Proactive scheduling of batch processes by a combined robust optimization and multiparametric programming approach

We address short‐term batch process scheduling problems contaminated with uncertainty in the data. The mixed integer linear programming (MILP) scheduling model, based on the formulation of Ierapetritou and Floudas, Ind Eng Chem Res. 1998; 37(11):4341–4359, contains parameter dependencies at multiple locations, yielding a general multiparametric (mp) MILP problem. A proactive scheduling policy is obtained by solving the partially robust counterpart formulation. The counterpart model may remain a multiparametric problem, yet it is immunized against uncertainty in the entries of the constraint matrix and against all parameters whose values are not available at the time of decision making. We extend our previous work on the approximate solution of mp‐MILP problems by embedding different uncertainty sets (box, ellipsoidal and budget parameter regulated uncertainty), and by incorporating information about the availability of uncertain data in the construction of the partially robust scheduling model. For any parameter realization, the corresponding schedule is then obtained through function evaluation. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4184–4211, 2013     

A multiparametric programming approach for mixed-integer quadratic engineering problems

In this work we propose algorithms for the solution of multiparametric quadratic programming (mp-QP) problems and multiparametric mixed-integer quadratic programming (mp-MIQP) problems with a convex and quadratic objective function and linear constraints. For mp-QP problems it is shown that the optimal solution, i.e. the vector of continuous variables and Lagrange multipliers, is an affine function of parameters. The basic idea of the algorithm is to use this affine expression for the optimal solution to systematically characterize the space of parameters by a set of regions of optimality. The solution of the mp-MIQP problems is approached by decomposing it into two subproblems, which converge based upon an iterative methodology. The first subproblem, which is an mp-QP, is obtained by fixing the integer variables and its solution represents a parametric upper bound. The second subproblem is formulated as a mixed-integer non-linear programming (MINLP) problem and its solution provides a new integer vector, which can be fixed to obtain a parametric solution, which is better than the current upper bound. The algorithm terminates with an envelope of parametric profiles corresponding to different optimal integer solutions. Examples are presented to illustrate the basic ideas of the algorithms and their application in model predictive and hybrid control problems.     

New MINLP formulation for the multiperiod pooling problem

The modeling of blending tank operations in petroleum refineries for the most profitable production of liquid fuels in a context of time‐varying supply and demand is addressed. A new mixed‐integer nonlinear programming formulation is proposed that using individual flows and split fractions as key model variables leads to a different set of nonconvex bilinear terms compared with the original work of Kolodziej et al. These are better handled by decomposition algorithms that divide the problem into integer and nonlinear components as well as by commercial solvers. In fact, BARON and GloMIQO can solve to global optimality all problems resulting from the new formulation and test problems from the literature. A tailored global optimization algorithm working with a tight mixed‐integer linear relaxation from multiparametric disaggregation achieves a similar performance. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3728–3738, 2015     

A new model and a reformulation for the crude distillation unit charging problem with oil blends and sequence-dependent changeover costs

In this paper, we address the problem of planning the crude distillation unit charging process with oil blend. It is well known that blending and splitting operations can lead together to both non-linearities and concavities in mathematical programming models. As result, many proposed models for this problem use simplifying assumptions to keep the formulation computationally tractable. However, we show the existence of splitting operations that can lead to inconsistencies in the solutions obtained by the previous MILP models from the literature. Then, we propose a way to address this issue through an aggregated inventory capacity combined with a disaggregation algorithm. Furthermore, we develop a mathematical reformulation that improves the solving efficiency of the method. Then, we report experiments that show that the reformulated MILP model presents significant gains concerning linear relaxation gaps and run times, and the disaggregation algorithm leads to feasible solutions for all the tested instances.     

Multi-bucket optimization for integrated planning and scheduling in the perishable dairy supply chain

This paper considers a dairy industry problem on integrated planning and scheduling of set yoghurt production. A mixed integer linear programming formulation is introduced to integrate tactical and operational decisions and a heuristic approach is proposed to decompose time buckets of the decisions. The decomposition heuristic improves computational efficiency by solving big bucket planning and small bucket scheduling problems. Further, mixed integer linear programming and constraint programming methodologies are combined with the algorithm to show their complementary strengths. Numerical studies using illustrative data with high demand granularity (i.e., a large number of small-sized customer orders) demonstrate that the proposed decomposition heuristic has consistent results minimizing the total cost (i.e., on average 8.75% gap with the best lower bound value found by MILP) and, the developed hybrid approach is capable of solving real sized instances within a reasonable amount of time (i.e., on average 92% faster than MILP in CPU time).     

多目标进化算法求解过程综合中的混合整数线性规划(MILP)与混合整数非线性规划问题(MINLP)

Steady-state non-dominated sorting genetic algorithm (SNSGA), a new form of multi-objective genetic algorithm, is implemented by combining the steady-state idea in steady-state genetic algorithms (SSGA) and the fitness assignment strategy of non-dominated sorting genetic algorithm (NSGA). The fitness assignment strategy is improved and a new self-adjustment scheme of crame is proposed. This algorithm is proved to be very efficient both computationally and in terms of the quality of the Pareto fronts produced with five test problems including GA difficult problem and GA deceptive one. Finally, SNSGA is introduced to solve multi-objective mixed integer linear programming (MILP) and mixed integer non-linear programming (MINLP) problems in process synthesis.     

