Multi-objective optimization of a plug flow reactor using a divide and conquer approach

Proceedings of the 20th World Congress Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th World Congress Proceedings of the 20th World The International Federation of Congress Automatic Control Toulouse, France,Federation July 9-14, 2017 Available online at www.sciencedirect.com The International of Automatic Control The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017

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Multi-objective optimization of a plug flow Multi-objective optimization of a plug flow Multi-objective optimization of a plug Multi-objective optimization ofconquer a plug flow flow reactor using a divide and reactor using a divide and conquer reactor using a divide and conquer reactor usingapproach a divide and conquer approach approach approach ∗ ∗ ∗ Ihab Ihab Ihab Ihab

Hashem ∗ Dries Telen ∗ Philippe Nimmegeers ∗ Hashem ∗ Dries Telen ∗ Philippe Nimmegeers ∗ ∗ ∗ Hashem Telen Nimmegeers ∗ Dries ∗ Philippe ∗ Filip Logist Van Impe ∗ Jan ∗ Hashem Dries Telen Nimmegeers Filip Logist Van Impe ∗ Jan Philippe ∗ Filip Logist Jan Van Impe Filip Logist ∗ Jan Van Impe ∗ ∗ ∗ KU Leuven, Chemical Engineering Department, BioTeC+ & ∗ KU Leuven, Chemical Engineering Department, BioTeC+ & Chemical Engineering Department, BioTeC+ & ∗ KU Leuven, OPTEC, Gebroeders De Smetstraat 1, 9000 Gent, Belgium (e-mail: KU Leuven, Chemical Engineering Department, BioTeC+ & OPTEC, Gebroeders De Smetstraat 1, 9000 Gent, Belgium (e-mail: OPTEC, Gebroeders De Smetstraat 1, 9000 Gent, Belgium (e-mail: [email protected]) OPTEC, Gebroeders [email protected]) De Smetstraat 1, 9000 Gent, Belgium (e-mail: [email protected]) [email protected]) Abstract: Lower profit margins and tighter environmental constraints push chemical companies Abstract: Lower profit margins and tighter environmental constraints push chemical companies Abstract: Lower profit and environmental constraints push companies to search for sustainable development and operation. To ensure sustainable development and Abstract: Lower profit margins margins and tighter tighter environmental constraints push chemical chemical companies to search for sustainable development and operation. To ensure sustainable development and to search for sustainable development and operation. ensure sustainable development and operation in practice, multiple and conflicting objectivesTo (e.g., maximize production vs. minimize to search for sustainable development and operation. To ensure sustainable development and operation in practice, multiple and conflicting objectives (e.g., maximize production vs. minimize operation in multiple objectives (e.g., vs. energy consumption) typically haveconflicting to be optimized simultaneously. The production frame of multi-objective operation in practice, practice, typically multiple and and conflicting objectives (e.g., maximize maximize production vs. minimize minimize energy consumption) have to be optimized simultaneously. The frame of multi-objective energy consumption) typically have optimized simultaneously. frame of optimization allows to systematically propose improvements and The evaluate trade-offs between energy consumption) typically have to to be be optimized simultaneously. The frametrade-offs of multi-objective multi-objective optimization allows to systematically propose improvements and evaluate between optimization allows to systematically propose improvements and evaluate trade-offs between different objectives. Solving a multi-objective optimization problem yields a set of solutions optimization allows to systematically propose optimization improvementsproblem and evaluate trade-offs between different objectives. Solving a multi-objective yields aa set of solutions different Solving a problem yields solutions called theobjectives. Pareto front, in which no improvementoptimization can be made in one objective without worsening different objectives. Solving a multi-objective multi-objective optimization problem yields without a set set of ofworsening solutions called the Pareto front, in which no improvement can be made in one objective called the Pareto which improvement can be made in objective another objective. Thein posteriori analysis of this a smart aimswithout to keepworsening only the called theobjective. Pareto front, front, inposteriori which no no analysis improvement canset be by made in one one filter objective without worsening another The of this set by aa smart filter aims to keep only the another objective. The posteriori analysis of this set by smart filter aims to keep only significant solutions for the decision maker who is interested in a specific level of trade-offs. another objective. The posteriori analysis ofwho this issetinterested by a smart aimslevel to keep only the the significant solutions for the decision maker in aafilter specific of trade-offs. significant solutions for the decision maker who is interested in specific level of trade-offs. However, this strategy suffers from the large overhead of insignificant solutions produced in significant solutions for suffers the decision maker who is interested in a specific level ofproduced trade-offs. However, this strategy from the large overhead of insignificant solutions in However, this strategy suffers from the large overhead of insignificant solutions produced in the original set. This situation makes applying this strategy to the complex multi-objective However, this strategy suffers from the large overhead of insignificant solutions produced in the original set. This situation makes applying this strategy to the complex multi-objective the original set. This makes this strategy the optimal control problems time consuming. In this paper, the to smart filter by multi-objective Mattson et al. the original set. problems This situation situation makes applying applying thispaper, strategy to the complex complex multi-objective optimal control time consuming. In this the smart filter by Mattson et al. optimal control problems time consuming. In this paper, the smart filter by Mattson al. (2004) is compared to a novel Divide and Conquer (D&C) algorithm to obtain a Pareto et front optimal control problems timeDivide consuming. In