# Multi-Objective Optimization Applications in Chemical Process Engineering: Tutorial and Review

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}(

**x**) is the objective or performance criterion or target to be minimized or maximized,

**x**is a vector of decision variables (also known as parameters), which can take any value between practical values selected as lower and upper limits of

**x**

^{L}and

**x**

^{U}, respectively, and g(

**x**) and h(

**x**) are inequality and equality constraints, respectively. There can be none or several inequality and/or equality constraints. The optimal values of

**x**are denoted by

**x***, and the optimal of the objective is F

_{1}(

**x***).

_{1}(

**x**), F

_{2}(

**x**), and F

_{3}(

**x**). In these, F

_{1}may be minimized or maximized, F

_{2}minimized or maximized, and so on. Whereas a SOO solution yields a single value for each component of

**x***, MOO often yields a series of values for each component of

**x***, which are referred to as a set of non-dominated solutions or Pareto optimal solutions. Corresponding to each of these solutions, there will be one set of values for objectives such as F

_{1}(

**x***), F

_{2}(

**x***), and F

_{3}(

**x***). A plot of these optimal values of objectives against each other in a two- or three-dimensional plot is referred to as a Pareto optimal front. For visualizing this front, more than one plot may be required if there are four or more objectives. Examples of MOO solutions are provided in Section 2 of this paper.

^{L}and x

^{U}, and also that the constraints g(x) and h(x) are convex), Providing the problem could be written so that it satisfies the requirements for convexity, the SOO solution could be obtained directly either using an in-built SOO solver (such as in Aspen Plus) or by running the modeling software from within a mathematical optimization software platform such as Matlab.

_{2}, the main exceptions being agriculture and land clearing. CPE process synthesis and design must weigh up economic performance against global warming potential, taking into account the whole life cycle rather than just a single process in isolation. Therefore, there is a need to widen the boundaries of process optimization. Whereas previously, it might have been around a single unit operation or a single plant, with global warming potential and other pervasive environmental problems, it is now necessary to calculate the results over the whole life cycle inventory, more than often using specialist Life Cycle Assessment software.

## 2. Procedure for MOO

**Step 1. Process Model Development and Simulation:**This is an essential and important step for any process optimization. The process model refers to a set of equations that correctly predict the response of the physical process under study. Equations can be algebraic and/or differential equations, and variables in them may be part of the system design and/or come from the operating conditions of the process. Examples of design variables are the reactor volume, heat exchanger area, number of stages in a distillation column, and membrane area in a membrane separation unit, whereas examples of operating variables are the reactor temperature, compressor outlet pressure, reflux ratio in a distillation column, and feed and permeate pressures in a membrane separation unit. Sometimes, design and operating variables together are referred to as design variables or parameters. Some of these variables (e.g., number of stages in a column) can be the discrete or integer type.

**Step 2. Formulation of the Optimization Problem:**This step involves the following tasks: (a) selection of objectives and the development of equations for them; (b) decision variables and their bounds; and (c) identification of constraints and the development of equations for them. Objectives are criteria that quantify the performance of the process. They can be related to economics, energy consumption, environmental impact, controllability, safety, etc. MOO allows the inclusion of any number of objectives. The selection of objectives is according to the desired aspect of process performance. Obviously, more than one objective can be chosen in MOO, whereas SOO allows the selection of only one objective.

**Step 3. Solution of the Optimization Problem:**This step requires the selection and use of a MOO technique for solving the formulated problem. As outlined in a later section, many programs for MOO are now available, and some of them are free. So, there is no need to develop a program for solving the formulated problem unless a new and/or improved technique for MOO is being studied. However, there may be a need to interface the available MOO program in one platform (e.g., MS Excel or Matlab) with the simulation model on another platform (e.g., Aspen Plus). The interfacing of Aspen Plus with Matlab and Aspen Hysys with MS Excel is described with an example in [4] and [2], respectively.

