Sustainability Enhancement under Uncertainty: A Monte Carlo Based Simulation and System Optimization Method† Zheng Liu and Yinlun Huang* Department of Chemical Engineering and Materials Science Wayne State University, Detroit, Michigan 48202 Abstract Known methods for sustainability enhancement are typically scenario based and the uncertainty surrounding available data and information is usually not addressed holistically, due to inherent problem complexity.
Thus the solutions identified by those methods could be not
sufficiently effective in many industrial applications.
In this paper, we introduce a Monte Carlo
based simulation and system optimization method for deriving sustainability enhancement strategies, where uncertainties are systematically taken into account.
The methodological
efficacy is illustrated through the study of an industrial sustainability enhancement problem involving a number of sectors.
Keywords: Industrial sustainability, uncertainty, system optimization, Monte Carlo simulation ______________________________________________________________________________ † For publication in Clean Technologies and Environmental Policy (in press). * All correspondence should be addressed to Prof. Yinlun Huang (Phone: 313-577-3771; Fax: 313-577-3810; E-mail:
[email protected]).
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1
Introduction
Sustainability, in a general sense, is the capacity to maintain a certain process or state indefinitely. As applied to the human society, “sustainable development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs” (WCED, 1987). The economic, environmental and social sustainability are normally accepted as the triple bottom lines, which should be systematically assessed and enhanced. Industrial sustainability is pursued to achieve the sustainable development (SD) of industrial organizations or systems. Decisions and strategies for SD are usually jointly derived by business leaders, planners, technical personnel, and stakeholders.
Decision making could be
a very challenging process, especially when an industrial system is large and where uncertainty is pervasive.
A large-scale industrial system may be composed of a few sectors, each of which
has a number of manufacturing plants. The plants have their own business development goals and may share supply chains.
Decisions and strategies for the SD of this type of hierarchically
structured systems must ensure a healthy development, for all entities, through synergistic collaboration and decision makers should be able to coordinate the SD objectives in a decision hierarchy under various constraints and uncertainties. Sustainability oriented decision-making is mostly scenario based in practice, where a scenario refers to a detailed plan or possibility. For instance, one scenario could call for improving profitability by increasing water reuse and chemical recovery, as well as reducing energy consumption in certain processes.
In a decision-making process, the first step is to
identify the possible scenarios, followed by a comparison against each other, and finally
2
selection of the most preferable scenario, based on group expertise.
In most decision making
processes, information and data uncertainty is not thoroughly considered, which could make the derived decisions not sufficiently effective in the implementation phase.
Note that the
uncertainty associated with sustainability problems is frequently pervasive, which appears due to incomplete or imprecise information and lack of deep knowledge about the interactions of participating industrial units within the system or surrounding environment (Piluso and Huang, 2009). According to Parry (1996), uncertainties can be classified into two types: aleatory and epistemic. Aleatory uncertainty refers to the inherent variations associated with the physical system and/or the environment under consideration; it is objective and irreversible. By contrast, epistemic uncertainty denotes a lack of knowledge and/or information; it is subjective and reducible. The uncertainties encountered in the study of large-scale industrial sustainability problems can be either aleatory or epistemic.
They are most likely related to parametric
uncertainties associated with models and/or due to limited understanding on sustainability problems (Dorini et al., 2011).
A variety of mathematical techniques and computer or
cognitive science based methods are available for handling uncertainties, such as those utilizing fuzzy logic theory, statistics theory, and artificial intelligence (Graham and Jones 1988; Ayyub and Gupta, 1997; Meinrath, 2001; Yang, 2001; Bilgic, 2003; Hanss, 2005; Kanovich and Vauzeilles, 2007; Cawleya et al., 2007). Despite the existence of numerous types of inherent uncertainties and uncertainty handling methods, this work is concerned with the uncertainties that appear in the sustainability problems of large-scale industrial systems.
We formulate sustainability problems as a class of
system optimization problem, which requires the identification of solution alternatives.
3
We
then introduce a Monte Carlo based simulation approach to evaluate solution alternatives when various uncertainties are incorporated.
Optimal solutions will be determined for achieving the
highest level of sustainability improvement while keeping below the given budget constraints. This work’s methodological efficacy is illustrated by analyzing sustainability issues and developing strategies to enhance the sustainability of a metal finishing industrial system that involves three sectors and six plants.
