Multistage Optimization for Chemical Process ... - ACS Publications

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Multistage Optimization for Chemical Process Sustainability Enhancement under Uncertainty Majid Moradi-Aliabadi and Yinlun Huang* Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, Michigan 48202, United States ABSTRACT: Chemical process sustainability is mainly concerned with energy and material efficiency, productivity and product quality, waste reduction, process safety, heath impact, etc. Due to these multiple factors as well as information and data uncertainty, development of optimal strategies and effective action plans for sustainability enhancement becomes a very challenging task. In this paper, we introduce a mathematical framework for optimal process sustainability performance enhancement. In this framework, we describe four types of optimization problems, which are defined based on decision makers’ different objectives for sustainability enhancement. In formulation, various economic, environmental, and social concerns and technical feasibilities are taken into account, where data uncertainty is dealt with by interval parameters. Solution search is performed by a genetic algorithm method and a Monte Carlo simulation technique. The methodological applicability is illustrated by a comprehensive case study on biodiesel manufacturing. The introduced methodology should be useful for decision makers to compare optimization methods and select the most suitable for their applications. KEYWORDS: Sustainability enhancement, Decision making, Uncertainty, Optimization, Genetic algorithm, Monte Carlo simulation

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INTRODUCTION

In order to reduce the environmental impact of chemical processes, many source reduction technologies and end-of-pipe solutions are proven effective in applications.7−11 However, environmental impact is only part of the sustainability problem. The idea of considering environmental issues as a part of the design objectives throughout the production chain was proposed by Cano-Ruiz and McRae,12 who defined it as a multiobjective optimization problem, and the system boundary was extended to consider the environmental impact from the whole life cycle of the product or process. The Life Cycle Assessment (LCA) methodology can be incorporated into other approaches such as optimization techniques for process design evaluation and operational improvement. Azapagic13 and Jacquemin et al.14 show that LCA is often used to generate environmental impact data for multiobjective optimization. These approaches place emphasis on the environment more than overall sustainable development. Sustainability has become a focal point of future development of manufacturing industries.15−17 The U.S. Department of Commerce defines sustainable manufacturing as “the creation of manufacturing products that use materials and processes that minimize negative environmental impacts, conserve energy and

Natural resource depletion, global climate changes, environmental pollution, excessive population growth, etc. have challenged industries globally. It is known that industries account for around 28% of final energy use1 and over 30% of total global GHG emission.2 The emission from industries is projected to increase by 50−150% by 2050, unless energy efficiency improvement is accelerated significantly.2 The manufacturing sector is responsible for about 98% of total direct CO2 emissions.3,4 Manufacturing more than 70 000 diverse products, the chemical industry is one of the most energy-intensive industrial sectors, consuming nearly 25% of energy used in the all manufacturing industries, or 6.8% of total energy consumption in the U.S. This sector is the largest user of primary energy and the second largest user of onsite energy and also releases more carbon emissions than any other sector, while ranking the second in onsite emissions.5 It also is one of the most regulated of all industries.6 These regulations are mostly related to its products and releases of chemical substances during manufacturing and processing. Our future challenges in environmental, economical, and societal sustainability demand more efficient and benign scientific technologies for working with chemical processes and products. This requires that chemical process designers incorporate environmental and societal concerns and objectives into technoeconomic process design and retrofitting techniques. Naturally, how to ensure industrial sustainability becomes a focal point. © XXXX American Chemical Society

Special Issue: Building on 25 Years of Green Chemistry and Engineering for a Sustainable Future Received: July 11, 2016 Revised: September 5, 2016

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DOI: 10.1021/acssuschemeng.6b01601 ACS Sustainable Chem. Eng. XXXX, XXX, XXX−XXX

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ACS Sustainable Chemistry & Engineering

process, especially due to the existence of uncertainties that appear in the available technical or nontechnical data, information, and possessed knowledge. 27 According to Parry,32 uncertainties can be classified into two types: aleatory and epistemic. Aleatory uncertainty refers to the inherent variations associated with the physical system or the environment under consideration, and it is objective and irreversible. By contrast, epistemic uncertainty is carried by the lack of knowledge and/or information, and it is subjective and reducible. The uncertainties encountered in the study of industrial sustainability problems could be either aleatory or epistemic. A variety of mathematical techniques and computer and cognitive science based methods are available for handling uncertainties, such as those utilizing statistics theory, fuzzy logic theory, and artificial intelligence.33−39 In this paper, we introduce a mathematical framework for optimal process sustainability performance enhancement. In this framework, we describe four types of optimization problems, which are defined based on decision makers’ different objectives for sustainability enhancement. In formulation, various economic, environmental, and social concerns and technical feasibilities are taken into account, where data uncertainty is dealt with by interval parameters. Solution search is performed by a genetic algorithm method and a Monte Carlo simulation technique. The methodological applicability is illustrated by a comprehensive case study on biodiesel manufacturing. The introduced methodology should be useful for decision makers to compare optimization methods and select the most suitable for their applications.