- Carefully read and follow the instructions that come with the MOO program.
- Learn how to use and also test the MOO program before using it for any application for the first time. For testing, choose a mathematical optimization problem with a known solution from the literature and/or the example provided with the MOO program and reproduce the known optimal solutions using the MOO program with default values in it for algorithm parameters.
- Be careful in correctly providing/entering the required inputs such as objectives, decision variables, bounds, and constraints of the application problem to the MOO program. Any wrong inputs may lead to failure of the program or incorrect results from the program. First, test the MOO software with a relatively narrow range of decision variables around a known solution. If successful, the range of decision variables can be widened as required.

**Step 4. Review of Optimization Results Obtained:**Two or more objectives in the formulated MOO problem are likely to be conflicting (i.e., improvement in one objective is accompanied by the worsening of another objective). Hence, unlike one or a few optimal solutions from SOO, MOO gives many optimal solutions that are known as Pareto optimal front (or solutions) and as non-dominated solutions. Results from MOO can be presented in three spaces, namely (1) objective space, (2) decision variable space, and (3) objective versus decision variable space; the first two spaces can be in more than two dimensions, depending on the number of objectives and decision variables. Objective space plots are not relevant in the case of SOO, since it involves only one objective.

_{2}removal from natural gas. Figure 2a depicts the optimal trade-off of the two objectives: maximize methane purity and maximize methane recovery, simultaneously. Figure 2b is the plot of decision variables—membrane 1 area and membrane 2 area—in the decision variable space. It shows that the optimal value of the membrane 1 area is at its upper bound of 10,000 m

^{2}, which means that both objectives improve if this bound is increased further, but within a practical limit. On the other hand, the optimal value of the membrane 2 area varies from a low value (but above its lower bound) to its upper bound of 10,000 m

^{2}. The variation of objectives with decision variables is presented in Figure 2c–f. For example, Figure 2c is a plot of methane purity versus the membrane 1 area. Figure 2c,d indicates that the membrane 1 area affects both objectives in the same direction, whereas Figure 2e,f show that the increase in the membrane 2 area increases methane purity but decreases methane recovery.

**Step 5. Selection of One Optimal Solution:**As noted above and assuming some conflict among the objectives used, MOO gives many non-dominated (Pareto optimal) solutions, which are equally good from the point of the objectives employed. However, only one of these solutions is required for implementation for the application under study. Hence, this final step in MOO involves the selection of one of the non-dominated solutions for implementation. Note that this selection is performed in Step 5 after reviewing the obtained non-dominated solutions in Step 4. It should not be confused with the selection operation used in evolutionary algorithms such as genetic algorithms and differential evolution. Some MOO methods (e.g., the weighted sum technique of SOO approach described later) find only one Pareto optimal solution. In such a case, this step of selection is not needed; however, such MOO methods need inputs on the relative importance of objectives in the earlier Step 3 itself.

- (i)
- normalization of values of each objective;
- (ii)
- choosing the weight for each objective;
- (iii)
- use of the MCA method for ranking or scoring of m Pareto optimal solutions;
- (iv)
- sensitivity analysis of Pareto ranking (i.e., ranking of Pareto optimal solutions) to normalization, weights, and/or MCA method used, and
- (v)
- choose the top-ranking solution for implementation.

**Summary:**The systematic application of MOO in CPE (in fact, any field) involves 5 steps in Figure 1. The first 4 steps should be carried out sequentially before repeating one or more of them, as required. Successful completion of the first 4 steps gives the Pareto optimal front (non-dominated solutions), which provides insights into quantitative trade-offs among different objectives used. In Step 5, one of the Pareto optimal solutions is chosen with or without inputs from decision maker(s) for implementation. With the exception of Step 3, the other four steps require knowledge and expertise in the domain of the application. Further, some computational background is necessary in all but Step 4 for a review of the optimal solutions found. The readily available software for MOO and Pareto ranking is covered later in Section 5.3.

## 3. MOO Application in CPE

#### 3.1. Detecting an Increasing Role of MOO in ChE

**In summary**, the number of papers with multi/bi and objective and optimization/optimisation in the title, abstract, and keywords in the journals of interest to ChE academicians and practitioners (within the subject areas of ChE and Energy in Scopus) from the year 2000 to the end of October 2019 is more than 4200. They are contributed by numerous researchers from 160 institutions in 90 different countries. A steady and significant increase in the number of papers on MOO applications in CPE can be clearly seen in Figure 3 and Figure 5. Hence, it can be concluded that MOO has been extensively employed for CPE applications and that the majority of future studies on optimization will be for two or more objectives.

