MILP reformulations for the design of biotechnological multi-product batch plants using continuous equipment sizes and discrete host selection
Introduction
Conventional multi-product batch process literature using an optimization-based approach model the design and synthesis of such plants with Mixed-Integer Non-Linear Programming (MINLP) formulations (Floudas, 1995). The usual objective is to minimize the investment cost subject to the fulfillment of the production targets of a given set of products. Major drawbacks are given by the combinatorial nature of mixed-integer programming and possible nonconvexities due to non-linearities. In computational optimization numerical issues of these formulations given by rounding errors, numerical instabilities and approximation errors are well-documented ([Goldberg, 1991], [Koch, 2004], [Margot, 2009], [Vielma, 2013]).
Since Robinson and Loonkar (1972) different procedures have been proposed to tackle these problems ([Reklaitis, 1990], [Rippin, 1993], [Barbosa-Póvoa, 2007], [Verderame et al., 2010], [Nikolopoulou and Ierapetritou, 2012]) but a method that is more efficient for a particular example is hardly predictable (Ponsich et al., 2007) and nowadays the development of effective solution approaches and algorithms remains very necessary (Grossmann and Guillén-Gosálbez, 2010).
The logarithmic change of variables proposed by Kocis and Grossmann (1988) linearizes most of the functions and leads to a convex MINLP problem, approach used by Ravemark and Rippin (1998) and Montagna et al. (2000) among others. Another approach chosen by Pinto et al. (2001) and Ponsich et al. (2007) among others is the use of specially designed solvers which can usually find good feasible solutions by the use of heuristic procedures (Grossmann et al., 2000). In practice the best off the shelf solvers for this kind of problems are the open source codes BONMIN and SCIP and the commercial solvers BARON and DICOPT that stand out in Mittelmann's benchmarks for optimization software (Mittelmann, 2013). Nevertheless none of them guarantee convergence to a global optimum, converging in some instances to local optima or not converging altogether. For the particular case of BARON and DICOPT performance failures are reported for non-convex models ([Ponsich et al., 2007], [Rebennack et al., 2011], [Li et al., 2012]); nevertheless even in cases where theoretically the algorithms work, we found that in practice, they do not converge to the global optimum. We have run precise experiments that demonstrate these failures in convex MINLP formulations (see Section 2).
It is a fact that there is a huge gap between Mixed-Integer Linear Programming (MILP or MIP) and MINLP solvers technology (Nowak, 2005). Nowadays mixed-integer linear techniques are fast, robust and able to provide solutions to problems with up to millions of variables (Geißler et al., 2012). Taking advantage of this Voudouris and Grossmann (1992) used reformulation schemes to develop MILP models for the preliminary design of multi-product batch plants, introducing binary variables for the selection of discrete available equipment sizes. From this point, to the design decisions other were included as synthesis, production planning and scheduling (Voudouris and Grossmann, 1993); design and planning in a multiperiod scenario (Moreno and Montagna, 2007); design of multi-product batch plants considering duplication of units in series (Moreno et al., 2009) and the design and planning of multi-product batch plants using mixed-product campaigns (Corsano et al., 2009). Most recently these MILP formulations have been used to account for the design and scheduling of this type of plants ([Fumero et al., 2011], [Fumero et al., 2012a], [Fumero et al., 2012b]) and for the design under uncertainty considering different types of decisions ([Durand et al., 2012], [Durand et al., 2014], [Moreno and Montagna, 2012], [Moreno-Benito et al., 2014]).
A key feature in these design problems is the use of Big-M constraints to account for selection decisions despite being problematic (Bosch and Trick, 2005). Some authors that have included this type of constraints in their formulations are [Gupta and Karimi, 2003], [Corsano et al., 2009], [Moreno et al., 2009], [Moreno and Montagna, 2012]. Obviously, these authors have found that the value of the Big-M parameters has a tremendous impact on the solution time; see for example Moreno et al. (2007). In addition it has been proven experimentally that other methods, as the convex hull formulation presented by Montagna et al. (2004) are better to account for selection decisions.
