Decision-Making for Sustainability Enhancement of Chemical Systems

Article Cite This: Ind. Eng. Chem. Res. 2019, 58, 12066−12079

pubs.acs.org/IECR

Decision-Making for Sustainability Enhancement of Chemical Systems under Uncertainties: Combining the Vector-Based Multiattribute Decision-Making Method with Weighted Multiobjective Optimization Technique Di Xu,*,† Weichen Li,‡ Weifeng Shen,‡ and Lichun Dong*,‡,§ †

College of Chemistry and Chemical Engineering, Chongqing University of Science & Technology, Chongqing 401331, China School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 400044, China § School of Chemistry and Chemical Engineering, Collaborative Innovation Center for Green Development in Wuling Moutain Area, Research Center for Environmental Monitoring, Hazard Prevention of Three Gorges Reservoir, Yangtze Normal University, Fuling, 408100 Chongqing, China

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‡

S Supporting Information *

ABSTRACT: The identification of the best solution for improving the sustainability of chemical systems is quite challenging because multiple objectives have to be considered and the trade-offs among the objectives must be addressed. This study aims to develop a novel framework to simplify multiobjective optimization (MOO) problems by incorporating a vector-based multiattribute decision-making (MADM) method. In the framework, the enhancement system is first defined and characterized on a case-by-case basis in the first phase, while the second phase adopts a unique Pareto solution for the MOO problems, which is characterized by not only comprehensively addressing the trade-offs among multiobjective via the aggregation of both decision makers’ preferences and system’s properties but also rigorously identifying the best solution via the integration of both absolute improvement degree and relative development balance of the multiobjective system. Besides, the uncertainty degrees of the original data can be wellpreserved in the whole framework, offering a rigorous approach for sustainability enhancement under uncertainties. The application of the framework was illustrated by a well-established case study, while the effectiveness and the advantages were verified by the results discussion.

1. INTRODUCTION Although representing 4% of the global economy, the chemical industry has often been taken as an unfavorable sector during the last decades for the growing burdens on high energyconsumption, severe environmental impacts, heavy investment, and various negative social effects. Accordingly, the incorporation of the concepts of “sustainable development” into chemical systems has been a hot topic for several years, which emphasizes triple-bottom-line (TBL) sustainability, i.e., the simultaneous optimization of economic prosperity, environmental performance, and social responsibility.1−4 Considering that numerous chemical projects with tremendous investments are already in operation, retrofitting actions with respect to the environmental, economic, and social concerns are also necessary to adapt the industry to sustainable development. Consequently, a variety of useful methods/tools have been developed for helping the decision makers to better understand the different sustainable enhancement alternatives. For instance, the process flow sheet decomposition-based methodology has been used for assessing the retrofitting actions of chemical systems.5−7 Jayswal et al.8 introduced a sustainability root cause analysis methodology for identifying the bottlenecks regarding the chemical © 2019 American Chemical Society

processes and offering opportunities for their sustainability improvement. Liu and Huang2 proposed a generic framework for technology assessment and decision making for enhancing the sustainability of chemical systems. Moradi-Aliabadi and Huang9 developed a mathematical methodology for the multistage optimization of chemical processes by incorporating the TBL concerns and technical feasibilities. Ruiz-Mercado et al.10 integrated three different tools for the sustainability investigation and improvement of chemical systems, i.e., WRA (waste reduction algorithm) for early stage design, GREENSCOPE for detailed analyze, and software of SustainPro for sustainability enhancement. The sustainability enhancement of an existing chemical system is a complex task that involves multiple steps and activities, e.g., defining the problem scope and the sustainability goal, evaluating the current sustainability status and solution options, selecting and implementing the preferred solution, and Received: Revised: Accepted: Published: 12066

March 19, 2019 May 19, 2019 June 18, 2019 June 18, 2019 DOI: 10.1021/acs.iecr.9b01531 Ind. Eng. Chem. Res. 2019, 58, 12066−12079

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Industrial & Engineering Chemistry Research assessing the outcomes after adopting the solution.11 More importantly, various complicated decision-making scenarios may arise from the multiple conflicting objectives with constraints that should be simultaneously satisfied.12 Moreover, when dealing with this issue, the parameters or variables involved in the decision making are usually imprecise due to the multiple uncontrollable factors,2,3,13 making the decisionmaking scenarios more complicated. For addressing the challenges for the multiobjectives sustainability enhancement, this paper aims to propose a systematic decision-support framework by incorporating a novel vector-based multiattribute decision-making (MADM) method into a weighted multiobjective optimization (MOO) technique under uncertainties. The framework can identify a unique Pareto solution of the MOO problem regarding a chemical system, which not only comprehensively reflects the decision makers’ preferences and the systems’ properties by using the combined weighs of the objectives but also escalates the modeling capability of the MADM-MOO combination via the integration of the absolute improvement degree and relative development balance regarding the multiple objectives of the system. Besides the introduction, the remainder of this study is organized as follows: Section 2 reviews the existing literature related to the problem. Section 3 proposes the decision-making framework. Section 4 presents a case study and the corresponding results and discussions. Section 5 offered the implications of the theory, and finally, this study was concluded in section 6.

for the process synthesis of a sustainable integrated biorefinery, in which the economic, environmental, safety, and health impacts were synthetically integrated. You et al.25 addressed the optimal design and planning of biofuel supply chains under economic, environmental, and social objectives by developing a multiobjective mixed-integer linear programming (MILP) model. While an optimization paradigm for the synthesis of chemical processes with product distribution optimization was introduced by Gong and You,26 in which, a multiobjective mixed-integer nonlinear program (MINLP) is used for designing the most sustainable process. Recently, Moradi-Aliabadi and Huang27 considered the technology-based retrofitting of the chemical systems as a MOO problem with the environmental, economic, and social objectives, where the Pareto set of the MOO model can be obtained by using the ε constraint method, offering the decision makers a feasible route to select the best solution for enhancing the system’s sustainability. Considering the complexity of the chemical systems, the MOO technique could be limited by various computational problems and the inability of human brains to balance a large number of objective simultaneously;28 in other words, it is not only time-consuming29 but also fails to sufficiently address the trade-offs between the multiple objectives.30 To solve this problem, the MADM methods have been suggested to be incorporated into the MOO technique for reducing the computation burdens of the MOO technique (by guiding the search toward a specific region of the Pareto front) and understanding the trade-offs between the objectives (through reflecting the decision makers’ preferences).29 For instance, the AHP method has been incorporated into MOO techniques for supporting the decision-making issues including the selection of industrial engineering sectors,31 the operation optimization of ethylene cracking furnace,32 and the design of resource network in an eco-industry park.24 Manzardo et al.12 developed a MOO approach by incorporating the philosophy of TOPSIS for the optimization of chemical pulp supply mix in the paper industry. Sehatpour and Kazemi14 introduced a hybrid MADM-MOO method for the sustainable fuel portfolio optimization, where the fuzzy multiobjective programming is supported by a MADM approach (i.e., the superiority and inferiority ranking method). Wheeler et al.29 simplified the MOO model in the design of biomass supply chains by using weighting factors that are derived from some existing MADM methods. Despite the emergence and development of the combination of MADM with MOO in the sustainability assessment and optimization, the following concerns have not been investigated sufficiently: (1). The previous studies rely heavily on the subjective opinions for weighting the multiple objectives, ignoring the involved objective properties of the investigated system. (2) The conventional MADM-MOO models identify the best solution only based on the absolute aggregated score regarding the assessment system, failing to address the relative balance among the objectives. (3) Different types of uncertainties with respect to the objective data and subjective preferences have not been tackled simultaneously in the existing sustainability enhancement frameworks. Therefore, this study aims to develop a novel framework for the decision making on the sustainability enhancement of chemical systems under uncertainties via the integration of a vector-based MADM method and a weighted MOO technique, in which, the objective data regarding the assessment system and the subjective preferences of the decision makers with different uncertainties are fully utilized to reflect the relative importance