natural resources, are safe for employees, communities, and consumers and are economically sound.”18 Sustainable chemical manufacturing specifically implements principles of sustainable chemistry and engineering to achieve the sustainability goals of the chemical industry.19,20 It encompasses design, manufacture, and use of efficient, effective, safe, and more environmentally benign chemical products and processes.21 This is a very challenging task as it entails complex decision-making, which can be accompanied by a number of difficulties, such as competing interests among stockholders, and multiple alternatives using multiple types of decision criteria of sustainability. Decision making for industrial sustainability has been investigated in several studies. Azapagic and Perdan22,23 proposed an integrated framework, based on Multiple Criteria Decision Analysis (MCDA), to support decision-making for sustainability. MCDA techniques are classified into two main groups, which are programming methods (e.g., multi-objective optimization, and goal programming) and multi-attribute decision analysis (MADA).23 Programming methods have found wider applications in operational types of decision making. Kim and Smith24 introduced a systematic approach for designing processes with multiple environmental objectives that can help the decision maker to determine design alternatives with respect to his or her preferences. Hoffmann et al.25 proposed a new approach for selecting promising process alternatives while uncertainties are taken into account explicitly; their approach is helpful when complex problems, such as multiobjective design problems under uncertainty, are solved. Guillén-Gosálbez et al.26 introduced a framework for optimal design of chemical processes with incorporation of environmental concerns based on the principals of LCA that finally leads to a multiobjective optimization problem. Piluso and Huang27 introduced a collaborative profitable pollution prevention (CP3) idea to aid in decision-making for the study of sustainable development of industrial zones. Liu and Huang28 presented a simple, yet systematic interval-parameterbased methodology for identifying quickly superior solutions under uncertainty for sustainability performance improvement. Moradi-Aliabadi and Hunag29 announced a vector-based sustainability analysis methodology to identify feasible strategies for process sustainability enhancement, but in their study, data uncertainty was not considered. Sustainability science has emerged in recent years as a vibrant field of research and innovation. Its core branch is to study a system’s “transition toward sustainability.” Kates et al.30 states that the purpose of sustainability assessment is to assist decision-makers in determining which actions should or should not be taken in an attempt to make a system sustainable. Any system’s transition consists of the following four principal stages and associated activities in investigation: the system characterization, sustainability assessment, sustainability enhancement, and system adaptation stages.31 In the first stage, the problem scope and context should be defined and a sustainability goal specified. In the next stage, the current status of sustainability and solution options are evaluated. In the succeeding stage, the preferred solution is selected and implemented. In the end, the outcomes are evaluated in order to keep the system on a sustainable trajectory in the longterm by an adaptive approach. Sustainability performance improvement may involve multiple stages to keep the system on a sustainable trajectory. Thus, this is a multistage decision problem, which is very challenging

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MULTISTAGE SUSTAINABILITY ENHANCEMENT Sustainability enhancement problems are usually complicated due to the requirement of achieving some preset economic, environmental, and social goals simultaneously. Thus, solving these kinds of problems may involve actions in multiple stages, which requires a sequence of decisions to be derived. Assume there are M stages involved. In the k-th stage, the system state transition is as depicted in Figure 1a, where the initial

Figure 1. Sustainability decision-making process: (a) single-stage and (b) multiple stages.

sustainability status is denoted by vector S⃗(k − 1), the decision variable is d(k), and the new state is vector S⃗(k). Thus, the stage transformation function can be expressed as S (⃗ k) = fk (d(k), S (⃗ k − 1)) B

(1)


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indicator value is more desirable or not. al, bm, and cn ∈ [1, 10] are the weights associated with different indicators, reflecting the relative importance of each individual indicator against others in the overall assessment. However, this is not a steadfast rule; it can be any range number. They could be equal or ranked. If the indicators are equally important, then each weight associated can be set to 1. Note that weighting factors can be determined by different methods.42 Equal weights can be used if there is no knowledge about the decision maker’s priorities and no data or information is available for differentiating the importance of indicators in assessment. The weights can also be set differently by a subjective weighting method (based on the decision maker’s preference), an objective weighting method (based on data), or a combination of both methods.43

According to Moradi-Aliabadi and Huang,29 the sustainability status vector is defined as a vector function whose components are composite economic sustainability index, composite environmental sustainability index, and composite social sustainability index. Here, a “stage” is defined as a sustainability status transition, which is resulted from an investment on technology adoption. The transition is realized at each stage if and only if the output of the stage meets the minimum requirement of each categorized sustainability index at that stage. To quantify the cost and other aspects of system transition (e.g., time), a function called the return function is introduced below: R(k) = rk(d(k), S (⃗ k − 1))

(2)

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Figure 1b gives a decision process involving M stages. Since the sustainability is triple-bottom-line-based, eq 1 can be detailed as ⎛ E(k) ⎞ ⎛ E(k − 1) ⎞ ⎛ ΔE(d(k)) ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ V (k)⎟ = ⎜ V (k − 1)⎟ + ⎜ ΔV (d(k))⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎝ L(k) ⎠ ⎝ L(k − 1) ⎠ ⎝ ΔL(d(k)) ⎠

MODELING OF OPTIMAL SUSTAINABILITY ENHANCEMENT In this work, we focus on technology-based sustainability performance improvement, although we understand that management and other measures are also critical to industrial sustainability. Assume that there are N technologies (decision variables) available for sustainability performance improvement. The task is to identify sets of technologies for implementation in M project stages in order to achieve a preset optimization objective. Therefore, a binary variable, yi,j, is defined as follows:

(3)

where ΔE(d(k)), ΔV(d(k)), and ΔL(d(k)) are, respectively, the change of the categorized economic, environmental, and social sustainability performance after implementing decision d(k). The closed form of eq 3 can be expressed as below: S (⃗ k) = S (⃗ k − 1) + ΔS (⃗ d(k))

⎧1, if technology j is selected at stage i ⎪ i = 1, 2, ..., M ; j = 1, 2, ..., N yi , j = ⎨ ⎪ ⎩ 0, otherwise

(4)

where ΔS⃗(d(k)) is the sustainability performance change vector resulted from adopting decision d(k). Thus, the objective of a multistage decision-making problem is to identify a set of optimal decisions, d(1), d(2), ..., d(M), for state transition toward a target goal. In this process, a set of return functions, R(1), R(2), ..., R(M), can be in turn determined. A variety of sustainability metrics systems, such as those by IChemE40 and AIChE,41 are available for sustainability assessment. Sustainability status of a system at kth stage can be represented by a state vector whose components are composite economic index (E(k)), composite environmental index (V(k)), and composite social index (L(k)). This vector shows sustainability state of the system, and it can be represented as follows: S (⃗ k) = ⟨E(k), V (k), L(k)⟩

(9)

Transition Function. Liu and Huang28 introduced a method for evaluating sustainability enhancement. On the basis of their approach, the improvement of economic, environmental, and social sustainability of system P after implementing technology Tj can be evaluated using individual sustainability indicators, such as ΔEl(Tj , P) = El(Tj) − El(P), l = 1, 2, ..., F ; j = 1, 2, ..., N (10)

ΔVm(Tj , P) = Vm(Tj) − Vm(P), m = 1, 2, ..., G ; j = 1, 2, ..., N

(5)

where

(11) F

E (k ) =

∑l = 1 alEl , N (k) F

∑l = 1 al

, l = 1, 2, ..., F

ΔLn(Tj , P) = Ln(Tj) − Ln(P), n = 1, 2, ..., H ; j = 1, 2, ..., N

(6)

(12)

G

V (k ) =

∑m = 1 bmVm , N (k) G

∑m = 1 bm

, m = 1, 2, ..., G

The above index-specific sustainability evaluation can then be used to calculate the categorized sustainability improvement level for system P as follows:

(7)

H

L (k ) =

∑n = 1 cnLn , N (k) H

∑n = 1 cn

, n = 1, 2, ..., H

F

ΔE(Tj , P) =

(8)

where El,N(k), Vm,N(k), and Ln,N(k) are, respectively, the individual normalized indices of economic, environmental, and social sustainability, each of which has a value between 0 and 1, with 0 the worst and 1 the best; the normalized indicator can take the value IN or 1 − IN depending on whether a higher

∑l = 1 al ΔEl , N F

∑l = 1 al

, j = 1, 2, ..., N , (13)

G

ΔV (Tj , P) = C

∑m = 1 bmΔVm , N G

∑m = 1 bm

, j = 1, 2, ..., N (14)


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where ε is a small positive number much less than 1. (3) The decision variables are binary:

H

ΔL(Tj , P) =

∑n = 1 cnΔLn , N H

∑i = 1 cn

, j = 1, 2, ..., N (15)

yi , j ∈ {0, 1}, i = 1, 2, ..., M ; j = 1, 2, ..., N

where ΔE(Tj, P), ΔV(Tj, P), andΔL(Tj, P) are, respectively, the economic, environmental, and social sustainability improvements of system P when technology Tj is adopted. In this study, we assume that each technology’s capability of sustainability enhancement is independent of other technologies, which is reasonable in most cases. Therefore, categorized sustainability development at the i-th stage can be evaluated using the following equations:

(4) Technology selection restriction. Note that each candidate technology can be selected at most once in the multistage improvement process. After adoption, it will be continuously used in the system in the following stages, although it may need further improvement later. Thus, the following logical constraint holds: M

∑ yi ,j ≤ 1,

N

E(i) = E(i − 1) +

∑ yi ,j ΔE(Tj , P),

i = 1, 2, ..., M

N

i = 1, 2, ..., M

j=1

(17) N

L(i) = L(i − 1) +

∑ yi ,j ΔL(Tj , P),

i = 1, 2, ..., M

j=1

(18)

(26)

yi , p + yj , q ≤ 1, i , j = 1, 2, ..., M

(27)

N

∑ f (ζi)yi ,j Bj ≤ Bup(i),

S (⃗ i) = ⟨E(i), V (i), L(i)⟩ = S (⃗ i − 1)

i = 1, 2, ..., M (28)

j=1

N

∑ yi ,j ΔS (⃗ Tj , P); i = 1, 2, ..., M

where f(ζi) is a coefficient showing a discount if more than one technology is used at the i-th stage; it is a function of ζi, which is the number of technologies adapted. Note that eq 23 will be used only in the minimum time and the minimum cost optimization problems; all other constraints are for all four types of optimization problems that are listed below. Optimization Types. Sustainability performance improvement may have the following types of objectives defined for optimization, which are investigated in this work. (a) Maximum sustainability improvement problem. This is a class of optimization aiming at achieving the maximum sustainability improvement in the final stage. Thus, the objective equation is defined as

(19)

j=1

Constraints. Optimization for sustainability performance improvement must be subjected to a number of physical, mathematical, and other types of major constraints, including the following major common ones: (1) A minimum requirement on sustainability performance at each stage, i.e., E(i) ≥ Emin(i), i = 1, 2, ..., M

