Pedro M. Castro

Scope for industrial applications of production scheduling models and solution methods

Abstract This paper gives a review on existing scheduling methodologies developed for process industries. Above all, the aim of the paper is to focus on the industrial aspects of scheduling and discuss the main characteristics, including strengths and weaknesses of the presented approaches. It is claimed that optimization tools of today can effectively support the plant level production. However there is still clear potential for improvements, especially in transferring academic results into industry. For instance, usability, interfacing and integration are some aspects discussed in the paper. After the introduction and problem classification, the paper discusses some lessons learned from industry, provides an overview of models and methods and concludes with general guidelines and examples on the modeling and solution of industrial problems.

Improvements for mass-exchange networks design

This paper addresses minimum utility cost mass-exchange network design by considering the special case of water minimisation. Two different situations are considered, re-use and regeneration re-use for single contaminants. For re-use, three different methods of targeting are presented, one of them being simultaneously a design method. This novel design method has the advantage of considering flowrate constraints only in the final stage of design. The concept of multiple pinches is introduced to prevent designing networks that do not lead to minimum cost distributed effluent treatment systems. For regeneration re-use, this paper presents the first known algorithm for targeting minimum water consumption in all possible situations. The targeted flowrate is then used to design the mass-exchange network, that almost always features splitting of operations.

Global optimization of bilinear programs with a multiparametric disaggregation technique

In this paper, we present the derivation of the multiparametric disaggregation technique (MDT) by Teles et al. (J. Glob. Optim., 2011) for solving nonconvex bilinear programs. Both upper and lower bounding formulations corresponding to mixed-integer linear programs are derived using disjunctive programming and exact linearizations, and incorporated into two global optimization algorithms that are used to solve bilinear programming problems. The relaxation derived using the MDT is shown to scale much more favorably than the relaxation that relies on piecewise McCormick envelopes, yielding smaller mixed-integer problems and faster solution times for similar optimality gaps. The proposed relaxation also compares well with general global optimization solvers on large problems.

Optimal scheduling of continuous plants with energy constraints

This work addresses the scheduling of continuous single stage multiproduct plants with parallel units and shared storage tanks. Processing tasks are energy intensive and we consider time-dependent electricity pricing and availability together with multiple intermediate due dates, handled as hard constraints. A new discrete-time aggregate formulation is proposed to rapidly plan the production levels. It is combined with a continuous-time model for detailed scheduling as the essential part of a rolling-horizon algorithm. Their computational performance is compared to traditional discrete and continuous-time full-space formulations with all models relying on the Resource-Task Network (RTN) process representation. The results show that the new models and algorithm can generate global optimal schedules much more efficiently than their counterparts in problems involving unlimited power availability. Under restricted power, the aggregate model underestimates the electricity cost, which may cause the rolling-horizon approach to converge to a suboptimal solution, becoming the discrete-time model a better approach.

An efficient MILP model for the short-term scheduling of single stage batch plants

This paper presents a multiple time grid continuous time MILP model for the short-term scheduling of single stage, multiproduct batch plants. It can handle both release and due dates and the objective can be either the minimization of total cost or total earliness. This formulation is compared to other mixed-integer linear programming approaches that have appeared in the literature, to a constraint programming model, and to a hybrid mixed-integer linear/constraint programming algorithm. The results show that the proposed formulation is significantly more efficient than the MILP and CP models and comparable to the hybrid model. For one large instance, both methods exceeded the time limit but the hybrid method failed to find a feasible solution. The results also show that a discrete-time formulation performs very efficiently even when a large number of time intervals are used.

Comparison of global optimization algorithms for the design of water-using networks

Abstract We address a special class of bilinear process network problems with global optimization algorithms iterating between a lower bound provided by a mixed-integer linear programming (MILP) formulation and an upper bound given by the solution of the original nonlinear problem (NLP) with a local solver. Two conceptually different relaxation approaches are tested, piecewise McCormick envelopes and multiparametric disaggregation, each considered in two variants according to the choice of variables to partition/parameterize. The four complete MILP formulations are derived from disjunctive programming models followed by convex hull reformulations. The results on a set of test problems from the literature show that the algorithm relying on multiparametric disaggregation with parameterization of the concentrations is the best performer, primarily due to a logarithmic as opposed to linear increase in problem size with the number of partitions. The algorithms are also compared to the commercial solvers BARON and GloMIQO through performance profiles.

Tightening piecewise McCormick relaxations for bilinear problems

Abstract We address nonconvex bilinear problems where the main objective is the computation of a tight lower bound for the objective function to be minimized. This can be obtained through a mixed-integer linear programming formulation relying on the concept of piecewise McCormick relaxation. It works by dividing the domain of one of the variables in each bilinear term into a given number of partitions, while considering global bounds for the other. We now propose using partition-dependent bounds for the latter so as to further improve the quality of the relaxation. While it involves solving hundreds or even thousands of linear bound contracting problems in a pre-processing step, the benefit from having a tighter formulation more than compensates the additional computational time. Results for a set of water network design problems show that the new algorithm can lead to orders of magnitude reduction in the optimality gap compared to commercial solvers.

Dynamic modelling and scheduling of an industrial batch system

Abstract This paper addresses the optimal schedule of a resource constrained four-batch digester system of an industrial acid sulphite pulp mill. This involves the development of two different models, one to model the scheduling operational problem and the other the batch digester operation—process model. The first model uses a discrete-time Resource Task Network (RTN) based representation leading to a Mixed Integer Linear Program (MILP) formulation. In this, the main operational limitation, steam availability, is modelled through the definition of a superstructure including the most relevant heating alternatives. The duration of the generated heating tasks was estimated through the use of the process model—a distributed heterogeneous dynamic model in gPROMS, with a heat-transfer resistance at the solid–fluid-interface—validated with experimental plant data. The optimal schedules obtained showed that an increase in the total available steam from the boiler is vital to allow a higher level of productivity and that not much can be done regarding steam sharing improvements.

Optimality-based bound contraction with multiparametric disaggregation for the global optimization of mixed-integer bilinear problems

We address nonconvex mixed-integer bilinear problems where the main challenge is the computation of a tight upper bound for the objective function to be maximized. This can be obtained by using the recently developed concept of multiparametric disaggregation following the solution of a mixed-integer linear relaxation of the bilinear problem. Besides showing that it can provide tighter bounds than a commercial global optimization solver within a given computational time, we propose to also take advantage of the relaxed formulation for contracting the variables domain and further reduce the optimality gap. Through the solution of a real-life case study from a hydroelectric power system, we show that this can be an efficient approach depending on the problem size. The relaxed formulation from multiparametric formulation is provided for a generic numeric representation system featuring a base between 2 (binary) and 10 (decimal).

Scheduling of a continuous plant with recycling of byproducts: A case study from a tissue paper mill

Abstract This paper considers an industrial scheduling problem. It involves profit maximization and the determination of the optimal cycle time, while meeting the minimum demands for the several products. Resource-Task Network-based formulations are employed and a detailed comparison between continuous- and discrete-time models is provided. Both have the improved capability of handling tasks with flexible proportions of input materials in order to consider the incorporation of different flowrates of byproducts that are recycled back to the first production stage. The continuous-time formulation is shown to be more efficient and the resulting mixed integer nonlinear program (MINLP) can be solved to optimality within reasonable computational time. A new recycling policy is proposed that achieves the double goal of making the process more profitable due to important savings on the more expensive raw-materials and also more environmentally friendly, due to the reduction of waste disposal requirements.