2. LITERATURE REVIEW There are many studies in the literature that focus on dealing with the complex information and identifying the best solution regarding chemical systems by resorting to the MADM methods or the MOO techniques.14 In their approaches, the MADM methods are usually used to screen and then rank a finite number of alternatives via the aggregation of performances of multiple criteria in terms of the weights regarding each criterion.15 For instance, Othman et al.16 presented an approach for the sustainability assessment and design selection by embedding the economic-environment-social criteria in the AHP (analytic hierarchy process) method. Ordouei et al.17 introduced a model for addressing the sustainability performance of chemical processes, in which, the sustainability indicators were weighted by using the AHP method and then aggregated to obtain the composite sustainability index for the decision making. Two MADM methods, i.e., the DEMATEL and AHP methods, were combined in the work of Serna et al.18 to calculate the weights and influence between the assessment indicators, which were then used for comparing chemical process alternatives. The other MADM methods like TODIM, TOPSIS, and PROMETHEE have also been adopted or adapted for supporting the decision making on chemical systems.3,19,20 Compared to the MADM methods, the MOO technique is much more favored in chemical engineering to handle sustainability issues because it can identify the best solution by considering the trade-off relationships under design constraints instead of enumerating the sustainability performance of each possible alternative as the MADM methods do.21,22 For instance, Liew et al.23 employed the MOO technique for the sustainability assessment of chemical processes with the consideration of inherent safety, health and environment, and economic performance. Leong et al.24 proposed a MOO model 12067

DOI: 10.1021/acs.iecr.9b01531 Ind. Eng. Chem. Res. 2019, 58, 12066−12079

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Industrial & Engineering Chemistry Research of the objectives and then to form the unique Pareto point with the consideration of both the absolute improvement degree and relative development balance among the multiple objectives. As a result, the proposed framework can simplify the MOO problem of the sustainability enhancement of chemical systems under uncertainties by combining the vector-based MADM algorithm, making three important contributions: (1) The trade-offs among the multiple objectives can be better addressed by adopting the combined weighting method to flexibly aggregated both of the subjective opinions and objective properties. (2) By developing a vector-based MADM-MOO model, both the absolute improvement degree and relative development balance regarding the multiobjective can be rigorously integrated for better identifying the best solution. (3) Uncertainties with respect to the subjective preferences as well as the objective data can be easily handled for better reflecting the complexities of the real-world decision issues by using the interval numbers.

3. DECISION-MAKING FRAMEWORK This study aims to propose a systematic decision-making framework for rigorously identifying the best solution for the technology-based sustainability enhancement of chemical systems. Since the accessible data and information regarding the enhancement system are usually uncertain, the interval numbers were incorporated into the proposed framework for handling the uncertainties, while some preliminary knowledge regarding the interval-based mathematical framework were offered in the Supporting Information (S1). As shown in Figure 1, a prestage of system definition should be conducted on a case-by-case basis regarding the investigated system before starting the mathematical framework, in which the enhancement objectives and the retrofit technologies should be identified, and the constraints such as technical feasibility and budget limit regarding the investigated system should be determined. Subsequently, the data with respect to each technology regarding a certain objective can be collected through specific analytic tools, where the uncertainties regarding the data can be preserved by interval numbers. The mathematical framework, acting as a generic optimization approach for identifying the best solution of the technologybased sustainability enhancement problem, embraces three stages, i.e., determination of the weighted objectives, development of the final goal, and generation of the best solution. The advantages and novelties of the developed framework in each stage can be summarized as follows: In Stage 1, the trade-offs among the multiple objectives can be better addressed by adopting the combined weighting method to flexibly aggregate the subjective judgments and objective data (steps 1−3); subsequently, the attainability of each enhancement objective can be guaranteed by introducing a maximization function (step 4). In Stage 2, both the absolute improvement degree and relative development balance regarding the multiobjective are integrated for offering a rigorous final goal with respect to the sustainability enhancement by modifying a vector-based algorithm (step 5− 7). In Stage 3, the identification of the best solution under uncertainties can be significantly simplified by introducing a maximization function (step 8) and a possibility approach-based searching strategy (step 9). 3.1. System Definition and Characterization (Prestage). Four actions are usually conducted in the prestage on

Figure 1. Decision-making framework for enhancing the sustainability of chemical systems.

a case-by-case basis, i.e., establishment of objectives system, recommendation of retrofit technologies, identification of implementation constraints, and collection of demand data; therefore, only the generic introduction of each action is offered below. 3.1.1. Establishment of Objectives System. For improving the sustainability of a chemical system, multiple concerns regarding economic prosperity, environmental impacts, and social responsibility should be taken as objectives according to the actual conditions of the system and the stakeholders’ preferences. In this study, Oi (i = 1, 2, ..., n) denotes the ith objective in an n-dimensional system. Notably, if too many concerns are involved in the decision issue, then they can be aggregated into three categorized (environmental, economic, and social) objectives by following the procedures reported by Liu and Huang.2 3.1.2. Recommendation of Retrofit Technologies. In this step, the candidate technologies were suggested as feasible solutions for the sustainability enhancement, which are commonly provided by professional experts via the approaches like experience-based methods, experimental tests, and computer simulation, and so on.5,33 Although this action relies heavily on the experts’ knowledge and experiences, a variety of tools7,8,10 and guidelines34,35 can be referred for systematically detecting the bottlenecks and then generating the possible retrofit technologies. In this study, the recommended technologies are decision variables, which can be directly defined as Tj (j = 1, 2, ..., m), referring to the jth technology 12068


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Industrial & Engineering Chemistry Research among a set of m recommendations. Tj is a binary variable that equals 1 if the jth technology is selected for the retrofit. Since different technologies can be employed and combined, the term of portfolio (P) was adopted here for representing the combined technologies, which is denoted as P = T1 + T2 + ... + Tm (Tj = 0 or 1); consequently, the total number of portfolios is 2m − 1. Apparently, limited technologies would be finally recommended into the shortlist decision variables with the help of expert consultation.2 3.1.3. Identification of Implementing Constraints. For conducting the technology-based sustainability enhancement, some constraints have to be established, which can be categorized into two types of equality and inequality. According to the literature,2,9 the equality constraints in this study come from the mutual conflictions and restrictions among the candidate technologies, e.g., some technologies cannot or must be adopted together, while the inequality constraints could be the additional costs for implementing a candidate technology, e.g., additional investments and construction time,9 or the minimum requirement for improving a certain objective.2 The detailed information and numerical properties of these constraints were offered in Supporting Information (S2). 3.1.4. Collection of Demand Data. In this step, the performance with respect to each technology regarding a certain objective would be collected through various analytic tools, like literature surveys, experimental tests, system simulations, and mathematical techniques, and so on.7,36 Generally, the information regarding a specific chemical system are collected by using some commercial simulations like Aspen Plus, HYSYS, and Pro/II, which can offer basic but detailed information regarding energy, materials, emissions, and equipment, among others.7,37 Subsequently, the simulation information can be utilized to generate the demand data by running corresponding calculation rules/techniques. On the basis of these specific activities, the data with respect to the system’s performances regarding the ith objective before and after implementing the jth technology can be determined, respectively, which are then used to generate the corresponding demand data via the subtraction. The collected data are denoted as the form of interval numbers as f ij = [f ijL, f ijU] for preserving the uncertainties,13 while for eliminating the effect of different physical units and scales associated with the data of different objectives, all the collected data should be normalized.2 For more details on how to use the normalized interval numbers to represent the demand data, the users can refer to the literature,36 and a simple example was also offered in the Supporting Information (S3). It is noteworthy that the above-mentioned four actions (in section 3.1) should be specifically conducted according to the investigated case, while the following sections 3.2−3.4 make up the novel and generic mathematical framework for supporting the decision making of the technology-based sustainability enhancement. Since the main motivation of this work is to develop a singlegoal function rather than a multiobjective one for finding a unique Pareto point, two key issues including the determination of the weights and the aggregation of the multiple objectives must be addressed, thereby avoiding the exhaustive exploration of a set of Pareto frontiers and consequently simplifying the entire analysis. Without losing the generality, a weighted MOO approach that could offer a unique Pareto point is given in eq 1.12,14,29

