Design of memetic algorithms for the efficient optimization of chemical process synthesis problems with structural restrictions

In Urselmann et al., 2011a, Urselmann et al., 2011b we presented a memetic algorithm (MA) for the design optimization of reactive distillation columns. The MA is a combination of a problem-specific evolutionary algorithm (EA) that optimizes the design variables and a mathematical programming (MP) method that solves the continuous sub-problems with fixed discrete decisions which are proposed by the EA to local optimality. In comparison to the usual superstructure formulation, the search space of the MA is significantly reduced without excluding feasible solutions. The algorithm computes many different local optima and can handle structural restrictions and discontinuous cost functions. In this contribution, a systematic procedure to modify the MA to solve more complex design problems is described and demonstrated using the example of a reactive distillation column with an optional side- or pre-reactor with structural restrictions on the number of streams. New concepts to handle connected and optional unit operations are proposed.     

Simultaneous mixed-integer dynamic optimization for integrated design and control

We consider strategies for integrated design and control through the robust and efficient solution of a mixed-integer dynamic optimization (MIDO) problem. The algorithm is based on the transformation of the MIDO problem into a mixed-integer nonlinear programming (MINLP) program. In this approach, both the manipulated and controlled variables are discretized using a simultaneous dynamic optimization approach. We also develop three MINLP formulations based on a nonconvex formulation, the conventional Big-M formulation and generalized disjunctive programming (GDP). In addition, we compare the outer approximation and NLP branch and bound algorithms on these formulations. This problem is applied to a system of two series connected continuous stirred tank reactors where a first-order reaction takes place. Our results demonstrate that the simultaneous MIDO approach is able to efficiently address the solution of the integrated design and control problem in a systematic way.     

Solving mixed integer nonlinear programming problems with line-up competition algorithm

Line-up competition algorithm (LCA), a global optimization algorithm proposed recently, is applied to the solution of mixed integer nonlinear programming (MINLP) problems. Through using alternative schemes to handle integer variables, the algorithm reported previously for solving NLP problems can be extended expediently to the solution of MINLP problems. The performance of the LCA is tested with several non-convex MINLP problems published in the literature, including the optimal design of multi-product batch chemical processes and the location-allocation problem. Testing shows that the LCA algorithm is efficient and robust in the solution of MINLP problems.     

Global optimization for scheduling refinery crude oil operations

In this work we present an outer-approximation algorithm to obtain the global optimum of a nonconvex mixed-integer nonlinear programming (MINLP) model that is used to represent the scheduling of crude oil movement at the front-end of a petroleum refinery. The model relies on a continuous time representation making use of transfer events. The proposed algorithm focuses on effectively solving a mixed-integer linear programming (MILP) relaxation of the nonconvex MINLP to obtain a rigorous lower bound (LB) on the global optimum. Cutting planes derived by spatially decomposing the network are added to the MILP relaxation of the original nonconvex MINLP in order to reduce the solution time for the MILP relaxation. The solution of this relaxation is used as a heuristic to obtain a feasible solution to the MINLP which serves as an upper bound (UB). The lower and upper bounds are made to converge to within a specified tolerance in the proposed outer-approximation algorithm. On applying the proposed technique to test examples, significant savings are realized in the computational effort required to obtain provably global optimal solutions.     

A Mixed Coding Scheme of a Particle Swarm Optimization and a Hybrid Genetic Algorithm with Sequential Quadratic Programming for Mixed Integer Nonlinear Programming in Common Chemical Engineering Practice

In this paper, mixed integer nonlinear programming (MINLP) is optimized by PSO_GA–SQP, the mixed coding of a particle swarm optimization (PSO), and a hybrid genetic algorithm and sequential quadratic programming (GA–SQP). The population is separated into two groups: discrete and continuous variables. The discrete variables are optimized by the adapted PSO, while the continuous variables are optimized by the GA–SQP using the discrete variable information from the adapted PSO. Therefore, the population can be set to a smaller size than usual to obtain a global solution. The proposed PSO_GA–SQP algorithm is verified using various MINLP problems including the designing of retrofit heat exchanger networks. The fitness values of the tested problems are able to reach the global optimum.     