#### 3.2. Selected Applications of MOO in ChE

_{2}emissions. The energy system consists of several energy sources including biomass and natural gas for the on-site generation of electricity and heat required in the building. The framework employed for this complex case study has three main stages of sampling, surrogate modeling, and optimization. Beykal et al. [23] found many Pareto optimal solutions for the bi-objective optimization problem of minimizing cost and CO

_{2}emissions by solving a series of SOO problems formulated using the ε-constraint technique.

_{2}emissions. The purpose of this network is to improve the integration of existing plants by adding new plants for capturing and converting CO

_{2}into the chemicals used by the existing plants. In a way, this is revamping an existing eco-industrial park by the addition of new plants. Panu et al. [30] found a number of trade-off solutions between cost and CO

_{2}emissions using the ε-constraint technique.

_{2}mole fractions, using NSGA-II. Before optimization, they estimated the kinetic parameters by fitting experimental data using a genetic algorithm and found them to be better than the two sets of values reported in the literature.

**In short**, there is a wide range of MOO applications in ChE, and the MOO techniques used for solving these application problems include evolutionary algorithms (metaheuristics), the ε-constraint technique, and the weight sum technique. These MOO techniques are elaborated later in Section 5.

## 4. Objectives Used for MOO

**Fundamental Criteria:**The fundamental criteria often relate to the physical and/or chemical performance of a reaction or material to quantify the performance of a reaction or separation process; some examples of fundamental criteria are the conversion, selectivity, recovery rate, product purity, production rate, etc. The conversion of a reactant is the ratio of the feed converted to the total feed used, whereas the selectivity of a product is the ratio of the desired product formed to the total reactant converted multiplied by the stochiometric factor [32]. Both of these are important to quantify the performance of an industrial reactor wherein a number of reactions are occurring. They have been employed to optimize the design of reactors such as steam methane reformer [33], a stirred tank reactor for simultaneously producing ethyl tert-butyl ether and tert-amyl ethyl ether using response surface methodology [34], and a fixed-bed reactor of methanol oxidation to formaldehyde [35].

_{2}product purity, recovery rate, and power consumption in the design of a vacuum/pressure swing adsorption process for separating CO

_{2}from flue gas. Estupiñan Perez et al. [37] used CO

_{2}product purity and the recovery rate as two objectives of MOO to validate experimentally the designed vacuum swing adsorption process for CO

_{2}capture. Reddy et al. [38] chose product yield and batch time as the objectives in the optimal design of a reactive batch distillation process. Yasari [39] applied the production rate and catalyst deactivation rate in the dynamic optimization of two types of tubular reactors for producing dimethyl ether. You et al. [40] defined the separation efficiency indicator to optimize extractive distillation processes; this particular indicator quantifies the ability of the extractive section to discriminate the desired product between the top and the bottom parts of the extractive section.

**Economic Criteria:**As expected, these criteria are often used objectives in the optimization of CPE applications, to design the process system having the most economic viability. The total capital cost (TCC) and annual operating cost (AOC) are the two important economic criteria for designing a new system or retrofitting/revamping an existing system. The former refers to the fixed cost for setting up a new system or modifying an existing system. TCC is often estimated using capital cost correlations, which are functions of the size and operating conditions of the equipment. The installed cost (and not purchase cost) of equipment must be used for a realistic estimation of TCC, since the installed cost can be 1.5 to 4 times the purchase cost (e.g., see Table 16.11 in the book by Seider et al. [41]). AOC refers to the sum of variable costs such as steam, coolant, electricity, raw material, and catalyst costs.

_{2}emission, and thermodynamic efficiency as the objectives. Shang et al. [47] conducted a tri-objective optimization with TAC, CO

_{2}emission, and separation efficiency as the objectives to optimize an extractive distillation for separating ethanol–water mixture using a deep eutectic as the entrainer. TAC as an economic objective, Eco-indicator 99 (ECO99) as an environmental objective, and Individual Risk (IR) as a safety criterion have been used in the MOO of alternate reactive distillation processes for ethyl levulinate production [48].

