The use of artificial intelligence technique for the optimisation of process parameters used in the continuous casting of steel

The productivity and quality of a continuous caster depend mainly on process parameters, i.e. casting speed, casting temperature, steel composition and cleanliness of the melt, water flow rates in the different cooling zones, etc. This work presents the development of an algorithm, which incorporates heuristic search techniques for direct application in metallurgical industries, particularly those using continuous casting process for the production of steel billets and slabs. This is done to determine the casting objectives of maximum casting rate as a function of casting constraints. These constraints are evaluated with the aid of a heat transfer and solidification model based on the finite difference technique, which has been developed and integrated with a genetic algorithm. The essential parts of continuous casting equipment, which must be subjected to monitoring, as well as a methodology of mathematical model and physical settlements in each cooling region, are presented. The efficiency of the intelligent system is assured by the optimisation of the continuous casting operation by maximum casting rate and defect-free products. This approach is applied to the real dimension of a steel continuous caster, in real conditions of operation, demonstrating that good results can be attained by using heuristic search, such as: smaller temperature gradients between sprays zones, reduction in water consumption and an increase in casting speed.     

直接配置法求解连续铸钢过程中的最优控制问题

利用描述连续铸钢过程二冷区喷水控制下钢的热传导的半离散化模型 ,我们构造一包含温度梯度约束的最优控制问题 .针对此最优控制问题 ,采用直接配置法进行数值求解 ,得出相应的近似最优控制 .     

Numerical solution of minimax optimal control problems by multiple shooting technique

This paper presents the application of the multiple shooting technique to minimax optimal control problems (optimal control problems with Chebyshev performance index). A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique, the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary-value problem with switching conditions, which can be treated numerically by a special version of the multiple shooting algorithm. The method is tested on the problem of the optimal heating and cooling of a house. This application shows some typical difficulties arising with minimax optimal control problems, i.e., the estimation of the switching structure which is dependent on the parameters of the problem. This difficulty can be overcome by a careful application of a continuity method. Numerical solutions for the example are presented which demonstrate the efficiency of the method proposed.     

Constrained group balancing: Why does it work

We consider a problem where a set of objects possessing multiple attributes must be partitioned into a certain number of groups so that the groups are as balanced as possible with respect to the number of objects possessing each attribute. This multi-criteria decision problem arises in a variety of practical applications, ranging from assigning students to study groups to designing level schedules for JIT assembly lines. A direct approach, enforcing balance through hard constraints, may lead to infeasibility, but works well in practice. We analyze this phenomenon from the worst-case and empirical perspectives, as well as through an in-depth analysis of one representative practical application – the design of student groups at the Rotman School of Management, University of Toronto. The goals of the analysis are to understand what classes of balancing problems may contain infeasible instances and how prevalent such instances are within these classes, as well as to synthesize practical managerial insights that a decision maker could follow in order to increase the chances that balanced groups can be found.     

D-optimal design of a monitoring network for parameter estimation of distributed systems

This paper addresses the design of a network of observation locations in a spatial domain that will be used to estimate unknown parameters of a distributed parameter system. We consider a setting where we are given a finite number of possible sites at which to locate a sensor, but cost constraints allow only some proper subset of them to be selected. We formulate this problem as the selection of the gauged sites so as to maximize the log-determinant of the Fisher information matrix associated with the estimated parameters. The search for the optimal solution is performed using the branch-and-bound method in which an extremely simple and efficient technique is employed to produce an upper bound to the maximum objective function. Its idea consists in solving a relaxed problem through the application of a simplicial decomposition algorithm in which the restricted master problem is solved using a multiplicative algorithm for optimal design. The use of the proposed approach is illustrated by a numerical example involving sensor selection for a two-dimensional convective diffusion process.     

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.     