Optimum F[wG i i(P)] (i = 1, 2, ..., n) s.t.

hl(P) = 0 (l = 1, 2, ..., r ) vk(P) ≤ 0 (k = 1, 2, ..., s) n

∑ wi = 1 i=1

(1)

where wi is the weight of the ith objective, P is a portfolio that represents a certain combination of technologies, Gi(P) refers to the ith objective, and hl(P) = 0 and vk(P) ≤ 0 indicate that a total of r equality constraints and s inequality constraints should be satisfied, respectively. In order to escalate the modeling capability of the weighted MOO model (eq 1), the key issue of weights determination (wi) is handled by fully utilizing both the subjective preferences and objective data, while a unique solution (rather than a set of Pareto frontiers) is identified by incorporating a vector-based MADM with the consideration of both the absolute improvement degree and the relative development balance among the multiple objectives. The combination of MADM-MOO was specified as follows. 3.2. Weighted Objectives Determination (Stage 1). As shown in eq 1, wi plays a critical role in the model for it reflects the trade-offs among the multiple objectives, which is then employed to form the final Pareto point in the decision making. Therefore, the accurate weighting of the objectives is the prerequisite for generating a reliable decision. Typically, the weighting methods in the literature can be categorized into three types: the subjective methods, such as AHP and ANP that allocate the weights according to the decision makers’ preferences;38 the objective methods, such as entropy technique and CRITIC method that assign the weights according to the numerical data associated with the investigated system;39 and the combined ones that integrate both the subjective and objective methods.40 Considering the complexity and specialty of chemical systems, the determination of weights should not only grasp the preferences of the decision makers but also reflect the natures regarding the numerical systems;40 moreover, the epistemic uncertainty in human’s judgments and the aleatory uncertainty in numerical data should also be addressed.13 Therefore, this framework combines the subjective AHP method with the objective entropy technique and extends them into interval conditions for the weights determination under different uncertainties. 3.2.1. Step 1. Obtaining the Subjective Weights. AHP, one of the most popular subjective methods, determines weights according to decision makers’ judgments through pairwise comparisons, offering a preferential-based result with good evaluation consistency.38 However, the conventional AHP method can only provide the users with crisp values to express their judgments, failing to tackle the epistemic uncertainty associated with the subjective information.3 Consequently, a variety of extended AHP methods have been developed to address vague and imprecise subjective judgments by incorporating interval numbers.41,42 In this study, the interval AHP method proposed by Zhang42 is used to generate the interval subjective weights from the interval comparison matrices with the consideration of uncertainties in human’s preferences. The adopted method has an excellent property to model the interval preference judgments with consistency, and its procedures are as follows.42 12069


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subjective weights [swLi , swUi ]. Then, the deterministic subjective weight can be calculated by eq 6.

3.2.1.1. Substep 1.1. The interval numbers in Satty’s 1−9 scale (Table 1) are used to establish the interval comparison Table 1. Comparison Scale38

wiS =

numerical scale

definition

1 3 5 7 9 2, 4, 6, 8 reciprocal

equal importance moderate importance essential importance very strong importance absolute importance intermediate value reciprocals of above

ÑÉ μ [a1Ln , a1Un] ÑÑÑÑ ÑÑ Ñ μ [a 2Ln , a 2Un]ÑÑÑÑ ÑÑ ÑÑ ÑÑ ∏ ∂ ÑÑ ÑÑ μ [1, 1] ÑÑÑÖ

[aijL,

(2)

fij

ÑÉ μ [r1Ln , r1Un] ÑÑÑÑ ÑÑ Ñ μ [rnL2 , rnU2] ÑÑÑÑ ÑÑ ÑÑ ÑÑ ∏ ∂ ÑÑ ÑÑ μ [0.5, 0.5]ÑÑÑÖ

É| ÄÅ É ÅÄÅ l ÄÅÅ m L L Ñ U Ñ Ño Å m U Ñ ÅÅ o − ÅÅÅ∑ j = 1 f ij̅ ln(f ij̅ )ÑÑÑ − ÅÅÅ∑ j = 1 f ij̅ ln(f ij̅ )ÑÑÑ o ÅÅ o o o o ÑÖ o Å Ñ Å Å o o L U Ç Ö Ç Å , [Hi , Hi ] = ÅÅÅmino }, m o o o ÅÅ o ln( m ) ln( m ) o o ÅÅ o o o ÅÇ n ~ É Ñ É Ä É Ä Ñ Ñ Å Ñ Å | l Ñ L L U U ÑÑ o ÅÅ m o ÅÅ m ÑÑ oÑÑÑ o o o o − ÅÅÅÇ∑ j = 1 f ij̅ ln(f ij̅ )ÑÑÑÖ − ÅÅÅÇ∑ j = 1 f ij̅ ln(f ij̅ )ÑÑÑÖ o ÑÑ o ÑÑ maxo , } m o o o o ln( m ) m ln( ) oÑÑÑÑ o o o oÑÑ o ~ÑÖ n

(3)

(4)

[HLi ,

3.2.2.3. Substep 2.3. The interval objective weight [owLi , owUi ] is obtained by using eq 9 and then obtaining the deterministic ones wOi according to eq 10. ÄÅ ÉÑ ÅÅ D L DiU ÑÑÑÑ ÅÅ i L U Ñ [ow i , ow i ] = ÅÅ n , ÅÅ ∑ D U ∑n D L ÑÑÑ ÅÇ i = 1 i (9) i=1 i Ñ Ö

i=1 n

sw iU ≥ 1

j = 1, j ≠ i n

sw iU +

∑

sw iL ≤ 1

j = 1, j ≠ i

where [DLi , DUi ] is the interval degree of diversification regarding the ith objective and DLi = 1 − HUi , DUi = 1 − HLi , while the obtained [owLi , owUi ] is used to generate the deterministic objective weight by eq 10.

