Application of multi-objective optimization in the design and operation of industrial catalytic reactors and processes

2.1 Concept of multi-objective optimization

The multi-objective optimization (MOO) concept originates from economics and was developed by the Italian economist, engineer and philosopher Vilfredo Pareto. First let us consider the definition of multi-objective optimization of a minimization problem (here and throughout the chapter we will discuss minimization MOO problems, since any maximization problem can be converted into a minimization one quite easily):

subject to:

where n is the number of objectives, K and J are the number of inequality and equality constraints respectively, x is a vector of decision variables in a search space S. A general solution for such an optimization problem is a set of points instead of a single one such as in SOO problems. However, in some special cases, a single point solution is also possible, which we will not consider for this trivial case. The set of points is called a Pareto set (front or distribution).

Definition of a Pareto optimal point

A point x is called a Pareto optimal point if and only if such a point x* doesn’t exist in a search space in which I_i (x*) is “better” than I_i (x) for all objectives simultaneously. By “better” it is necessary to assume mathematical operators ≤ or ≥ depending on the particular problem formulation.

A Pareto set can be presented in terms of decision variables (set of x) or objectives (set of I(x)). For a better understanding, let’s illustrate a Pareto concept for a two objective function problem (Fig. 3). If the problem requires simultaneous maximization of both objectives A and B, Fig. 3 describes the Pareto set obtained with respect to decision-variable limits and equality and inequality constraints. If we move from point 1 to point 2, objective A is increasing (desired) while objective B is decreasing (undesired). It is said that these two points like any other points on the curve are non-inferior (non-superior or equally good) to each other. If we move from point 3 in the direction of Pareto, one can see that both objectives A and B are improving, thus point 3 is not a Pareto point.

Fig. 3

Pareto set for two conflicting objectives.

2.2 MOO methods

When we have determined a solution for a MOO problem in the form of a Pareto set, let us turn our attention to methods utilized for its search. There are a number of different techniques; later, we will provide the accepted classification of them for better understanding [2–4]. The classification is based on a decision maker’s (DM) role in the optimization search. Here the DM is a person familiar with the formulated problem; he/she can impact on the preference of objectives or solutions. Therefore, methods are divided into:

1. no-preference;

2. a priori;

3. a posteriori;

4. interactive.

The first group excludes any influence of the DM on a search of Pareto points while the next two take it into account. The latter case is a group of currently developing methods in which DM is directly involved in the optimization search and is able to alter the preferences until the best solution is found. Just note here as a remark that the classification is not strict because the same methods can be referred to by more than one group; this will be shown later. From here we provide further a concise review of introduced methods.

3.1 Neutral compromised solution

The neutral compromised solution method allows for finding optimal solutions “somewhere in the middle” of a Pareto set. To apply this technique, it is required to define the norm in an objective domain, which will be a measure of distance for the “middle” solution and the selection of a reference point from which the distance shall be minimized [5]. However, we can say that the DM expresses preferences by choosing the norm and reference point, but he/she is not doing it in an explicit way. For MOO problems, it is necessary for all objectives to be of the same dimension or dimensionless. The commonly used norms are p-norm, Chebyshev norm or an augmented Chebyshev norm. The following problems are to be minimized respectively:

where ε is a small number > 0. The denominator in each term plays the role of a scaling factor for minimizing the distance between upper I_i,up and lower I_i,lower values for each objective function. Also, a method of global criterion is one of such methods but will be considered in section on a priori methods below with some remarks.

4.1 Method of Weighted global criterion

This method with some variations is the most popular technique for a MOO. The idea is to transform objective functions into a single one, thereby scalarizing the search space. In the most general form, this method can be written as

A scalarized function represents the sum of composite functions of objective I_i (x) and weighting factor w_i. The latter itself is a measure of the DM’s preferences for a particular objective. Usually weighting factors are assigned in such a way that ∑j=1nwj = 1 and w_j > 0. In a simplest form, the expression (3) can be written as a weighted exponential sum [4]:

Note that in the case of Eq. (2) with p = 1 (because of its simplicity), it is called the method of a weighted sum and widely used in applied chemical engineering problems [6].