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System Modeling and Alternative Scenario Identification
Industrial sustainability is largely reflected by material and energy efficiencies, product quality and productivity, environmental cleanness, safety, etc.
A variety of methods can be used
to generate the system information that is needed for a sustainability assessment, such as Material Flow Analysis (MFA), Total Material Requirement (TMR), Material Intensity Per Unit Service (MIPS), and Substance Flow Analysis (SFA). is Input-Output Analysis (IOA) and its variants.
The other type of flow analysis method
Bailey et al. (2004) provided a comprehensive
review on sector-based IOA approaches that are integrated with materials, energy, and/or environmental impacts, and introduced an Ecological Input-output Analysis (EIOA) method. Piluso et al. (2008) extended the EIOA method (named e-EIOA method here) by separating the output streams into two sets, the product stream set and the waste stream set, which facilitates a system performance analysis.
An e-EIOA-based system model can provide the necessary
information for decision makers to develop alternative scenarios, each of which may adopt one or more technical approaches, or simply called technologies here, that can be further studied for system performance improvement.
The most effective scenario will then be selected for
4
implementation.
Detailed information of the e-EIOA method can be obtained in Piluso et al.
(2008). In this work, the e-EIOA based system modeling method is adopted due to its capability of providing comprehensive system information. System description.
According to Piluso et al. (2008), a complex industrial system,
named R (for example, an industrial region), is assumed to have n sub-systems (e.g., plants), each of which is named Hi, and it is a basic element in input-output flow analysis. As shown in Fig. 1(a), the sub-system’s inputs include the raw material and/or energy streams (denoted as zi), the internal flow from sub-system Hj to sub-system Hi (symbolized as fi,j), and flow from Hi to Hk (i.e., fk,i), while the sub-system’s outputs include the product stream (pi) and the waste stream (wi). For the system, R, composed of n sub-systems, the system matrix is shown in Fig. 1(b), which can be expressed as:
0 R R21 0
0 R22 R32
0 0 0
(1)
where
R21 Diag zi ,
i 1,, n
(2)
R22 f i , j n n ,
i, j 1,, n
(3)
R32 Diag yi ,
i 1,, n
(4)
yi pi
i 1,, n
(5)
wi
T
The throughflow ( Tk ) of sub-system H k is defined as the rate of material flow through the sub-system and is defined as follows: 5
n
Tk f k , j zk ,
k 1, ,n
(6)
k 1, ,n
(7)
j 1
n
Tk f i ,k pk wk , i 1
Note that throughflow is either the sum of all inflows to a sub-system (Eq. 6, which equals the sum of the rows in R) or the sum of all outflows from the sub-system (Eq. 7, which equals the sum of the columns in R); the two throughflows are equal to each other. Given the system matrix, R, and throughflows, an inflow analysis can be performed, which allows one to trace the system outputs back to their origins by determining the amount of direct and indirect flows within the system needed to generate that outflow. The detailed mathematical description can be found in Piluso et al (2008). Assessment of system sustainability.
It is assumed that a set of sustainability metrics is
selected by decision makers, each of which contains three subsets and can have a number of specific indices (Zhou et al., 2012):
S E , V , L,
(8)
where
E Ei i 1, 2, , M E , the set of economic sustainability indices, V Vi i 1, 2, , M V , the set of environmental sustainability indices,
L Li i 1, 2, , M L , the set of social sustainability indices. For convenience, all indices are normalized (0 to 1).
By using the selected sustainability indices,
the status quo of the sustainability of system R can be assessed using the data collected from the system.
This data can be used to estimate the categorized sustainability for the system, i.e., E R ,
6
V R , and LR , each of which are called the composite sustainability indices and can be evaluated using the following formulas: Me
E R
a E R i 1
i
i
,
Me
a
(9)
i
i 1
G
V R
b V R i 1
i i
G
b
,
(10)
,
(11)
i
i 1
H
L R
c L R i 1
i
i
H
c i 1
i
where ai, bi, and ci [1, 10] are weighting factors associated with the individual indices, reflecting the relative importance of an individual index against others in the sustainability assessment (Sikdar, 2009). Piluso et al. (2010) introduced a sustainability status representation scheme, which is called the “sustainability cube” and is shown in Fig. 2.
The three coordinates are designated for the
composite economic index, the composite environmental index, and the composite social index. Each composite index is set to have a value between 0 (meaning no sustainability) and 1 (meaning complete sustainability).