0 ≤ sw iL ≤ sw iU ≤ 1

yz ij n z jj jj∑ rijL − n + 0.5zzzsw iL + zz jj = j 1 { k

yz jij n U jj∑ r − n + 0.5zzzsw U + zz i jj ij z j j=1 { k ciU , ciL , diU , diL ≥ 0

n

∑

(8)

HUi ]

where represents the entropy value for the ith objective, m is the number of the recommended technologies, L U m m and f ij̅ = (fijL )α /∑ j = 1 (fijU )α , f ij̅ = (fijU )α /∑ j = 1 (fijU )α .

n

∑

ij

(7)

∑ (ciU + ciL + diU + diL)

sw iL +

ij

where 0 < α ≤ 1. A larger α indicates that the interval number is more precise, while a smaller α suggests that the result can be realized with higher confidence.44 3.2.2.2. Substep 2.2. Calculating the interval entropy value for each objective according to eq 8.

3.2.1.3. Substep 1.3. The interval range of the subjective weights [swLi , swUi ] is calculated by using eq 5 (based on the work of Zhang)42 and then obtaining the deterministic ones wiS via running eq 6.

s.t.

(6)

[min{fij ∈ R |μ f (fij ) ≥ α}, max{fij ∈ R |μ f (fij ) ≥ α}]

U

rijL = aijL /(aijL + aijU), rijU = aijU /(aijL + aijU)

Min Q =

+ sw iU)

[(fijL )α , (fijU )α ] =

In eq 2, there are n objectives; aij ] denotes the interval relative importance of the ith objective comparing with the jth one, and [ajiL, ajiU] = [1/aijU, 1/aijL]. 3.2.1.2. Substep 1.2. The matrix A is converted into the preference relation matrix R by using eq 3. ÅÄÅ L U ÅÅ[0.5, 0.5] [r12 , r12 ] ÅÅ ÅÅ ÅÅ [r L , r U ] [0.5, 0.5] Å R = ÅÅÅÅ 21 21 ÅÅ ∂ ∂ ÅÅ ÅÅ ÅÅÅ [r L , r U ] [r L , r U ] n2 n2 ÅÇ n1 n1

sw iU

3.2.2. Step 2. Obtaining the Objective Weights. In order to reflect the numerical variations regarding the information in real situations, Shannon43 shifted the idea of entropy from thermodynamics into the information theory to measure the disorder degree of the information, i.e., an objective with a higher disorder degree should be given a larger weight. In this study, the interval entropy technique developed by Lotfi and Fallahnejad44 is adopted for determining the objective weights with the consideration of the aleatory uncertainties existing in the numerical data, which is characterized by making a trade-off between the precision and confidence regarding the objective weighting result by means of Shannon’s entropy at certain α-cut levels.44,45 3.2.2.1. Substep 2.1. The precision/confidence level of the numerical information is set by using eq 7 based on the work of Lotfi and Fallahnejad.44

matrix (A) as given in eq 2. For instance, the interval number of [5, 7] denotes that the relative importance of one objective over another is between the degrees of essential importance (5) and very strong importance (7), while [1, 1] implies that the two objectives are of equal importance. ÅÄÅ L U ÅÅ [1, 1] [a12 , a12 ] ÅÅ ÅÅ ÅÅ[a L , a U ] [1, 1] Å A = ÅÅÅÅ 21 21 ÅÅ ∂ ∂ ÅÅ ÅÅ ÅÅ L U L ÅÅÇ [an1, an1] [an2 , anU2]

sw iL + n ∑i = 1 (sw iL

rij L sw iU−ciU + ciL = 0

j = 1, j ≠ i

wiO =

n

∑

rijU sw iL−diU

+

diL

=0

j = 1, j ≠ i

ow iL + n ∑i = 1 (ow iL

ow iU + ow iU)

(10)

3.2.3. Step 3. Obtaining the Combined Weights. Equation 11 (minimum-relative-entropy) is used to integrate the subjective and objective weights,46 in which the relative priority between the objective weights and subjective ones can be

(5)

where cUi , cLi , dUi , and dLi are the positive and negative deviations of the function; the optimal solution of eq 5 is the interval 12070


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Industrial & Engineering Chemistry Research flexibly determined by the users via setting the combined coefficient u (u ∈ [0, 1]). wi =

(wiS)u (wiO)(1 − u) n

∑i = 1 (wiS)u (wiO)(1 − u)

(11)

3.2.4. Step 4. Setting Each Weighted Objective. According to the determined weights, eq 12 is used to set each weighted objective under constraints, in which the optimal solution is denoted as Gi*, representing the maximized improvement of the system regarding the ith objective by implementing retrofit technologies under constraints. ÄÅ ÉÑ ÅÅ ÑÑ ÅÅ Ñ U * Å max Gi = wi × ÅÅ ∑ (Tj × fij )ÑÑÑÑ ÅÅ ÑÑ ÑÑÖ ÇÅÅ j ∈ P

s.t.

Tj(1 − Tj) = 0 P∈C

Figure 2. Principle of the vector-based algorithm for a biobjective scenario.

(12)

where f ijU is the upper bound of the collected data, representing the most desirable degree of improvement of the system regarding the ith objective after implementing the jth technology; Tj(1 − Tj) = 0 implies that 1 or 0 can be assigned to the jth technology for judging whether it is adopted; P is a portfolio embracing one or more technology candidates, P = ∑j ∈ P (Tj); P ∈ C implies that the adopted portfolio (technologies combination) should satisfy the constraints. Apparently, in different weighted objectives, the corresponding portfolios (P) could be different from each other. 3.3. Final Goal Development (Stage 2). As stated in section 2, a single-goal model, offering a unique Pareto point, can be generated by incorporating MADM into the MOO technique, which is quite useful to simplify the original MOO problem associating with a set of Pareto frontiers. The effectiveness of the hybrid models for supporting the decision making has been demonstrated by combining a variety of wellknown MADM methods, e.g., WSM (weighted sum method), AHP, and TOPSIS, with the MOO techniques.12,24,29,31,32 However, the previous models identify the best solution only according to the final absolute score that is derived from the (additive/multiplicative) aggregation of preference ratings of feasible solutions regarding each objective, failing to reflect the relative balance among the multiple objectives, which may lead to an unreliable solution especially regarding the sustainability planning issues. To be specific, when dealing with the sustainability enhancement of chemical systems, the optimal solution should not only have a larger absolute improvement degree of sustainability but also be in line with a relative balanced development direction.47 As illustrated in Figure 2, although exhibiting similar performance in the absolute improvement degree of sustainability, solution 1 is superior to solution 2 since its development direction regarding the biobjective is more balanced. Finding that a feasible solution can be expressed as a vector with both magnitude and direction,47 a vector-based MADM method and its variants have been recently proposed,4,13 for offering a more rigorous sustainability assessment with the consideration of both the absolute improvement score and the relative development balance. Therefore, the vector-based MADM method was introduced in this stage for developing the final goal via the following steps.