Some other modifications are required for the idea of a utopia point, I^utopia(x); the imaginary point in a search space where all the objectives reach a minimum value simultaneously. The aim is to minimize the weighted distance between the objectives and that point [7]. Different metrics can be used as a distance measure. Often this group of techniques is called weighted metrics [3]. Here we provide some of them:

Note here that instead of a utopia point, the DM can determine a set of objectives that one desires to reach. This makes sense from a practical point of view, or when the real utopia point is unknown.

Remarks: A group of weighted global criterion methods is also a popular a posteriori technique. By varying the weights, it is possible to obtain a Pareto set instead of a single point. These methods always converge to a Pareto optimal solution, but an entire Pareto set can not be found if the problem is not convex [3]. If one assigns all weights, w_i, equal to 1, the approach can be classified as no-preference; but the drawback remains the same. In addition, the magnitudes of objective values should be commensurable with each other to avoid overemphasis of one over the other. Hence, normalization is required for applying this technique [3].

4.2 Lexicographic method

Lexicographic methods require the DM to sequentially organize objectives from 1 to N in terms of preferences [8]. The following problem has to be solved [4]:

subject to:

where k is the function order in a preference list, Ik(xk*) the constraint’s limit received at k^th step. The first objective in the list should be minimized with the original constraints. If the DM obtains a single solution, one can accept it as an optimum. If not, the new constraint Ik(xk*) has to be accepted to keep the k^th objective’s optimal values. The procedure continues with the next objective function (e.g. second function in a list, third function in a list, etc.), until the optimum is reached.

In reality, it is often difficult for a DM to distinctly organize objectives in an order of importance on account of the complexity of a MOO problem. Another drawback with this technique is that a unique solution is often found before the best optimal solution is reached. It means that some of the objectives are not taken into consideration at all [3].

4.3 Goal Programming (GP)

This method was developed by Charnes and Cooper [9]. The DM defines a set of goals G that should be achieved for each objective I_i (x). Even if all these goals are unattainable simultaneously, it is still desired to reach them “as close as possible”. It is proposed to minimize the distance between vectors I(x) and G. Such a weighted GP problem formulation is written as:

where w_i is a weighting factor for objective I, is a deviation of objective I_i (x) from goal g_i. The formulation of this goal programming problem doesn’t necessarily require the solution to be Pareto optimal. The solution obtained can be referred to as: (a) efficient; (b) inefficient; or (c) an unbounded solution. An efficient solution belongs to a Pareto front while an inefficient solution can be improved for two or more objectives simultaneously. The latter case is a solution located too far from a Pareto front [10].

Setting goals is a clear approach for a DM (unlike, for example, the use of a utopia point in the global criterion method). However, the further procedure for an optimum search is not necessarily easy, e.g. weights assignment can be more difficult. Some GP methods are combined with a lexicographic method, where deviations are structured in preference order and then minimized. The DM has to be aware of all the drawbacks of GP methods and choose the proper technique for finding an optimal solution.

5.1 ε-Constraint Method

The ε-constraint method is a non-scalarizing approach. The original idea was reported by Yacov Haimes [11]. The more comprehensive explanation is provided by Chankong and Haimes [12]. It is proposed to solve the following n-objective problem (Eq. (2)) to define a Pareto set:

subject to:

where ε_m are user defined constraints. Note that any of the objective functions can be chosen for minimization. By varying ε_m, the Pareto set can be reached. It is reported by authors that the current method can deal with non-convex problems. However, drawbacks still exist. The choice of ε_m is not as easy for DM; the technique also significantly increases computation time if the total number of equations (objectives and constraints) is relatively high.