this (D&C) paper, the smart to filter by Mattson et al. (2004) is compared to a novel and Conquer algorithm obtain aa Pareto front (2004) is compared to a novel Divide and Conquer (D&C) algorithm to obtain Pareto front with adaptive resolution for a Divide case study of a plug (D&C) flow reactor. The to new algorithm depends (2004) is compared to a novel and Conquer algorithm obtain a Pareto front with adaptive resolution for a case study of a plug flow reactor. The new algorithm depends with adaptive resolution for case of flow The depends on obtaining Pareto front terminating the exploration of a Pareto front with adaptivethe resolution for a a recursively case study study while of aa plug plug flow reactor. reactor. The new new algorithm algorithm depends on obtaining the Pareto front recursively while terminating the exploration of aa Pareto front on obtaining the Pareto front recursively while terminating the exploration of Pareto front segment as soon as an insignificant point is found. It is shown that the D&C algorithm produces on obtaining theasPareto front recursively while terminating the exploration of a Pareto front segment as soon an insignificant point is found. It is shown that the D&C algorithm produces segment as soon as an insignificant point is found. It is shown that the D&C algorithm produces representations with similar quality to the smart filter with higher speed and a more intuitive segment as soonwith as ansimilar insignificant point is found. It is shown that the D&Cand algorithm produces representations quality to the smart filter with higher speed a more intuitive representations with similar quality trade-off oriented solution procedure. representations with similar quality to to the the smart smart filter filter with with higher higher speed speed and and aa more more intuitive intuitive trade-off oriented solution procedure. trade-off oriented solution procedure. trade-off oriented solution procedure. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Multi-objective optimisation, Optimal control, Recursive algorithms, Process Keywords: Multi-objective Keywords: Multi-objective optimisation, optimisation, Optimal Optimal control, control, Recursive Recursive algorithms, algorithms, Process Process industry Multi-objective Keywords: optimisation, Optimal control, Recursive algorithms, Process industry industry industry 1. INTRODUCTION the solutions that are expected to be more interesting 1. the solutions that are expected to be more interesting 1. INTRODUCTION INTRODUCTION the solutions are expected be more interesting to the Decisionthat Maker Oneto techniques is 1. INTRODUCTION the solutions that are (DM). expected toof bethese more interesting to the Decision Maker (DM). One of these techniques is to the Decision Maker (DM). One of these techniques is the smart filter (Mattson et al. (2004)) which works by The existence of multiple conflicting objectives for engi- to the Decision Maker (DM). One of these techniques is the smart filter (Mattson et al. (2004)) which works by The existence of multiple conflicting objectives for engithe smart filter (Mattson et al. (2004)) which works by filtering a dense Pareto front in order to provide a Pareto The existence of multiple conflicting objectives for engineering optimization problems is a situation that is frethe smart filter (Mattson et al. (2004)) which works by The existence of multiple conflicting objectives for engifiltering a dense Pareto front in order to provide a Pareto neering optimization problems is situation that is is frefre- front aa dense Pareto front order to provide a with adaptive resolution, at neering optimization problems is a situation that quently optimization encountered in the chemical industry, e.g., the filtering filtering dense Pareto front in in i.e., orderthe to points providedensity a Pareto Pareto neering problems is aa situation that is frefront with adaptive resolution, i.e., the points density at quently encountered in the chemical industry, e.g., the front with adaptive resolution, i.e., the points density at any segment is proportional to its slope. This way the quently encountered in the chemical industry, e.g., the profit versus the environmental impact of a process. In front with adaptive resolution, i.e., the points density at quently encountered in the chemical industry, e.g., the any segment is proportional to its slope. This way the profit versus the environmental impact of a process. In any segment is proportional to its slope. This way the more interesting ”knee” regions of the Pareto front are profit versus the environmental impact of a process. In such cases, the setenvironmental of optimal points is called the Pareto any segment is proportional to its slope. This way the profit versus the impact of a process. In more interesting ”knee” regions of the Pareto front are such cases, the set points is Pareto more interesting regions of front represented”knee” than the less interesting flat segments. such cases, the set of of optimal optimal points is called calledinthe the Pareto front,cases, whichthe is defined as the set of solutions which no better more interesting ”knee” regions of the the Pareto Pareto front are are such set of optimal points is called the Pareto better represented than the less interesting flat segments. front, which is defined as the set of solutions in which no better represented than interesting flat segments. One major drawback ofthe theless smart filter strategy is the front, which isbedefined defined as the the set of solutions in no objective canis improved without worsening at which least one better represented than the less interesting flat segments. front, which as set of solutions in which no One major drawback of the smart filter strategy is the objective can improved without worsening at one drawback the strategy need major to obtain a denseof frontfilter for the filter is to the be objective can be be The improved without worsening at least leastusing one One other objective. Pareto front can be obtained One major drawback of Pareto the smart smart filter strategy is the objective can be improved without worsening at least one need to obtain aa dense Pareto front for the filter to be other objective. The Pareto front can can be