In the cube, the corner coordinate of (0, 0, 0) represents the system’s
status of no sustainability, while the opposite corner having the coordinate (1, 1, 1) indicates complete sustainability.
The point, S(t), represents the system’s overall sustainability status at time
t, which is determined by the values of three categorized sustainability, i.e., E(t), V(t), and L(t), as displayed in the figure.
The sustainability cube can be used to represent both the categorized and
overall sustainability status at any time instant. In an industrial organization, the personnel at a
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lower administrative level may pay the most attention to the values of the individual sustainability indices (i.e., Ei(t), Vj(t), and Lk(t)), while those at a higher administration level may be more interested in knowing the values of the three categorized sustainability (i.e. E(t), V(t), and L(t) and the overall sustainability performance (S(t)).
S R
E R , V R , LR , ,
The overall sustainability can be estimated as follows: ,
(12)
where , , and each have a value of 1 (default) to 10, and S R is normalized.
System state transition.
For operational convenience, we can restructure the system
matrix, R(t), in Eq. 1 by removing all “zero” sub-matrices. The new matrix, namely R’(t), can be obtained as follows:
R' t 0
I
0 R 0 0
I R0 0
I 0 0 T
I
T
R21t R22 t R32 t
T
(13)
Note that the values of the parameters, zi, fi,j, pi, and wi, in matrix R’(t) will be used to evaluate the system sustainability performance by using the indices in Eq’s. 9-13. Thus, E R' t g E zi t , f i , j t , pi t , wi t ,
i, j 1,, n
(14)
V R' t gV zi t , f i , j t , pi t , wi t ,
i, j 1,, n
(15)
LR' t g L zi t , fi , j t , pi t , wi t ,
i, j 1,, n
(16)
S R' t g S zi t , fi , j t , pi t , wi t ,
i, j 1,, n
(17)
and,
Sustainability performance can be improved if appropriate actions, e.g., new technology implementation, are taken, which normally requires some form of an investment. vector of investments at time t, and X(t) be the system parameter vector. transition model can be expressed as: 8
Let U(t) be a
A system state
X tk 1 h X tk , U tk ,
(18)
where X tk hX zi tk , fi , j tk , pi tk , wi tk
i, j 1,, n
U tk u1 tk u2 tk uJ u tk
System optimization model.
T
(19) (20)
Alternative scenarios for sustainability enhancement can
be identified by solving a sustainability optimization problem (Wang et al., 2010).
The
optimization problem, formulated below, is designed to achieve the best sustainability performance under a number of system, investment, and other relevant constraints.
J Max S R' tk 1
(21)
X tk 1 h X tk , U tk
(18)
U t k
s.t .
0 u j tk u max j tk JU
j 1, , JU
(22)
u t U t
(23)
E R' tk 1 1 E E R' tk
(24)
V R' tk 1 1 v V R' tk
(25)
LR' tk 1 1 L LR' tk
(26)
i 1
max,total
i
k
k
i 1, ,n
(27)
i 1, ,n
(28)
i 1, ,n
(29)
Ei tk 1 1 Ei Ei tk
Vi tk 1 1 Vi Vi tk Li tk 1 1 LL Li tk
In the above model, the objective function in Eq. 21 is to maximize the total sustainability of the
9
system, and the decision variables are the investment vector, U(tk). Equation 18 is the system model. The investment constraints are listed in Eq’s. 22 and 23, including the upper limit of each investment type and the total budget constraints. sustainability improvement constraints.
There are two sets of categorized
The first set (Eq’s. 24 to 26) ensures that the
categorized sustainability performance of the whole system is at least i 100% better than the status before improvement, while the second set (Eq’s. 27 to 29) requires at least i 100% of a categorized sustainability improvement for each sub-system after taking certain actions. Alternative scenario identification.
The optimization model listed above is a
non-linear optimization model. While there are a number of available mathematical programming techniques to solve the problem, in this work, the model is solved using a Genetic Algorithm (GA) technique, by which a number of local optimal solutions can be readily identified.
GA application methods have been implemented in many other works (e.g.,
Ruszczyński, 2006; Bartholomew and Michael, 2005).
These local optima are used as
alternative scenarios that are to be further studied by incorporating uncertainties, which will be discussed in a later section.