In this paper we develop a robust methodology to solve the design problem of a biotechnological multi-product batch plant in situations where equipment can be manufactured according to customer needs, as fermentors or tanks in general. To do that, we develop a MILP formulation which does not rely on the use of Big-M constraints and does not use a discrete range of equipment sizes. To do that we use four basic techniques (see Fig. 1): First, an extension of the non-linear (but convex) formulation proposed by Kocis and Grossmann (1988) is applied. Secondly, to deal with non-linear convex inequalities a priori we constructed linear outer (or inner) approximations of them which allow us to compute (a posteriori) true feasible solutions and lower (or upper) bounds. Thirdly, to deal with integer variables, we used advanced reformulation techniques coming from the mixed-integer-programming literature (clique constraints). Finally, once the initial problem is transformed into a standard mixed-integer programming problem, it is possible to take advantage of mature commercial MIP solvers.
This approach, at least in our experiments, is more stable numerically, scalable, and faster to solve than current alternatives and can deal with the more general problem of jointly selecting equipment sizes and alternative production paths for multiple products. Using our approach, it is possible to quickly and accurately compute solutions at any desired precision level. In our extensive computational experiments (see Fig. 11) we found that current non-linear solvers only solved 43% of the instances generated for this study, while our approach was able to solve over 95% of the studied instances in a running time that, on average, was more than ten times faster than MINLP solvers in equivalent and standard MINLP formulations. To make these comparisons we introduce the performance profiles; a methodology borrowed from the optimization literature.
The rest of this paper is organized as follows. In Section 2 typical drawbacks found by a commonly used MINLP solver and the standard MINLP formulation is presented. In Section 3 classic and novel formulations for the design problem are described. Relevant information about the methodology used to benchmark different formulations and to avoid numerical instabilities is given in Section 4 and computational results are presented and discussed in Section 5. Finally, the conclusions are presented in Section 6.
Section snippets
Current limitations
Our main objective is finding a robust and scalable methodology for the design of biotechnological multi-product batch plant considering equipment sizing (design decisions) and selecting the downstream processing stages (synthesis decisions). Given the complexities that to date have been added to the original design problem we decided to go back to the problem studied by Iribarren et al. (2004) where only design and synthesis decisions are modeled. In their paper they designed a
Problem formulation
Two major contributions are presented in this section. First, clique constraints are introduced to formulate the discrete part of the model allowing the selection of the production path without the use of Big-M constraints, in models (P2) and (P4). Second, a new approach, in Section 3.2, to handle non-linearities using standard reformulation techniques from the optimization field that permits the use of linear solvers leading to more reliable results and faster computing time. The relation
Solvers and modelling language
For MINLP problems the open source BONMIN 1.5 and SCIP 3.0.1 solvers were studied. In our computational tests SCIP uses SoPlex 1.7.1 as the LP solver and BONMIN (with its default algorithm, B-Hyb) uses Cbc 2.7.1 as the MIP solver and Ipopt 3.10.0 with MUMPS as linear solver. For the case of BONMIN we tested 3 over 5 available algorithms: B-Hyb the default algorithm, B-Ecp a specific parameter setting of B-Hyb that can be faster in some cases (Bonami and Lee, 2013) and B-OA using CPLEX as the
Results and discussion
In this section we show the robustness of our proposed MILP transformations and its superiority over classic MINLP formulations with Big-M constraints using performance profiles, a methodology borrowed from the optimization literature. Our approach is not only able to find correct solutions in realistic situations unlike MINLP formulations but also in a small fraction of the time required by those approaches. Major implications of these features are the exactness of the solutions that make this
Conclusions
In this work we present a scalable approach to solve, within reasonable running times and quality assurance requirements, the problem of designing a biotechnological multi-product batch plant that support continuous equipment sizes and discrete host and/or process selection, up to sizes of real instances and that can be applicable to any kind of multi-product batch plant.
The proposed method was proved to be more numerically stable than other alternative approaches for the same problem giving
Acknowledgements
This work was supported by a CONICYT scholarship for doctoral studies, FONDECYT Grant 1110024, Núcleo Milenio Información y Coordinación en Redes P10-024-F, and CONICYT for funding of Basal Centre, CeBiB, FB0001.
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