obtained obtained using need to obtain dense Pareto front for the filter to be applied on. The result is a large overhead of wasted points other objective. The Pareto front be using two classes of techniques: vectorization (Deb (2001))using and need to obtain a dense Pareto front for the filter to be other objective. The Pareto front can be obtained applied on. The result is a large overhead of wasted points two classes of techniques: vectorization (Deb (2001)) and applied on. The result is a large overhead of wasted points that will get removed in the final representation, and two classes of techniques: vectorization (Deb (2001)) and scalarization methods (Miettinen (1999)). Scalarization applied on. The result is a large overhead of wasted points two classes of techniques: vectorization (Deb (2001)) and that will get removed in the final representation, and scalarization methods (Miettinen (1999)). Scalarization will removed the representation, and hence, longget computational (Hancock et al. (2015)). scalarization methods (Miettinen (1999)). Scalarization methods havemethods been shown to be more convenient when that that will get removed in in times the final final representation, and scalarization (Miettinen (1999)). Scalarization hence, long computational times (Hancock et al. (2015)). methods have been shown to be more convenient when hence, long computational times (Hancock et al. (2015)). This disadvantage worsens with computationally expenmethods have been been shown to be be more more convenient when hence, long computational times (Hancock et al. (2015)). handling optimal control problems (Logist et al. (2013)). methods have shown to convenient when This disadvantage worsens with computationally expenhandling optimal control problems (Logist et al. (2013)). This disadvantage worsens with MOOPs. Typical examples of such problemsexpenthat handling optimal control control problems (Logist et etthe al. Multiple (2013)). sive These algorithms work by parameterizing This disadvantage worsens with computationally computationally expenhandling optimal problems (Logist al. (2013)). sive MOOPs. Typical examples of such problems that These algorithms work by parameterizing the Multiple sive MOOPs. Typical examples of such problems that are frequently encountered in the chemical industry are These algorithms work by parameterizing the Multiple Objectives Optimization Problem (MOOP) into a set of sive frequently MOOPs. Typical examples ofchemical such problems that These algorithms work by parameterizing the Multiple are encountered in the industry are Objectives Optimization Problem (MOOP) into a set of are frequently encountered in the chemical industry are Multi-Objective Optimal Control Problems (MOOCPs). Objectives Optimization Problem (MOOP) into a set of Single Objective Optimization Problems (SOOPs), such are frequently encountered in the chemical industry are Objectives Optimization Problem (MOOP) into a set of Multi-Objective Optimal Control Problems (MOOCPs). Single Objective Optimization Problems (SOOPs), such Multi-Objective Optimal Control Problems (MOOCPs). A MOOCP involves finding the optimal time or space Single Objective Optimization Problems (SOOPs), such that the solving of each SOOP corresponds to obtaining Multi-Objective Optimal Control Problems (MOOCPs). Single Objective Optimization Problems (SOOPs), such A MOOCP involves finding the optimal time or space that the solving of SOOP corresponds to A MOOCP involves finding the optimal time or a control variable a system that the on solving of each eachfront SOOP corresponds to obtaining obtaining a point the Pareto (Marler and Arora (2004)). trajectory A MOOCPof involves finding the for optimal time described or space space that the solving of each SOOP corresponds to obtaining trajectory of aa control variable for aa system described aa point on the Pareto front (Marler and Arora (2004)). trajectory of control variable for system described by differential equations in order to optimize multiple point on the Pareto front (Marler and Arora (2004)). Moreover, several techniques exist to analyze the obtained trajectory of a control variable for a system described a point on the Pareto front (Marler and Arora (2004)). by differential equations in order to optimize multiple Moreover, several techniques exist to the obtained differential equations to objectives. A novel divide in andorder conquer algorithmmultiple aims to Moreover, several techniques exist to analyze analyze the obtained by solution set a posteriori for exist the purpose of the emphasizing by differential equations in order to optimize optimize multiple Moreover, several techniques to analyze obtained objectives. A novel divide and conquer algorithm aims to solution set a posteriori for the purpose of emphasizing objectives. A novel divide and conquer algorithm aims to produce a Pareto front with adaptive resolution with solution set a posteriori for the purpose of emphasizing objectives. A novel divide and conquer algorithm aimsless to solution set a posteriori for the purpose of emphasizing produce a Pareto front with adaptive resolution with less  This work was supported by KU Leuven [PFV/10/002 produce a Pareto front with adaptive resolution with less insignificant solutions than the smart filter strategy. Its produce a Pareto front with adaptive resolution with less  This work was supported by KU Leuven [PFV/10/002 insignificant solutions than the smart filter strategy. Its  insignificant solutions than Its Center-of-Excellence in Engineering (OPTEC), FWO This work was Optimization supported by KU Leuven [PFV/10/002 concept depends on obtaining thesmart Paretofilter frontstrategy. recursively.  insignificant solutions than the the smart filter strategy. Its This work was Optimization supported by KU Leuven [PFV/10/002 Center-of-Excellence in Engineering (OPTEC), FWO concept depends on obtaining the Pareto front recursively. [G.0930.13] and BelSPO (DYSCO) in [IAP VII/19]. (OPTEC), FWO Center-of-Excellence Optimization Engineering concept depends on obtaining the Pareto front recursively. Center-of-Excellence Optimization in Engineering (OPTEC), FWO concept depends on obtaining the Pareto front recursively. [G.0930.13] and BelSPO (DYSCO) [IAP VII/19].