**Energy Criteria:**The quantity of energy (in the form of steam, electricity, and/or fuel) is used to maximize the energy efficiency of process systems via MOO. Energy used in different forms can be combined into a single quantity using the efficiency of producing them. For example, Singh and Rangaiah [50] used efficiencies of 0.9 and 0.36, respectively, for steam and electricity production, and they justified these values. Even then, the quantity of energy, which is based on the first law of thermodynamics, does not explicitly consider the quality of energy based on the second law of thermodynamics, which can quantitatively measure the available and unavailable energy of the system and thus enhance the energy-based analysis. Hence, energy consumption and exergy loss (or efficiency) are two alternate criteria in the optimization of chemical processes. Details for calculating exergy and exergy efficiency can be found in the book by Seader and Henley [51]. Belfiore et al. [52] optimized the natural gas regasification process for minimizing investment and maximizing exergy efficiency simultaneously. Safari and Dincer [53] employed total exergy efficiency and methane production rate as the criteria to optimize an integrated wind power system for hydrogen and methane production.

**Environmental Criteria:**With the increasing concern on environmental pollution and global warming, sustainability and life cycle analysis (LCA) have become important for designing sustainable and environmentally friendly processes. Goedkoop et al. [56] proposed ECO99 for evaluating the sustainability and quantifying the environmental impact of the process, which is consistent with the philosophy of LCA and sustainability in the design of chemical processes. This methodology is based on evaluating three major damage categories: human health, ecosystem, and quality and resources depletion; and each category is divided into 11 sub-categories. ECO99 has been used as an objective in MOO studies. For example, Sánchez-Ramírez et al. [57] formulated a tri-objective optimization problem with ECO99, TAC, and individual risk as objectives for optimizing hybrid intensified downstream separation of biobutanol. The same three objectives have been employed to optimize reactive distillation processes for the eco-efficient production of ethyl levulinate [48]. Xu et al. [58] developed a vector-based multi-attribute decision-making method with weighted MOO for enhancing the sustainability of process systems.

_{2}emissions contribute significantly to global warming, and hence it is commonly employed as the criterion to quantify environmental impact and for the MOO of chemical processes. In this approach, the energy consumption including heating, cooling, and electricity duties is converted into equivalent fuel oil consumption to estimate CO

_{2}emission; more detailed calculations can be found in [59]. As mentioned above, exergoenviromental analysis does not measure the economic performance of the process, but it quantifies the impact of the designed process on environmental sustainability. Hence, it has become an important criterion used in MOO problems to design or retrofit a process [55].

**Control Criteria:**Traditionally, chemical processes are first designed based on steady-state simulation and economic criteria followed by the synthesis of their control structures. Accordingly, the control system design begins from the designed process, which may lead to poor dynamic operability under process disturbances due to the trade-off between the optimal steady state and controllability of the process. Hence, this sequential approach may require iterations between process design and control system design for resolving conflicting objectives. Alternatively, process design and control can be conducted simultaneously to resolve the trade-off between design and control.

**Other Criteria:**Process safety is an important aspect of process synthesis and design. Among safety criteria, the Inherent Safety Index (ISI) developed by Heikkilä [66] ranks the inherent safety level of the chemical process based on the main and side reactions, parameters including pressure, temperature, yield, heat of reaction, inventory, flammability, toxicity, explosiveness, corrosion, equipment type, and process structure. Hassim et al. [67] developed the Inherent Occupational Health Index (IOHI) to evaluate the process in the initial stages of research and development based on the potential of working activities and process conditions that may harm workers. Later, Teh et al. (2019) [68] extended it to the Health Quotient Index (HQI) to quantify the health risk from fugitive emissions. Further, they utilized ISI and HQI together with the Potential Environmental Index evaluated by Waste Reduction (WAR) algorithm to formulate an MOO problem for optimizing the 1,4-butanediol production process. Eini et al. [69] proposed an MOO framework that employs Quantitative Risk Assessment (QRA) together with economic cost to optimize chemical processes.