On the global and efficient solution of stochastic batch plant design problems

The presence of uncertainty in product demands of batch plant design formulations with fixed structure and continuous equipment sizes transforms them into large-scale nonconvex nonlinear programs. This paper describes recent developments towards the efficient solution of such mathematical models. Two global optimization algorthms, a specialized GOP algorithm and a reduced space branch and bound algorithm, are presented and applied to this class of batch plant design models. It is shown that, by taking advantage of the special structure of the resulting mathematical formulations, encouraging computational results can be obtained from both algorithms for problem sizes that would otherwise be practically unsolvable with conventional global optimization techniques. An efficient, specialized Gaussian quadrature technique is also described for the case of product demands following normal probability distribution functions with which reduced model size and improved estimation of the expected profit integral are achieved. These developments are tested on example problems from the literature covering single batch plant configuration with various scheduling policies and flexible configurations with alternative production sequences.     

Optimal storage design for a multi-product plant: A non-convex MINLP formulation

We discuss a tank design problem for a multi product plant, in which the optimal cycle time and the optimal campaign size are unknown. A mixed-integer nonlinear programming (MINLP) formulation is presented, where non-convexities are due to the tank investment cost, storage cost, campaign setup cost and variable production rates. The objective of the optimization model is to minimize the sum of the production cost per ton per product produced. A continuous-time mathematical programming formulation is proposed and several extensions are discussed. The model is implemented in GAMS and computational results are reported for the two global MINLP solver BARON and LINDOGlobal as well as several nonlinear solvers available in GAMS.     

Optimal design of reverse-osmosis networks for wastewater treatment

An optimization study of reverse-osmosis networks (RON) for wastewater treatment has been carried out by describing the system as a nonconvex mixed-integer nonlinear problem (MINLP). A mixed-integer linear problem (MILP) is derived from the original nonlinear problem by the convex relaxation of the nonconvex terms in the MINLP to provide bounds for the global optimum. The MILP model is solved iteratively to supply different initial guesses for the nonconvex MINLP model. It is found that such a procedure is effective in finding local optimum solutions in reasonable time and overcoming possible convergence difficulties associated with MINLP local search methods. Examples of water desalination and wastewater treatment from the pulp and paper industry are considered as case studies to illustrate the proposed solution strategy.     

An optimization‐based approach for structural design of self‐assembled DNA tiles

DNA tiles are self‐assembled nanostructures, which offer exciting opportunities for synthesis of novel materials. A challenge for structural design of DNA tiles is to identify optimal locations for so‐called crossovers, which are bridges between DNA double helices formed by pairs of single‐stranded DNA. An optimization‐based approach is presented to identify optimal locations for such crossovers. Minimization of a potential‐energy model for a given structural design demonstrates the importance of local minima. Both deterministic global optimization of a reduced model and multistart optimization of the full model are applied successfully to identify the global minimum. MINLP optimization using a branch‐and‐bound algorithm (GAMS/SBB) identifies an optimal structural design of a DNA tile successfully with significant reduction in computational load compared to exhaustive enumeration, which demonstrates the potential of the proposed method to reduce trial‐and‐error efforts for structural design of DNA tiles. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1804–1817, 2017     

多目的间歇化工过程最优设计——SA/LP算法

提出了具有多条生产路线的多目的间歇化工过程最优设计混合整数非线性规划(MINLP)模型,该模型允许同时设立同步、异步平行单元以及中间储罐,并允许设备尺寸离散变化.在结合模拟退火(SA)和线性规划(LP)的基础上提出了可求解上述MINLP问题的SA/LP算法,该算法结合了SA全局收敛性好和LP可处理连续变量与约束方程的优点.计算表明,上述模型与算法实施简便,得到了文献算例中未得到的全局最优解,且在计算速度、内存占用上都远远优于文献中的方法.     

基于全局正交配置的非线性预测控制算法

针对非线性预测控制(NMPC)在线优化计算量大这一关键问题,提出一种基于全局正交配置的非线性预测控制算法。该算法以高阶插值正交多项式为基函数同时配置优化时域内的状态变量和控制变量,将连续动态优化问题转化为非线性规划问题(NLP)求解。全局正交配置可以使用较少的配置点而获得较高的逼近精度,这样即使NMPC使用很长的优化时域,离散化后得到的NLP问题的规模也比较小,能够有效地降低在线优化计算量。最后,以连续聚合反应过程为例验证了算法的有效性。     

Energy optimization of water supply system scheduling: Novel MINLP model and efficient global optimization algorithm