[G.0930.13] and BelSPO (DYSCO) [IAP VII/19]. [G.0930.13] and BelSPO (DYSCO) [IAP VII/19]. Copyright © 2017, 2017 IFAC 9056Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 9056 Copyright ©under 2017 responsibility IFAC 9056Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 9056 10.1016/j.ifacol.2017.08.1712

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Hence, the solution process can get terminated when the algorithm converges to a low slope segment of the front. The aim of this paper is to compare the performance of the novel divide and conquer algorithm to the smart filter with respect to speed and clarity of the solution process. The case study used is the multi-objective optimal control of a plug flow reactor (Logist et al. (2008)). The paper is structured as follows: Section 2 introduces the MOOCP formulation and an overview of the optimization algorithms used in this study. Section 3 presents the novel algorithm’s concept. The case study formulation, simulation and results are illustrated in Section 4 . Finally, Section 5 summarizes the conclusions.

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algorithms are characterized by being deterministic in nature. They work by converting the MOOP into a series of parametrized SOOPs using a weights vector. There have been several successful applications of scalarization methods for solving high dimensional optimization problems with complex constraints, examples are: de Hijas-Liste et al. (2014); Nimmegeers et al. (2016). One of the most widely used scalarization algorithms that is able to obtain a uniform representation of the Pareto front is introduced hereafter.