(20)

V (i) ≥ Vmin(i), i = 1, 2, ..., M

(21)

L(i) ≥ Lmin(i), i = 1, 2, ..., M

(22)

J = max

where Emin(i), Vmin(i), and Lmin(i) are categorized, respectively, the economic, environmental, and social sustainability goals at the i-th stage of sustainability development. These should be provided by decision makers based on their understanding of their organization’s plan, current status, and many other factors. In many cases, decision makers do not have a clear goal or goals. They may prefer to compare the consequences if different goals are set and then select a reasonable goal and identify an optimal pathway toward the goal. This involves a trial and error process, because these goals might be unreachable at the first attempt. (2) Tolerance of sustainability goal achievability. A system’s sustainability performance at stage M, after implementing different technologies in different stages, is acceptable if it reaches the preset goal, Sg(M), within a small range, i.e., g

yi , p + yi , q ≤ 1, i = 1, 2, ..., M

(6) Budget limit for technology adoption at each stage, i.e.,

Correspondingly, the sustainability status vector can be expressed as

+

(25)

(5) Restriction on the combined use of certain technologies. Some technologies may not be allowed to be used together either at one stage (e.g., i-th) or at different stages (e.g., i-th and j-th). For example, for two technologies, namely, p and q, the following constraints hold:

(16)

∑ yi ,j ΔV (Tj , P),

j = 1, 2, ..., N

i=1

j=1

V (i) = V (i − 1) +

g

(1 − ε)|| S ⃗ || ≤ || S (⃗ M )|| ≤ (1 + ε)|| S ⃗ ||

(24)

yi , j

||⟨αE(M ), βV (M ), γL(M )⟩|| ||⟨α , β , γ ⟩||

(29)

where α, β, and γ are weighting factors, each having a value of 1 (default) to 10. (b) Minimum cost problem. It is likely that an industrial organization has a preset sustainability goal, Sg⃗ (M), and then seeks technologies to implement to achieve that at the lowest total cost in the M-stage sustainability performance improvement process. Note that this is in fact the minimization of the summation of the return functions (costs) over the stages. The objective function is expressed as M

N

J = min ∑ ∑ f (ζi)yi , j Bj yi , j

i=1 j=1

(30)

where Bj is the cost associated for the sustainability state transition using technology Tj. Note that the cost for the

(23) D


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ACS Sustainable Chemistry & Engineering performance improvement is path dependent; there is a return function at each stage as shown in Figure 1. (c) Maximum efficiency problem. An organization may want to know the best scenario, which shows the most efficient way to invest for sustainability performance improvement under different constraints including budget limit and minimum sustainability improvement requirements in different stages. Thus, the objective function can be written as J = max yi , j

J = min[max{t p , z 2(t p + 12),

where tPi is the total time of project i that can be evaluated using eq 33. Note that the time needed for project implementation can be reduced by assigning more labor or providing more resources (material, equipment, etc.), but these could cost more. In this study, we do not consider these situations. Optimization Model Summary. As discussed above, there are four types of optimization models. Table 1 summarizes each

(31)

Table 1. Optimization Types and Model Formulations optimization types

i=1

(32)

(33)

Scenario 2. Assume that funds are available annually for the M-stage sustainability improvement problem. In this scenario, M means the number of years that funds are available, and thus the minimum time optimization model is formulated based on the following assumptions: (i) a new project can be started at the beginning of each year when funds are available; (ii) the technologies can be implemented at the same time; (iii) while a project implementation may require more than one year, its use for sustainability improvement is assumed at the beginning of the following year. Since the number of stages needed to achieve the sustainability goal is yet to be determined, a new binary variable, zi, is defined as follows: ⎧1, if stage i is needed zi = ⎨ ⎩ 0, otherwise

eq 30 eq 31 scenario 1: eq 32 scenario 2: eq 35

X̅ = [x L , x U] L

decision variables eq 24

model constraints

eqs 10−18, 24−28 eq 24 eqs 10−18, eq 24 eqs 10−18, 24−28 scenario 1: eq 24 eqs 10−18,

20−22, 20−28 20−22, 20−28

scenario 2: eqs 24, 34

(36)

U

where x and x are the lower bound and the upper bound of the interval, respectively. In fact, the interval shows a uniform distribution for an index with the parameters of xL and xU. Note that a parameter interval can be set based on the analysis of historical data, literature, and an expert’s knowledge and experience. The sustainability status of the system P can be assessed using the available data collected from the system. Also, it is very possible that technology inventors, providers, and users can provide some technology assessment information based on their tests and experience for new technologies. In the case of missing technical data, a reliable system simulator can be used to generate reasonable performance data. Since much data are expressed as intervals, the index-specific assessment results,