This article is concerned with global optimization of water supply system scheduling with pump operations to minimize total energy cost. The scheduling problem is first formulated as a non‐convex mixed‐integer nonlinear programming (MINLP) problem, accounting for flow rates in pipes, operation profiles of pumps, water levels of tanks, and customer demand. Binary variables denote on–off switch operations for pumps and flow directions in pipes, and nonlinear terms originate from characteristic functions for pumps and hydraulic functions for pipes. The proposed MINLP model is verified with EPANET, which is a leading software package for water distribution system modeling. We further develop a novel global optimization algorithm for solving the non‐convex MINLP problem. To demonstrate the applicability of the proposed model and the efficiency of the tailored global optimization algorithm, we present results of two case studies with up to 4 tanks, 5 pumps, 5 check valves, and 21 pipes. © 2016 American Institute of Chemical Engineers AIChE J, 62: 4277–4296, 2016     

Global optimization of large‐scale mixed‐integer linear fractional programming problems: A reformulation‐linearization method and process scheduling applications

Mixed‐integer linear fractional program (MILFP) is a class of mixed‐integer nonlinear programs (MINLP) where the objective function is the ratio of two linear functions and all constraints are linear. Global optimization of large‐scale MILFPs can be computationally intractable due to the presence of discrete variables and the pseudoconvex/pseudoconcave objective function. We propose a novel and efficient reformulation–linearization method, which integrates Charnes–Cooper transformation and Glover's linearization scheme, to transform general MILFPs into their equivalent mixed‐integer linear programs (MILP), allowing MILFPs to be globally optimized effectively with MILP methods. Extensive computational studies are performed to demonstrate the efficiency of this method. To illustrate its applications, we consider two batch scheduling problems, which are modeled as MILFPs based on the continuous‐time formulations. Computational results show that the proposed approach requires significantly shorter CPU times than various general‐purpose MINLP methods and shows similar performance than the tailored parametric algorithm for solving large‐scale MILFP problems. Specifically, it performs with respect to the CPU time roughly a half of the parametric algorithm for the scheduling applications. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4255–4272, 2013     

Scheduling of crude oil operations under demand uncertainty: A robust optimization framework coupled with global optimization

Scheduling of crude oil operations is an important component of overall refinery operations, because crude oil costs account for about 80% of the refinery turnover. The mathematical modeling of blending different crudes in storage tanks results in many bilinear terms, which transform the problem into a challenging, nonconvex, mixed‐integer nonlinear programming (MINLP) optimization model. In practice, uncertainties are unavoidable and include demand fluctuations, ship arrival delays, equipment malfunction, and tank unavailability. In the presence of these uncertainties, an optimal schedule generated using nominal parameter values may often be suboptimal or even become infeasible. In this article, the robust optimization framework proposed by Lin et al. and Janak et al. is extended to develop a deterministic robust counterpart optimization model for demand uncertainty. The recently proposed branch and bound global optimization algorithm with piecewise‐linear underestimation of bilinear terms by Li et al. is also extended to solve the nonconvex MINLP deterministic robust counterpart optimization model and generate robust schedules. Two examples are used to illustrate the capability of the proposed robust optimization approach, and the extended branch and bound global optimization algorithm for demand uncertainty. The computational results demonstrate that the obtained schedules are robust in the presence of demand uncertainty. © 2011 American Institute of Chemical Engineers AIChE J, 58: 2373–2396, 2012     

多产品间歇过程的分层次在线调度--一种数学规划与遗传算法的混合算法

In this contribution we present an online scheduling algorithm for a real world multiproduct batch plant. The overall mixed integer nonlinear programming (MINLP) problem is hierarchically structured into a mixed integer linear programming (MILP) problem first and then a reduced dimensional MINLP problem, which are optimized by mathematical programming (MP) and genetic algorithm (GA) respectively. The basis idea relies on combining MP with GA to exploit their complementary capacity. The key features of the hierarchical model are explained and illustrated with some real world cases from the multiproduct batch plants.     