2. MATHEMATICAL FORMULATION AND METHODS 2.1 Problem formulation Consider the following formulation of a MOOCP, (Logist et al. (2010)): min{J1 , J2 , ..., Jm } (1) y

subject to: dx = d 0= 0≥ 0≥

F (x(), u(), p, ) bi (x(0), p) bt (x(f ), p) cp (x(), u(), p, )

(2)

with y defined as the optimization variables vector y = (u(·), x(·), p, f ),  as the independent variable, usually time or space, over [0, f ]. x and u are the state and control variables respectively, p are the system parameters. Vectors bi and bt represent the initial and terminal conditions, while vector cp represents the path constraints. Objective Ji is formulated as the sum of the Mayer M and Lagrange L terms as follows:  f

Ji = Mi (x(f ), p, f ) +

Li (x(), u(), p, )d

(3)

0

with the Mayer term Mi representing the function’s terminal cost, e.g., yield and the Lagrange term Li being the objective function’s integral cost, e.g., the total heat removal during a process. 2.2 Multi-objective optimization algorithms In case of conflicting objectives, an infinite set of solutions exists for a MOOP called the Pareto set. As formulated by Miettinen (1999) , a solution y ∗ is said to be Pareto optimal iff there exist no other y ∈ S such that Ji (y) ≤ Ji (y ∗ ) for i = 1, 2, ..., n and Jj (y) < Jj (y ∗ ) for at least one objective j. According to a review by Marler and Arora (2004) , there exist two classes of algorithms to obtain a representation of the Pareto front: vectorization methods and scalarization methods. Vectorization methods use a stochastic approach to tackle the MOOP directly. Due to their time consuming nature and inability to handle complex constraints, these methods are limited to low dimensional search spaces and are not suitable for solving the typically high dimensional MOOCPs (Logist et al. (2013)). The other major class is the scalarization methods. These

Fig. 1. Illustration of the Normal Boundary Intersection method, adopted from Das and Dennis (1998) Normal Boundary Intersection (NBI) This method has been proposed by Das and Dennis (1998). The main advantage of this technique is that an even distribution of input parameters (weights) produces a uniform distribution of points on the Pareto front. This is in contrast with the widely used (convex) weighted sum. Before proceeding to the algorithm structure, the concepts of the Utopia point and the Convex Hull of Individual Minima need to be introduced. The Utopia point J ∗ is the vector that consists of all the individual minima Ji∗ of the individual objects as follows (Das and Dennis (1998)): ∗  ] (4) J ∗ = [J1∗ , J2∗ , ..., Jm The Convex Hull of Individual Minima (CHIM ) is a hyperplane in the objective space that connects all the individual minima of the different objectives. In Figure 1, the CHIM can be observed as a line in 2D space. The method starts by rescaling the objective functions as follows (Das and Dennis (1998)):

J(y) ← J(y) − J ∗

(5)

Subsequently, the Utopia point is shifted to the graph origin. The next step is the construction of a set of quasinormal lines to the CHIM. The MOOP is reformulated such that the distance λ between a point on the CHIM and the Utopia point is sought to be maximized. The problem can be formulated as follows (Das and Dennis (1998)): max λ (6) subject to: φw − λe = J (7) with φ the pay-off matrix defined as an n × n matrix ∗ whose m i-th column is Ji −J . w is the weights vector where w = 1 and e is a vector filled with ones. Equation i=1 i (7) represents the additional constraint of having the

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solution along the quasi-normal line. Through a uniform distribution of the weights vector w on the CHIM, a uniform representation of the Pareto front can be obtained. 2.3 Posteriori Pareto front analysis A simple way to facilitate the analysis of the obtained Pareto front by the DM is to apply a smart filter to the obtained representation (Mattson et al. (2004)). The smart filter aims to reduce the Pareto set size by removing the points that are deemed insignificant to the DM. The solution process proceeds as following: first, a dense Pareto front is obtained using a MOO algorithm such as NBI. Then, the user defines the level of trade-offs he/she is interested in, t. The smart filter proceeds by making pairwise comparisons on the Pareto set, keeping only the solutions that satisfy the following significance condition in bi-objective problems: | PiJ1 − PkJ1 |≥ t

and

| PiJ2 − PkJ2 |≥ t

(8)

with Pi and Pk any two points in the filtered set. This way, a Pareto front with adaptive resolution can be obtained. Segments of low level of trade-offs are less represented compared to high slope segments. Thus, a representation with only the points interesting to the DM is produced. However, a disadvantage of this approach is that it is applied a posteriori to a large set of solutions. Hence, to produce a smart set of solutions, a lot of computational effort is used to produce an overhead of insignificant points that are not needed for the final representation (Hancock et al. (2015)). 3. A NOVEL DIVIDE AND CONQUER ALGORITHM FOR MULTI-OBJECTIVE OPTIMIZATION 3.1 Algorithm concept

