A Multiobjective Optimization-Based Approach for Optimal Chemical

Article pubs.acs.org/IECR

A Multiobjective Optimization-Based Approach for Optimal Chemical Product Design Lik Yin Ng, Nishanth G. Chemmangattuvalappil,* and Denny K. S. Ng Department of Chemical and Environmental Engineering/Centre of Excellence for Green Technologies, The University of Nottingham, Malaysia Campus, Broga Road, 43500 Semenyih, Selangor Malaysia ABSTRACT: Over the past decades, chemical products have been constantly evolving to satisfy the demands and requirements of market. Hence, there is a continuous search for new and improved products which possess enhanced properties for specific applications. The conventional approach to produce new chemical products with specific functionality is based on design heuristics, experimental studies, and expert judgments. Other than the traditional methods, chemical products can also be designed by identifying the properties of the product by using the reverse engineering approach. This approach first identifies the needs to fulfill, and then searches for molecule/s that possess properties which can meet the target needs. On the basis of the approach, an optimal chemical product can be designed by identifying the molecule/s with the best properties that correspond with the target functionalities of the product. Most of the established chemical product design methodologies focus on optimizing a single property of a product. It is mindful that in some situations, there are several important product properties to be considered in order to design an optimized chemical product. In cases where more than one product property is to be considered and optimized, a multiobjective optimization design problem is needed to optimize all the important targeted properties simultaneously. In the previous works, the product properties were optimized based on the weighting factors assigned by decision makers. This method tends to be biased as it depends heavily on expert judgments or personal preferences. To address this problem, a systematic methodology for the design of chemical products with optimal properties has been developed in this paper. A fuzzy optimization approach and molecular design techniques are adapted to address the above-mentioned design problem. Max−min aggregation and two-phase approaches are incorporated into fuzzy optimization approach, and the solutions generated from both approaches are compared. A case study on the design of a solvent used in a gas sweetening process is presented to illustrate the developed methodology in chemical product design.

1. INTRODUCTION Chemical product design is the process of choosing the optimal product/s to be made for a specific application.1 The established techniques to generate ideas of novel products are natural product screening, random molecular assembly, and combinatorial chemistry.1 These conventional techniques to identify suitable molecules for a specific application are heavily dependent on expert knowledge, heuristics, and costly experimentation rather than any scientific reasoning. Thus, these techniques are usually very expensive and timeconsuming.2−4 In addition, it is challenging to produce nonintuitive solutions as these approaches are largely reliant on available knowledge and information. This often results in difficulty in searching for a new product with optimal performance. According to Gani and O’Connell5 and Eden et al.,6 a chemical product design process can be considered as an inverse property prediction problem in which the desired attributes of the product are represented in terms of physical properties of the molecule/mixture. As stated by Stephanopoulos,7 one of the important sources of product specification and requirements in product design is customers’ needs. Therefore, the ability of translating the qualitative attributes into measurable quantitative parameters is vital in order to design a chemical product. This process of representing product attributes by using measurable parameters is often done by using computer-aided molecular design (CAMD) techniques. © 2014 American Chemical Society

CAMD techniques are being developed as powerful techniques in the field of chemical product design. These techniques are important for chemical product design because they are able to predict, estimate, and design molecules with a set of predefined target properties.8 Thus, CAMD techniques are often used for screening possible molecular structures during the predesign stage of a chemical product to select promising designs and filter redundant designs of the product.9−11 Various CAMD techniques have been developed to provide reliable and accurate solutions based on the given properties. Over the recent decades, applications of CAMD techniques have included the design of different chemical products. For example, Camarda and Maranas12 developed an algorithm that includes a structure−property correlation of a polymer repeat unit in an optimal polymer design problem. Sahinidsi et al. 13 proposed a mixed integer nonlinear programming (MINLP) model which can handle a large number of preselected molecular groups and search for global optimal solutions in an alternative refrigerants design problem. The design of pharmaceutical compounds is also investigated.14 A two-step method in designing novel pharmaceutical products is introduced by developing structure-based correlations for Received: Revised: Accepted: Published: 17429

July 23, 2014 September 30, 2014 October 10, 2014 October 10, 2014 dx.doi.org/10.1021/ie502906a | Ind. Eng. Chem. Res. 2014, 53, 17429−17444

Industrial & Engineering Chemistry Research

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follows the principles in chemical graph theory which considers the molecules as vertices and edges in a graph. The atoms in the graph are named vertices while the bonds used to connect them are called edges.31 This method allows molecular information such as types of atoms, bonds, total number of atoms and bonds between the atoms to be known. With such information, the interactions among different atoms/molecular groups and their effects can be utilized in describing a molecular graph as an index. The index can then be used to correlate the chemical structure to physical properties of a molecule. These correlated relationships are called quantitative structure property/activity relationships (QSPR/QSAR).32,33 To demonstrate how a property prediction model can be modeled by using QSPR, a general schematic representation is shown in Figure 1.

physical properties and identifying the molecules having the desired properties. Karunanithi et al.15 developed a framework in designing crystallization solvents by solving the problem with a decomposition-based solution approach. Later, Karunanithi et al.16 proposed a systematic methodology in crystallization solvent design of carboxylic acids by combining targeted bench scale experiments, CAMD approach, and database search approach. The design of biofuels is explored by Hechinger et al.17 The work identifies the molecular structures of the biofuels by using CAMD techniques while searching the alternative production pathways by utilizing the Reaction Network Flux Analysis (RNFA). Samurda and Sahinidis18 utilized the CAMD approach to identify efficient refrigerant components that fulfill process targets and environmental regulations as well as safety guidelines. Recently, the design of ionic liquids has also been studied via CAMD techniques.19,20 As only limited structure− property correlations are currently available for ionic liquids, the research is focusing on the development of such correlations for the physical and chemical properties of these ionic liquids. Note that most of the CAMD techniques described earlier generally utilize property models to predict the properties of the product. Property models are computational tools developed to estimate the physical properties from molecular structure, which are quantified in terms of structural descriptors such as chemical bonds and molecular geometry.21 At present, to ensure the designed structures satisfy the property constraints, most CAMD techniques use property models developed from group contribution (GC) methods to verify whether the generated molecules possess the specified set of desirable properties.22 These methods consider a molecule as a collection of various molecular groups such as −CH3, −NH2, −OH, and -Cl. The properties of the molecule can then be estimated by summing up the contributions of the molecular groups according to their frequency in the molecule.23,24 The typical property models developed by using GC methods can be represented by the following equation: f (X ) =

∑ NC i i i

Figure 1. Property prediction by using QSPR model.