⎪

⎪

eq 29

type of optimization model with a list of its objective functions and constraints. These models are called binary programming (BP) formulations, as decision variables are all binary. The models can be solved using classical techniques (e.g., branch and bound technique, cutting planes technique, outer approximation technique) or stochastic techniques (e.g., simulated annealing, controlled random search, genetic algorithms). Interval-Parameter-Based Uncertainty Handling. Uncertainties are inevitable at the sustainability assessment stage due to incomplete and imprecise information about the industrial system. There also exist much external uncertain information related to raw materials and product pricing, market demand, environmental regulation, etc.44 Optimization under uncertainty must incorporate an approach for uncertainty handling in order to ensure the feasibility and robustness of the derived solutions. Note that each sustainability indicator is evaluated using collected technical and nontechnical data, which are available frequently in a certain range. Thus, we use an interval parameter method45 to deal with such uncertainties, which can be expressed as follows:

where tPi is the project time (e.g., month) that is determined by the selected project which requires the longest implementation, as compared with others; this can be expressed as t Pi = max{yi ,1t1 , yi ,2 t 2 , ..., yi , N tN }, i = , 1, 2, ..., M ,

objective function

maximum sustainability minimum cost maximum efficiency minimum time

M yi , j

(35)

M

(d) Minimum time problem. Any transition toward sustainability is associated with time. In order to evaluate the total time needed to achieve a sustainability goal, the time for each stage of sustainability transition should be evaluated. The time length of each transition depends on an organization’s plan for technology adoption; it may want to implement one or more technologies at each time, depending on budget availability, production and operational constraints, etc. Here, we discuss two different scenarios for the minimum-time optimization problem. Before that, we define the “project” as a set of technologies (one or more), whose adoption meets the budget constraints and leads to an improvement. Scenario 1. Assume that the budget limits, Bup(i), i = 1, 2, ..., M, are known for the M-stage sustainability improvement task. In this scenario, a new project should start only after the completion of the existing project. Thus, M means the total number of projects. In each project period, if there is more than one technology to adopt, the technologies should be implemented simultaneously. For this scenario, the objective function should be defined as follows: J = min ∑ t Pi

2

..., zM(t p + (M − 1) × 12)}]

|| S (⃗ M )|| M N ∑i = 1 ∑ j = 1 f (ζi)yi , j Bj

1

yi , j

(34)

On the basis of the above assumptions, the objective function can be expressed as E


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ACS Sustainable Chemistry & Engineering i.e., E̅ i, V̅ i, and L̅i, will be also expressed as intervals, which are exemplified in the case study. Because of the use of internal parameters, a Monte Carlo (MC) simulation technique is adopted to derive solutions and calculate the expected value of the objective function. Monte Carlo simulation is a numerical stochastic process involving calculation of a sequence of random events. It starts with sampling a random value from each input distribution and runs the model using the resulting value. After repeating this for a large number of times, it estimates the output value by taking the average of the results. This method is widely used for uncertainty analysis46,47 and stochastic programming.48 Solution Method. In this work, we use genetic algorithm (GA) with Monte Carlo (MC) simulation techniques to solve all four types of optimization models. A flowchart of the procedure for uncertainty incorporation and optimal strategy development is shown in Figure 2, and an algorithm for solution derivation is briefly described as follows:

the survival-of-the-fittest mechanism on the candidate solutions. This selection step allows the preference of better solutions over worse ones. Step 4 − Crossover. This step combines two or more parental solutions to create new, possibly better solutions (i.e., offspring). Competent performance depends on a properly designed recombination mechanism. Step 5 − Mutation. While the last step operates on two or more parental chromosomes, this step modifies a solution locally but randomly. There are many variations of mutation, but it usually involves one or more changes made to an individual’s trait. Note that the mutation performs a random walk in the vicinity of a candidate solution. Step 6 − Of fspring Evaluation. The offspring population created in the past step is to be evaluated in this step, and the expected value of the objective function is estimated by Monte Carlo simulation. Note that the number of iterations in the Monte Carlo simulation should be large, e.g., 1000. Step 7 − Stopping Criteria. Repeat steps 3 to 6 until a terminating condition is met.

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CASE STUDY We consider a sustainability enhancement problem with biodiesel manufacturing. Figure 3 shows an alkali-catalyzed manufacturing process, with a production capacity of 8000 tons/yr. The process produces biodiesel (alkyl ester) by transesterification of a lipid feedstock, such as vegetable oil or animal fat, and its reaction is listed in Figure 4. A catalyst is used to improve the reaction rate and yield. Because the reaction is reversible, excess alcohol is used to shift the equilibrium to the products side. More information about the process and the technologies considered for sustainability improvement in this study can be found in Zhang et al.,49 Glisic and Skala,50 and West et al.51 A manufacturing company plans to identify the most suitable technologies for economically feasible waste reduction, energy recovery, and product quality based on the four types of optimization problems discussed above. Information and Data Collection. The system and sustainability related technical data are adopted from Liu and Huang.28 Ten candidate technologies can be classified as two groups for this sustainability enhancement problem. The first group of technologies is related to source waste reduction technologies. The four identified technologies are (1) T1, to separate the methanol in the waste stream from the glycerol purification column and then to recycle it to the transesterificaiton reactor, (2) T2, to recycle the unconverted oil as part of the feedstock after pretreatment, (3) T3, to recycle the waste stream of the glycerol purification column to the liquid− liquid extraction column as a washing solvent to replace fresh water, and (4) T4, to recovery the solid waste from the catalyst removal separator as a type of fertilizer. Note that technologies T1 and T3 cannot be adapted simultaneously because they work on the same stream (waste stream from the glycerol purification column). The second group of technologies is related to energy efficiency and product performance improvement technologies. They are (1) T5, to redesign the product purification sequence, (2) T6, to pretreat waste cooking oil as a new feedstock, (3) T7, to adopt a new catalyst for the transesterification reactor in order to improve its conversion rate, (4) T8, to recover the energy from the glycerol purification process, (5) T9, energy recovery from the transesterification reaction process, and (6)

Figure 2. Flowchart of the procedure for uncertainty-incorporated optimal strategy development.