Optimal Control of Continuous Casting by Nondifferentiable Multiobjective Optimization

A new version of an interactive NIMBUS method for nondifferentiable multiobjective optimization is described. It is based on the reference point idea and the classification of the objective functions. The original problem is transformed into a single objective form according to the classification information. NIMBUS has been designed especially to be able to handle complicated real-life problems in a user-friendly way.The NIMBUS method is used for solving an optimal control problem related to the continuous casting of steel. The main goal is to minimize the defects in the final product. Conflicting objective functions are constructed according to certain metallurgical criteria and some technological constraints. Due to the phase changes during the cooling process there exist discontinuities in the derivative of the temperature distribution. Thus, the problem is nondifferentiable.Like many real-life problems, the casting model is large and complicated and numerically demanding. NIMBUS provides an efficient way of handling the difficulties and, at the same time, aids the user in finding a satisficing solution. In the end, some numerical experiments are reported and compared with earlier results.     

A Practical Solution to a Cargo Loading Problem at Bethlehem Steel

The optimum assignment of structural steel shapes to rail cars is an important logistical problem in the steel industry. In this paper, we discuss an application at Bethlehem Steel that not only involves weight and dimensional constraints, but also customer unloading constraints. The formulation is a generalized bin packing problem which is solved by modifying and extending the first fit decreasing algorithm. The solution algorithm, SOLID (for Structural Optimal Loading IDentification), has been used extensively for one of Bethlehem's high tonnage customers providing very good practical (implementable) results that achieve the desired goals. Bethlehem has enhanced this approach for use with other customers.     

The coastal seaspace patrol sector design and allocation problem

In this paper, we model and solve the problem of designing and allocating coastal seaspace sectors for steady-state patrolling operations by the vessels of a maritime protection agency. The problem addressed involves optimizing a multi-criteria objective function that minimizes a weighted combination of proportional measures of the vessels’ distances between home ports and patrol sectors, the sector workload, and the sector span. We initially assure contiguity of each patrol sector in our mixed-integer programming formulation via an exponential number of subtour elimination constraints, and then propose three alternative solution methods, two of which are based on reformulations that suitably replace the original contiguity representation with a polynomial number of constraints, and a third approach that employs an iterative cut generation procedure based on identifying violated subtour elimination constraints. We further enhance these reformulations with symmetry defeating constraints, either in isolation or in combination with a suitable perturbation of the objective function using weighted functions based on such constraints. Computational comparisons are provided for solving the problem using the original formulation versus either of our three alternative solution approaches for a representative instance. Overall, a model formulation based on Steiner tree problem (STP) constructs and enhanced by the reformulation-linearization technique (RLT) yielded the best performance.     

Multiple crane scheduling in a batch annealing process with no-delay constraints for machine unloading

In this work, we focus on the scheduling of multi-crane operations in an iron and steel enterprise for a two-stage batch annealing process. The first stage is the heating process, and the second stage is the cooling process. To start the heating (cooling) stage, a special machine called a furnace (cooler) must be loaded. The real constraints of no-delay machine unloading are defined as follows: once the heating (cooling) is completed, the furnace (cooler) must be unloaded by crane immediately. The goal is to schedule limited machines (furnaces and coolers) operated by multiple cranes to minimize the completion time of the last annealed coil (makespan). We formulate a mixed-integer linear programming model to address this problem. Certain feasible properties are identified to avoid crane conflicts and ensure that the machine unloading no-delay constraints are met. Based on these necessary conditions, we then present a heuristic algorithm with running time in connection with the number of cranes, coils and machines. A lower bound to the problem is also developed. Through theoretical analysis, we show the worst-case bound of our heuristic algorithm. The average performances of the solution approaches are computationally evaluated. The computational results show that the proposed heuristic algorithm is capable of generating good quality solutions.     