finding the two individual minima, anchor points, of the Pareto front. After that, a NBI sub-problem is solved to obtain a Pareto point using the weight point at the center of the weight line, CHIM. The weight line continues to be divided recursively while calculating the corresponding Pareto points (divide). At each step, the obtained Pareto point is checked if it meets the significance condition. This occurs by comparing the new point to its two parent points using Equation (8). If the point is found to be insignificant relative to one of its neighbors, it is not added to the final representation and the algorithm stops exploring the Pareto segment involved (conquer). Figure 2 shows an illustration of the principle of the D&C algorithm where the most right weight cell at the second recursive level yields an insignificant point with respect to its parents. For an overview of the algorithm structure see Algorithm 1. Algorithm 1 A 2D divide and conquer algorithm Input: Significant trade-off level t. Output: Pareto set with adaptive resolution S Step 1: Initialization of solution set S = {}. Step 2: Construction of anchors weight cell Cin :   1 0 Cin = 0 1 Step 3: Initialization of weight cell Cw : Cw = Cin . Step 4: Using weights in Cin , solve the NBI optimization sub-problems and add the anchor points Pin1 , Pin2 to S. Points Pi , Pj are initialized such that Pi = Pin1 , Pj = Pin2 . Step 5: Start the recursive function using weight cell Cw and Pareto points Pi , Pj as inputs. Solve the quasi-normal line maximization problem with the weight (Cw [0] + Cw [1])/2 to obtain the Pareto point Pm . Then the following condition is checked: if Pi , Pm and Pm , Pj satisfy significance condition do (1) Add Pm to S. (2) Divide: two sets of daughter weight cells Cd1 and Cd2 are constructed as follows:   Cw [0] + Cw [1] Cd1 = Cw [0] 2   Cw [0] + Cw [1] Cd2 = Cw [1] 2 (3) Call the recursive function twice by updating inputs to step 5 such that: Cw = Cd1 , Pi = Pi , Pj = Pm and Cw = Cd2 , Pi = Pm , Pj = Pj . else Conquer : exit, stop exploring current segment. Step 6: When all recursive calls are exited, produce S.

Fig. 2. Illustration of the D&C algorithm. The CHIM is divided recursively. For each weight, a problem of maximizing the quasi-normal line is solved to obtain the corresponding Pareto point, the most right recursive branch is shown. A divide and conquer approach aims to obtain a Pareto front with adaptive resolution while producing less insignificant solutions, overhead, than a smart filter. Instead of obtaining a dense representation using NBI and then filtering it, the D&C algorithm explores the Pareto front recursively. In two dimensions, the algorithm starts by

In three dimensions, a square based recursive approach is proposed to distribute weights. An initial weight cell is constructed such that it encloses the triangular CHIM, see Figure 3. Then the square cell is divided recursively into four daughter square cells. Every weight cell consists of four weights, if a weight lies in the CHIM, it is used to find the corresponding Pareto point as by solving an NBI sub-problem. The recursive process is terminated when the Pareto segment corresponding to a certain weight cell is

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Algorithm 2 A 3D divide and conquer algorithm 2 2 Input: Stopping criterion specifications σlow , σhigh Output: Pareto set with adaptive resolution S Step 1: Initialization of solution set S = {}. Step 2: Construction of a weight cell Cin =  [w1 w2 w3 w4 ] such that Cin is a square cell enclosing the CHIM. Step 3: Initialization of weight cell Cw : Cw = Cin . Step 4: Start the recursive function using weight cell Cw as an input: for i = 1:4 do Fig. 3. Illustration of the initial square weight cell enclosing the CHIM. The square cell is divided recursively. When a weight is found to be in the triangular CHIM, it is used to find the corresponding point on the Pareto curve by solving a NBI sub-problem estimated to have a low level of trade-offs. This is done using the following stopping criterion. Stopping criterion With respect to each objective, the variance of the Pareto points corresponding to a weight cell is calculated. Consider a solution cell consisting of N Pareto points, CP = [P1 , ..., Pi , ... , PN ]T , each point consists of M components, Pi = [PiJ1 , PiJ2 , PiJk .... , PiJM ], where M is the number of objectives. The set of values for a specific objective k across all cell points is defined as Sjk = {x | x = PiJk , i = 1, 2, ..., N }, where N is the number of points in the cell, N ≥ 2. The information criterion can be defined as a vector whose components are the individual variances for SJ1 , SJ2 , ..., SJM : IC = [σS2J1 , σS2J2 , ..., σS2JM ]