Step 1 − Initialization. An initial population of candidate solutions is generated randomly across the search space. Step 2 − Evaluation. Once a population is initialized or an offspring population is created, the fitness values of the candidate solutions are evaluated. In this step, Monte Carlo simulation is used to calculate the following expected value, E[f(y,ω)], of the fitness function: E[f (y , ω)] =

1 NMC

NMC

∑ f (y , ωi) i=1

(37)

where NMC refers to the number of iterations in a Monte Carlo simulation and ω is the set of uncertainty parameters. Step 3 − Selection. This step allocates more copies of the candidate solutions with higher fitness values and thus imposes F


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Figure 3. Flowsheet of an alkali-catalyzed biodiesel manufacturing process.

(V2), and (3) fraction of raw materials recycled (V3). In the social sustainability category, the selected indices are (1) lost time accident frequency (L1) and (2) number of complaints per unit value added (L2). For a multistage improvement, the organization should specify its sustainability goal for each stage. Here, we describe our methodology for a three-stage sustainability improvement. The assessment results of the status quo of process system P and the two groups of technologies and also the total normal cost and the total normal time needed for implementing each technology are listed in Table 2. Note that the exact values shown in the table are also intervals with the same upper and lower bounds or very small difference between them (e.g., < 0.001). The minimum requirement of each categorized sustainability and the budget limit for each stage and model parameters are shown in Table 3. In this case, the plant

Figure 4. Transesterification of triglycerides with alcohol.

T10, to recover the energy from the biodiesel purification system. Seven indicators from the IChemE Sustainability Metrics system are used in this study. The economic indices include (1) value added (E1) and (2) gross margin per direct employee (E2). The environmental sustainability category contains three indices: (1) total raw materials used per pound of product produced (V1), (2) hazardous solid waste per unit value added

Table 2. Index-Specific Sustainability Assessment of the System and Candidate Technologies system category econ. (E)

environ. (V)

index E̅1

soc. (L)

P

E̅2

[0.550, 0.570] 0.450

V̅ 1

0.400

V̅ 2

[0.350, 0.380] 0.420

V̅ 3 L̅1

technologies in group 1

[0.335, 0.340] L̅2 [0.370, 0.380] cost for technology use B(Tj) (K $) time for technology use tj (month)

T1 0.620

T2

[0.500, 0.530] 0.430

[0.600, 0.620] [0.480, 0.490] 0.450

0.380

0.360

0.430

[0.420, 0.430] 0.340

[0.355, 0.360] 0.400

0.380

technology in group 2

T3 0.570 [0.460, 0.470] [0.410, 0.420] [0.360, 0.380] 0.400

T4

T5

[0.600, 0.610] [0.490, 0.510] [0.430, 0.440] 0.370 0.430

0.330

0.350 [0.380, 0.385] 80 5

100

50

[0.378, 0.380] 50

8

4

3

G

[0.600, 0.610] 0.510

T6

120

[0.750, 0.7700] [0.600, 0.620] [0.600, 0.650] [0.380, 0.400] [0.420, 0.430] [0.390, 0.410] [0.420, 0.430] 300

7

24

[0.440, 0.450] [0.360, 0.380] 0.450 [0.310, 0.315] 0.400

T7 0.610 [0.460, 0.470] 0.430 [0.360, 0.370] [0.430, 0.440] 0.330

T8 [0.620, 0.630] 0.460 [0.460, 0.470] 0.350 [0.430, 0.440] 0.440

T9 [0.600, 0.620] [0.520, 0.530] [0.410, 0.420] 0.400

[0.380, 0.390] 100

[0.360, 0.370] 90

[0.420, 0.430] [0.390, 0.400] [0.400, 0.410] 60

3

5

6

T10 [0.580, 0.590] [0.460, 0.480] [0.460, 0.470] [0.400, 0.410] 0.410 [0.350, 0.365] 0.370 80 7


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ACS Sustainable Chemistry & Engineering Table 3. Model Parameters Used in Optimization Models parameters min. economic sustainability requirement min. environmental sustainability requirement min. social sustainability requirement budget limit (K$) weighting factor of selected indices weighting factors discount function sustainability goal and tolerance level total number of candidate technologies

stage 1

stage 2

stage 3

0.650 0.450

0.750 0.500

0.800 0.550

Table 4. Summary of the Optimal Solutions of Different Sustainability Optimization Problems optimization problem

category

0.400 0.450 0.500 300 200 100 (a1 = 1, a2 = 1), (b1 = 2, b2 = 1, b3 = 1), (c1 = 3, c2 = 1) α = 1, β = 1, γ = 1 f(ζ) = −0.05ζ + 1.05 Sg⃗ (M) = 0.65 and ε = 0.02 10