## 5. Computational Aspects

#### 5.1. MOO Techniques

_{1}, x

_{2}, …

_{,}x

_{n}), which can be continuous or discrete, lower and upper bounds on them (Equation (9)), m equality constraints (Equations (10)), and (p–m) inequality constraints (Equation (11)). For simplicity, all three objectives (Equations (6)–(8)) are taken to be for minimization; maximization objectives, if present, can be changed to the minimization type or MOO techniques (some of which are described below) can be suitably modified. Inequality constraints (Equation (11)) are assumed to be more than or equal to type; if some inequality constraints are less than or equal type, then they can be changed to more than or equal to type.

#### 5.1.1. ε-Constraint Technique

#### 5.1.2. Weighted Sum Technique

_{1}, w

_{2}and w

_{3}are suitable weights for the three objectives. Except for this combined objective, the rest of the MOO problem (i.e., Equations (9)–(11)) remains the same. Weights are usually less than 1.0, and the sum of all weights is unity (i.e., w

_{1}+ w

_{2}+ w

_{3}= 1.0). Hence, values for at least n-1 (here, 3–1 = 2) weights have to be provided. With the weighted sum of objectives, the MOO problem in Equations (6)–(11) becomes an SOO problem (with Equation (19) as the objective, and Equations (9)–(11) as bounds and constraints. Any suitable SOO technique can be employed for solving this resulting SOO problem.

#### 5.1.3. Metaheuristics/Stochastic Optimization Techniques

**Step 1—Initialization of the population**: An initial population of N individuals (or trial solutions or simply solutions) is generated, often randomly in the entire search space defined by lower and upper bounds. Values of all the objectives and constraints (both inequality and equality) are calculated at each of these N trial solutions. These N individuals form the parent population for the next step.

**Step 2—Generation of Child Population:**This step of generating a child population varies from one evolutionary algorithm to another, and it has several sub-steps. In general, it consists of selecting a few individuals from the parent population, crossover among these selected individuals, and the mutation of resulting individuals. Thus, N new individuals are generated as the child population, and the values of all objectives and all constraints are calculated at each of these N new solutions.

**Step 3—Choosing Individuals for Next Generation:**For this, parent and child populations are combined. Then, N individuals from this combined population are chosen for the next generation, considering feasibility (to ensure satisfaction of constraints), dominance (to achieve improvement in objective values), and crowding distance (for the wider spread) of solutions. One popular strategy for this selection is the non-dominant sorting based on the dominance concept (i.e., a solution is dominant over another if the former has objective values that are not inferior to those of the latter and has better value for at least one objective). In case constraints are handled by the penalty function approach, then N individuals are chosen for the next generation based on dominance and crowding distance only. In short, the chosen N individuals are better than others in the combined population and form the parent population for the next generation.

**Step 4—Checking Stopping Criteria:**The most common stopping or termination criterion in metaheuristics is the maximum number of generations (iterations). Alternate criteria based on search progress (i.e., improvement in objective values) are possible. For example, two such criteria and their evaluation for the MOO of complex processes modeled by process simulators are presented by Rangaiah et al. [86]. If the specified stopping criteria are satisfied, iterations stop; else, iterations continue from Step 2.

#### 5.1.4. Single versus Multi-Objective Approach

#### 5.1.5. Surrogate-Assisted Multi-Objective Optimization

#### 5.2. Selection or Ranking Techniques

#### 5.3. Software

## 6. Discussion

^{®}[109]) has many optimization algorithms, including MOO algorithms. This software availability and this paper should motivate and lead to the beneficial utilization of MOO in industrial practice.

- Novel applications of MOO in CPE such as for parameter estimation and control
- MOO of CPE applications with many objectives (and not just two or three objectives) and comprehensive analysis of the optimal results obtained
- Surrogate-assisted MOO for computationally intensive applications in CPE
- Efficient and reliable codes including properly parallelized codes for MOO
- Implementation of MOO codes in process simulators
- Effective techniques for constraints, particularly, equality constraints in stochastic optimization techniques (metaheuristics) for MOO
- Application and analysis of techniques for selecting one optimal solution from the Pareto optimal front obtained for CPE applications

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Steps in the systematic procedure for the multi-objective optimization (MOO) of processes; curved arrow on the left emphasizes the need for repetition from an earlier step.