Let be the set of identified alternative scenarios, which can be
expressed as:
S* R' tk 1 i , U * tk i i 1, 2, , NGA
(30)
where N GA is the total number of alternative scenarios; S* R' tk 1 i is the sustainability value of the system in the i-th scenario; U * tk i is the identified investment strategy for the i-th scenario.
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3
Uncertainty Incorporation and Optimal Strategy Development
As stated earlier, the uncertainty associated with sustainability problems is usually pervasive, which appears due to incomplete and imprecise information and a lack of deep knowledge about the interactions of participating industrial members within the system and the surrounding environment.
For instance, raw materials and product pricing, market demand,
technology uncertainty, environmental regulations, and organizational investment and development capabilities could all change over time. Thus, in the development of sustainability improvement strategies, the evaluation of alternative scenarios must consider various forms of uncertainty in order to identify practical and effective solutions.
Khajuria and Pistikopoulos
(2013) also expressed this concern when solving optimization problems under uncertainty, and proposed a post-dynamic feasibility test after solving an optimal design problem of pressure swing adsorption processes.
Similarly, Zhou et al. (2013) proposed a two-stage stochastic
programming model for the optimal design of distributed energy systems, where the second stage employs Monte Carlo based sampling to estimate an average value under uncertainty for each model parameter that is used by the programming at the first stage. In this work, uncertainties are introduced to the system by utilizing a Monte Carlo simulation technique, which is widely used for uncertainty analysis (Chettouh et al., 2014; Kazantzi et al., 2013). The simulation will then reevaluate the sustainability performance of each candidate scenario after the uncertainties have been introduced.
Parameters that involve
uncertainty, each is required to specify a parameter variation range.
For instance, if the
anticipated sale price of a product is $100/lb, its fluctuation range may be set at $80/lb to $120/lb. Note that there could be infinitive combinations of uncertain parameter values, which adds to the
11
difficulty of evaluating scenarios under uncertainty.
Here, the Monte Carlo based method
developed by Geltle (1998) and then Kolos and Whitlock (2008) is used to randomly sample a large number of uncertainty combinations for scenario evaluation.
Such a Monte Carlo
sampling approach was also successfully used by Shastri and Diwekar (2006) for solving stochastic nonlinear programming (SNLP) problems, and for performing the stochastic simulation that was applied to the GREET Model (Subramanyan et al., 2008) under uncertainty. Optimal scenario determination.
An optimal strategy for sustainability performance
enhancement can be derived using the following procedure, which is shown in Fig. 3. Step 1. Collect all of the necessary system information, budget availability data, sustainability goals for individual sub-systems and for the entire system, etc.
Identify the
uncertain parameters and define their variation ranges. Select sustainability indices for each of the three categorized sustainability groups (see Eq. 8) and assign preferred values for coefficients, ai, bj, and ck in Eq’s. 9 through 11, and α, β, and γ in Eq. 12. Step 2. Create a system model using the e-EOIA modeling method. This will lead to the generation of Eq’s. 1 through 7 and, therefore, Eq’s. 14 through 20. Step 3. Generate a sustainability-oriented optimization model (see Eq’s. 21 through 29). Step 4. Use the given parameter uncertainty data to conduct the Monte Carlo simulation to generate NMC sets of random parameter values; NMC should be large, e.g., 1,000. Step 5. Calculate the total sustainability for each alternative scenario using each of the NMC sets of parameter values and obtain an average total sustainability value for each scenario. This step will continue until all alternative scenarios have been evaluated, which leads to the generation of the strategies listed in Eq. 30. Step 6. Rank all NMC scenarios using Eqs. 31 and 32, which are listed below.
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Step 7. Output the best scenario for sustainability enhancement, for which the optimal budget allocation and specific sustainability assessment results are included. Note that in Step 5, for each scenario, the calculated value of total sustainability is averaged based on all NMC simulations.
Due to the stochastic nature of a Monte Carlo
simulation, among these simulations, there could be a number of calculations that result in the largest total sustainability value as compared with other scenarios. There could also be some other number of calculations that result in the second largest value of the total sustainability compared with the simulations for all remaining scenarios, and so on.. This results in many options for the decision makers and some decision makers may prefer to take this into account in their scenario ranking.