An efficient reformulation of the multiechelon stochastic inventory system with uncertain demands

It is shown that by reformulating the three‐stage multiechelon inventory system with specific exact linearizations, larger problems can be solved directly with mixed‐integer linear programming (MILP) without decomposition. The new formulation is significantly smaller in the number of continuous variables and constraints. An MILP underestimation of the problem can be solved as part of a sequential piecewise approximation scheme to solve the problem within a desired optimality gap.     

Design,synthesis and scheduling of multipurpose batch plants via an effective continuous-time formulation

Design, synthesis and scheduling issues are considered simultaneously for multipurpose batch plants. An earlier proposed continuous-time formulation for scheduling is extended to incorporate design and synthesis. Processing recipes are represented by the State-Task Network (STN). The superstructure of all possible plant designs is constructed according to the potential availability of all processing/storage units. The proposed model takes into account the trade-offs between capital costs, revenues and operational flexibility. Computational studies are presented to illustrate the effectiveness of the proposed formulation. Both linear and nonlinear models are included, resulting in MILP and mixed-integer nonlinear programming (MINLP) problems, respectively. The MILP problems are solved using a branch and bound method. Globally optimal solutions are obtained for the nonconvex MINLP problems based on a key property that arises due to the special structure of the resulting models. Comparisons with earlier approaches are also presented.     

Generic mathematical programming formulation and solution for computer-aided molecular design

This short communication presents a generic mathematical programming formulation for computer-aided molecular design (CAMD). A given CAMD problem, based on target properties, is formulated as a mixed integer linear/non-linear program (MILP/MINLP). The mathematical programming model presented here, which is formulated as an MILP/MINLP problem, considers first-order and second-order molecular groups for molecular structure representation and property estimation. It is shown that various CAMD problems can be formulated and solved through this model.     

Temporal and spatial Lagrangean decompositions in multi-site, multi-period production planning problems with sequence-dependent changeovers

We address in this paper the optimization of a multi-site, multi-period, and multi-product planning problem with sequence-dependent changeovers, which is modeled as a mixed-integer linear programming (MILP) problem. Industrial instances of this problem require the planning of a number of production and distribution sites over a time span of several months. Temporal and spatial Lagrangean decomposition schemes can be useful for solving these types of large-scale production planning problems. In this paper we present a theoretical result on the relative size of the duality gap of the two decomposition alternatives. We also propose a methodology for exploiting the economic interpretation of the Lagrange multipliers to speed the convergence of numerical algorithms for solving the temporal and spatial Lagrangean duals. The proposed methods are applied to the multi-site multi-period planning problem in order to illustrate their computational effectiveness.     

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.     

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.     

Multi-week MILP scheduling for an ice cream processing facility

This paper presents a multi-week mixed integer linear programming (MILP) scheduling model for an ice cream processing facility. The ice cream processing is a typical complex food manufacturing process and a simplified version of this processing has been adapted to investigate scheduling problems in the literature. Most of these models only considered the production scheduling for a week. In this paper, multi-week production scheduling is considered. The problem has been implemented as an MILP model. The model has been tested on a set of cases from the literature, and its results were compared to the results of problems solved using hybrid MILP-heuristics methods in the literature. The inclusion of clean-up session, weekend break and semi-processed product from previous week were also assessed with two additional sets of experiments. The experiments result show that the proposed MILP is able to handle multi-week scheduling efficiently and effectively within a reasonable time limit.     

A quadratic approximation‐based algorithm for the solution of multiparametric mixed‐integer nonlinear programming problems

An algorithm for the solution of convex multiparametric mixed‐integer nonlinear programming problems arising in process engineering problems under uncertainty is introduced. The proposed algorithm iterates between a multiparametric nonlinear programming subproblem and a mixed‐integer nonlinear programming subproblem to provide a series of parametric upper and lower bounds. The primal subproblem is formulated by fixing the integer variables and solved through a series of multiparametric quadratic programming (mp‐QP) problems based on quadratic approximations of the objective function, while the deterministic master subproblem is formulated so as to provide feasible integer solutions for the next primal subproblem. To reduce the computational effort when infeasibilities are encountered at the vertices of the critical regions (CRs) generated by the primal subproblem, a simplicial approximation approach is used to obtain CRs that are feasible at each of their vertices. The algorithm terminates when there does not exist an integer solution that is better than the one previously used by the primal problem. Through a series of examples, the proposed algorithm is compared with a multiparametric mixed‐integer outer approximation (mp‐MIOA) algorithm to demonstrate its computational advantages. © 2012 American Institute of Chemical Engineers AIChE J, 59: 483–495, 2013     

Source-based discrete and continuous-time formulations for the crude oil pooling problem

The optimization of crude oil operations in refineries is a challenging scheduling problem due to the need to model tanks of varying composition with nonconvex bilinear terms, and complicating logistic constraints. Following recent work for multiperiod pooling problems of refined petroleum products, a source-based mixed-integer nonlinear programming formulation is proposed for discrete and continuous representations of time. Logistic constraints are modeled through Generalized Disjunctive Programming while a specialized algorithm featuring relaxations from multiparametric disaggregation handles the bilinear terms. Results over a set of test problems from the literature show that the discrete-time approach finds better solutions when minimizing cost (avoids source of bilinear terms). In contrast, solution quality is slightly better for the continuous-time formulation when maximizing gross margin. The results also show that the specialized global optimization algorithm can lead to lower optimality gaps for fixed CPU, but overall, the performance of commercial solvers BARON and GloMIQO are better.     