The model as shown in Figure 1 is formed from a first order connectivity index (CI). It is to be noted that the molecule is first converted into a hydrogen-suppressed graph (molecular graph without considering hydrogen vertices), where vertices and edges of the molecule are identified at the same time. Next, each edge is characterized by the reciprocal square root product of the vertex valencies. The index can then be determined by summing up the values obtained, which can be used to develop the QSPR model to express the target property as a function of CI. It can be seen that the formation of the model involves a mathematical formulation which relates the vertices (atoms) and edges (bonds). Currently, a variety of properties can be predicted by using these QSPR/QSAR models which utilize their own property prediction methods. Some of the commonly used TIs are edge adjacency index,34 Wiener index,35 Hosoya TI,36 and shape indices.37 As these relationships systematically categorize and relate atoms or molecules to their properties, property models developed based on TI have been used in molecular design in various research works. Some of the property models developed are affinity, mobility and retention,38 octanol−water partition coefficient, melting point,14 water solubility,14 and flash point.39 Note that the focus of most of the current product/molecular design algorithms is on designing molecule/s with a single optimum target property. To design an optimal product, there can be a situation where multiple product properties need to be considered and optimized simultaneously. For example, during

(1)

where f(X) is a function of property X to be determined, Ni is the frequency of molecular group i in the molecule, and Ci is the contribution of the molecular group i. This approach was then improved by Constantinou and Gani25 to distinguish different functional groups and isomers by considering different types of interactions among the molecular groups. The work defined the molecular groups as first- and second-order groups in which the second-order groups are built by having first-order groups as their building blocks. Later, Marrero and Gani26 further extended the approach to focus on polyfunctional groups, multiring compounds, and fused ring compounds by introducing a third-order molecular group. GC methods are vastly utilized in developing different property models especially in estimating thermodynamic properties of organic compounds.24,26,27 Other than thermodynamic properties, property models based on GC methods are also developed in predicting nonthermodynamic properties such as acute toxicity,28 surface tension,29 and viscosity29 of organic chemical compounds. Alternately, other established methods in developing property models include the use of topological indices (TI) which are molecular descriptors calculated on the basis of the molecular graph of a chemical compound.30 This approach 17430

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been introduced and extended for solving and improving decision-making problems. Some of the recent applications of the approaches were in the fields of banking and finance,52 project management,53 and transportation.54 In addition, fuzzy set theory is also widely used in supply chain network development. Liang55 developed a fuzzy multiobjective linear programming model to solve an integrated multiproduct and multitime period production/distribution planning decisions problem. Campuzano et al.56 developed a simulation model with fuzzy estimation of demand for production planning in a two-stage, single-item, multiperiod supply chain based on system dynamics. Furthermore, the application of fuzzy set theory can be seen in environmental management systems. Aviso et al.57 presented a model for optimizing water and wastewater reuse for encouraging cooperation among several plants contained in an eco-industrial park. Tan et al.58 extended fuzzy set theory in addressing the environmental and economic goals, as well as the uncertainties in emerging technologies for optimal carbon sequestration retrofit in the power sector. Tay et al.59 adapted fuzzy mathematical programming to address the conflicting nature of economic performance and environmental impact in synthesis of an integrated biorefinery. Ubando et al.60 proposed fuzzy fractional programming in designing trigeneration plants that gives the minimum carbon footprint for each energy stream. Zhang et al.61 developed an inexact fuzzybounded programming approach to tackle independent uncertainties in the constraints for agricultural water quality management. Most recently, Ng et al.62 extended the fuzzy optimization approach in addressing the uncertainties of property estimation methods while solving computer-aided chemical product design problem. As shown in the above-mentioned works, several approaches have been extended and developed based on fuzzy set theory owing to its flexibility and efficacy in solving multiobjective optimization problems. One of the widely utilized approaches is the max−min aggregation approach.51,63 Tan et al.64 utilized the max−min approach and presented a fuzzy multiple objective approach in determining an optimal bioenergy system configuration while simultaneously considering the land use, water, and carbon footprints. Aviso et al.65 applied the approach in fuzzy mathematical programming to address the contradictory objectives from several decision makers while designing eco-industrial water exchange networks. Later, Kasivisvanathan et al.66 adapted the approach to retrofit a palm oil mill into a sustainable biorefinery which fulfills the conflicting objectives of economic performance and environmental impact. In this work, the max−min approach is adapted into the fuzzy optimization approach for simultaneous optimization of multiple properties in a chemical product design problem. In addition to the max− min aggregation approach, the two-phase approach developed by Guu and Wu67,68 is also widely applied. Liang69 developed a two-phase fuzzy mathematical programming approach for attempting to simultaneously minimize project cost, completion time, and crashing cost in a project management decision problem. Aviso et al.70 presented an approach to design ecoindustrial resource conservation networks while considering the individual fuzzy goal of participating plants in the presence of incomplete information. Later, Lu et al.71 proposed an inexact two-phase fuzzy programming approach for municipal solid waste management in which a solution with high satisfactory level is obtained through relaxation of objective functions and constraints. Recently, Ng et al.72 incorporated the approach in the synthesis of an integrated biorefinery while optimizing

the design of fuel, one might want the fuel to give maximum performance while producing minimum pollutants as well as possessing other physical properties so that it can be used safely. Refrigerant design is another example where multiple product properties are important. To design an effective refrigerant, the volumetric heat capacity for the designed refrigerant should be high so that the amount of refrigerant required is reduced for the same refrigeration duty. Besides, the designed refrigerant should have a low viscosity to achieve a low pumping power requirement. Since more than one design objective is involved in designing these products, the design problems have to be solved as a multiobjective optimization problem. According to Kim and de Weck,40 the most commonly used approach in solving the multiobjective optimization problem is the weighted sum method. This method can be explained mathematically as below:41 Aweighted sum = b1A1 + b1A1 + ··· + bmA m

(2)

where Aweighted sum is the overall objective function and bm is the weighting factor for the individual objective function Am. This method converts multiple objectives into an aggregated scalar objective function by first assigning each objective function with a weighting factor, and then summing up all the contributors to obtain the overall objective function. Methods which utilize the logic of weightage allocation include the goal-programming technique42 and normal boundary intersection method.43 By using these techniques, each of the objectives is given a weight to differentiate their relative importance during the aggregation of the overall objective function. The major drawback of these methods is that a decision maker is required in finding the appropriate weighting factors to be assigned to each objective.44 As a result, these methods tend to be biased as the weighting factors assigned to the objective are based on expert knowledge or personal subjective preferences of the decision maker.45 In the context of chemical product design, while considering the design problem as a decision-making problem, the weighting factors for each property are assumed to be deterministic/crisp when the conventional multiobjective optimization methods are used.46,47 However, the relative importance of each property to be optimized in a design problem is not always definable. Hence, the significance of each product property to design an optimal product in a design problem is normally uncertain/fuzzy. Furthermore, these objectives might be “unclear” or contradictory in nature. To solve a decision-making problem under such an environment, fuzzy set theory was developed by Zadeh.48 As the theory systematically defines and quantifies vagueness and uncertainty, it is possible to solve problems which require decision-making under the fuzzy environment. On the basis of the fuzzy set theory, Bellman and Zadeh49 developed a fuzzy optimization approach that is able to select the preferred alternative in a fuzzy environment by solving an objective function on a set of alternatives given by constraints. Zimmermann50 then adapted fuzzy set theory into linear programming problems by solving the problems under fuzzy goals and constraints. Later, Zimmermann51 extended the approach to address linear programming problems which involve multiple objectives. This extended fuzzy optimization approach integrates several objectives into a single objective and solves the overall objective based on the predefined fuzzy limits to obtain an optimized solution in a multiobjective optimization problem. Since the development of fuzzy set theory and fuzzy programming techniques, fuzzy optimization approaches have 17431



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Figure 2. Illustration of atomic signatures.75

Figure 3. Graph coloring of atomic signature.