7.1 About binary-coded variables

Preceding the explanation of GAs’ working principles, one has to know about binary-coded variables. The most common representation of a variable utilized by GAs is a binary string. That variable is simply a certain length sequence of ones and zeros (e.g. 1001). If a user deals with continuous variable (e.g. length, product yield, time, etc.), it is required to discretize the variable. The procedure is quite simple. For example, the decision variable x ∈ [X_min, X_max] has to be mapped into a binary string. The user decides to use 4 bits for each variable, in other words, the length of binary string is set to 4 digits. Thereby, we have 2⁴ = 16 possible combinations of strings. Lower and upper bounds are assigned with the values X_min → 0000 and X_max → 1111. All other values are mapped in between these two values (Fig. 4).

Fig. 4

Mapping of real value into binary variable.

The precision of discretization of variables is directly dependent on the string length; the longer the length, the more binary variables can be mapped between the lower and upper limit. The precision π may be calculated as [17]:

7.2 Simple Genetic Algorithm (SGA)

For a better understanding the GAs’ principle, let us consider a simple genetic algorithm (SGA) first. The main components of a SGA include: (a) reproduction; (b) crossover; and (c) mutation of genetic operators. At the beginning, the initial population is generated randomly. The population is a set of individuals; each of which represents a single decision variable (or a vector). The reproduction operator is applied to the population to create a “mating pool”. Individuals with a higher objective function value have a higher chance of being copied into a matting pool. Classical and simple way to perform a reproduction operator is a roulette wheel [17].

Once the mating pool is formed, crossover and mutation operations are executed. In a single point crossover, two individuals (called parent chromosomes) are chosen randomly to exchange “information” with each other. They swap binary sequences after the arbitrary position p (which is randomly selected) and then generate “daughter chromosomes” (Fig. 5).

Fig. 5

Representation of single point crossover between two binary strings.

Mutation is also aimed at altering the daughter chromosomes’ binaries but in a different manner. Like mutation in nature, it occurs with a very small probability. Mathematically, it alters one cell in a sequence each time from 0 to 1 or vice versa. It is absolutely necessary to keep diversity in the population [20]. For example, let’s assume a case where all individuals in a population have 0 at k^th position, under these conditions the crossover operator cannot create 1 at this point. Mutation allows one to overcome this issue.

The best n daughter individuals are taken to form a new mating pool where crossover and mutation are carried out again. This procedure repeats until the termination criterion is satisfied. Below we provide a generalized scheme of SGA (Fig. 6).

Fig. 6

Simple genetic algorithm.

7.3 Use of GA in MOO

If one has a SOO problem, it is easy to choose the best solutions from the population by comparing the single objective values of individuals. When one deals with multiple objectives, it is not clear how to compare them. To deal with this, Goldberg introduces the concept of non-dominated vectors [17]. Vector a is said to be less than vector b if and only if these two conditions are satisfied simultaneously:

1. all components of a are less or equal to corresponding components of b;

2. at least one component of a is strictly less than corresponding element of b;

or, in other words (for a minimization MOO problem), a dominates b. If for the vector a there is no such vector c that dominates it, vector a is called non-dominated. From this point of view, a Pareto set is a non-dominated set.

Recent GAs’ modifications are more complex than a SGA. There is diversity of different algorithms presented in the open literature: Vector Evaluated Genetic Algorithm (VEGA) [21], Multi-objective GAs (MOGA) [22], Strength Pareto Evolutionary Algorithm (SPEA) [23], Niched-Pareto GAs [24], Predator-Prey Evolution Strategy [25], Rudolph’s Elitist Evolutionary Algorithm [26], NSGA-II [27], Differential Evolution (DE) [28] based methods and many others. We will not discuss most of them here, but refer readers to original sources.

We would like to emphasize one of the state-of-the-art algorithms: the non-dominated sorting genetic algorithm II (NSGA-II). The reader can note that this algorithm was used in a majority of MOO problems solved in the literature (Table 1). After development by Deb [27], it has been widely propagated in optimization problems for chemical engineering as well as for many other fields. NSGA-II is notable for its characteristics, especially its ability to find diverse solutions close to a real Pareto set and the speed of convergence [27]. Here are the elements that contribute to its high performance.