management sets the plant’s goal for categorized sustainability SP to SSP 1 (0.65, 0.45, 0.40) for the first stage, S2 (0.75, 0.50, 0.45) SP for the second stage, and S3 (0.80, 0.55, 0.50) for the third stage. The budget limit is $300K, $200K, and $100K for the first, second, and third stages, respectively. The norm of the categorized sustainability vector is evaluated by eq 29, where parameters α, β, and γ take the default value of 1. For the discount function, we estimate it as a linear function shown in Table 3; it is a function of the number of technologies to be used. Three Stages of Sustainability Development. Four different optimization problems are studied, which are the minimum cost, the maximum sustainability, the maximum efficiency, and the minimum time as objectives. Our target is to find the optimal decision (i.e., a set of technologies) at each stage for each type of optimization problem. For the minimum cost and minimum time problems, the sustainability goal (∥Sg⃗ ∥) and tolerance (ε) are set to 0.650 and 0.02, respectively. Results and Discussions. The four optimization models are solved using the integrated approach, the genetic algorithm with Monte Carlo simulation method, discussed in the last section. In solution search, the number of samples in each Monte Carlo simulation should be large enough so that the expected value term does not change significantly with the sample size. Therefore, we set this parameter to 1000 for each Monte Carlo simulation to ensure the accuracy of the modeling. The results for four different scenarios are shown in Table 4, which are discussed below. (a) Maximum sustainability problem. The result of this optimization problem is shown in the second column of Table 4. As shown, the best strategy for maximizing sustainability through a three-stage improvement effort is to adopt technology set {T6} at the first stage, technology set {T1,T2, T9} at the second stage, and technology set {T8} at the third stage, which costs $579K in total to achieve the maximum sustainability of ∥S⃗( 3)∥ = 0.701. In detail, the final sustainability status vector of the process after adoption of this strategy is S⃗(3) = (0.890, 0.613, 0.556). (b) Minimum cost problem. The result of this optimization model is shown in the third column of Table 4. It shows that using technology set {T1,T2,T4,T9,T10} at the first stage and technology set {T7,T8} at the second stage is the optimal strategy. As shown, there is no need of the third stage because the system sustainability has already reached the final sustainability goal and all constraints have been satisfied. After two stages, the system’s sustainability performance is S⃗(2) = (0.813, 0.573, 0.518) or ∥S⃗(2)∥ = 0.647, at the overall cost of $476.5K. In this optimization, a discount for adoption of more

max. sustainability

min. cost (K$)

max. efficiency (1/ MM$)

min. time (month)

{T1, T2, T9} {T8} 0.701

{T1, T2, T4, T9, T10} {T7, T8} N/A 476.5

{T2, T8, T9} N/A 1.38

{T1, T2, T4, T9, T10} {T7, T8} N/A 13

0.890

0.813

0.827

0.813

0.613

0.573

0.591

0.573

0.556

0.518

0.534

0.518

579 0.701

476.5 0.647

480 0.660

476.5 0.647

37

13

30

13

1.17

1.36

1.38

1.36

stage 1

{T6}

stage 2 stage 3 objective function value economic sustainability environmental sustainability social sustainability overall cost (K$) overall sustainability total time (month) efficiency (1/ MM$)

{T6}

technologies is included in the cost model. This is another reason why only two stages are needed. (c) Maximum efficiency problem. The optimal solution of this optimization problem is shown in the fourth column of Table 4. Like the minimum cost problem, only two stages are needed to achieve the minimum requirement of each sustainability index set for the third stage. The optimal solution shows that using technology set {T6} at the first stage and technology set {T2, T8, T9} at the second stage maximizes the efficiency objective function and brings the initial sustainability status of the system to the status of S⃗(2) = (0.827, 0.591, 0.534) with the overall cost of $480K; the final sustainability value is ∥S⃗(2)∥ = 0.660. (d) Minimum time problem. Two scenarios are considered for this optimization. For the first scenario, it is assumed that the second project can start immediately after the first project. The optimal solution for this scenario is shown in the fifth column of Table 4. The result shows that using technology set {T1,T2,T4,T9,T10} in the first stage and technology set {T7,T8} in the second stage can bring the initial sustainability status of the system to the final sustainability state of S⃗(2) = (0.813, 0.573, 0.518), with a minimum total time of 13 months, for which all constraints are satisfied. The first and second stage sustainability development lasts 8 and 5 months, respectively, in this scenario. Note that if using the same plan for the other problems, the total time for the maximum sustainability problem, the minimum cost problem, and the maximum efficiency problem would be 37 months, 13 months, and 30 months, respectively. The same optimal solution is obtained for the second scenario with a total time of 17 months without a need of the third stage. Since the first project lasts less than one year (8 months), the projects do not have overlap and the second project starts 4 months later when the fund is available and lasts 5 months. Using the same plan for the other problems, the total time for the maximum sustainability problem, the minimum cost problem, and the maximum efficiency problem would be 29 months, 17 months, and 24 months, respectively. As stated earlier, the total time needed for sustainability enhancement depends on the resources available H