in a molecule by using the extended valencies to a predefined height. On the basis of this approach, it is possible to represent different types of structural descriptors on the same platform by utilizing molecular signatures. It is particularly useful for a chemical product design problem which involves multiple target properties.77 Since target properties in a product design problem might not be able to be estimated by using only a single class of property model, the application of molecular signature allows the utilization of different classes of property models in a chemical product design problem. In this work, molecular signature descriptors are used to represent property models developed based on GC methods and TIs for solving a chemical product design problem for which distinct property prediction models are required. To utilize molecular signature descriptors, the molecular structures of chemical products are represented in terms of molecular signatures, which serve as the building block of the molecule. Figure 2 shows how a signature for a molecule is formed. The procedure for obtaining the atomic signature of atom N up to height 3 in the molecule is explained as follows. To construct an atomic signature for an atom x (atom N in Figure 2) in a molecule, the molecule is first transformed into a molecular graph G, and the atoms in the molecule are labeled/ numbered to distinguish them from each other. The atomic signature of a distance h from x can be represented as a subgraph containing all atoms that are at height (distance) h of x in the molecular graph G. This subgraph is the atomic signature at a different height, annotated as hG(x). For example, since a signature up to height 3 is constructed, all the atoms at height 3 from atom N are extracted, as shown in Figure 2. To obtain the signature of height 1, only the atoms bonded to atom N (atoms y for example/atom C no. 2 and C no. 3 in Figure 2) are considered; to obtain signature of height 2, atoms y and atoms bonded to atoms y (atoms z for example/atom C no.4, C no.5 and C no.6 in Figure 2) are included; to obtain signature up to height 3, all the atoms y, z and atoms bonded to atoms z (atom C no. 7, C no. 8 and C no. 9 in Figure 2) are taken into account. By representing a molecule with atomic signature, signature of a molecule can be obtained as a linear combination of its atomic signatures.75,76 In Figure 2, it can also be seen that graph coloring has been used to differentiate between different types of atoms. Here, the graph coloring function used is the valency of each atom at all levels. A more detailed explanation is shown in Figure 3.

economic, environmental and inherent safety and inherent occupational health performances simultaneously. To address the above-mentioned chemical product design problem in which multiple product properties have to be optimized without any favoritism and prejudice against the property/ies, different fuzzy optimization approaches are adapted. In this work, a fuzzy optimization-based inverse design technique is proposed. This work offers a systematic methodology that aims to utilize different classes of property prediction models to design the best molecule/s with optimized product properties subject to incomplete information about the product requirements.

2. SIGNATURE-BASED MOLECULAR DESIGN As mentioned earlier, a chemical product design process can be considered as an inverse property prediction problem in which the molecular structure of a chemical product can be estimated from the desired product properties by using property prediction models. Although property models are useful to estimate the product property, applying different classes of property prediction models in a molecular design problem is computationally challenging.12,13,73 First, the mathematical formulation for each of the property prediction models is different and exclusive. While GC methods estimate property by summing the frequency of each molecular group occurring in the molecule times its contribution, estimations of TIs involve the operations on a vertex-adjacency matrix.74 Hence, it is difficult to utilize these different models simultaneously by using a similar calculation method. Moreover, most of property prediction models are nonlinear in nature which leads to mixedinteger nonlinear programming (MINLP) formulation. Besides, second- and third-order groups of GC methods are formed by using first-order groups as their building blocks. This makes it very challenging to consider the property contributions and identify the presence of second- or third-order molecular groups without the knowledge of the complete molecular structure. In addition, according to Trinajstic,30 the inverse relationships between TIs are unable to provide a unique molecular structure as a final solution, therefore resulting in high degeneracy in the approach. These restrictions make it complicated to utilize different property models in inverse design problems. To overcome the above-mentioned problems, a descriptor known as molecular signature descriptor is utilized.75,76 The signature is a systematic coding system to represent the atoms 17432



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Figure 4. Expression of higher order molecular groups with signatures.

available first order molecular groups, the signature distinctly represents second order molecular groups of CH3COCH and CH3CHCH as illustrated in Figure 4. Therefore, by utilizing molecular signature descriptors, higher order molecular groups in GC methods can now be differentiated from the first order molecular groups and represented with signatures. More importantly, property models developed by GC methods can be applied together with property models developed from TIs in inverse design problems. To utilize molecular signature descriptors in a molecular design problem, Chemmangattuvalappil et al.77 developed a signature-based algorithm for molecular design. For a molecular design problem, the target properties of the desired product are estimated using property models, which can be represented as a function of TI, θ as shown in eq 4:

As shown in Figure 3, the graph coloring starts from the root atom (atom N in Figure 3) to all atoms up to level h − 1. By using graph coloring on the atoms in signatures, atomic signatures can be discriminated uniquely. Therefore, any molecule can be distinctly represented with its atomic signatures. As a molecular signature descriptor systematically characterizes the atoms in a molecule, it is employed in an inverse chemical product design problem. To express different classes of property prediction models simultaneously on a common platform, both molecular groups in GC methods and TIs are written in terms of signature. Note that TI can be obtained from molecular signatures by using the following relationship:76 TI(G) = khαG· TI((root(hΣ))

(3)

As discussed earlier that the molecular signature can be represented as the linear combinations of all atomic signatures, the TI value of a molecule represented by a molecular graph G, TI(G) can be obtained as the dot product between the vector of the occurrence number of atomic signature of height h, hαG and TI calculated for each of the atomic signature, TI((root(hΣ)). k in eq 3 is a constant specific to TI. While signature descriptors represent individual building blocks for a complete molecule, they are related to the rest of the building blocks as they carry information on their neighboring atoms. Hence, TIs can be defined and derived from the signature of a molecule.76 By being written in terms of a signature, GC methods and TIs with different mathematical formulations can now be expressed and used on a common platform. Another important application of a signature descriptor is its ability to account for the contributions of second- and third-order molecular groups in the property model developed based on GC methods. The procedure to identify the contributions of higher order molecular groups is explained in Figure 4: In the first step, the signatures are generated based on first order molecular groups without considering higher order molecular groups. Signatures that carry higher order group contributions can then be identified among the generated signatures. For those identified higher order molecular groups, property contributions of the actual molecular groups as well as the contributions of the higher order groups are assigned. Figure 4 shows two examples of expressing higher order molecular groups by using signatures. After generating the height three signatures correspond to the first order molecular groups (shown in dotted squares), signatures which carry the contributions corresponding to the second order molecular groups (shown in dotted circles) are identified. With the

θ = f (TI)

(4)

Note that the property models θ can be linear or nonlinear. To employ signatures into a molecular design problem, the analogy or property operator is extended. Property operators are functions of the original properties that obey linear mixing rules.78 Thus, regardless of the linearity nature of the property models, the property operator will follow simple mixing rules irrespective of the original property. By following eqs 3 and 4, molecular property operator as shown in eq 5 can now be used to estimate the property of the molecule, where i is the molecules index. N

ψ (P ) =

∑ xi TI(root(hΣ)) i=1

(5)

In a product design problem, customers’ needs can be expressed in terms of property specifications. These property specifications are written as a set of property constraints bounded by upper and lower limit:79 vipL ≤ Vip ≤ vipU

i = 1, 2, ...; p = 1, 2, ...P

(6)

where Vip is the value of the target property, i is the molecules index, and p is the properties index. This target property range is bounded by lower (vLip) and upper (vUip) limits. Hence, eq 5 is utilized to predict the property, which is bounded by the target property range as shown in eq 6. While property constraints guarantee that the properties of the signature combination obtained from the design problem falls within the product specification, structural constraints are imposed to make sure that complete molecular structures will be formed from a collection of molecular signatures.77 Consider 17433



