Optimal use of mixed catalysts for two successive chemical reactions

Recently, Gunn and Thomas showed that—when the conversion of a feedstock to a product takes place in two chemically distinct steps, each of which is promoted by a different catalyst—there are advantages to be gained by mixing the catalysts in a single reactor rather than carrying out the two reaction steps separately. In this paper, the maximum principle is applied to the problem of determining the optimal variation in catalyst blend along the reactor and, for a simple first-order kinetic scheme, it is shown to lead to a complete solution in closed form.     

A simulated annealing approach to mine production scheduling

Increasing global competition, quality standards, environmental awareness and decreasing ore prices impose new challenges to mineral industries. Therefore, the extraction of mineral resources requires careful design and scheduling. In this research, simulated annealing (SA) is recommended to solve a mine production scheduling problem. First of all, in situ mineral characteristics of a deposit are simulated by sequential Gaussian simulation, and averaging the simulated characteristics within specified block volumes creates a three-dimensional block model. This model is used to determine optimal pit limits. A linear programming (LP) scheme is used to identify all blocks that can be included in the blend without violating the content requirements. The Lerchs–Grosmann algorithm using the blocks identified by the LP program determines optimal pit limits. All blocks that lie outside of the optimal pit limit are removed from the system and the blocks within the optimal pit are submitted to the production scheduling algorithm. Production scheduling optimization is carried out in two stages: Lagrangean parameterization, resulting in an initial sub-optimal solution, and multi-objective SA, improving the sub-optimal schedule further. The approach is demonstrated on a Western Australian iron ore body.     

Periodic switching strategies for an isoperimetric control problem with application to nonlinear chemical reactions

This paper deals with an isoperimetric optimal control problem for nonlinear control-affine systems with periodic boundary conditions. As it was shown previously, the candidates for optimal controls for this problem can be obtained within the class of bang-bang input functions. We consider a parametrization of these inputs in terms of switching times. The control-affine system under consideration is transformed into a driftless system by assuming that the controls possess properties of a partition of unity. Then the problem of constructing periodic trajectories is studied analytically by applying the Fliess series expansion over a small time horizon. We propose analytical results concerning the relation between the boundary conditions and switching parameters for an arbitrary number of switchings. These analytical results are applied to a mathematical model of non-isothermal chemical reactions. It is shown that the proposed control strategies can be exploited to improve the reaction performance in comparison to the steady-state operation mode.     

Optimal catalyst concentration profile for bifunctional catalysts

The optimal catalyst profile for a consecutive reaction scheme (where the first reaction is promoted by a catalyst of one type, and the second by a catalyst of a different type) is determined by use of the Pontryagin maximum principle. Solutions are obtained for nonlinear reaction rate expressions. The optimal concentration profile can be obtained by relatively simple numerical computation and does not necessitate the solving of a two-point boundary-value problem.Professor C. Crowe, McMaster University, suggested several inprovements to the original draft of this paper. This paper is published with the approval of the Director of the National Institute for Metallurgy.     

Simulation of fluid catalytic cracking operation

Since its introduction in 1942 the fluid catalytic cracking (FCC) has been the most important and widely used process for the production of gasoline from heavy distillates. In most refineries the capacity of the FCC unit is second only to that of the crude distillation unit. Often an FCC unit is referred to as the heart of a modern refinery oriented toward maximum production of gasoline.The basic step in the FCC process is the recirculation of the catalyst through the reactor, stripping and regenerator. In the reactor system the hydrocarbon feed is heated and cracked. Coke (or carbon) may be produced and may deposit on the catalyst reducing its activity and selectivity. When the catalyst is circulated to the regenerator carbon is burned off causing the heating of the catalyst before its return to the reactor part. The products from the reactor are separated in a main fractionator into gas and liquid streams normally including a recycle feed to the reactor.The operation of an FCC unit requires the manipulation of a large number of controlled variables affecting its performance. Major process variables such as reactor temperature, catalyst circulation rate, catalyst inventory and recycle feed rate can be varied to influence the product yields and to accomodate widely different feedstocks. Unpredictable variation can occur in feed stock, catalyst quality and equipment performance. Most normal variation can be accomodated by a small change in operating conditions.For a new plant, comparison of actual versus predicted performance provides a valuable check on the validity of the design correlations and a guide for future laboratory and engineering research.The objectives of the present work are to simplify the complicated FCC process variables and to develop a computer model to simulate the operation of an FCC at different conditions. This includes the prediction of the effects of the operating variables on the reactor product yields. These products include fuel gas, C₃, C₄ gasoline, light gas oil and coke. The model provides a good base for troubleshooting and debottlenecking and may be useful in optimal control of the FCC.     