if Cw [i] in CHIM do Solve an NBI sub-problem using Cw [i] to obtain corresponding Pareto point Pi end for Step 5: Using obtained Pareto points, construct a Pareto cell CP : CP = [P1 ... Pn ], n ≤ 4. Then the following condition is checked: if Stopping criterion is not activated do (1) Add CP points to S (2) Divide: construct four daughter cells Cd1 , Cd2 , Cd3 , Cd4 such that: Cd1 Cd2 Cd3

(9)



  Cw [0] + Cw [1] Cw Cw [0] + Cw [3] = Cw [0] 2 4 2    Cw [1] + Cw [2] Cw [0] + Cw [1] Cw = Cw [1] 2 2 4   Cw [2] + Cw [3] Cw Cw [1] + Cw [2] = Cw [2] 4 2 2    Cw [0] + Cw [3] Cw Cw [2] + Cw [3] = Cw [3] 2 4 2

These variances reflect the level of trade-offs with respect to every objective in the Pareto segment concerned. One way to characterize the trade-off levels in a problem with 2 multiple objectives is that the algorithm accepts σlow and 2 σhigh from the user. For a Pareto segment corresponding 2 2 to a weight cell, if min IC < σlow and max IC < σhigh , the stopping criterion is activated and the weight cell is considered ”conquered”.

(3) Call the recursive function four times by updating the input to step 4 such that: Cw = Cd , Cd ∈ {Cd1 , Cd2 , Cd3 , Cd4 }. else Conquer : exit, stop exploring current segment. Step 6: When all recursive calls are exited, produce S.

It should be noted that the stopping criterion can be extended to M objectives problem, in such case the weight cell shape is a hypercube of 2M −1 points. Moreover, the criterion can be used for alternative recursive schemes for 3D problems, e.g., triangular schemes. Pareto points outside the CHIM can be missed depending on the choice of the weight cell shape. Finally, for an overview of the algorithm structure, see Algorithm 2.

optimization problems are solved simultaneously. Also, Pomodoro contains a collection of algorithms for solving MOOPs including NBI (Vallerio et al. (2015)). Obtaining the gradients in the solution process is done by CasADi (Andersson et al. (2012)), an open source software for solving ODEs automatic differentiation. Pomodoro is written in Python while CasADi is written in C++ with a frontend to Python.

Cd4

3.2 Software

4. RESULTS

The software tool used in this study for tackling the MOOCP is Pomodoro (Bhonsale et al. (2016)). It contains a specialized library for solving dynamic optimization problems. For this case study, the orthogonal collocation method (Biegler (2010)) has been applied for solving the optimal control problem. It is a first discretize and then optimize approach in which the simulation and the

4.1 Case study: temperature control of a plug flow reactor In this case study, the aim is to find the optimal profile of the jacket fluid temperature for a tubular reactor in which the exothermal, irreversible reaction ( A → B) occurs. The dynamic system is described by the following equations (Logist et al. (2008)):

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α dx1 = (1 − x1 )eγ/(1+x2 ) dz v (10) αδ dx2 = (1 − x1 )eγ/(1+x2 ) + β(u − x2 ) dz v with x1 and x2 the dimensionless reactant concentration and the reactor temperature, and u the temperature profile of the reactor’s jacket. Full values of the parameters are available in Logist et al. (2008). The objectives of the problem are maximizing the conversion, J1 , and the heat recovery, J2 . The problem is transformed to a minimization problem by multiplying the objective functions by -1. Consequently, they are formulated as follows (Logist et al. (2008)): J1 = CF(1 − x1 (zf )) β zf (11) J2 = (u(z) − x2 )dz zf 0 The MOOCP can then be formulated as follows: min{J1 , J2 } u

(a) Pareto front representa- (b) Pareto front representation tion generated using NBI, 199 after being filtered using a points smart filter with a normalized specification of 0.005

(12) (c) Pareto front representation produced directly using a D&C algorithm of normalized specification of 0.005

subject to system dynamics and following constraints: umin ≤ u(z) ≤ umax x2,min ≤ x2 (z) ≤ x2,max x(0) = [0, 0]