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ACS Sustainable Chemistry & Engineering to the company and how much its operation and production planning allow. If a company could invest more money at the first stage, the total time could be shortened. In order to study this, a sensitivity analysis is performed on the first stage budget limit for both scenarios. The result shows that by increasing the budget limit at the first stage from $300K to $500K, the total time is shortened from 17 months to 8 months without a need of even the second stage by using technologies {T1,T2,T4, T7,T8,T9,T10} all together in the first stage. Note that we have the same optimal solutions for the minimum cost and the minimum time problems, but this happens to be in this case; it should not be general. For example, if there was a technology, namely T11 (with the same sustainability improvement capacity as technology T1), that demands a higher cost but a shorter implementation time, that would be in the optimal solution of minimum time problem. Also, we consider the normal time and normal cost of technology adoption for sustainability enhancement problems. However, the time duration of each technology adoption or a project can be reduced by assigning more resources, but this could lead to a higher total cost. As the results of the different problems show, different objective functions for systems’ sustainability enhancement gives rise to different optimal solutions and consequently different improvement strategies. The maximum sustainability problem uses the funds of all three stages to achieve the maximum sustainability at the final stage, which leads to the most expensive plan for sustainability enhancement. This plan uses technology T6, which has the highest sustainability improvement capacity among all technologies in the first stage, where the budget is available for its implementation; however, it needs a higher cost and a longer time for implementation in comparison to the minimum time and minimum cost problems. In the second stage, this plan uses the technology set {T1, T2, T9}, which also has the highest sustainability improvement capacity and also the highest implementation cost and the longest time respect to other technology sets in the same stage of other plans. The minimum cost problem uses the funds available at the first and second stages to achieve the sustainability goal; it does not need the third stage budget. However, increasing the sustainability goal value for this problem can change the optimal solution and involve the third stage. The technology sets used in the first stage of this plan have the lowest cost among technology sets of other plans, but the cost in the second stage is approximately the same as the maximum efficiency problem. The maximum efficiency problem combines the maximum sustainability and minimum cost objective functions. As we expected, the final sustainability status of the system resulted from the maximum efficiency problem is placed somewhere between the final sustainability status of the minimum cost problem and the final sustainability status of maximum sustainability problem. For the minimum time problem, increasing the budget at the first stage reduces the total time for achieving the sustainability goal. Figure 5 depicts the increment of the categorized sustainability (ΔE, ΔV, and ΔL) and the overall sustainability (∥ΔS⃗∥) achieved by the four different problems. All the scenarios show their capacity of significant improvement of the system’s sustainability performance. It shows that the maximum sustainability scenario provides a better sustainability performance over other scenarios, mainly because of the economic index improvement. This kind of information can be valuable for decision makers to make a better decision for sustainability

Figure 5. Comparison of four different scenarios for sustainability performance improvement.

improvement. Piluso et al.44 introduced a representation scheme to depict the status of triple-bottom-line-based sustainability and its transition. Here, we use this representation scheme to show the sustainability status of the system and transition toward sustainability for different scenarios. Figure 6

Figure 6. Sustainability development paths for different scenarios.

shows the initial sustainability status of the system, sustainability status transition paths of the system for all different optimization problems, and the final sustainability status of the system. The green and red paths show the two-stage sustainability development of the system for the minimum cost and the maximum efficiency scenarios, respectively. The blue path shows the three-stage sustainability improvement for the maximum sustainability scenario. The green path also shows the sustainability transition for the minimum time problem, while the purple path shows the one stage sustainability transition for the minimum time problem when the budget is increased to $500K.

■

CONCLUDING REMARKS Strategic planning for sustainability enhancement of an industrial system usually involves multiple stages, where triple-bottom-line-based sustainability principles should be applied in each improvement stage. This is a very challenging task involving different optimization objectives. It becomes more challenging when it encounters various types of uncertainties that appear in the available technical or nontechnical data, information, and possessed knowledge. I


Research Article

ACS Sustainable Chemistry & Engineering Subscripts

In this paper, we have introduced a sustainability performance improvement methodology by resorting to optimization and uncertainty theories. We have explained the multistage sustainability enhancement problem and introduced four types of sustainability optimization problems with different objective functions over a period of the project development phase. The data uncertainty is dealt with using interval parameters that results in uniform distribution for each sustainability index. However, the uniform distribution provides one of the simplest means of representing our uncertainty in a model input. The results of the model can be more reliable, if suitable probability distribution functions can be used, but in most cases, there is always a lack of sufficient data in real-world problems. In solution identification, we used GA and MC simulation techniques. The methodology is particularly useful for identifying optimal strategies for the multistage sustainability enhancement problem under different objective functions, which can provide valuable information for decision makers to make a better decision in their projects. The case study on a biodiesel manufacturing problem has demonstrated the efficacy of the methodology.

■

■

a = weighting factor of economic sustainability indicators b = weighting factor of environmental sustainability indicators c = weighting factor of social sustainability indicators i = index of stages j = index of technologies l = index of economic sustainability indicators m = index of environmental sustainability indicators n = index of social sustainability indicators

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*Phone: 313-577-3771. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work is supported in part by NSF (No. 1434277 and No. 1604756). NOMENCLATURE Bj = cost for adopting technology Tj Bup = budget limit E = categorized economic sustainability index Emin = minimum requirement for the categorized economic sustainability index F = total number of economic sustainability indicators G = total number of environmental sustainability indicators H = total number of social sustainability indicators L = categorized social sustainability index Lmin = minimum requirement for the categorized social sustainability index M = number of stages P = plant S⃗ = sustainability status vector of the system T = technology t = time tj = total time for implementing adopted technology Tj V = categorized environmental sustainability index Vmin = minimum requirement for the categorized environmental sustainability index y = binary variable

Greek

α = weighting factor of the categorized economic sustainability index β = weighting factor of the categorized environmental sustainability index γ = weighting factor of the categorized social sustainability index ε = tolerance parameter for the final sustainability state J


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