One-dimensional equilibrium model and source parameter determination for soil-column experiment

One-dimensional equilibrium soil-column experiment models with source (sink) reaction terms are discussed in this paper. In the case of occurring high-order chemical reactions, the zero production term in traditional models should be modified to a nonlinear term related with time (or space) and solute concentration, and then a mathematical model with nonlinear terms is put forward. Furthermore, an actual soil-column experiment in Zhangdian, Zibo is investigated. By applying an optimal perturbation algorithm, the source coefficient in the model is determined both in the cases of accurate data and inaccurate data. The inversion results show that for such inverse source coefficient problems with limited additional data, some optimal methods could be more efficient than regularization strategies, and for some real equilibrium soil-column experiments, the process of source (sink) reactions could be a key factor in the solute transportation.     

Coal blending models for optimum cokemaking and blast furnace operation

An important problem at an integrated steel-producing plant is the blending of different types of coals to make coke for the blast furnace operation. Historically, linear blending models were not appropriate because coal properties important for both optimum cokemaking and blast furnace operation do not combine linearly and are not completely understood. In this paper, a solution methodology is developed that utilizes two techniques: (1) a mixed integer linear programming model for blending the candidate coals to produce coke at a minimum cost and (2) binary decision tree analyses and results that are converted into model constraints to ensure the production of high-quality coke. Subsequently, the model results are used at the pilot-scale oven for testing and for validating the new, improved blend(s) that have been recommended by the model. This is an on-going need that is dictated by changing availabilities in both coal prices and sources. These steps reduce costs by both minimizing the number of blends to be tested at the pilot-scale facility and ensuring a minimum cost coal blend that is useable for the operating facilities. Hypothetical, but realistic, data are used to illustrate how the model performs.     

Coexistence results for catalysts

Summary In this paper we consider a modification of Ziff, Gulari and Barshad's (1986) model of oxidation of carbon monoxide on a catalyst surface in which the reactants are mobile on the catalyst surface. We find regions in the parameter space in which poisoning occurs (the catalyst surface becomes completely occupied by one type of atom) and another in which there is a translation invariant stationary distribution in which the two atoms have positive density. The last result is proved by exploiting a connection between the particle system with fast stirring and a limiting system of reaction diffusion equations.     

Optimal prediction of human postural response under anterior–posterior platform tilting

Previous studies have suggested that human beings movements can be related to the problem of cost function minimization. But at the present time it is not clear that which objective function(s) and constraints are used by central nervous system (CNS) to produce optimal reactions under perturbations. Present study has been done experimentally and by numerical simulations to explore the stability constraints which should be used in combination with energy based cost function (weighted minimum torque) to estimate the motor planning criterion is used by CNS for disturbance rejections. The influence of three stability criterions (ZMP, extrapolated center of mass and a vertical force criterion) in combination with minimum torque model on the optimal trajectory formation is investigated. First, the response of 10 male healthy persons to platform oscillation was recorded by motion analysis system and the hip, knee and ankle angular trajectories were derived from recorded data. Second, the dynamic simulation of a four-segment, three actuated degrees of freedom mechanical model of the human body was performed using predictive dynamic method which leads to an optimization problem. The simulated trajectories were then compared to the experimental data. With comparison between experimental results, the weighting coefficients of the objective function were found to achieve best estimation. It was seen that the minimum torque objective function with weighting coefficients gives trajectories that are mostly matched with experimental observation. Moreover, the results showed that between stability criterions, the ZMP predictions are near to experimental results. Although by using vertical force criterion some nearness to experimental results are lost (in comparison with ZMP criterion) but a secured flat-foot posture for the model is obtained which this posture is more applicable than others in humanoid implementations.     

