends on the windshear intensity, the initial altitude, and the power setting rate; it increases as the windshear intensity increases and the initial altitude increases; and it decreases as the power setting rate increases; (iv) the peak altitude drop of the optimal abort landing trajectory is less than the peak altitude drop of comparison trajectories, for example, the constant pitch guidance trajectory and the maximum angle of attack guidance trajectory; (v) the survival capability of the optimal abort landing trajectory in a severe windshear is superior to that of comparison trajectories, for example, the constant pitch guidance trajectory and the maximum angle of attack guidance trajectory.Portions of this paper were presented at the IFAC 10th World Congress, Munich, Germany, July 27–31, 1987 (Paper No. IFAC-87-9221).This research was supported by NASA Langley Research Center, Grant No. NAG-1-516, by Boeing Commercial Airplane Company (BCAC), and by Air Line Pilots Association (ALPA). Discussions with Dr. R. L. Bowles (NASA-LRC) and Mr. C. R. Higgins (BCAC) are acknowledged.     

Infrastructure development for conversion to environmentally friendly fuel

An important concern for any nation wishing to convert to alternate, environmentally friendly energy sources is the development of appropriate fuel distribution infrastructure. We address the problem of optimally locating gas station facilities for developing nations, like India, which are in the process of converting from leaded to unleaded fuel. Importantly, a similar approach may be used in developed countries, which are in the process of converting to automobiles using hydrogen or electrical energy. An integer-programming model with the objective of balancing the perspectives of coverage and cost is presented for this facility location problem. Given the existing network of roads, the model considers the traveling population, the location of existing facilities and the cost of, either converting these facilities to carry unleaded fuel, or of installing new facilities in an attempt to minimize cost and simultaneously maximize coverage of population. We develop a heuristic solution procedure for this problem. The methodology is applied to data sets obtained from Current et al. [J.R. Current, C.S. ReVelle, J.L. Cohon, Decision Sciences 19 (1988) 490] representing the Ohio state limited access highway network, and to the Indian national highway network. Additionally, extensive simulations are carried out in order to examine where our approach compares with the maximum weighted spanning tree approach. This work extends the Maximum Covering/Shortest Path problem (MCSPP) formulated by Current et al. [J.R. Current, C.S. ReVelle, J.L. Cohon, European Journal of Operational Research 21 (1985) 189] to accommodate multiple sources and destinations.     

Hybrid heuristics for a short sea inventory routing problem

We consider a short sea fuel oil distribution problem where an oil company is responsible for the routing and scheduling of ships between ports such that the demand for various fuel oil products is satisfied during the planning horizon. The inventory management has to be considered at the demand side only, and the consumption rates are given and assumed to be constant within the planning horizon. The objective is to determine distribution policies that minimize the routing and operating costs, while the inventory levels are maintained within their limits. We propose an arc-load flow formulation for the problem which is tightened with valid inequalities. In order to obtain good feasible solutions for planning horizons of several months, we compare different hybridization strategies. Computational results are reported for real small-size instances.     

Rate of consolidation of stone column-improved ground considering change in permeability and compressibility during consolidation

In the available mathematical formulations to determine the rate of radial consolidation of stone column-improved ground, the soil properties especially permeability and compressibility are assumed to be constant during consolidation process. However, permeability changes with void ratio (void ratio also changes with consolidation) and compressibility of soil varies along the consolidation curve (compressibility is a function of effective stress and void ratio). Thus, these properties are not constant during consolidation period. In the present paper, mathematical formulation is developed to determine the rate of consolidation of stone column-improved ground due to radial flow considering change in permeability and compressibility of soft soil during consolidation period. Equal strain approach has been considered in the analysis. The parabolic variation in permeability and compressibility within smear zone are incorporated in the formulation. It is observed that the variation of degree of consolidation due to change in stress concentration ratio and diameter ratio reduces when variable soil properties are considered. The difference between the degree of consolidation obtained considering variable and constant soil properties is almost constant due to variation of smear zone parameters. The time required to achieve 90% degree of consolidation increases or decreases (depending on the properties of soil) by around 50%–100% or up to 25%, respectively, when change in soil properties during consolidation is considered.     

Monotone iterative method of upper and lower solutions applied to a multilayer combustion model in porous media

This paper presents a new nonlinear reaction–diffusion–convection system coupled with a system of ordinary differential equations that models a combustion front in a multilayer porous medium. The model includes heat transfer between the layers and heat loss to the external environment. A few assumptions are made to simplify the model, such as incompressibility; then, the unknowns are determined to be the temperature and fuel concentration in each layer. When the fuel concentration in each layer is a known function, we prove the existence and uniqueness of a classical solution for the initial and boundary value problem for the corresponding system. The proof uses a new approach for combustion problems in porous media. We construct monotone iterations of upper and lower solutions and prove that these iterations converge to a unique solution for the problem, first locally and then, in time, globally.     