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and the connectivity rules of signatures are discussed in detail by Chemmangattuvalappil and Eden.80

the design of molecules which involve multiple bonds, cyclic compounds, and the maximum valency (combining power with other atoms) of four in a hydrogen-suppressed graph, handshaking lemma as shown in eq 7 must be followed in order to ensure the connectivity of signatures without any free bonds in the structure:30 n1

n2

n3

3. OPTIMIZATION MODEL The chemical product design goal is to determine the structure of the chemical product while considering multiple design targets/objectives simultaneously. Molecular design techniques incorporated with fuzzy logic have been used to achieve this objective. To solve a chemical product design problem under different fuzzy environments such as imprecise or unclear objective, fuzzy optimization as well as bilevel optimization are adapted and discussed in the following sections. 3.1. Fuzzy Optimization. Fuzzy mathematical programming is incorporated in this work to address the multiobjective optimization problems. To trade-off the targeted properties, a degree of satisfaction λ is introduced. These property targets are linear functions bounded by upper and lower limits, as shown by eqs 9 and 10:

n4

∑ xi + 2 ∑ xi + 3 ∑ xi + 4 ∑ xi i=1

n1

⎡⎛ 1 = 2⎢⎜⎜∑ xi + ⎢⎣⎝ i = 1 2 N

n2

n3

NDi

NMi

NTi

⎞

⎤

i=0

i=0

i=1

⎠

⎥⎦

∑ xi + ∑ xi + ∑ xi⎟⎟ − 1 + R ⎥ (7)

where n1, n2, n3, and n4 are the number of signatures of valency one, two, three, and four, respectively, N is the total number of signatures in the molecule, NDi, NMi, and NTi are the signatures with one double bond, two double bonds, and one triple bond; R is the number of circuits in the molecular graph. Other than the handshaking lemma, the handshaking dilemma31 should also be fulfilled to make sure the number of bonds in each signature matches with the bonds in the other signatures. This di-lemma is represented mathematically in eq 8:

∑ (li → lj)h = ∑ (lj → li)h

if Vp ≤ vpL

1 λp =

(8)

where (li → lj)h is one coloring sequence li → lj at a level h. Equation 8 must be obeyed for all color sequences at each height. This is explained in Figure 5.

vpU − Vp vpU − vpL

if vpL ≤ Vp ≤ vpU ∀ p ∈ P

0

if Vp ≥ vpU

0

if Vp ≤ vpL

λp = 1

vpU − Vp vpU − vpL

(9)

if vpL ≤ Vp ≤ vpU ∀ p ∈ P if Vp ≥ vpU

(10)

where λp is the degree of satisfaction for property p. Values of 0 and 1 of λp denote the fuzzy limits of the property to be optimized. Higher λp values indicate higher satisfaction of each objective function in fuzzy optimization. For a property to be minimized, as lower values are desired, when the property approaches the lower bound, the value of λp approaches 1; when the property approaches the upper bound, the value of λp approaches 0, as shown in Figure 6a. The opposite trend is observed when the optimization objective is maximized as higher values are desired. Hence, eq 9 is used for a property to be minimized while eq 10 is used for a property to be maximized. A pictorial representation is shown in Figure 6. Note that the degree of satisfaction (Figure 6) can be split into below satisfactory, satisfactory, and above satisfactory

Figure 5. Explanation of handshaking di-lemma.

In Figure 5, the edges of the signatures have the colors of 1 and 2. The reading of coloring sequence for signature S1 will be 1 → 2, for signature S2 it is 2 → 2, and 2 → 1; for signature S3 it is 2 → 3 and 2 → 1; for signature S4 it is 2 → 4 and 2 → 1. Hence, by following eq 8, the handshaking di-lemma can be written. Each color sequence (e.g., 1 → 2) has to be complemented with another coloring sequence in reverse order (e.g., 2 → 1) to ensure linkage and consistency of the signatures. The use of molecular signatures in molecular design

Figure 6. Fuzzy degree of satisfaction (λ) of the inequalities: (a) property to be minimized; (b) property to be maximized. 17434



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minimize Vp*

regions. To ensure a non-negative degree of satisfaction, the values of λp are set to be more than 0: λp ≥ 0

∀p∈P

subject to

(11)

vpL ≤ Vp ≤ vpU

Hence, the property to be optimized can only be more than satisfactory of customers’ requirements, but not under satisfactory. 3.2. Bi-Level Optimization. In a chemical product design problem, there are times where some of the product property target ranges are unknown. Besides, there are also situations where product property is bounded by either an upper or lower limit as only one of them is significant. For instance, toxicity is one of the properties where only the lower limit is important. In this case, as long as the LC50/LD50 is above the safety limit, it is nonhazardous. Hence, the upper limit for LC50/LD50 is meaningless. However, to utilize fuzzy optimization model in the algorithm, both upper and lower limits for a property target range are required. As discussed in section 3.1, the satisfactory level for each property to be optimized is defined by a linear membership function consisting of upper and lower limits of the property target range. The situation in which the information about the target property ranges is incomplete can be modeled as a bilevel optimization problem. Introduced based on a static Stackelberg game with leader−follower strategy, the concept of bilevel optimization is to obtain an optimized solution for the main optimization problem while independently optimizing the second-level optimization problem.81 In other words, to optimize the main objective (multiobjective chemical product design problem), the second-level objectives (properties with incomplete target property ranges) must be optimized beforehand. This approach has been utilized in various research fields. Takama and Umeda82 developed an algorithm for solving a water allocation and wastewater treatment problem. By using a bilevel optimization approach, two independent subsystems that concern water allocation and control system reliability are solved and the overall optimal solution is determined. Cao and Chen83 proposed a two-level decision-making process in a capacitated plant selection problem under a decentralized manufacturing environment. By considering opportunity cost and the independent relationship between the principal firm and the selected plants, an optimal solution is obtained by solving the two-level nonlinear programming model which is transformed and linearized into an equivalent single level model. Roghanian et al.84 adapted a bilevel multiobjective programming model to solve a supply enterprise-wide chain planning problem which considers market demand, production capacity, and resources availability for each plant. Later, Aviso et al.65 presented a bilevel fuzzy optimization model to explore the effect of charging fees for the purchase of freshwater and the treatment of wastewater in a water exchange network of an eco-industrial park. As proven in the literature for its suitability and efficacy in solving similar problems, in order to obtain lower and/or upper limit/s for target properties which are required for the formulation of afuzzy optimization model, bilevel optimization is adapted in this work to determine the product target property ranges which are not well-defined and do not have exact values. Properties where the range of limits is indefinite are modeled as the second-level objective in bilevel optimization problem by eqs 12−14: maximize Vp*

(13)

(14)

vUp

L

where Vp is the value for property p, and v p are the upper and lower bounds, respectively, for property p. While setting the property with known property target ranges Vp as constraints, eqs 12 and 13 are solved for the property with unknown property target ranges Vp*. The solution attained from eq 12 would serve as the upper limit for the respective property, while the solution obtained from eq 13 would serve as the lower limit. With the identification of all property target ranges, the leader’s objective, which is the optimization of multiple objectives, can now be solved. This would be modeled as a fuzzy optimization problem. 3.3. Approaches in Fuzzy Optimization. To use the fuzzy optimization model in solving multiobjective optimization problems, an overall objective on the individual objectives which are defined in terms of fuzzy quantities are required. The overall objective defines the ultimate goal to be achieved in the decision problem. In this work, two different approaches are chosen for their advantages and suitability in the chemical product design problems. These two approaches are discussed in detail in the following sections. 3.3.1. Max−Min Aggregation Approach. The objective of the max−min aggregation approach is to make sure that every fuzzy constraint will be satisfied partially to at least the degree λ. Therefore, each individual objective/constraint has an associated fuzzy membership function and the optimum overall objective is obtained by maximizing the least satisfied objective.51,63 As mentioned in section 1, the application of the max−min aggregation approach can be seen in several research works.57,64,66 To optimize the properties of the product simultaneously without any bias, each of the properties is treated with equal importance. To achieve this, there would be no compensation between the levels of satisfaction of each of the properties to be optimized. Therefore, the objective is to optimize the “weakest” or “worst” property among all the properties to be optimized. The formulation of the proposed model is shown as follows: maximize λ λ ≤ λp