3.3.1. Step 5. Representing the Alternative Solutions. By referring to the previous studies,4,13,47,48 the vector function of the ideal solution that owns the best performance regarding each weighted objective (denoted as S (⃗ PId)), and that of a feasible solution by implementing the kth portfolio (denoted as S (⃗ Pk)) can be presented by eqs 13 and 14, respectively. S (⃗ PId) = [G1*, G2* ..., Gn*]

(13)

S (⃗ Pk) = [S (⃗ PkL), S (⃗ PkU)] = {[G1(PkL), G1(PkU)] , [G2(PkL), G2(PkU)] , ..., [Gn(PkL), Gn(PkU)]}

(14)

3.3.2. Step 6. Characterizing the Vectors. According to the characteristic of the vector function that has both magnitude and direction,47 the absolute degree of sustainability improvement regarding each alternative is presented by the corresponding vector’s magnitude (M) (eqs 15 and 16 for the ideal solution and for a feasible solution, respectively), while the relative balance of the sustainable development is quantified by the cosine angle of the relative deviation between the ideal solution and the feasible solution (cos(A)) (eqs 17 and 18 for the ideal solution and for a feasible solution, respectively). n

M(PId) = ||S (⃗ PId)|| =

∑ (Gi*)2 (15)

i=1

M(Pk) = [M(PkL), M(PkU)] = [||S (⃗ PkL)|| , ||S (⃗ PkU)||] ÄÅ ÉÑ n ÅÅ n ÑÑ ÅÅ Ñ L 2 U 2 = ÅÅÅ ∑ (Gi(Pk )) , ∑ (Gi(Pk )) ÑÑÑÑ ÅÅ i = 1 ÑÑ i=1 ÅÇ ÑÖ

ij S (⃗ PId) ·S (⃗ PId) yz zz cos(A(PId)) = jjjj ⃗ PId)||S (⃗ PId)|| zz || S ( k { n * * ∑i = 1 [Gi × Gi ] = =1 n n ∑i = 1 (Gi*)2 × ∑i = 1 (Gi*)2 12071

(16)

(17)


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Industrial & Engineering Chemistry Research cos(A(Pk)) = [cos(A(PkL)), cos(A(PkU))] ÄÅ ÅÅÅ l o oij S (⃗ PId) ·S (⃗ PkL) yz ij S (⃗ PId) ·S (⃗ PkU) yz| o jj zz, jj zzo = ÅÅÅÅminm , } j z j z o o L U j z j z o o ⃗ ⃗ ÅÅ o ||S (⃗ PId)||||S (⃗ Pk )|| { k ||S (PId)||||S (Pk )|| {o ÅÇ nk ~ É |ÑÑÑ l L U o oÑÑÑ oijj S (⃗ PId) ·S (⃗ Pk ) yzz ijj S (⃗ PId)·S (⃗ Pk ) yzzo jj z j zoÑÑ maxm o j ||S (⃗ P )||||S (⃗ P L)|| zz, jj ||S (⃗ P )||||S (⃗ PU)|| zz} o oÑÑ o Id Id k k k { k {o n ~ÑÖ ÅÄÅ li yz n oj ÅÅÅ o ∑i = 1 [Gi* × Gi(PkL)] jj zz o o Å j zz, Å = ÅÅminm ojjj n n 2 L 2 zzz ÅÅ o * j o ∑ ∑ × [ ] G G P ( ) ( ) ÅÅ o i i k i=1 i=1 { Ç nk | ij n y o zzo ∑i = 1 [Gi* × Gi(PkU)] jj zzo o jj } z o, jjj ∑n (G*)2 × ∑n [G (PU)]2 zzzo o i i k i 1 i 1 = = k {o ~ l i yz n oj o ∑i = 1 [Gi* × Gi(PkL)] jj zz o o j zz, maxm ojjj n n 2 L 2 zzz o * j o ∑ ∑ × [ ] G G P ( ) ( ) ok i i k i=1 i=1 { n ÑÉÑ | Ñ ij yzo n oÑÑÑÑ ∑i = 1 [Gi* × Gi(PkU)] jj zzo jj zzo } jj zoÑÑÑÑ j ∑n (Gi*)2 × ∑n [Gi(PkU)]2 zzo oÑÑ i=1 i=1 k {o (18) ~ÑÑÖ 3.3.3. Step 7. Setting the Final Goal of GAD (Goal Achievement Degree). Both the absolute enhancement degree and the relative development balance among the multiple objectives can be integrated into a unique formulation (eqs 19 and 20) by projecting the vector function of the solution on that of the ideal solution; therefore, the final goal of GAD (ranging from 0 to 100%) can be introduced by eq 21.

Pro(Pk) = [Pro(PkL), Pro(PkL)] = [M(PkL) × cos(A(PkL)), M(PkU) × cos(A(PkU))] (20)

GAD(Pk) = [GAD(PkL), GAD(PkU)] ÄÅ É ÅÅ Pro(P L) Pro(PU) ÑÑÑ k k Ñ Å Å ÑÑ × 100% = ÅÅ , ÅÅÅ Pro(PId) Pro(PId) ÑÑÑÑ Ç Ö

3.4. Generation of the Best Solution (Stage 3). 3.4.1. Step 8. Formulating the MADM-MOO Model. In eq 21, a higher value of GAD implies a better solution; therefore, eq 22 was formulated for identifying the best solution by maximizing the GAD under the same constraints as given in eq 12. Max GAD(Pk) s.t. Pk ∈ C

∑ (Gi*)2 i=1

(22)

where GAD is the proposed function (see eq 21) for measuring the degree of sustainability improvement after the technologyretrofit, while Pk is the decision variable that refers to the kth set of technology combination (portfolio). 3.4.2. Step 9. Developing a Searching Strategy. The outputs of eq 22 are still interval numbers, implying that the final decision for implementing the portfolio should also be considered under uncertainty; therefore, the possibility approach (eq 23)49 is referred for developing a searching strategy.

n

Pro(PId) = [M(PId) × cos(A(PId))] =

(21)

(19)

ÄÅ l ÅÅ o GAD(PpU) − GAD(PqL) o ÅÅ o Å R (q > p) = maxm 1 − maxÅÅ , o o ÅÅ GAD(PpU) − GAD(P pL) + GAD(PqU) − GAD(PqL) o ÅÇ n

In eq 23, any two intervals like GAD(Pp) = [GAD(PpL), GAD(PpU)] and GAD(Pq) = [GAD(PqL), GAD(PqU)] can be compared through the degree of possibility of GAD(Pq) > GAD(Pp), denoted as R(q>p), while R(q>p) > 0.5 implies GAD(Pq) is more likely to be superior to GAD(Pp), and vice versa. Apparently, if GAD(PpU) < GAD(PqU) and GAD(PpL) < GAD(PqL), then the judgment that GAD(Pq) is prior to GAD(Pp) can be confirmed; otherwise, the interval numbers should be compared by using eq 23. Therefore, the search procedure is outlined as below to identify the best portfolio that has the greatest possibility for maximizing the system’s sustainability. 3.4.2.1. Substep 9.1. The portfolios are identified that have the largest GAD values in the lower bound and the upper bound by referring to eq 22, respectively. If the same portfolio is identified for maximizing both the lower and upper bound, then it is the final decision; otherwise, go to substep 9.2. 3.4.2.2. Substep 9.2. The corresponding interval values for the identified portfolios (obtained in substep 9.1) are obtained by using eq 21.

ÉÑ ÑÑ Ñ 0ÑÑÑÑ, ÑÑ ÑÖ

| o o 0o } o o o ~

(23)

3.4.2.3. Substep 9.3. The pair of interval numbers obtained in substep 9.2 are compared by using eq 23, preserving the prior portfolio (denoted as PPreserved) for further comparison while deleting the other one. 3.4.2.4. Substep 9.4. Another possible portfolio (denoted as PIdentified) is identified that has the potential to be superior to the preserved one (in substep 9.3) according to the following two scenarios: (a) When the portfolio with a higher upper bound is preserved after substep 9.3, we identify the portfolio (excluding the preserved and deleted ones) that has the largest GAD value in the lower bound by using eq 22. If GAD(PIdentifiedL) > GAD(PPreservedL), then we go back to substep 9.2; otherwise the preserved portfolio is the final decision. (b) When the portfolio with a higher lower bound is preserved after substep 9.3, we identify the portfolio (excluding the preserved and deleted ones) that has the largest GAD value in the upper bound by using eq 22. If GAD(PIdentifiedU) > GAD(PPreservedU), then we go back to substep 9.2; otherwise, the preserved portfolio is the final decision. 12072


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Figure 3. Ten recommended technologies and five enhancement objectives regarding the case study. Adapted from ref 2. Copyright 2019 American Chemical Society.