Modeling and computer simulation of the identification problem related to the sludge concentration in a settler

The mathematical model of sludge particles settling in the water treatment plant (settler) is considered. In the case of the residence time of sludge particles in the settler the model leads to a nonlinear age-dependent transport–diffusion with a nonlocal additional condition. This problem is formulated as an identification/optimal control problem, where the sludge concentration is assumed to be a control. For the case of constant (“average”) velocity, as a characterization of the optimal control problem two necessary conditions are obtained. These conditions permit reducing the nonlinear coupled two-dimensional problem to the two-point boundary value problem for the second order nonlinear ordinary differential equation, and then, to a nonlinear equation, with respect to sludge concentration. For the solution of the problem an iteration algorithm is derived. Convergence of the iteration algorithm is analyzed theoretically, as well as on test examples. The numerical procedure for the considered problem is demonstrated on concrete examples.     

Global optimality of extremals: An example

The question of the existence and the location of Darboux points (beyond which global optimality is lost) is crucial for minimal sufficient conditions for global optimality and for computation of optimal trajectories. Here, we investigate numerically the Darboux points and their relationship with conjugate points for a problem of minimum fuel, constant velocity, horizontal aircraft turns to capture a line. This simple second-order optimal control problem shows that ignoring the possible existence of Darboux points may play havoc with the computation of optimal trajectories.The authors are indebted to G. Moyer for his constructive comments. This research was supported, for the first author, by a National Research Council Associateship at NASA Ames Research Center.on leave from the Technion, Israel Institute of Technology, Haifa, Israel.     

Lie‐group method for solving the problem of fission product behavior in nuclear fuel

Calculation of the gas atom concentration is an important feature of all physical models of fission gas release. We apply Lie‐group method for determining symmetry reductions to the diffusion equation describing the fission gas release from nuclear fuel. The resulting nonlinear ordinary differential equation is solved numerically using nonlinear finite difference method. Effects of the dimensionless group constant, the time, and the grain radius on the concentration diffusion function have been studied, and the results are plotted. It is found that the concentration of gas atoms increases as the dimensionless group constant, the power index, and the time increase, and it decreases with increase of the grain radius. Copyright © 2013 John Wiley & Sons, Ltd.     

Reaction-Diffusion Model for Combustion with Fuel Consumption: I. Dirichlet Boundary Conditions

A model is derived for the reaction between a gaseous oxidantand a solid porous fuel, which allows for the consumption ofthe fuel. A single exothermic reaction, first-order in the concentrationof the oxidant and fuel with an Arrhenius temperature dependenceof the reaction rate constant is assumed. The solid reactantis taken to have an infinite slab geometry and be surroundedby a reservoir of constant oxidant concentration and fixed ambienttemperature. Numerical solutions of the governing equationsare obtained and the effects of varying the dimensionless parameters,especially the Frank-Kamenetskii parameter and the Lewis numberL, are examined. Analytic solutions are then derived for theinitial development of the solution and for large values of. This latter solution shows the development of a reaction-diffusionwave which starts at the edge and propagates across the slab,the structure of which depends on the Lewis number.     

Timetabling optimization of a single railway track line with sensitivity analysis

The paper deals with the timetabling problem of a single-track railway line. To solve the timetabling problem, we propose a three-stage approach combining several optimization criteria. Initially and mainly, the maximum relative travel time (ratio of travel time to minimum possible travel time) is minimized subject to a set of constraints, including departure time, train speed, minimum and maximum dwell time, and headway at track segments and stations. Since this problem has many solutions, the process is repeated for other trains, keeping the relative travel times of the critical train fixed, until all trains have been assigned their optimal relative travel times. In the second stage, the prompt allocation of trains is a secondary objective, and finally, in the third stage, the one minimizing the sum of the station dwell times of all trains, keeping the relative travel times constant, is selected to reduce fuel consumption, as a tertiary objective. To consider the user preferences in the optimization problems, the user preference departure time is used instead of the actual planned departure times. In order to guarantee that the exact or a very good approximate global optimum is attained, an algorithm based on the bisection rule is used. This method allows the computation time to be reduced in at least one order of magnitude for 42 trains. The problem of sensitivity analysis is also discussed, and closed form formulas for the sensitivities in terms of the dual variables are given. Several examples of applications are presented to illustrate the goodness of the proposed method. The results show that an adequate selection of intermediate stations and of the departure times are crucial in the good performance of the line and that inadequate spacings between consecutive trains can block the line. In addition, it is shown that, in order to improve performance, regional trains must be scheduled just ahead of or following the long distance trains, rather than having independent schedules. The sensitivities are shown to be very useful in identifying critical trains, segments, stations, departure times, and headways and in suggesting line infrastructure changes.     

On the Definition of a Minimum in Parameter Optimization

For the unconstrained minimization of an ordinary function, there are essentially two definitions of a minimum. The first involves the inequality \(\le \), and the second, the inequality <. The purpose of this Forum is to discuss the consequences of using these definitions for finding local and global minima of the constant objective function. The first definition says that every point on the constant function is a local minimum and maximum, as well as a global minimum and maximum. This is not a rational result. On the other hand, the second definition says that the constant function cannot be minimized in an unconstrained problem. It must be treated as a constrained problem where the constant function is the lower boundary of the feasible region. This is a rational result. As a consequence, it is recommended that the standard definition (\(\le \)) for a minimum be replaced by the second definition (<).