∀p∈P

(15) (16)

where λp is the degree of satisfaction for property p determined from eqs 9 and 10 depending on whether the property is to be maximized or minimized. To optimize the “weakest” or “worst” property among all degrees of satisfaction for properties to be optimized λp, the least satisfied degree of satisfaction λ is maximized, as shown in eqs 15 and 16. To generate different feasible solutions for a multiobjective optimization problem, integer cuts are utilized. When a solution has been obtained, an integer cut is applied by adding an extra constraint to the mathematical programming model to make sure that the generated solution (same combination of signatures) will not appear again. This process can be continued until no feasible solution is found, which indicates that all possible combinations of signatures that make up the molecules which satisfy all property constraints have been identified. While the max−min aggregation approach aims to maximize the least satisfied property so that the disparity in degrees

(12) 17435



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satisfaction among all properties would be reduced, it is noted that this approach is unable to discriminate between solutions that vary in attained levels of satisfaction other than the least satisfied goal.85,86 While the least satisfied goal is maximized, since the other goals might be overly curtailed or relaxed, there is still room to search for better solutions in terms of degree of satisfaction. Thus, other than the max−min aggregation approach, the two-phase approach proposed by Guu and Wu67,68 is adapted into the product design algorithm. 3.3.2. Two-Phase Approach. In the first phase, the multiobjective optimization problem is solved by using the max−min aggregation approach to obtain the degree of satisfaction of the least satisfied property. In the second phase, a two-phase approach is utilized to solve the optimization problem. The overall objective for the two-phase approach is maximizing the summation of all degrees of satisfaction. This means that all of the problem objectives or goals are optimized as whole. To adapt the two-phase approach in the product design problem, the optimization objective is set as the maximization of the summation of all degrees of satisfaction for each product property. This can be described by eq 17. maximising ∑ λp* p

(17)

where λp* is the degrees of satisfaction determined from the second phase. The main purpose of applying the two-phase approach is to distinguish solutions with identical least satisfied goals and search for better/improved solutions if there are any. To achieve the goal in differentiating solutions with a similar least satisfied goal and identifying a better solution, it is required to ensure that the solution obtained in the second phase will not be any worse than that initially obtained in the first phase. Hence, the degree of satisfaction obtained by using the two-phase approach should not be lower than the degree of satisfaction of the least satisfied goal determined by using the max−min approach. This is achieved by adding eq 18 in the model.

λp* ≥ λp

∀p∈P

(18)

where λp is the degree of satisfaction of the least satisfied property obtained by using max−min aggregation. As shown in eq 18, the satisfactory levels of the properties in the second stage λ*p will not be lower than the least satisfied property identified in the first stage, λp. For its ability to distinguish solutions with similar least satisfied property, the two-phase approach is utilized in this work. 3.4. Solution Procedure. This work presents a product design technique that identifies the optimal product that satisfies customer requirements by optimizing multiple product properties. Bilevel and fuzzy optimization approaches are adopted to address the chemical design problem. The comprehensive methodology aims to solve multiobjective chemical product design problems efficiently and systematically without the presence of decision maker and biased weighting factor. Figure 7 is a flowchart that shows the systematic procedure to design a chemical product by incorporating fuzzy optimization approaches into signature-based inverse design techniques. The procedure is designed specifically for a chemical product design problem in which the different classes of property prediction models are used and multiple product properties are optimized.

Figure 7. Procedure for solving a multiobjective chemical product design problem.

Step 1: Define the objective and collect related information for the chemical product design problem. Step 2: Identify and analyze target properties for the design problem. On the basis of the product specification, these properties can be optimized or utilized as constraints during the design. Step 3: Identify the appropriate property prediction models that can estimate the determined properties of interest. The 17436



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Figure 8. Flow sheet of gas sweetening process.

the acid gases via a counter-current absorption process. The gas stream free of acid gases (sweet gas) exits from the absorber as a top product while the amine solution rich in absorbed acid gases (rich amine) exits as the bottom product. The “rich amine” stream is then sent into a regenerator for the regeneration process. The resultant bottom product, regenerated amine (lean amine) is recycled to the absorber, while concentrated acid gases (H2S and CO2) exit as stripped overhead gas. This process uses methyl diethanolamine (MDEA) as the feed into the acid gas removal unit. The design goal is to identify a solvent that will replace MDEA as the absorbent that will reduce the usage of amine solution. Hence, the new solvent must be designed to possess similar functions of MDEA so that it can be used in the existing gas sweetening process without changing and rectifying the process. Previously, Eljack et al.88 solved the same design problem via property clustering technique. As shown in Eljack et al.,88 the targeted solution meets the required properties within the given property target range. The proposed technique solves the design problem as a property matching problem without performing property optimization. Without optimizing any property that can contribute to the design goal of reducing the usage of amine solution, it cannot be guaranteed that the designed product is the optimal product. The same design problem is solved by Chemmangattuvalappil and Eden80 as a single objective optimization problem. In the work presented by Chemmangattuvalappil and Eden,80 only one of the properties of interest is optimized during the design stage while matching the other properties of interest within the predefined target property ranges. Other than targeting only one property, there are other properties which play an equally important part in the overall design goal to be considered. In this work, the solvent design problem is solved as a multiobjective optimization problem by addressing and optimizing a number of important properties simultaneously during the design stage. 4.2. Solution of Design Problem. As the main objective of the design problem is to reduce the losses of MDEA, properties that contribute in attaining the overall design objective are chosen as the target properties the design problem. To reduce the losses of MDEA, the designed solvent should possess a high heat of vaporization (Hv) and low vapor pressure (VP) to reduce evaporation losses of solvent.

property models chosen can be developed based on GC methods or different TIs. For target properties for which property prediction models are unavailable, models which combined experimental data and available property prediction models can be developed to estimate the respective property. Step 4: Determine the property target ranges based on the product specification, which can be obtained from customer preferences or process requirements. For target properties for which the property target ranges are unknown, utilize a bilevel optimization approach as discussed in section 3.2 to identify the particular property target ranges. Step 5: Introduce degree of satisfaction λ to target properties to be optimized by expressing the properties as fuzzy linear functions, as shown in eqs 9 and 10. Step 6: On the basis of the nature of the chemical product target molecule, select the appropriate molecular building blocks (possible functional groups, types of bonds and atoms) and enumerate the molecular signatures. Step 7: Form the property prediction models as normalized property operators, which are expressed as linear combinations of atomic signatures. Step 8: Develop structural constraints from eqs 7 and 8 to ensure the formation of a complete molecular structure. Step 9: Define the objective function and solve the overall model by using the max−min aggregation approach as discussed in section 3.3.1 to obtain an initial solution with a least satisfied fuzzy goal. Integer cuts may be utilized to generate different alternatives. Step 10: Utilize the two-phase approach as discussed in section 3.3.2 to achieve an improved solution.