(13)

The problem can be treated as a three objectives optimization problem by considering the reactor length zf as a variable and aiming to minimize it, such that J3 = zf The problem can then be formulated as follows: (14) min{J1 , J2 , J3 } u,zf

The following constraints are added to the bi-objective case: 0.5 ≤ zf ≤ 1 (15) x1 zf ≥ 0.9 4.2 Numerical results A Pareto front with adaptive resolution is obtained both by smart filtering a dense representation and by the divide and conquer algorithm for the plug flow reactor case study. As observed in Figure 6(a), the Pareto front of this MOOCP is characterized by a relatively sharp change in slope in the middle of the curve, indicating a high tradeoffs level in this region of the Pareto front. This should be reflected by high density of points in this segment in the filtered representation. Using a smart filter of a normalized specification 0.005, a 199 points Pareto front is filtered. This Pareto front is the least dense Pareto front for the smart filter to be functional; at least one point has been removed for every pair of neighboring points in the filtered representation. Figure 6(c) shows that a representation with similar quality can be obtained directly using a D&C algorithm with the same specification. The main advantage of using the D&C algorithm as seen in Figure 5 is that a lower amount of overhead points is produced, 51 versus 144 points, resulting in an improvement of 193% in speed when using a smart filter with the least dense Pareto representation. The gain in speed when using the D&C algorithm compared to using a smart filter increases linearly as the density of the initial representation for the filter increases. Moreover, the D&C algorithm

Fig. 4. Comparing Pareto front representations produced by smart filter and D&C algorithm for the plug flow reactor has a notably simpler solution procedure as the user only needs to enter the trade-offs level he/she is interested in to obtain the representation. On the other hand, when using a smart filter, the user has to ”guess” the number of points to produce a sufficiently dense Pareto front in order to apply the smart filter successfully. The gain in speed and the clear solution procedure make the D&C algorithm a more attractive option for tackling MOOCPs with significant slope variations. The 3D extension of the

Fig. 5. Gain in speed when using a D&C algorithm compared to a posteriori filtering of a dense Pareto front with the smart filter D&C algorithm is tested on the plug flow reactor case study after adding a third objective of minimizing the reactor length. The stopping criterion specifications is set 2 2 = 0.02 and σhigh = 0.2 Figure 6 reflects higher to be σlow concentration of points in the curved region at the center of the Pareto front than the flat regions near the anchors. It has to be recognized that while the D&C algorithm shares

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as well as the intuitive, trade-off oriented, solution procedure make the D&C concept an attractive option for future interactive MOO algorithms. REFERENCES

(a) Pareto front representation (b) Pareto front representation generated using NBI after being filtered using a smart filter

(c) Applying the D&C algorithm to the three objectives plug flow reactor case study

Fig. 6. Comparing Pareto front representations produced by smart filter and D&C algorithm for the plug flow reactor the same significance condition with the smart filter in 2D, this cannot be generalized to higher dimensions. In 3D, a smart filter works by pairwise comparison of points in a dense NBI representation. On the other hand, the D&C algorithm works by recursively dividing the Pareto front while checking the trade-off levels using the variance based stopping criterion. The D&C algorithm still keeps the advantages of the 2D case, an improvement in speed as less insignificant solutions are produced and a simple single step, trade-off oriented solution procedure. However, a quantitative comparison of the two techniques is not feasible. A representation produced by the smart filter is provided for reference. 5. CONCLUSION In this paper, the MOOCP of finding the temperature profile of a plug flow reactor has been solved with the aim of generating a Pareto front with adaptive resolution. Two approaches are compared, the posteriori analysis of a dense representation using a smart filter and the D&C algorithm. It has been found that the D&C approach has two main advantages over the smart filter approach: less overhead produced and a more clear solution procedure. It should be noted that the D&C algorithm is based on the assumption that all points produced during the solution process are global Pareto points. This might not be true in some rare cases when non-Pareto points can be produced by the MOO algorithm used. Finally, The divide and conquer approach can be most suitable for MOOCPs in chemical industry as the high computational cost associated with solving the individual SOOPs make the gain in speed by the D&C algorithm more essential. This algorithm is based on the simple mathematical idea that Pareto solutions contain information about the geometry of the curve. This idea will be exploited further in future work. Also, the enhanced computational efficiency

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