A penalty function-based random search algorithm for optimal control of switched systems with stochastic constraints and its application in automobile test-driving with gear shifts

Practical industrial process is usually a dynamic process including uncertainty. Stochastic constraints can be used for industrial process modeling, when system sate and/or control input constraints cannot be strictly satisfied. Thus, optimal control of switched systems with stochastic constraints can be available to address practical industrial process problems with different modes. In general, obtaining an analytical solution of the optimal control problem is usually very difficult due to the discrete nature of the switching law and the complexity of stochastic constraints. To obtain a numerical solution, this problem is formulated as a constrained nonlinear parameter selection problem (CNPSP) based on a relaxation transformation (RT) technique, an adaptive sample approximation (ASA) method, a smooth approximation (SA) technique, and a control parameterization (CP) method. Following that, a penalty function-based random search (PFRS) algorithm is designed for solving the CNPSP based on a novel search rule-based penalty function (NSRPF) method and a novel random search (NRS) algorithm. The convergence results show that the proposed method is globally convergent. Finally, an optimal control problem in automobile test-driving with gear shifts (ATGS) is further extended to illustrate the effectiveness of the proposed method by taking into account some stochastic constraints. Numerical results show that compared with other typical methods, the proposed method is less conservative and can obtain a stable and robust performance when considering the small perturbations in initial system state. In addition, to balance the computation amount and the numerical solution accuracy, a tolerance setting method is also provided by the numerical analysis technique.     

A Multi-criteria Decision Support System for R&D Project Selection

This paper describes the development of a decision support system for the selection of a portfolio of R&D projects, which was carried out for a large electricity utility corporation. The DSS was constructed round a reference point approach for the underlying multi-criteria decision problem. The application of this approach did require a less usual form of scalarizing function as well as a heuristic algorithm for solving a non-linear knapsack problem. Practical aspects of the implementation of the multi-criteria approach in a DSS operating on a PC are also discussed.     

Numerical solution to the optimal feedback control of continuous casting process

Using a semi-discrete model that describes the heat transfer of a continuous casting process of steel, this paper is addressed to an optimal control problem of the continuous casting process in the secondary cooling zone with water spray control. The approach is based on the Hamilton–Jacobi–Bellman equation satisfied by the value function. It is shown that the value function is the viscosity solution of the Hamilton–Jacobi–Bellman equation. The optimal feedback control is found numerically by solving the associated Hamilton–Jacobi–Bellman equation through a designed finite difference scheme. The validity of the optimality of the obtained control is experimented numerically through comparisons with different admissible controls. Detailed study of a low-carbon billet caster is presented.     

Solving a bi-objective nurse rerostering problem by using a utopic Pareto genetic heuristic

Nurse rerostering arises when at least one nurse announces that she will be unable to undertake the tasks previously assigned to her. The problem amounts to building a new roster that satisfies the hard constraints already met by the current one and, as much as possible, fulfils two groups of soft constraints which define the two objectives to be attained. A bi-objective genetic heuristic was designed on the basis of a population of individuals characterised by pairs of chromosomes, whose fitness complies with the Pareto ranking of the respective decoded solution. It includes an elitist policy, as well as a new utopic strategy, introduced for purposes of diversification. The computational experiments produced promising results for the practical application of this approach to real life instances arising from a public hospital in Lisbon.     

A transformation technique for optimal control problems with partially linear state inequality constraints

This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector.A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector.In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-76-3075, and by the National Science Foundation, Grant No. MCS-76-21657.     

Maximum principle for optimal control problems with intermediate constraints

We consider optimal control problems with constraints at intermediate points of the trajectory. A natural technique (propagation of phase and control variables) is applied to reduce these problems to a standard optimal control problem of Pontryagin type with equality and inequality constraints at the trajectory endpoints. In this way we derive necessary optimality conditions that generalize the Pontryagin classical maximum principle. The same technique is applied to so-called variable structure problems and to some hybrid problems. The new optimality conditions are compared with the results of other authors and five examples illustrating their application are presented.     