1. This algorithm uses the concept of elitism. After mating pool formation, N parents and N daughters’ chromosomes are united into a single group of 2N. Selection is carried out over this pool and not only from the original mating pool. If parents are better than their daughters, it allows them to not be excluded them from population, but carry on in the next generation. This allows diversity.

2. The Non-dominated Sorting Approach is used as a selection procedure. It divides an entire population into groups of non-dominated individuals (non-dominated fronts). Any solution in front 1 is superior to any solution in front 2, and so on.

3. To maintain the diversity of the population, authors introduced crowding distance. If some region in an objective domain is too populated with individuals, it is reasonable to exclude some of them from the population. The crowding distance of point i represents an average side length of n-dimensional cuboid in objective space, drawn out around point i where two neighboring points are taken as vertices. The higher the crowding distance, the less crowded a region. Points from the same front but with less values than this parameter have less chance to carry on into the next generation. A step-by-step guide to execute NSGA-II, a performance of the algorithm in test problems or other characteristics can be found elsewhere [19, 27].

7.4 Constraint handling in GA

There are different techniques aimed at constraint handling in GAs. Constraints impose extra conditions on a MOO problem, thereby limiting the search space. Based on this, solutions are divided into feasible and infeasible regions. An infeasible solution cannot be neglected in GAs in order to maintain diversity. Even if a particular solution violates constraints, it should have a chance to remain in the population in order to have a chance to move to a feasible region [19]. To do this, many techniques evaluate the extent of violation from a feasible region. Two noteworthy techniques are discussed below, which have been utilized more frequently while solving applied MOO problems in chemical engineering.

4.9.1 Petroleum Processing Engineering

Noticeable contribution to GA and its application to MOO in petroleum processing was made by Kasat et al. [35]. They introduced a genetic operator called a Jumping Gene (JG). A JG (or transposon) mimics the real nature phenomena discovered in 1987 by McClintock. The main point of the discovery was that transposon is a DNA sequence which can randomly migrate among chromosomes and replace existing sequences. One of the transposon roles is providing for diversity in genotype. The jumping gene was introduced as a binary sequence that can replace a part of the original individual. First, the chromosome is checked for carrying JG out with some probability P_JG. If the condition is satisfied, two positions of binaries, p and q, are randomly chosen in the current chromosome with a total length l_str (p < q ≤ l_str). The random binary string of length (q − p − 1) is generated and inserted between p and q. Another alternative for JG is to inverse binary sequences between chosen locations. It is reported that these two modifications have the same performance. Authors suggested to implement a JG operator after mutation and combine this operator with NSGA-II (NSGA-II-JG). Using benchmark problems, they demonstrated that the proper choice of P_JG (≈ 0.5 or more) provides a faster convergence to a Pareto front and better distribution of the population along it. The JG concept was developed in some later works [36–38] where new modifications with improved characteristics were introduced. We will not describe all of them in detail; we refer readers to the original articles.

In their work, NSGA-II-JG was applied for multi-objective optimization of an industrial fluidized catalytic cracking unit (FCCU). The FCC is very relevant for the petroleum processing industry since it is the main process for gasoline production. Industrial FCCUs consist of a reactor-riser and catalyst regenerator. Authors used a five-lump kinetic scheme with two steady state models of these units, previously developed and verified by Arbel et al. [39] and Krishna and Parkin [40]. The two objectives were to maximize the yield of gasoline from the FCCU and minimize coke content on the catalyst. The decision variables used were feed temperature and the catalyst flow rate into the reactor, as well as air temperature and flow rate into the regenerator with lower and upper bounds based on process technology. Again, the problem was solved with both NSGA-II and NSGA-II-JG. The obtained results were compared with their previous work [41], where optimization was carried out with original NSGA-II. The generated Pareto fronts ware similar but with a wider distribution of solutions for NSGA-II-JG. Authors emphasized computational efficiency and the speed of convergence and proposed methods for MOO problems in chemical engineering.