For instance, the following scenario selection rule may be used:
i ς θi ξ i υ ρi
i = 1, 2, ..., NGA
(31)
where i is the score for the i-th alternative scenario; i , i , and ρi are the number of times that the sustainability values are the largest, the second largest, and the third largest among NMC Monte Carlo simulations for the i-th scenario as compared with other scenarios; ϛ, ξ, and υ are constants to be selected based on a decision maker’s preference. For example, one may set ϛ = 10, ξ = 6, and υ = 2. The best scenario can be easily selected using the following formula:
Max1 , 2 , , N
4
MC
(32)
Case Study
System and alternative scenarios.
Piluso et al. (2008) studied an interesting
sustainability enhancement problem for a large-scale industrial system.
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As shown in Fig. 4, the
system is a surface-finishing-centered network involving three sectors: a chemical supply sector consisting of two chemical solvent plants, a surface finishing sector containing two electroplating plants, and an automotive manufacturing sector with two OEM plants.
The
parameter values for the system inputs, outputs, and internal material flow streams for sub-matrices R21, R22, and R32 in Eqs. 1 through 5 are listed in Table 1, where each stream’s unit price information is also shown. Using the e-EIOA modeling method, four alternative scenarios for sustainability performance improvement were identified. While their method is systematic and sound, data and information uncertainty issues were not considered.
In this section, we
study the same industrial system, but take various uncertainties into account in the strategy development process. For the system shown in Fig. 4, Piluso et al. (2008) proposed four candidate scenarios for sustainability performance enhancement. These are: Scenario 1: Improve material recycling for Chemical Supplier #1 ( H 3 ).
The process
stream involved is f 3,Zn3 . Scenario 2: Improve material recycling for Chemical Supplier #2 ( H 4 ).
The process
stream involved is f 4,Zn4 . Scenario 3: Improve material recycling from Automotive OEM #1 ( H 5 ) to Chemical Supplier #1 ( H 3 ) as well as Chemical Supplier #2 ( H 4 ). The recycle streams are f 3,Zn5 and
f 4,Zn5 . Scenario 4: Improve process efficiency of Chemical supplier 2 ( H 3 ) to reduce its waste generation that is quantified by stream w3Zn . Sustainability assessment of existing system. 14
A number of sustainability metrics can
be used for studying this problem, such as AIChE Sustainability IndexTM (Sikdar et al., 2011) and IChemE Sustainability Metrics System (IChemE, 2002). For simplicity, in this case study, only one index from each sustainability category is selected and they are listed below. Economic sustainability indicator (E).
This indicator is defined as the total profit,
which is the difference between the revenue gained from the sale of products and the sum of the raw material cost, production cost, and waste treatment cost. E R Revenue from product Raw material cost Operating cost Waste treatment cost
(33)
Environmental sustainability indicator (V). Only the mass intensity of the system is considered, which is the ratio between the materials used for the final products and the raw materials consumed in the plant (Tang et al., 2013; Smith et al., 2013).
Materials for products Raw material consumed
(34)
Social sustainability indicator (L).
The collaboration level among all six plants is taken
V R
as the index for measuring their synergistic efforts. This collaboration level is quantified by the amount of materials recycled and reused in the entire industrial system.
LR Mass recycled and reused
(35)
Using these indices, the sustainability performance of the existing system, S(R(t0)), can be evaluated using Eq. 12, where weighting factors, α, β, and are all set to 1 for convenience. The sustainability assessment results are summarized in Table 2. Budget availability for scenario based system improvement. each of the four scenarios requires an investment.
The implementation of
The cost parameters are designated as
ui t0 , i 1, 2,, 4 , each of which has an upper limit, uimax t0 . The cost required for
15
implementing each scenario is related to the level of material recycling (for the increase of f 3,Zn3 (for Scenario 1), f 4,Zn4 and f 3,Zn5 (for Scenario 2), f 4,Zn5 (for Scenario 3) or the decrease of w3Zn (for Scenario 4)). As an illustration, the following relationships are used in this study.
Note
that in the existing system, material recycles as well as waste reduction actions have already been taken, but not in an optimal way. The following relationships are expressed on an incremental basis.
9uk t0 Zn max Δf i Zn 1 , j t1 log max u t Δf i , j , k 0
i =3 or 4; j =3, 4, or 5; k = 1, 2, or 3
9u t max Δw3Zn t1 log max4 0 1Δw3Zn u4 t0 Assume that Δf 3Zn ,3
max
Δf
Zn max 4 ,5
(36)
(37)
= 4 103 lbs/yr, Δf 4Zn ,4
max
= 0.8 103 lbs/yr, and Δw3Zn ,4 te
max
= 2 103 lbs/yr, Δf 3Zn ,5
= 4.2 103 lbs/yr.
max
= 1.2 103 lbs/yr,
The budget limits for the
implementation of the four scenarios are: u1max t0 = $750 K, u2max t0 = $900 K, u3max t0 = $1,000 K, and u4max t0 = $500 K, which refer to complete material recycle or zero waste discharge. Optimization model.