4. CASE STUDY: SOLVENT DESIGN FOR GAS SWEETENING PROCESS 4.1. Problem Description and Problem Statement. To illustrate the proposed methodology, a solvent design problem for a gas sweetening process taken from Kazantzi et al.87 is solved. The gas sweetening process for the solvent design problem is given in Figure 8. The gas sweetening process aims to remove acid gases (hydrogen sulfide, H2S, and carbon dioxide, CO2) from a gas stream. It is commonly used in refineries, petrochemical, and natural gas processing plants. A typical gas sweetening process consists of mainly an absorber and a regenerator as well as accessory equipment. In the absorber, amine solution removes 17437



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Table 1. Upper and Lower Bounds for Solvent Design property targets

property operators targets

property

Ωp

lower bound

upper bound

lower bound

upper bound

Koc VP Vm Hv TLC

log (Koc) − 0.66 exp (Tb/tb0) (Vm − 30.67)/33.52 Hv − hv0 [4.204 − log(TLC)]/1.385

unknown 0.07 mmHg 60 cm3/mol 28 kJ/mol 10 ppm

unknown 515.00 mmHg 216 cm3/mol 66 kJ/mol

unknown 4.50 0.89 16.81

unknown 12.00 5.50 53.46 2.31

the molecular connectivity index of molecular nonpolar structure. For Hv, a reliable GC model is available as given in eq 21,26 where f(X) is the function of target property X; w and z are binary coefficients, Ni, Mj, Ok are the number of occurrence of first-, second-, and third-order group contribution correspondingly, and Ci, Dj, Ek are the contribution of first-, second- and third-order groups subsequently.

Furthermore, to encourage efficient removal of acid gases, the molar volume (Vm) of the solvent should be low so that there will be more solvent present in a fixed volume. Other than the properties which contribute to reducing the usage of MDEA, the designed solvent should have a minimum soil sorption coefficient (Koc), which is a measure of the tendency of a chemical to bind to soils to avoid accumulation of the escaping solvent in one place. Instead of estimating aqueous solubility of the solvent, the soil sorption coefficient is chosen as the measure of solubility to prove the flexibility of the developed approach in utilizing different classes of property models in a product design problem. This is further discussed in the following paragraph. Other than the above-mentioned properties, the designed solvent should also be environmentally benign and satisfy the environmental constraints. Thus, the toxic limit concentration (TLC) for the solvent should be high. Note that on the basis of the description, more than one property of interest must be considered and optimized while designing the new solvent. Hence, this problem should be formulated as a multiobjective optimization problem to design an optimal new solvent which fulfills the design goal. In this case study, Koc, VP, and Vm are optimized simultaneously while designing the new solvent as they quantify the performance and safety of the solvent. Hv and TLC are target properties which are used as constraints to be fulfilled. The values of these properties are made sure to fall within the property target range or above/below the lower/upper limit. As there is no obvious connection and relationship between these properties to be optimized, this solvent design problem can be regarded and formulated as a fuzzy multiobjective optimization problem. Following the proposed procedure, after identifying the properties of interest for the solvent, property prediction models for each property of interest are identified. To demonstrate the ability of the developed approach in utilizing different classes of property models in a product design problem, property models of GC methods and different TIs are chosen to estimate the target properties. Hence, log(Koc) which can be estimated by using a QSAR model90 that consists of different classes and heights of CIs is chosen instead aqueous solubility, which can be estimated by the GC method89 and CI.14 Note that eq 19 shows the QSAR model that makes use of different classes and heights of CIs such as CI of order 1 and valence CI of order 0 and 1, as well as delta CI. Note that as this property model considers the polarity between different atoms, the difference between polar and nonpolar molecular structure is addressed.

f (X ) =

i

(22)

To demonstrate the ability of the signature-based inverse design technique in handling different classes of property prediction models, a property model based on the edge adjacency index is utilized in this case study as well. The edge adjacency index is a TI developed by considering the interaction between the bonds (edges) in a molecule. In this work, the edge adjacency index will be used to estimate the Vm of the solvent, where ε is the edge adjacency index:92

Vm = 33.52ε + 30.67 For TLC, a valence CI of order two will be utilized: log(TLC) = 4.204 − 1.385(2 χ v )

(23) 93

(24)

With the property prediction models identified, the next step is to determine the property target ranges for the properties of solvent to be designed. The upper and lower bounds of the properties of interest have been identified in the previous steps. Property models (eqs 19, 21, 23, 24) and property target range for each of the property are transformed into their respective normalized property operators. The property target ranges and their normalized property target ranges are shown in Table 1. As shown in Table 1, the property target range for Koc is unknown at this stage. To determine the lower and upper bounds for Koc, the bilevel optimization approach is utilized. The design problem is formulated as single objective optimization problem at this stage where only Koc is optimized while other properties are made sure to fall within the identified property target ranges. Since the property target range for VP, Vm, Hv and TLC is known, these properties take constraints while Koc is optimized. The objective function is shown in eq 25:

(20)

where χ is CI of order 1, χ and χ are the valence CI of order 0 and 1, respectively; Δχ is the delta connectivity index, χnp is 0 v

(21)

k

⎛ T ⎞1.7 log VP = 5.58 − 2.7⎜ b ⎟ ⎝T ⎠

(19)

1

j

As there is no CI or GC relationship available for VP, an empirical relationship presented in eq 22 is used to calculate vapor pressure from the boiling point.91 T is the temperature where VP is evaluated and Tb is the boiling point of the liquid, which will be expressed by using a group contribution model as shown in eq 20.

log(Koc) = 0.53(1 χ ) − 1.25(Δ1 χ v ) − 0.72(Δ0 χ v ) + 0.66

Δχ = χnp − χ

∑ NC i i + w ∑ MjDj + z ∑ Ok Ek

1 v

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Table 2. Property and Property Operators Target Ranges property targets

property operators targets

property

Ωp

lower bound

upper bound

lower bound

upper bound

Koc VP Vm Hv TLC

log(Koc) − 0.66 exp(Tb/tb0) (Vm − 30.67)/33.52 Hv − hv0 [4.204 − log(TLC)]/1.385

100 0.07 mmHg 60 cm3/mol 28 kJ/mol 10 ppm

1000 515.00 mmHg 216 cm3/mol 66 kJ/mol

1.34 4.50 0.89 16.81

2.34 12.00 5.50 53.46 2.31

maximize/minimize Ω Koc

for the design of the new absorbent. These potential groups are OH, CH3, CH2, CH3N, CH2NH2, CH2NH, NHCH It is assumed that by designing a new absorbent with the chosen molecular groups as building blocks, the designed product will possess the properties and functionalities of an amine solution capable of removing acid gases. Signatures corresponding to these molecular groups are then generated so that only those which can form identical structures to the existing absorbents are selected. 4.3. Results and Discussion. 4.3.1. Max−Min Aggregation Approach. First the design problem is solved with the max−min approach. The optimization objective is to maximize the least satisfied property among Koc, VP and Vm. The multiobjective MILP model is shown as below:

(25)

Together with structural constraints (eqs 7 and 8), the design problem is solved as a single objective MILP model. Koc is minimized and maximized to obtain the lower and upper bound for the property, respectively. This property target range represents the lowest and highest values achievable for Koc, while satisfying other property and structural constraints is shown in Table 2. As shown in Table 2, the identified lower and upper bound of Koc are 100 and 1000, respectively. These values are then transformed into the normalized property operators, with the value of 1.34 and 2.34 in Table 2. Note that a bilevel optimization approach can be applied to identify the property target ranges even when the property target ranges for all properties of interest are unknown. This is because other property constraints, the objective function of the design problem, are solved together with structural constraints. Hence, without the presence of any property target ranges, the bilevel optimization approach identifies the lower and upper limits for each property of interest that satisfies the structural constraints. Following the procedure, the properties to be optimized are written as linear membership functions as shown by eqs 26−28 for Koc, VP, and Vm respectively. For properties which are not optimized but used as constraints, this step is unnecessary. The property constraints for such properties will remain the same, bounded by lower and upper bounds as shown in eq 6. 2.34 − Ω KOC 2.34 − 1.34