多阶段生产系统最优生产批量和原材料订购决策模型

Sarker和Parija(1996)建立了生产系统最优生产批量和原材料订购决策模型。然而他们的模型仅局限于单阶段生产系统,本文将他们的模型扩展到多阶段生产系统,我们首先建立了使整个多阶段生产系统总成本最小的各阶段最优生产批量、原材料订购批量及阶段之间的运输批量模型,然后分析了原材料订购费、半成品运费及设备安装费的敏感性。最后,我们结合实例综合分析了原材料订购费、半成品运输费和设备安装费的变化及最小值点取整后对原材料订购决策、最优生产批量和总成本的影响。     

博弈参与人的偏好对最优反应的影响分析

在现实社会中，每个人都有自己的偏好，因此在博弈过程中，参与人的不同偏好在其选择策略时起着不同的作用。本文运用多目标决策方法研究了总需求不确定情况下具有风险偏好的企业决策者关于一种产品价格博弈模型和均衡，并进一步讨论了参与人具有相同偏好和不同偏好情况下风险厌恶程度、价格对需求影响程度等参数对参与人最优反应的影响。     

On the optimal harvesting of size-structured population dynamics

This work is concerned with a kind of optimal control problem for a size-structured biological population model. Well-posedness of the state system and an adjoint system are proved by means of Banach’s fixed point theorem. Existence and uniqueness of optimal control are shown by functional analytical approach. Optimality conditions describing the optimal strategy are established via tangent and normal cones technique. The results are of the first ones for this novel structure.     

A computer software package for the simulation and optimization of an industrial ammonia converter based on a rigorous heterogeneous model

A computer package for the modelling, simulation and optimization of an ammonia reactor is developed for use on personal computers. The package is easy to use with user friendly interface utilizing the graphical capabilities of the personal computer. The model used in the package is a heterogeneous model, which has been checked against an industrial reactor having three adiabatic beds with interstage cooling, and gave excellent agreement with the industrial data. The two-point boundary value differential equation for the catalyst particles is solved using the orthogonal collocation method and three internal collocation points are necessary to obtain accurate results. The model is used to investigate the optimal behavior of the reactor.     

Optimal sales force compensation plans: an operational procedure

This paper describes an operational procedure for identifying optimal sales force compensation plans featuring salary, commissions and/or quota/bonus. Utility-maximising salespeople's behaviours and reactions to given compensation plans are simulated, and the resulting sales, costs and long-term expected profits are assessed. Then, a search technique attempts to identify the long-term profit-maximising compensation plan structure. Operationally, the simulation model parameters are calibrated so as to reflect those of an actual sales force, and consequently the optimal compensation scheme for this specific sales force can be identified. The concept is illustrated in an actual case study.     

Economic design of an inventory policy for non-instantaneous deteriorating items under permissible delay in payments

In this paper, Economic Order Quantity (EOQ) based model for non-instantaneous deteriorating items with permissible delay in payments is proposed. This model aids in minimizing the total inventory cost by finding an optimal replenishment policy. In this model shortages are allowed and partially backlogged. The backlogging rate is variable and dependent on the waiting time for the next replenishment. Some useful theorems have been framed to characterize the optimal solutions. The necessary and sufficient conditions of the existence and uniqueness of the optimal solutions are also provided. An algorithm is designed to find the optimal replenishment cycle time and order quantity under various circumstances. Numerical examples are given to demonstrate the theoretical results. Sensitivity analysis of the optimal solution with respect to major parameters of the system has been carried out and the implications are discussed in detail. In the discussions, suggestions are given to minimize the total cost of the inventory system.     