4. CASE STUDY AND RESULTS DISCUSSION 4.1. Case Study. The effectiveness of the developed framework was verified by a sustainability enhancement problem regarding a biodiesel production system reported by Liu and Huang,2 who made the first effort to provide recommendations on technology adoption by using a generic decision-making tool. In this study, the definition and characterization of the case study were adapted from their study (see the literature2 for more details), while the procedures regarding the mathematical framework were fully implemented for illustrating the efficacy of the developed model. 4.1.1. Prestage System Definition and Characterization Regarding the Case Study. As summarized in Figure 3, 10 retrofit technologies were recommended to improve the sustainability of the system, while five objectives from the IChemE Sustainability Metrics50 were considered by integrating the economic (O1, O2), environmental (O3, O4), and social (O5) concerns. For illustrating the effect of constraints on the optimization model, two restrictions were assumed in the case study by referring to the study of Liu and Huang,2 i.e., the budget limit (a total cost of 550 K$ for implementing the retrofit technologies), and the technical feasibility (T1 and T2 cannot be adopted simultaneously). As shown in Table 2, the demand data with both positive and negative values were already normalized in the previous work,2 where a positive one implies that the corresponding performance of the system would benefit from implementing a certain technology, otherwise, the categorized sustainability of the system would be reduced. In Table 2, the last row gives the budget for implementing each technology (denoted as BDj). 4.1.2. Stage 1. Weighted Objectives Determination Regarding the Case Study. 4.1.2.1. Step 1. Subjective Weights. After constructing the interval comparison matrix regarding the priority of the objectives according to the decision makers’ preferences (Table 3), eqs 2−6 were used to generate the subjective weights, i.e., [w1S, w2S, w3S, w4S, w5S] = [0.401, 0.084, 0.241, 0.161, 0.113], which implies that the improvement of O1 (the value added) of the system is regarded as the prior objective for the sustainability enhancement. 4.1.2.2. Step 2. Objective Weights. On the basis of the data set in Table 2, the objective weights were generated by running the interval entropy technique (eqs 7−10). It is worth pointing out that the α-cut level of 0.5 was set when running eq 7 for balancing the levels of precise and confidence among the objective weighting result.44 Since [w1O, w2O, w3O, w4O, w5O] = [0.132, 0.211, 0.209, 0.246, 0.202], the objective of O4 (hazard solid waste per unit value added) has the greatest effect on the sustainability enhancement from the objective perspective. 4.1.2.3. Step 3. Combined Weights. Subsequently, eq 11 was used for integrating the subjective and objective weights, in

which the coefficient u = 0.5 was set for minimizing the information losses during the combination.40 As depicted in Figure 4, the subjective and objective weights are very different from each other, while the combination could provide a wellrounded rather than one-sided weighting result. Notably, the comparison matrix in step 1, the α-cut level in step 2, and the coefficient u in step 3 can be determined by the users according to the actual enhancement requirements and their preferences regarding the system, offering a flexible and comprehensive means for weighting the multiple objectives. 4.1.2.4. Step 4. Weighted Objectives. According to the collected data, the identified constraints, and the combined weights, each weighted objective can be determined by running eq 12. Taking the objective to maximize G*1 as an example, the model is coded in eq 24. ÄÅ ÉÑ ÅÅ ÑÑ ÅÅ Ñ U max G1* = w1 × ÅÅÅ∑ Tj × f1j ÑÑÑÑ ÅÅ ÑÑ ÅÅÇ j ÑÑÖ s.t.

Tj(1 − Tj) = 0, j = 1, 2, ..., 10 T1 × T2 = 0

∑ BDjTj ≤ 550, j

j = 1, 2, ..., 10 (24)

By running eq 24, the obtained result of max G1* = 8.096 × 10−2 was contributed by the technologies combination of (T1 + T4 + T5 + T7 + T8 + T9) with the total cost of 550 K$. Similarly, the maximum potential improvements regarding the other four objectives and their corresponding portfolios were summarized in Table 4; apparently, the best improvement in each objective was contributed by different portfolios. 4.1.3. Stage 2. Final Goal Development Regarding the Case Study. Considering that steps 5 and 6 are mainly used for demonstrating the mathematical philosophy of the vector-based algorithm, and then supporting the multiobjective optimization in step 7, only the portfolio of P1 = (T1 + T4 + T5 + T7 + T8 + T9) in Table 4 and the ideal solution (PId) contributed by the aggregation of [G1*, G2*, G3*, G4*, G5*] were taken as examples for the demonstration. 4.1.3.1. Step 5. Vector-Represented Solutions. On the basis of Table 4, the vector-represented ideal solution can be denoted as S (⃗ PId) = [8.096, 4.689, 5.029, 2.541, 4.517] × 10−2, which cannot be realized since the five weighted objectives are contributed by different portfolios. While for any feasible solution, taking P1 as an example, the corresponding vector function is S (⃗ P1) = {[5.880, 8.096], [3.408, 4.402], [0.239, 3.346], [1.060, 1.098], [2.415, 2.737]} × 10−2. 4.1.3.2. Step 6. Characteristic of the Vector-Represented Solution. By running eqs 15−16 and 17−18, the vector’s 12073


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[0.030, 0.040] [0.010, 0.030] [0.040, 0.060] [−0.010,−0.010] [0.015, 0.025] 80

Table 3. Interval Comparison Matrix for Determining the Subjective Weights of the Objectives Regarding the Case Study

[0.060, 0.070] [0.010, 0.010] [−0.030, 0.000] [0.010, 0.020] [0.100, 0.105] 90 [0.040, 0.060] [0.010, 0.020] [−0.010, 0.010] [0.010, 0.020] [−0.010,−0.005] 100

[0.010, 0.030] [0.070, 0.080] [0.020, 0.050] [0.000, 0.010] [0.055, 0.060] 60

O1 O2 O3 O4 O5

O2

O3

O4

O5

[1, 1] [1/7, 1/5] [1/3, 1] [1/5, 1/3] [1/5, 1/3]

[5, 7] [1, 1] [3, 5] [1, 3] [1, 3]

[1, 3] [1/5, 1/3] [1, 1] [1/4, 1/5] [1/3, 1]

[3, 5] [1/3, 1] [2, 4] [1, 1] [1, 2]

[3, 5] [1/3, 1] [1, 3] [1/2, 1] [1, 1]

Table 4. Weighted Objectives and the Corresponding Portfolios in the Case Study objective

[0.040, 0.050] [0.060, 0.060] [0.000, 0.010] [0.030, 0.030] [−0.025,−0.025] 120

G1* G*2 G*3 G*4 G*5

max score −2

8.096 × 10 4.689 × 10−2 5.029 × 10−2 2.541 × 10−2 4.517 × 10−2

portfolio

cost (K$)

(T1 + T4 + T5 + T7 + T8 + T9) (T1 + T4 + T5 + T7 + T9 + T10) (T1 + T4 + T6 + T9 + T10) (T3 + T5 + T7 + T8 + T9) (T3 + T6 + T8 + T9 + T10)