Based on the given system information, sustainability assessment
results, relationships between system improvement measures and investments, as well as budget constraints and minimum expectations on sustainability improvement, a specific system optimization model, described in Eqs. 21 through 29, can be established and is shown below. The goal of the optimization is to identify the best strategy for budget allocation and to achieve the highest level sustainability. Note that at this stage, no uncertainty is considered. J
Max
ui t0 , i 1, , 4
1 E Rt1 , V Rt1 , LRt1 3 16
(38)
s.t. 9u1 t0 Zn max 9u2 t0 Zn max Zn Δf 3Zn ,3 t1 log u max t 1Δf 3,3 ; Δf 4 ,4 t1 log u max t 1Δf 4 ,4 ; 1 0 2 0 9u3 t0 Zn max 9u3 t0 Zn max Zn Δf 3Zn ,5 t1 log u max t 1Δf 3,5 ; Δf 4 ,5 t1 log u max t 1Δf 4 ,5 ; 3 0 3 0
(39)
9u t max Δw3Zn t1 log max4 0 1Δw3Zn u4 t0 0 u1 t0 4 105 ; 0 u2 t0 3 105 ;
(40)
0 u3 t0 2 105 ; 0 u4 t0 3 105 4
u t 510
(41)
E Rt1 0.8772; V Rt1 0.7691; LRt1 0.5921
(42)
5
i 1
1
0
E1 te 0.8351; V1 te 0.9300; L1 te 0.5000;
E2 te 0.7842; V2 te 0.8800; L2 te 0.5000; E3 te 0.8572; V3 te 0.8500; L3 te 0.5051;
E4 te 0.8731; V4 te 0.8300; L4 te 0.6682;
(43)
E5 te 0.8473; V5 te 0.9000; L5 te 0.6851; E6 te 0.6561; V6 te 0.9200; L6 te 0.8820
The above optimization problem is solved using a Genetic Algorithm (GA) technique. The solution derivation process involves a total of 100 generations, each of which includes 100 populations. Finally, a set of 10 local optimal cases (i.e., N GA 10 in Eq. 30) are obtained, which consist of 10 sets of alternative budget distribution strategies, listed in Table 3, where the evaluated system sustainability performance for each set of budget distribution strategies is also included. Uncertainty incorporation via Monte Carlo based simulation.
The 10 sets of
alternative budget distribution strategies developed should be further investigated by
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incorporating uncertainties that appear in system parameters and performing a Monte Carlo based simulation.
The parameters experiencing uncertainties are listed in Table 4.
In the Monte Carlo based simulation, a set of parameter values are randomly generated from their variation range. These values are used to evaluate the sustainability performance for each of the 10 sets of investment strategies. For instance, one set of parameter values generated randomly from the domains are: z1Zn = 0.58 $/lb, z2Zn = 0.54 $/lb, p5Zn = 6.06 $/lb, p6Zn = 2.99 $/lb, u1max t0 = $795 K, u2max t0 = $928 K, u3max t0 = $984 K, and u4max t0 = $473 K.
The
evaluated sustainability performance for each of the 10 sets of strategies is listed in Table 5. Note that the Monte Carle simulation involves 1,000 random samples.
These results,
similar to the case in Table 5, should be averaged (see the averaged sustainability values, S * Rt1 , in the 6th column of Table 6).
sets of alternative strategies.
Applying Eqs. 31 and 32, we are able to rank the 10
In this case, the coefficients in Eq. 31, i.e., ς , , and , are set
to 10, 6, and 2, respectively. Column 7 in Table 6 lists the ranking results. As shown, the alternative strategy set, No. 5, is the best, which receives the highest score (9,004); the optimal budget distribution of the total funds of $500,000 for u1* t0 , u*2 t0 , u*3 t0 , and u*4 t0 are $190K, $91K, $122K, and $97, respectively.
The categorized and the overall sustainability
values, E * Rt1 , V * Rt1 , L* Rt1 , and S * Rt1 are 0.9231, 0.8011, 0.7670, and 0.8331, respectively.