**Figure 2.**Pareto optimal (non-dominated) solutions for optimizing a two-stage membrane separation process for CO

_{2}removal from natural gas: (

**a**) objective space showing the trade-off between two objectives, (

**b**) decision variable space showing optimal values of decision variables, and (

**c**) to (

**f**) variation of one objective function (methane purity or recovery) with decision variables (membrane 1 and 2 areas).

**Figure 3.**Number of journal papers containing multi/bi and objective and optimization/optimisation in the title, abstract, and keywords, from the year 2000 to the end of October 2019 in the journals related to CPE in the subject area of Chemical Engineering in Scopus. The number for the year 2019 is for 10 months only, and there is one paper for the year 2020.

**Figure 4.**Researchers who (co-)authored 10 or more papers in Figure 3.

**Figure 5.**Number of journal papers containing multi/bi and objective and optimization/optimisation in the title, abstract, and keywords, from the year 2000 to the end of October 2019, in the journals likely of interest to chemical engineers in the subject area of energy in Scopus. The number for the year 2019 is for 10 months only, and there are already 27 papers for the year 2020.

**Figure 6.**Researchers who (co-)authored 10 or more journal papers/reviews in Figure 5.

Trade-Off in CPE Problems | Aspect 1 | Aspect 2 |
---|---|---|

Efficiency in use of capital | Operating costs | Capital costs |

Raw material efficiency | Revenue from sales | Capital costs |

Environmental benefits | Harmful emissions | Capital costs |

Process safety | Risk profile | Capital costs |

Reliability | Plant downtime | Capital costs |

**Table 2.**Titles of journals (which published at least 40 relevant papers) and number of papers in each of them from the year 2000 to October 2019: subject area of chemical engineering (ChE) in Scopus.

Title of Journals | Number of Papers |
---|---|

Computers and Chemical Engineering | 247 |

Computer Aided Chemical Engineering | 90 |

Industrial and Engineering Chemistry Research | 85 |

Applied Sciences (Switzerland) | 80 |

International Journal of Heat and Mass Transfer | 68 |

Chemical Engineering Science | 64 |

Chemical Engineering Research and Design | 63 |

Chemical Engineering Transactions | 52 |

Desalination | 47 |

AIChE Journal | 41 |

**Table 3.**Titles of journals (which published at least 15 relevant papers) and number of papers in each of them in the subject area of energy in Scopus from the year 2000 to October 2019.

Title of Journals | Number of Papers |
---|---|

Applied Energy | 366 |

Energy Conversion and Management | 339 |

Journal of Cleaner Production | 310 |

Energies | 279 |

Applied Thermal Engineering | 236 |

Renewable Energy | 143 |

Sustainability (Switzerland) | 123 |

Energy | 96 |

International Journal of Hydrogen Energy | 76 |

Renewable and Sustainable Energy Reviews | 65 |

Solar Energy | 58 |

Journal of Renewable and Sustainable Energy | 49 |

Name | Brief Description | Reference |
---|---|---|

ParadisEO-MOEO | A C++-based open source objective-oriented framework, providing visualization facilities and on-line definition of parameters. | [97] |

jMetal | An object-oriented java-based MOO library framework. | [95] |

EMOO | MS Excel based NSGA-II program for non-linear constrained MOO. | [98] |

MOSQP | Sequential Quadratic Programming (SQP) method in Matlab for differentiable constrained MOO. | [99] |

OTL | A C++ library for solving MOO problems, and it is then extended to Python modules in PyOTL software. | [100] |

PyGMO | Parallel global multi-objective optimizer coded in Python for non-linear constrained MOO problems. | [101] |

IMODE | Integrated Multi-Objective Differential Evolution program in MS Excel for non-linear constrained MOO. | [5] |