550 540 460 520 520

[0.040, 0.050] [0.040, 0.060] [−0.010, 0.020] [0.010, 0.010] [0.010, 0.015] 80

magnitude, the angle between the ideal vector, and the investigated one can be respectively obtained as below. M(PId) = ||S (⃗ PId)|| = 11.82 × 10−2 , M(P1)

[0.070, 0.070] [0.030, 0.040] [−0.020, 0.010] [0.030, 0.040] [0.045, 0.050] 150

= [||S (⃗ P1L)|| , ||S (⃗ P1U)||] = [7.29, 10.36] × 10−2

(25)

cos(A(PId)) = 1, cos(A(P1)) = [cos(A(P1L)), cos(A(P1U))] = [0.909, 0.982]

[0.010, 0.030] [0.010, 0.030] [0.020, 0.040] [−0.020,−0.020] [−0.010,−0.005] 50 [0.050, 0.070] [0.050, 0.080] [0.040, 0.050] [−0.010, 0.000] [0.020, 0.020] 100 O1 O2 O3 O4 O5 BD (K$)

O1

Figure 4. Objective, subjective, and combined weights.

[0.030, 0.030] [0.020, 0.030] [0.020, 0.030] [−0.020,−0.010] [0.035, 0.040] 140

T8 T4 T3 T2 T1 [f ijL, f ijU]

Table 2. Collected Data with Respect to Each Technology for Each Objective

T5

T6

T7

T9

T10


(26)

Since each possible portfolio can be represented by a certain vector function, previous studies that rely on the enumeration method2 or genetic algorithm9 would definitely need lots of time and complex operations when multiple technologies can be adopted. On the contrary, the MOO model can address these burdens by using a maximization function to identify the best solution as follows. 4.1.3.3. Step 7. Final Goal of GAD. According to eqs 19−21, Pro(PId) = 11.82 × 10−2 was determined; then, the final goal of GAD for a promising solution can be denoted, and it is ÄÅ ÉÑ ÅÅÅ Pro(PkL) Pro(PkU) ÑÑÑ ÑÑ × 100% GAD(Pk) = ÅÅÅ , ÅÅÅ 11.82 × 10−2 11.82 × 10−2 ÑÑÑÑ Ç Ö (27) 12074


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Industrial & Engineering Chemistry Research 4.1.4. Stage 3. Best Solution Generation Regarding the Case Study. 4.1.4.1. Step 8. MADM-MOO Model. Here, the MADM-MOO model can be coded in eq 28.

employed for obtaining the subjective, objective, and combined weights, respectively, while eq 12 should be implemented n times for setting a number of n objectives. In section 3.3, eqs 13 and 14 in step 5, eqs 15−18 in step 6, and eqs 19 and 20 in step 7 were used to illustrate the procedures of the development of the final goal, and all of them should be aggregated into eq 21 in step 7. Herein, the real required equation in section 3.3 is eq 21. In section 3.4, the involved equations for identifying the best solution would vary with the actual conditions of the investigated systems, taking the case study as an example (see Figure 5), if Pq is the same as Pq after implementing substep 9.1, the final result of the best solution can be offered by only repeating eq 28 twice, which is quite different from the original searching procedures as presented in Figure 5. As for implementing the model in the case study, MS Excel was used for generating the objective and combined weights, while the software Lingo 11.0 was employed for obtaining the subjective weights, the weighted objectives, and the final decision. Since the proposed model can be implemented step by step and only algebraic calculations are involved in each step, the computational time of the model using a usual personal computer should be no more than a few seconds. Notably, the computation time scale of the proposed model would not be an issue for solution identification, since the technology candidates should be screened by the experts before the recommendation, while various concerns could be aggregated into the three categorized objectives (environmental, economic, and social). 4.2. Results Discussion. In order to provide an insight into the obtained results as well as the developed framework, three analysis items were conducted in this subsection, i.e., a vectorbased analysis of the results, sensitivity analysis by varying the weights, and qualitative and quantitative comparisons between the developed model and previous methods. 4.2.1. Vector-Based Analysis. Since two promising portfolios (Pp and Pq) with very similar GAD values were identified in the case study, we took one step backward by using the vector-based MADM algorithm (eqs 15−21) to compare them, in which the weighted objectives regarding each portfolio (the ideal portfolio, and the upper/lower bounds of the identified ones) were given in Table 5 (columns 2−6). Subsequently, by running eqs 15−16 and eqs 17−18, respectively, the corresponding absolute improvement degree and relative development balance can be determined, which were integrated into the projection value by using eqs 19 and 20, and finally, the GAD values can be generated by using eq 21. The obtained values were summarized in Table 5 (columns 7−10). It can be observed from Table 5 that the two portfolios exhibit similar performances in both absolute improvement degree and relative development balance, resulting in similar projection values and GAD results. More importantly, from this analysis, it can be concluded that a real desirable solution should not only have a higher absolute improvement but also a more balanced development direction, because the final decision is based on the multiplication of M and cos(A).

ÄÅ ÉÑ ÅÅ Pro(P L) Pro(PkU) ÑÑÑÑ k Max GAD(Pk) = ÅÅÅÅ , Ñ × 100% ÅÅ 11.82 × 10−2 11.82 × 10−2 ÑÑÑ ÅÇ ÑÖ

s.t.

Tj(1 − Tj) = 0, j = 1, 2, ..., 10 T1 × T2 = 0

∑ BDjTj ≤ 550,

j = 1, 2, ..., 10

j

(28)

4.1.4.2. Step 9. Searching Strategy. As illustrated in Figure 5, by running the searching strategy, the final decision can be

Figure 5. Procedures of the searching strategy and the corresponding results.

offered by implementing the technologies combination of Pp = (T1 + T4 + T5 + T8 + T9 + T10), which has the largest chance for maximizing the sustainability enhancement, and its GAD is [57.89%, 88.47%]. In summary, the utilization of equations in the developed framework can be clarified as follows: In section 3.2, eqs 2−6 in step 1, eqs 7−10 in step 2, and eq 11 in step 3 should be

Table 5. Parameters for the Interval Vector-Based MADM Method

PId Pp Pq

G1 × 10−2

G2 × 10−2

G3 × 10−2

G4 × 10−2

G5 × 10−2

M × 10−2

cos(A)

Pro × 10−2

GAD

8.096 [5.64, 7.61] [5.64, 7.85]

4.689 [3.41, 4.55] [2.70, 3.98]

5.029 [0.96, 4.55] [0.72, 4.55]

2.541 [0.64, 1.27] [0.21, 1.06]

4.517 [2.82, 3.23] [3.07, 3.55]

11.82 [7.26, 10.55] [7.01, 10.58]

1 [0.942, 0.991] [0.923, 0.990]

11.82 [6.84, 10.46] [6,46, 10.48]

100% [57.89%, 88.47%] [54.65%, 88.64%]