Optimal production run length for products sold with warranty

This article studies the optimal production run length for a deteriorating production system in which the products are sold with free minimal repair warranty. The deterioration process of the system is characterized by a two-state continuous-time Markov chain. For products sold with free minimal repair warranty, we show that there exists a unique optimal production run length such that the expected total cost per item is minimized. Since there is no closed form expression for the optimal production run length, an approximate solution is derived. In addition, three special cases which provide bounds for searching the optimal production run length are investigated and some sensitivity analysis is carried out to study the effects of the model parameters on the optimal production run length. Finally, a numerical example is given to evaluate the performance of the optimal production run length.     

Optimal control problem of the uncertain second‐order circuit based on first hitting criteria

First hitting criteria of a system are to initially achieve some performance indeces of the target state set. This paper primarily investigates the optimal control problem of the uncertain second‐order circuit based on first hitting criteria. First, considering time efficiency and different from the ordinary expected utility criterion over an infinite time horizon, two first hitting criteria which are reliability index and reliable time criteria are innovatively proposed. Second, assuming the circuit output voltage as an uncertain variable when the historical data is lacking, we better model the real circuit system with the uncertain second‐order differential equation which is essentially the uncertain fractional‐order differential equation. Then, based on the first hitting time theorem of the uncertain fractional‐order differential equation, the distribution function of the first hitting time under the second‐order circuit system is proposed and the uncertain second‐order circuit optimal control model (reliability index and reliable time‐based model) is transformed into corresponding crisp optimal problem. Lastly, analytic expressions of the optimal control for the reliability index model are obtained. Meanwhile, sufficient condition and guidance for parameters for the optimal solution of the reliable time‐based model are derived, and corresponding numerical examples are also given to demonstrate the fluctuation of our optimal solution for different parameters.     

基于时变需求的集成多级供应链生产订货策略研究

考虑一个时变需求环境下集成多级供应链问题,在有限的规划时间内销售商以固定周期订货,而生产商以不同的周期生产,目的是寻找销售商最优的订货周期和生产商最佳的生产策略,从而使供应链系统的总运营成本最少.建立了该问题的混合整数非线性规划模型,求解该模型分为两步：先求对应一个订货周期的最佳生产策略,再求最优的订货周期,第一步用到了图论里求最短路方法.给出了两个步骤的算法和程序,实验证明它们是有效的.通过算例对模型进行了分析,研究了各参数对最优解及最小费用的影响.     

Optimal lot size and inspection policy for products sold with warranty

This paper develops a mathematical model to jointly determine the optimal lot size and product inspection policy for a deteriorating production system, when products are sold with free minimal repair warranty. Due to system deterioration, a last-K product inspection scheme is proposed, under which the last K products in a production lot are inspected and nonconforming products found are reworked. Based on the model, we show that there exist a unique optimal lot size and a corresponding inspection policy such that the expected total cost per unit time is minimized. Since there is no closed-form expression for the optimal lot size, an upper bound and approximate solutions are obtained to facilitate the search process. Furthermore, an algorithm is provided to efficiently search for the optimal policy and the performance of the optimal policy is evaluated through numerical examples.     

An optimal birth control problem for a dynamical population model with size-structure

This work is concerned with an optimal control problem for a size-structured population model, which takes fertility as the control variable. The existence and uniqueness of solutions to the basic state system and the dual system are proven via the Banach fixed point theorem. Necessary optimality conditions of first order are established in the form of an Euler-Lagrange system by the use of tangent-normal cone technique. The existence of a unique optimal controller is established by means of Ekeland’s variational principle. An example and some comments are presented.