Long-term scheduling of a single-unit multi-product continuous process to manufacture high performance glass

In this paper we address the long-term scheduling of a real world multi-product single stage continuous process for manufacturing glass. This process features long minimum run lengths, and sequence dependent changeovers of the order of days, with high transition costs. The long-term scheduling involves extended time horizons that lead to large scale mixed-integer linear programming (MILP) scheduling models. In order to address the difficulties posed by the size of the models, three different rolling horizon algorithms based on different models and time aggregation techniques are developed. The models are based on the continuous time slot MILP model, and on the traveling salesman model proposed by Erdirik-Dogan and Grossmann (2008). Due to the particular characteristics of the process under study, several new features, including minimum run lengths and changeovers across due dates, are proposed. The performance and characteristics of the proposed rolling horizon algorithms are discussed for one industrial example.     

MILP-based initialization strategies for the optimal design of water-using networks

The optimal design of water-using systems involves necessarily the exploitation of all possible water reuse and recycling alternatives. The general problem can be formulated as a non-convex nonlinear program (NLP), but due to the presence of bilinear terms, it may be difficult for local optimization solvers to attain global optimal solutions. To overcome this difficulty, this paper presents two mixed integer linear programming (MILP)-based procedures to generate a few structurally different starting points for the NLP. In both, the problem is decomposed into calculation stages by assuming that the water streams progress in series through the water-using units, with the binary variables selecting which unit belongs to a certain stage. Their main difference concerns the way fixed flowrate units are handled, either separately or in conjunction with a fixed load operation, since the former comprise a linear subsystem. The two algorithms are compared to a closely related LP-based method taken from the literature and to the one employed by the global optimization solver BARON. The results from a large set of example problems confirm their effectiveness in avoiding local solutions despite the small number of starting points. In contrast to the previous method they are easily scalable and, for some of the larger problems, could find better solutions than BARON with significantly fewer computational resources. The results have also shown that the option of tackling one unit at a time is the most favorable.     

Bi-objective optimization approach to the design and planning of supply chains: Economic versus environmental performances

Traditionally the design of supply chains has been based on economic objectives. However, as societal environment concerns grows, environmental aspects are also emerging at academic and industry levels as decisive factors within the supply chain management context. The investment towards logistics structures that considers both economic and environmental performances is nowadays an important and current research topic.This paper addresses the planning and design of supply chain structures for annual profit maximization, while considering environmental aspects. The latter are accounted for through the Eco-indicator methodology. Profit and environmental impacts are balanced using an optimization approach adapted from symmetric fuzzy linear programming (SFLP), while the supply chain is modelled as a mixed integer linear programming (MILP) optimization problem using the Resource-Task-Network (RTN) methodology. The obtained model applicability is validated through the solution of a set of supply chain problems.     

A rule-based genetic algorithm for the scheduling of single-stage multi-product batch plants with parallel units

This paper presents a heuristic rule-based genetic algorithm (GA) for large-size single-stage multi-product scheduling problems (SMSP) in batch plants with parallel units. SMSP have been widely studied by the researchers. Most of them used mixed-integer linear programming (MILP) formulation to solve the problems. With the problem size increasing, the computational effort of MILP increases greatly. Therefore, it is very difficult for MILP to obtain acceptable solutions to large-size problems within reasonable time. To solve large-size problems, the preferred method in industry is the use of scheduling rules. However, due to the constraints in SMSP, the simple rule-based method may not guarantee the feasibility and quality of the solution. In this study, a random search based on heuristic rules was proposed first. Through exploring a set of random solutions, better feasible solutions can be achieved. To improve the quality of the random solutions, a genetic algorithm-based on heuristic rules has been proposed. The heuristic rules play a very important role in cutting down the solution space and reducing the search time. Through comparative study, the proposed method demonstrates promising performance in solving large-size SMSP.     

石化企业蒸汽动力系统优化设计建模及应用

在石化企业蒸汽动力系统(SPS)热力学分析与数学规划集成优化策略的指导下,建立了多周期设计与运行协同优化的混合整数线性规划模型(MILP)。通过热力学效率、电热供应比2个指标筛选组合子系统,采用设计负荷离散化的方法确定SPS初始超结构,大大降低了优化设计问题模型的规模和复杂度。案例分析部分采用提出的优化设计方法和建模求解策略,得到了最优设计和运行方案,同传统设计方法相比,模型求解时间大大降低,方案可操作性大大提高,证实了文中提出的优化设计策略、筛选规则和优化设计模型的可行性和有效性。