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Industrial & Engineering Chemistry Research 4.2.2. Sensitivity Analysis. In the developed framework, the relative importance regarding the multiobjective is derived from the combination of the subjective and objective weighting methods. In order to test the effect of the weights on the identification of the best solution, a sensitivity analysis has been conducted by varying the combined coefficient u in eq 11 from 1 to 0 with a step-change of 0.20, offering different sets of weights as depicted in Figure 6. Subsequently, by running the same

performances as discussed in section 4.2.1. Therefore, it can be concluded that the obtained result regarding this case study is reliable and the developed method is effective. 4.2.3. Comparison with Previous Models. This section provides both qualitative and quantitative comparisons between the proposed MADM-MOO model and some existing approaches. From the qualitative view, this study is inspired by the studies of Liu and Huang2 and Moradi-Aliabadi and Huang,9 both of them provide decision-making platforms for the technology-based sustainability enhancement. However, the first2 has to generate a complete list of technology portfolios and their performances through enumerating the combinations of m technology candidates; in other words, there are 2m − 1 different portfolios existing in the list that should be assessed, making the evaluation quite complex and time-consuming. While the second platform9 relies on the genetic algorithm (seven steps with inner loops) to identify the best solution, which also suffers from the implementing complexities since the genetic algorithm is related to the choice of multiple parameters like the size of the population, mutation rate, crossover rate, selection method, and its strength. More importantly, none of them considered the relative balance among multiple objectives when making the decision. In one step forward, a quantitative comparison was offered between the model developed in this study and the most-used weighted MOO technique.14,22,29,51 The weighted MOO prefers to employ the WSM method and its variants to generate the Pareto point as given in eq 29, which is known as the WSMMOO model. For comparison, eq 29 was solved by using the same search logic as stated in section 3.2.3; the corresponding result was illustrated in Figure 8.

Figure 6. Variation of the combined weights with respect to different combined coefficients.

procedures specified in the framework, each set of weights was used to identify the best solution according to the corresponding weighted objectives as shown in Figure 7. The results demonstrate that the weighted objectives in different scenarios are quite different from each other (presented by the radar charts) implying the necessity for adopting an accurate weighting method for the decision making; in contrast, the identified best portfolios with respect to the scenarios are relatively robust, where the portfolio of (T1 + T4 + T7 + T8 + T9 + T10) is the best solution for u = 1−0.8, while that of (T1 + T4 + T5 + T8 + T9 + T10) is the final decision for u = 0.6−0. This variation is reasonable since these two portfolios exhibit quite similar

Figure 7. Identified portfolios and the corresponding GAD values with respect to different weighted objectives. 12076


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ÄÅ n ÉÑ n ÅÅ ÑÑ ÅÅ L UÑ Ñ Max WSM(pk ) = ÅÅÅ∑ wp ÑÑ i k , ∑ wp i k Ñ ÅÅÅ i = 1 ÑÑÑ i = 1 Ç Ö É ÅÄÅ n i y yzÑÑÑÑ ÅÅ jj z n ijj Å Ñ Lz Uz j z j z Å = ÅÅÅ∑ wijjj∑ Tj×fij zzz, ∑ wijjj∑ Tj×fij zzzÑÑÑÑ z i=1 j j zÑÑ ÅÅÅÅ i = 1 jk j { k {ÑÑÖ Ç


s.t.

degree and relative development balance among multiobjective is incorporated into the weighted MOO technique, which can shed new light on the MADM-MOO combination for the sustainability enhancement by recognizing the significance of relative balance among multiple objectives.

6. CONCLUSIONS This study aims to simplify the MOO problem regarding the technology-based sustainability enhancement of chemical systems under uncertainties by incorporating a novel vectorbased MADM method. The proposed framework can not only rigorously identify the best solution in accordance with both the decision makers’ subjective preferences and the system’s objective properties but also escalate the modeling capability of the MOO method by integrating the absolute improvement degree and relative development balance among the multiple objectives. The application of the proposed framework was illustrated by a well-established case study regarding a biodiesel production system reported by Liu and Huang,2 and the effectiveness and advantages of the framework were verified by analyzing the obtained results via the vector-based analysis, sensitivity analysis, and result comparison. Remarkably, the developed model in the framework has the following advantages: (1) The developed model adopts a unique Pareto point instead of a set of Pareto frontiers for the multiple objectives optimization, which can not only address the tradeoffs between the objectives but also simplify the MOO problem. (2) Both of the decision makers’ preferences regarding the relative importance of each objective and the nature of numerical data within the sustainability enhancement system can be flexibly aggregated for better addressing the trade-offs among the multiple objectives. (3) The final Pareto point regarding the single-goal model is comprehensively and reliably determined by the integration of the absolute improvement degree and relative development balance regarding the multiple objectives. (4) Different types and degrees of uncertainties regarding the input numerical data as well as the subjective judgments can be properly handled by using the interval parameters; moreover, the uncertainties are incorporated in the model as well as the final result for better capturing the complexity of the chemical systems in real-world issues. Although this framework provides a novel and systematic route for sustainability enhancement of chemical systems under uncertainties, some limitations need to be addressed in the future: (1) The framework follows the assumption that each technology’s capability of sustainability enhancement is independent of others,9 which fails to depict the cooperative relationships among the combination of multiple technologies. (2) It directly adopts the interval parameters to handle the uncertainties, which could be expressed more explicitly in the future works by combining with the specific distribution and/or Monte Carlo simulation. (3) The proposed framework currently focuses on the sole manufacturing stage, which could be extended into the life cycle perspective for deeply enhancing the sustainability of the chemical systems.

Tj(1 − Tj) = 0, j = 1, 2, ..., 10 T1 × T2 = 0

∑ BDjTj ≤ 550,

j = 1, 2, ..., 10

j

(29)

Figure 8. Procedure of the searching strategy and the corresponding result of the WSM-MOO model.

As can be seen in Figure 8, Pp = (T1 + T4 + T5 + T8 + T9 + T10) was identified as the portfolio that has the largest values of WSM(PpL) = 0.135 and WSM(PpU) = 0.212 simultaneously, directly offering the final decision. Apparently, the result derived from the WSM-MOO combination is the same as the decision offered by the vector-based MADM-MOO model developed in this study, verifying the effectiveness of the developed method. However, in the developed model, Pq = (T1 + T4 + T7 + T8 + T9 + T10) was identified as another promising portfolio that has the largest value in the upper bound of GAD, resulting in a further comparison between Pp and Pq. This difference could be attributed to the fact that the relative development balance among the multiobjective is innovatively incorporated into the MOO model, which can affect the final decision. Since balancing the performances of multiobjective is quite favored by the nature of the sustainability,4,13,47 the developed model can shed significant light on the combination of the MADM and MOO methods for addressing the sustainability issues by integrating the absolute rating and relative balance among the multiple objectives.

5. THEORETICAL IMPLICATIONS This paper proposed a systematic decision-making framework for optimizing the technology-based sustainability enhancement of chemical systems. Compared to the existing MADM-MOO for addressing the sustainability issues, the developed mathematical framework could offer three theoretical implications: (1) Different types and degrees of uncertainties with respect to both the subjective judgments and objective information can be properly handled by adopting the interval parameters, which introduce an easy but generic way for the sustainability enhancement of the complex systems. (2) Both the decision makers’ subjective preferences and the system’s objective properties can be fully utilized for assigning comprehensive weighs to the objectives with the consideration of uncertainties, while the combined weighting method is able to offer a flexible way to adjust the priority of the subjective weights over the objective ones. (3) A novel vector-based MADM method via the aggregation of both the absolute improvement

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.9b01531. Preliminary knowledge for the interval-based model; constraints for the technology-based sustainability 12077


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enhancement; numerical example regarding the generation of the demand data (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (D.X.). *E-mail: [email protected] (L.D.). ORCID

Di Xu: 0000-0003-1441-0337 Weifeng Shen: 0000-0002-0418-6848 Lichun Dong: 0000-0002-9876-0133 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research is supported by the National Science Foundation of China (21776025). REFERENCES

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