A new approach to multi-criteria sorting based on fuzzy outranking relations: The THESEUS method

In this paper, we propose the THESEUS method, a new approach based on fuzzy outranking relations to multi-criteria sorting problems. Compared with other outranking-based methods, THESEUS is inspired by another view of multi-criteria classification problems. It utilizes a new way of evaluating the assignment of an object to an element of the set of ordered categories that were previously defined. This way is based on comparing every possible assignment with the information from various preference relations that are derived from a fuzzy outranking relation defined on the universe of objects. The appropriate assignment is determined by solving a simple selection problem.The capacity of a reference set for making appropriate assignments is related to a good characterization of the categories. A single reference action characterizing a category may be insufficient to achieve well-determined assignments. In this paper, the reference set capacity to perform appropriate assignments is characterized by some new concepts. This capacity may be increased when more objects are added to the reference set. THESEUS is a method for handling the preference information contained in such larger reference sets.     

A mixed integer programming model for scheduling orders in a steel mill

The problem of scheduling orders at each facility of a large integrated steel mill is considered. Orders are received randomly, and delivery dates are established immediately. Each order is filled by converting raw materials into a finished saleable steel product by a fixed sequence of processes. The application of a deterministic mixed integer linear programming model to the order scheduling problem is given. One important criterion permitted by the model is to process the orders in a sequence which minimizes the total tardiness from promised delivery for all orders; alternative criteria are also possible. Most practical constraints which arise in steelmaking can be considered within the formulation. In particular, sequencing and resource availability constraints are handled easily. The order scheduling model given here contains many variables and constraints, resulting in computational difficulties. A decomposition algorithm is devised for solving the model. The algorithm is a special case of Benders partitioning. Computational experience is reported for a large-scale problem involving scheduling 102 orders through ten facilities over a six-week period. The exact solution to the large-scale problem is compared with schedules determined by several heuristic dispatching rules. The dispatching rules took into consideration such things as due date, processing time, and tardiness penalties. None of the dispatching rules found the optimal solution.     

Minimum-Time Control of a Crane with Simultaneous Traverse and Hoisting Motions

The problem of the minimum-time control of a crane having simultaneous traverse and hoisting motions is considered. We propose an approach that converts this problem into a finite-dimensional optimization problem via control parametrization with an appropriate basis function. Such an approach simplifies the treatment of the constraints and allows for the easy satisfaction of the endpoint constraints. This optimization problem is solved using a novel two-stage optimization process. Under additional conditions, the solution obtained from this process can be shown to be the optimum. When these conditions are not met, a near-optimal solution is obtained. Several numerical examples are provided, including the case where there is unequal cable length at the endpoints. The validity of the solution is verified experimentally on a test rig.     

Large scale stochastic inventory routing problems with?split delivery and service level constraints

A stochastic inventory routing problem (SIRP) is typically the combination of stochastic inventory control problems and NP-hard vehicle routing problems, which determines delivery volumes to the customers that the depot serves in each period, and vehicle routes to deliver the volumes. This paper aims to solve a large scale multi-period SIRP with split delivery (SIRPSD) where a customer??s delivery in each period can be split and satisfied by multiple vehicle routes if necessary. This paper considers SIRPSD under the multi-criteria of the total inventory and transportation costs, and the service levels of customers. The total inventory and transportation cost is considered as the objective of the problem to minimize, while the service levels of the warehouses and the customers are satisfied by some imposed constraints and can be adjusted according to practical requests. In order to tackle the SIRPSD with notorious computational complexity, we first propose an approximate model, which significantly reduces the number of decision variables compared to its corresponding exact model. We then develop a hybrid approach that combines the linearization of nonlinear constraints, the decomposition of the model into sub-models with Lagrangian relaxation, and a partial linearization approach for a sub model. A near optimal solution of the model found by the approach is used to construct a near optimal solution of the SIRPSD. Randomly generated instances of the problem with up to 200 customers and 5 periods and about 400 thousands decision variables where half of them are integer are examined by numerical experiments. Our approach can obtain high quality near optimal solutions within a reasonable amount of computation time on an ordinary PC.