Some other petroleum processing MOO are presented in the open literature. Various researchers investigate MOO problems for different types of naphtha catalytic reformers, such as conventional catalytic naphtha reactor (CR) or the thermally coupled fluidized bed naphtha reactor (TCFBNR) [42–45]. Besides the designation for feed conversion into products, the naphtha reforming process could be aimed at a refinery’s hydrogen supply. Because of this, objectives can vary from one reformer to another, depending on their roles in particular productions. In the majority of research, it was proposed to maximize the production of aromatic compounds and hydrogen while other objectives differed. However, only Weifeng et al. [42] treated objectives directly with a Neighbourhood and Achieved Genetic Algorithm (NAGA) to generate an entire Pareto set. Other researchers used summation method to form the SO function, and solve it with methods of differential evolution. All of them could provide improved objectives and propose a better operation conditions for naphtha reformers than current ones.

4.9.2 Steam Reforming

The first multi-objective optimization of a side-fired steam reformer was performed by Rajesh et al. [46]. They combined the kinetic model of main reactions, a heat transfer model through a furnace tube wall and the diffusion model in a catalyst pellet. The complex model was utilized for optimization. Authors assumed that the rate of hydrogen production was kept at a required level. The main operational costs of steam reforming are: (a) methane feed; (b) furnace fuel; and (c) steam. The first objective used was the minimization of the methane feed rate. The second objective used was the maximization of CO in the reformer outflow. The reason for this was that the higher the CO % in outflow, the more heat can be generated at the shift converter and, consequently, more steam can be produced in heat exchangers at the exit of the unit. Decision variables used were the temperature and pressure of the feed flow and its rate, steam/methane ratio (S/C), recycled hydrogen/methane ratio (H/C) and temperature of the furnace gas. Additionally, the process was constrained by a maximum possible furnace wall temperature. Thus, they came up with two objective problem formulations subjected to lower and upper boundaries for decision variables based on process technology and one constraint. The objectives and constraints were treated in the form of a penalty function. A Pareto set was obtained. It was noticed that most of the decision variables didn’t differ significantly for an entire Pareto, but that the S/C ratio makes a significant contribution to the objectives value and for the Pareto distribution. They also studied the effect of catalyst deactivation on the change in optimal parameters. This change wasn’t important due to thermodynamically controlled reactions. Generally it was shown in the work how to apply MOO with GAs to optimize the steam reformer. More precise problem formulations (e.g. constraints, process parameters limits, etc.) for a particular steam reformer can bring different results.

The work of Nandasana et al. [47] extended the optimization of a steam reformer dynamic regime. The existing model was modified as a non-steady state to study the effect of disturbances on the reformer. The objectives of MOO were to minimize the reduction of loss of the total (a) hydrogen and (b) steam production caused by a sudden change in some process parameters. Two disturbances were independently introduced to the system: a step decrease of methane feed; and a drop in feed temperature. Authors reported the high computational intensity of MOO problems. They could carry out 9 and 18 generations for the problem respectively.

In more recent research, Ebrahimi et al. [48] performed MOO of a steam reforming arrangement for the synthesis gas production (mixture CO and H₂). They modeled two combinations of top-fired methane steam and auto-thermal reformers, parallel and in series. They formulated objectives similar to Rajesh et al. [46]: (a) maximize production of syngas; (b) minimize furnace fuel consumption; and (c) minimize CO₂ releases. Like in previous research, the main constraint for the steam reforming operation was the maximum tube wall temperature. Obtained Pareto sets showed that a parallel arrangement is superior for higher syngas production while the configuration in series allows for a decrease of fuel consumption and CO₂ release.