= λ KOC

Ω VP − 4.50 = λVP 12.00 − 4.50

5.50 − ΩVm 5.50 − 0.89

= λVm

maximize λ

(29)

subject to λ≤

λ≤

λ≤

(26)

2.34 − Ω Koc 2.34 − 1.34

(30)

Ω VP − 4.50 12.00 − 4.50

(31)

5.50 − ΩVm 5.50 − 0.89

(32)

16.81 ≤ Ω Hv ≤ 53.46

(33)

Ω TLC ≤ 2.31

(34)

Together with structural constraints shown in eqs 7 and 8, the design problem is solved and an optimum solution is obtained in terms of signatures. By utilizing integer cuts, the best five solutions are obtained in terms of signatures and summarized in Table 3. With the signatures obtained from solving the design problem, a molecular graph can now be generated based on the graph signature enumeration algorithm by Chemmangattuvalappil and Eden.80 By using the graph enumeration algorithm, molecular structures are generated from the list of signatures, and the names of the new solvents are identified. The best five solutions are summarized in Table 4: From Table 4, it can be seen that all of the solvent properties fall between the boundaries that represent customer requirements (see Table 2). Note that these molecules are targeted on the basis of the given properties and structural constraints. Therefore, these molecules are capable of replacing MDEA as the solvent for the gas sweetening process. Compared to MDEA, most of the generated solvents possess better performance of Koc while having comparable values with MDEA for Vm and Hv. However, VP and TLC for the generated solvents are not as good as those possessed by MDEA as the VP for most of the generated solvents are higher than that for

(27)

(28)

Note that Koc, VP, and Vm are to be minimized in this case study. As shown in eq 22, ΩVP is expressed in term of Tb. From the equation, it is known that a higher Tb value leads to a lower value of VP. Hence, ΩVP (expressed in terms of Tb) is maximized in order to obtain minimum VP as shown in eq 28. Next, molecular building blocks which are suitable for the design problem are determined. The molecular building blocks have to be selected such that the properties and molecular structure of the new solvent are similar to the available amine solutions. Absorbents which are available in market and widely used industrially for the gas sweetening process are monoethanolamine NH 2 CH 2 CH 2 OH, diethanolamine OHCH 2 CH 2 NHCH 2 CH 2 OH, methyl diethanolamine OHCH2CH2NCH3CH2CH2OH, diisopropylamine CH3CH3CHNHCHCH3CH3 On the basis of the chemical formula of these existing amine solutions, molecular groups were chosen as the building blocks 17439



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Table 3. List of Solutions in Terms of Signatures signature

occurrence Solution A

C1(C2(C1(C)C2(CC))) C2(C1(C2(CC))C2(C2(CC)C2(CC))) C2(C2(C1(C)C2(CC))C2(C2(CC)C2(CC))) C2(C2(C2(CC)C2(CC))C2(C2(CC)C2(CC))) C2(C2(C2(CC)C2(CC))C2(C2(CC)C2(CO))) C2(C2(C2(CC)C2(CC))C2(C2(CC)O2(CC))) C2(C2(C2(CC)C2(OC))O2(C1(O)C2(CO))) O2(C1(O2(CC))C2(C2(CC)O2(CC))) C1(O2(C1(O)C2(CO))) Solution B C1(C2(C1(C)C2(CC))) C2(C1(C2(CC))C2(C2(CC)C2(CC))) C2(C2(C1(C)C2(CC))C2(C2(CC)C2(CC))) C2(C2(C2(CC)C2(CC))C2(C2(CC)C2(CO))) C2(C2(C2(CC)C2(CC))C2(C2(CC)O2(CC))) C2(C2(C2(CC)C2(OC))O2(C2(CO)C2(CO))) O2(C2(C1(C)O2(CC))C2(C2(CC)O2(CC))) C2(C1(C2(OC))O2(C2(CO)C2(CO))) C1(C2(C1(C)O2(CC))) Solution C C1(C2(C1(C)C2(CC))) C2(C1(C2(CC))C2(C2(CC)C2(CC))) C2(C2(C1(C)C2(CC))C2(C2(CC)C2(CC))) C2(C2(C2(CC)C2(CC))C2(C2(CC)C2(CC))) C2(C2(C2(CC)C2(CC))C2(C2(OC)C2(CC))) C2(C2(C2(CC)C2(CC))C2(C2(CC)O2(CC))) C2(C2(C2(CC)C2(OC))O2(C2(CO)C2(CO))) O2(C2(C1(C)O2(CC))C2(C2(CC)O2(CC))) C2(C1(C2(CO))O2(C2(CO)C2(CO))) C1(C2(C1(C)O2(CC))) Solution D C1(C2(C1(C)C2(CC)) C2(C1(C2(CC))C2(C2(CC)C2(CC))) C2(C2(C1(C)(2(CC))C2(C2(CC)C2(CC))) C2(C2(C2(CC)C2(CC))C2(C2(CC)C2(CC))) Solution E C1(C2(C1(C)C2(CC)) C2(C1(C2(CC))C2(C2(CC)C2(CC))) C2(C2(C1(C)(2(CC))C2(C2(CC)C2(CC)))

1 1 1 1 1 1 1 1 1

minimize Ω KOC

(35)

maximize Ω VP

(36)

minimize ΩVm

(37)

From Table 5, the solution with minimized Koc is solution B (ethylhexyl ether), the solution with minimized VP is solution C (ethyl heptyl ether), while the solution with minimized Vm is solution E (hexane). Although the generated solutions with a single optimized property are within the solutions generated by using the multiobjective optimization approach, it is very difficult to ensure that the generated solutions are Pareto optimal solutions. As the single objective optimization approach identifies solutions by optimizing only one of the properties of interest, there is a possibility that the other properties of interest might be improved to produce a better solution. Hence, Pareto optimal solutions cannot be guaranteed while solving a product design problem with multiple properties of interest by using the single objective optimization approach. Table 6 shows the comparison of degrees of satisfaction between the solutions generated by using the max−min aggregation approach. The relative importance of each property of interest, wp, which can be determined by using eq 38, is shown in Table 6.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

wp =

λp ∑ λp

(38)

wp denotes the weighting factors that a decision maker has to assign to each property of interest if the design problem is solved by using the conventional weighted sum method. As shown in Table 6, the property with the highest weighting factors is not always the same for the generated solutions. Since all properties of interest are treated without any bias, it is very difficult in identifying the optimal solutions by using the conventional weighted-sum method. From Table 6, it can be seen that each of the properties of the solvent is being treated justly as solutions are ranked accordingly to their least satisfied property. It is shown that the least satisfied property is not always the same property for all the five generated solutions. This indicates that the developed algorithm identifies the relative importance of each property to be optimized without the presence of a decision maker. As long as the degree of satisfaction for the least satisfied property is maximized, the generated solution is a feasible molecule capable of replacing the available solvent. However, it can also be noticed from Table 6 that the least satisfied property for solution A and solution B is the same: the VP at 0.2807 is the same for both. Therefore, it is not possible to distinguish which solution is more superior at this stage by solely referring to the least satisfied property. This is the limitation of the max−min aggregation approach. As discussed

2 2 2 1 2 2 2

MDEA, while the TLC for most of designed solvents are lower than that of MDEA. To compare the solutions with the solutions generated by solving the design problem as a single objective optimization problem, the MILP model is solved as a single objective optimization problem to generate solutions with a single optimized property. Equations 35−37 are solved separately subject to the corresponding property and structural constraints. Table 5 shows the solutions with minimized Koc, minimized VP, and minimized Vm. Table 4. Possible Design of Solvents solution

solvent

Koc

VP (mm Hg)

Vm (cm3/mol)

Hv (kJ/mol)