MOEA Framework | A Java library for developing and experimenting with single and MOO algorithms. | [96] |

PlatEMO | A powerful Matlab-based software for solving MOO problems, and it includes more than 50 multi-objective evolutionary algorithms. | [102] |

TSEMO | Thompson sampling efficient MOO algorithm coded in Matlab for constrained non-linear MOO problems. | [103] |

MOGOA | Multi-Objective Grasshopper Optimization program in Matlab for constrained non-linear MOO problems. | [104] |

NAGA-III | Implementation of NSGA-III algorithm in Matlab. | [105] |

PISA | For solving constrained MOO problems in Matlab, C, and java. | [106] |

DAKOTA | A C++-based toolkit for solving MOO and single-objective optimization problems. | [107] |

NGPM | Reference-point based NSGA-II Program in Matlab for solving MOO problems. | [108] |

MO-HERPA | A commercial software for solving large engineering problems. | [109] |

Feature | Similarity |
---|---|

Decision Variables | Continuous and/or integer variables with or without bounds on them. |

Constraints | The problem may or may not have equality and/or inequality constraints. |

Type of Equations | Objective(s) and/or constraints can be linear, non-linear, differential, and/or integral equations. |

Solution Techniques | Both deterministic and stochastic (metaheuristics) optimization techniques can be used. |

Optimal Solutions | An SOO problem can have unique or multiple optimal solutions (such as local and global optima). Similarly, an MOO problem can have local and global Pareto optimal fronts (with each front having multiple optimal solutions). |

Feature | SOO | MOO |
---|---|---|

Number of Objectives | Only one objective to minimize or maximize. | Two or more objectives, which can be the minimization and/or maximization type. |

Number of optimal solutions | Usually only one optimal solution. | Many optimal solutions, which are known as Pareto optimal or non-dominated solutions. One single optimal solution only when all objectives are not conflicting. |

Multi-dimensional Spaces | Only one multi-dimensional space for decision variables. | Two multi-dimensional spaces: one for objectives and another for decision variables. |

Development of Techniques | Optimization techniques were originally developed for SOO problems with or without constraints. | There are two approaches for solving an MOO problem. One approach is to convert an MOO problem into an SOO problem for solution by an SOO technique. Another approach is to modify stochastic optimization techniques (metaheuristics) to handle multiple objectives for solving an MOO problem. |

Computational Time for Finding Optimal Solutions | Deterministic techniques are faster compared to stochastic techniques, but the latter are more likely to find the global solution. | Deterministic techniques are faster if only a few Pareto optimal solutions are required. Stochastic techniques may be faster for finding many Pareto optimal solutions. In general, computational time is expected to increase with the number of objectives. |

Knowledge on Optimal Solutions | Usually, limited to one optimal solution, either local or global optimum. | Quantitative variation and trade-off of objectives from the many optimal solutions in the Pareto optimal front, which can be a local or global optimal front. |

Selection of a Solution for Implementation | It is straightforward, since only one or a few optimal solutions are found. Global optimum is often preferred. | Additional preferences and techniques are required for selecting one of the non-dominated (optimal) solutions, which are equally good from the perspective of objectives in the MOO problem. Owing to many choices for normalization, weighting, and ranking, the selection of one of the non-dominated solutions is itself another optimization problem. |

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**MDPI and ACS Style**

Rangaiah, G.P.; Feng, Z.; Hoadley, A.F.
Multi-Objective Optimization Applications in Chemical Process Engineering: Tutorial and Review. *Processes* **2020**, *8*, 508.
https://doi.org/10.3390/pr8050508

**AMA Style**

Rangaiah GP, Feng Z, Hoadley AF.
Multi-Objective Optimization Applications in Chemical Process Engineering: Tutorial and Review. *Processes*. 2020; 8(5):508.
https://doi.org/10.3390/pr8050508

**Chicago/Turabian Style**

Rangaiah, Gade Pandu, Zemin Feng, and Andrew F. Hoadley.
2020. "Multi-Objective Optimization Applications in Chemical Process Engineering: Tutorial and Review" *Processes* 8, no. 5: 508.
https://doi.org/10.3390/pr8050508