Operational strategy and planning for raw natural gas refining complexes: Process modeling and global optimization

Optimal operational strategy and planning of a raw natural gas refining complex (RNGRC) is very challenging since it involves highly nonlinear processes, complex thermodynamics, blending, and utility systems. In this article, we first propose a superstructure integrating a utility system for the RNGRC, involving multiple gas feedstocks, and different product specifications. Then, we develop a large‐scale nonconvex mixed‐integer nonlinear programming (MINLP) optimization model. The model incorporates rigorous process models for input and output relations based on fundamentals of thermodynamics and unit operations and accurate models for utility systems. To reduce the noncovex items in the proposed MINLP model, equivalent reformulation techniques are introduced. Finally, the reformulated nonconvex MINLP model is solved to global optimality using state of the art deterministic global optimization approaches. The computational results demonstrate that a significant profit increase is achieved using the proposed approach compared to that from the real operation. © 2016 American Institute of Chemical Engineers AIChE J, 63: 652–668, 2017     

A novel adaptive surrogate modeling‐based algorithm for simultaneous optimization of sequential batch process scheduling and dynamic operations

A novel adaptive surrogate modeling‐based algorithm is proposed to solve the integrated scheduling and dynamic optimization problem for sequential batch processes. The integrated optimization problem is formulated as a large scale mixed‐integer nonlinear programming (MINLP) problem. To overcome the computational challenge of solving the integrated MINLP problem, an efficient solution algorithm based on the bilevel structure of the integrated problem is proposed. Because processing times and costs of each batch are the only linking variables between the scheduling and dynamic optimization problems, surrogate models based on piece‐wise linear functions are built for the dynamic optimization problems of each batch. These surrogate models are then updated adaptively, either by adding a new sampling point based on the solution of the previous iteration, or by doubling the upper bound of total processing time for the current surrogate model. The performance of the proposed method is demonstrated through the optimization of a multiproduct sequential batch process with seven units and up to five tasks. The results show that the proposed algorithm leads to a 31% higher profit than the sequential method. The proposed method also outperforms the full space simultaneous method by reducing the computational time by more than four orders of magnitude and returning a 9.59% higher profit. © 2015 American Institute of Chemical Engineers AIChE J, 61: 4191–4209, 2015     

Global optimization of multiphase flow networks using spline surrogate models

A general modelling framework for optimization of multiphase flow networks with discrete decision variables is presented. The framework is expressed with the graph and special attention is given to the convexity properties of the mathematical programming formulation that follows. Nonlinear pressure and temperature relations are modelled using multivariate splines, resulting in a mixed-integer nonlinear programming (MINLP) formulation with spline constraints. A global solution method is devised by combining the framework with a spline-compatible MINLP solver, recently presented in the literature. The solver is able to globally solve the nonconvex optimization problems. The new solution method is benchmarked with several local optimization methods on a set of three realistic subsea production optimization cases provided by the oil company BP.     

Fair profit allocation in supply chain optimization with transfer price and revenue sharing: MINLP model and algorithm for cellulosic biofuel supply chains

A mixed‐integer nonlinear programming (MINLP) formulation to simultaneously optimize operational decisions as well as profit allocation mechanisms in supply chain optimization, namely material transfer prices and revenue share policies among the supply chain participants is proposed. The case of cellulosic bioethanol supply chains is specifically considered and the game‐theory Nash bargaining solution approach is employed to achieve fair allocation of profit among the collection facilities, biorefineries, and distribution centers. The structural advantages of certain supply chain participants can be taken into account by specifying different values of the negotiation‐power indicators in the generalized Nash‐type objective function. A solution strategy based on a logarithm transformation and a branch‐and‐refine algorithm for efficient global optimization of the resulting nonconvex MINLP problem is proposed. To demonstrate the application of the proposed framework, an illustrative example and a state‐wide county‐level case study on the optimization of a potential cellulosic bioethanol supply chain in Illinois are presented. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3211–3229, 2014     

A new superstructure optimization paradigm for process synthesis with product distribution optimization: Application to an integrated shale gas processing and chemical manufacturing process

We propose a novel process synthesis framework that combines product distribution optimization of chemical reactions and superstructure optimization of the process flowsheet. A superstructure with a set of technology/process alternatives is first developed. Next, the product distributions of the involved chemical reactions are optimized to maximize the profits of the effluent products. Extensive process simulations are then performed to collect high‐fidelity process data tailored to the optimal product distributions. Based on the simulation results, a superstructure optimization model is formulated as a mixed‐integer nonlinear program (MINLP) to determine the optimal process design. A tailored global optimization algorithm is used to efficiently solve the large‐scale nonconvex MINLP problem. The resulting optimal process design is further validated by a whole‐process simulation. The proposed framework is applied to a comprehensive superstructure of an integrated shale gas processing and chemical manufacturing process, which involves steam cracking of ethane, propane, n‐butane, and i‐butane. © 2017 American Institute of Chemical Engineers AIChE J, 63: 123–143, 2018