TLC (ppm)

A B C D E

heptyl methyl ether ethylhexyl ether ethyl heptyl ether heptane hexane

139 108 198 295 160

22.82 22.82 9.13 151.85 470.01

169 171 188 128 112

47 47 52 37 32

14 58 19 22 69

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Table 5. Design of Solvents with Single Optimized Property optimized property

solution

Koc

VP (mm Hg)

Vm (cm3/mol)

Hv (kJ/mol)

TLC (ppm)

minimized Koc minimized VP minimized Vm

B C E

108 198 160

22.82 9.13 470.01

171 188 112

47 52 32

58 19 69

Table 6. Comparison of λp between Different Designs of Solvent solution

solvent

λKoc

λVP

λVm

wKoc

wVP

wVm

A B C D E

heptyl methyl ether ethylhexyl ether ethyl heptyl ether heptane hexane

0.8572 0.9680 0.7030 0.5304 0.7954

0.2807 0.2807 0.3759 0.1025 0.0073

0.2983 0.2863 0.1779 0.5606 0.6690

0.60 0.63 0.56 0.44 0.54

0.19 0.18 0.30 0.09 0.01

0.21 0.19 0.14 0.47 0.45

Table 7. Comparison of Σλ*p between Different Designs of Solvent

earlier, the primary weakness of a max−min aggregation approach is its lack of discriminatory power to distinguish between solutions which have different levels of satisfaction other than the least satisfied goal. It is noted that the max−min aggregation approach models flexible constraints rather than objective functions. Some of the objectives might be overly relaxed or curtailed in order to maximize the least satisfied objective. Because of this limitation, the max−min aggregation approach does not guarantee to yield a Pareto optimal solution.94,95 To discriminate these solutions to refine the order of solutions and at the same time to ensure Pareto optimality to identify the best solvent, a two-phase approach is utilized. 4.3.2. Two-Phase Approach. The aim for two-phase approach is to maximize all the properties as a whole, as shown by eq 18. As the degree of satisfaction for the least satisfied property identified by using max−min aggregation approach is 0.2807, the degree of satisfaction for the second stage should be equal or higher than this value to ensure that the solution obtain in the second stage is not worse than that obtained in the first stage. Hence, eq 40 is included in the optimization model. maximize λ

λ K*OC =

* = λVP λV*m =

solvent

λ*Koc

λVP *

λ*Vm

∑λp*

B E A C D

ethylhexyl ether hexane heptyl methyl ether ethyl heptyl ether heptane

0.9680 0.7954 0.8572 0.7030 0.5304

0.2807 0.0073 0.2807 0.3759 0.1025

0.2863 0.6690 0.2983 0.1779 0.5606

1.5350 1.4716 1.4362 1.2568 1.1934

although both heptyl methyl ether and ethylhexyl ether have the same least satisfied property, heptyl methyl ether is now the third solution, while ethylhexyl ether is now the first solution. The degree of satisfaction in terms of least satisfied property is still the highest for solution 1 by using the two-phase approach. This indicates that even though the two-phase approach is able to discriminate the solutions with similar least satisfied property, the approach does not compromise the degree of satisfaction of that property. Hence, utilization of the two-phase approach after the max−min aggregation approach ensures the generation of optimal results without worsening any property in terms of degree of satisfaction. This can be further confirmed by illustrating all possible optimal solutions via the Pareto frontier as shown in Figure 9. Figure 9 shows the Pareto frontier for the solvent design problem. The optimization objective used to generate the Pareto frontier is given as below:

(39)

subject to

λp* ≥ 0.2807

solution

(40)

2.34 − Ω KOC 2.34 − 1.34

(41)

Ω VP − 4.50 12.00 − 4.50

(42)

5.50 − ΩVm 5.50 − 0.89

(43)

16.81 ≤ Ω Hv ≤ 53.46

(44)

Ω TLC ≤ 2.31

(45)

The above model is solved and the best five solutions are obtained. These are shown in Table 7. The solutions are ranked according to their summation of all degrees of satisfaction. It is noted that the ranking changes significantly compared to that obtained in the first stage (Table 6). More importantly, the solutions with similar least satisfied property are distinguished. According to their summation of all levels of satisfaction,

Figure 9. Pareto frontier for the solvent design problem. 17441


Industrial & Engineering Chemistry Research objective function = aKoc + b VP + cVm

Article

(46)

where a, b, c are the weighting factors manually assigned to Koc, VP, and Vm to represent the property contributions decided by a decision maker. From Figure 9, it can be seen that ethylhexyl ether is on the Pareto frontier. This shows that although the max−min aggregation approach maximizes the least satisfied property, it does not guarantee to yield a Pareto optimal solution. However, by utilizing the two-phase approach after the max−min approach, a Pareto optimal solution can be obtained. By seeking for maximum utilization of each product property, this guarantees a Pareto optimal product while optimizing multiple product properties.

5. CONCLUSIONS This paper introduces a novel methodology in chemical product design by incorporating fuzzy and bilevel optimization into the molecular design techniques. By utilizing signaturebased molecular design, different classes of property models can be expressed in terms of molecular signatures and utilized in a chemical product design problem. By incorporating fuzzy optimization into the methodology, property weighting factors in a multiobjective optimization problem are able to be addressed systematically without bias and the optimal product can be identified. Bilevel optimization is utilized to determine the property target ranges which are undefined. The product properties are first maximized on the basis of a max−min aggregation approach. The two-phase approach is then employed to discriminate the products with similar least satisfied property. In this case study, different topological indices are transformed into molecular signatures with different height which are required to describe the final molecules. Apart from that, by incorporating bilevel optimization and both a max−min aggregation approach and two-phase approach into the fuzzy optimization algorithm, the developed method is able to address the properties of the molecules and optimize them simultaneously without any bias. Future efforts will be on enhancing the robustness of this methodology by studying the uncertainty of property models. In addition, effectiveness of the methodology could be improved by developing sensitivity analysis on the property target ranges. This is possible by investigating the influence of product target ranges on the identification of optimal product. Furthermore, to improve the completeness of the methodology, the applicability of the methodology may be extended by considering practical product design factors such as the cost of raw material or operating cost in overall process and product design.

■

■

QSPR/QSAR = quantitative structure property/activity relationships TI = topological index χ = connectivity index ε = edge adjacency index G = molecular subgraph h = height of signature vLp = lower limit for target property p vUp = upper limit for target property p Vp = value for target property p Vp* = value for target property p* with unknown property target ranges λp = degree of satisfaction for target property p λ*p = degree of satisfaction for target property p in two-phase approach N = total number of signatures in a molecule n1 = number of signatures of valency one n2 = number of signatures of valency two n3 = number of signatures of valency three n4 = number of signatures of valency four NDi = signatures with one double bond NMi = signatures with two double bonds NTi = signatures with one triple bond Ni = number of occurrence of first order group of type-i Mj = number of occurrence of second order group of type-j Ok = number of occurrence of third order group of type-k Ci = contribution of the first order group of type-i Dj = contribution of the second order group of type-j Ek = contribution of the third order group of type-k R = number of circuits in a molecular graph Ψ = property operator θ = property function α = number of each signature x = number of signatures Ωp = normalized property operator for target property p Hv = heat of vaporization VP = vapor pressure Vm = molar volume Koc = soil sorption coefficient TLC = toxic limit concentration Tb = boiling point

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the University of Nottingham Research Committee via the Dean’s Ph.D. Scholarship is gratefully appreciated.

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NOMENCLATURE CI = connectivity index GC = group contribution 17442



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A Multiobjective Optimization-Based Approach for Optimal Chemical

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