4.9.3 Polymer industry

MOO in polymer manufacturing has been an intensive research field in so far as such processes with multiple objectives result in more meaningful solutions. Many works had been aimed at the optimization of polymerization reactors’ and processes’ performances. One of the first MOO problems was solved for the Nylon 6 reactor by Mitra et al. [49]. They utilized a kinetics scheme combined with a batch reactor model. Objectives to minimize were (a) reaction time and (b) undesired product concentration. Constraints implemented were desired monomer conversion and the degree of polymerization. For a solution of a current MOO problem, they used NSGAs, and constraints were handled by penalty functions. Authors varied different GA parameters (e.g. number of generations, crossover probability) to show the stability of obtained Pareto sets because no significant changes were observed. Earlier authors tried to carry out SOO for the same system with Pontryagin’s principle; this failed due to some numerical complexity. It was emphasized that NSGA allowed for the overcoming of previous problems and generation of a reasonable set of optimal solutions.

An interesting discovery was found in Bhaskar et al. [50]. The authors carried out MOO for a polyethylene terephthalate wiped-film reactor. They chose to minimize the (a) acid and (b) vinyl end group concentration in the polymer product for a better polymer quality. Optimization with NSGA showed that the problem had a unique solution instead of a Pareto set. To confirm the results, they carried out the SOO problem for each objective independently; it resulted in the same solution. Later, in another work by Bhaskar et al. [51], it was pointed out that the unique solution was dependent on the seed random generator (a number used in computer code to execute randomization). By varying this number, they always obtained different single optimal points. Also they showed that NSGAs didn’t obtain an optimal point if more than one decision variable was used. The conclusion was made that NSGAs failed to converge to global optimal solutions and some other search technique is required. Current issues were resolved in Babu et al. [52], in which the authors used a multi-objective differential evolution (MODE) for the optimization of the same system. Different MOO cases were considered and MODE converged to a Pareto front in each of them.

Many other similar works for optimization of industrial continuous or batch polymerization processes are made. In general, it can be noted that the main objectives in MOO problems could be:

1. maximization of monomer conversion;

2. minimization of the concentration of side products or some functional groups; and

3. maintainence of quality-related parameters on a desired level (e.g. molecular weight, degree of polymerization).

Some MOO problems include a design stage and significant improvement in the reactor’s performance is reported. For example, in the work by Agrawal et al. [53] the operation of another type of polyethylene reactor, tubular, was optimized. Two objectives were to maximize monomer conversion and minimize the concentration of side products. The reactor and jacket diameter, and length of reactor zones were included as decision variables. So they carried out optimization of both design-stage and operation-stage optimization. They reported improved results in objectives when compared to the case when design variables are not included into the MOO problem [54].

For MOO problems in polymerization reactors and processes, researchers mentioned significant computational issues such as: to obtain global optimal solutions, large computation time is required. The mathematical models are relatively complex for such processes, because they include mass, heat and momentum balance equations that could include comprehensive equations involving partial derivatives or other intensive mathematical variables involving highly non-linear equations. To carry out MOO using GAs, it is required to perform a number of simulation runs to evaluate objectives for one population, while a MOO search requires a number of generations to obtain convergence to a global optimal Pareto front. All together, it increases computational time up to some hours or even days and hence necessitates the use of supercomputers.

Besides the polymerization processes, there are works made in monomer production. Most commonly used objectives used in these MOO problems are monomer’s yield and selectivity. Among the works, there is a group of researchers who optimized styrene production. They carried out various MOOs for different reactor types using different algorithms. Firstly, Yee et al. [55] performed two-objective optimization for the operation of adiabatic and steam-injected reactors. The same work was done by Li et al. [56], but including reactor design parameters into decision variables. Both used NSGAs with a penalty function approach and obtained smooth Pareto fronts. Babu et al. [57] performed a MOO for an adiabatic styrene reactor with the same problem formulation but used a multi-objective differential evolution. They reported a better Pareto front. However, it can be noted that MODE hadn’t affected the Pareto front significantly, but there are still improvements in objective values for some MOO cases. Tarafder et al. [58] compared performance of three types, single-, double-bed and steam-injected reactors, with NSGA-II in a three-objective problem formulation (maximization of yield and selectivity of styrene plus minimization of heat-exchanger duty). They showed better objective values for the double-bed reactor. It can provide better productivity for styrene with a higher selectivity at the same time.