By comparing the sustainability performance of the existing system, the economic,
environmental, social and overall sustainability improvement after implementing technologies are 5.2%, 4.2%, 29.5%, and 10.3%, respectively.
This comparison is clearly depicted in Fig. 5.
Prioritized sustainability improvement. In principle, a balanced sustainability development based on the three-pillars is always preferred, however, there could be cases where an organization, during a certain period of time, emphasizes one categorized sustainability (e.g., 18
environmental sustainability) more than another categorized sustainability.
The model in Eq.
12 provides such an opportunity, as decision makers can assign different values to the weighting factors, α, β, and . As an example, if we let α, β, and be 5, 2, and 1 respectively, then we can re-evaluate the 10 alternative budget distribution strategies.
In this scenario, it is found that the
best strategy for the distribution of $50K is as follows: u1* t0 , u*2 t0 , u*3 t0 , and u*4 t0 are 255, 68, 51, and 91, and the overall sustainability, S * Rt1 , is increased to 0.8760. Table 7 provides a comparison between this unbalanced development strategy and the balanced strategy. 5
Concluding Remarks
The improvement of the sustainability performance of industrial systems is always very challenging, especially when systems are complex and the available data and accessible information are uncertain. The introduced Monte Carlo simulation and system optimization methodology is effective in developing sustainability improvement strategies where uncertainties are systematically taken into account.
This methodology is particularly useful for evaluating
candidate solutions and identifying the best solution, as well as determining the optimal investment strategy for achieving the highest level of sustainability improvement under pre-specified system and budget constraints and performance improvement requirements.
Acknowledgment
This work was supported in part by National Science Foundation Grants (No. 1140000, 1322172, and 1434277).
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Table 1. Stream parameter values of cost data of the system before modification Variable z1Zn
Flow ( 103 lbs/yr) 50.00
Unit value ($/lb) 0.58
z2Zn
70.00
0.55
f 3,Zn1
46.50
0.89
f 3,Zn2
27.72
0.88
f 3,Zn3
4.04
0.40
f 3,Zn5
2.61
0.35
f
Zn 4, 2
33.88
0.88
f
Zn 4, 4
4.03
0.45
f
Zn 4,5
1.74
0.37
f 4,Zn6
0.60
0.42
f 5,Zn3
68.75
2.93
f 5,Zn4
18.37
2.51
f 6,Zn4
15.03
2.51
p5Zn
78.41
5.93
p6Zn
13.83
2.93
w1Zn
3.50
0.25
Zn 2 Zn 3
w
8.40
0.27
w
8.09
0.29
w4Zn
2.82
0.29
w5Zn
4.36
0.35
Zn 6
0.60
0.35
w
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Table 2. System sustainability assessment result before implementing improvement strategy Interested System Z H1 H2 H3
E Rt0
V Rt0
LRt0
S Rt0
0.8772 0.8351 0.7842 0.8572
0.7690 0.9300 0.8800 0.8500
0.5921 0 0 0.5051
0.7553 0.7216 0.6805 0.7555
H4 H5
0.8731 0.8473
0.8300 0.9000
0.6682 0.6851
0.7953 0.8159
H6
0.6561
0.9200
0.8820
0.8276
24
Table 3. Budget distribution strategies with sustainability evaluations for ten alternative strategies Strategy No. 1 2 3 4 5 6 7 8 9 10
Budget distribution (×103 $) u1* t0 u*2 t0 u*3 t0 u*4 t0 161 125 89 125 178 79 86 156 178 85 120 118 209 132 93 67 190 91 122 97 173 106 172 48 208 143 83 65 127 129 140 104 191 108 126 75 171 133 116 80
25
Sustainability evaluation result E Rt1 V Rt1 LRt1 S Rt1 0.9269 0.8046 0.7526 0.8312 0.9292 0.8062 0.7500 0.8318 0.9272 0.8046 0.7556 0.8323 0.9236 0.8019 0.7650 0.8329 0.9261 0.8037 0.7597 0.8328 0.9205 0.7996 0.7618 0.8300 0.9233 0.8017 0.7646 0.8326 0.9241 0.8027 0.7493 0.8286 0.9242 0.8023 0.7626 0.8325 0.9239 0.8023 0.7595 0.8315
Table 4. List of uncertain parameters Parameter z1Zn
Nominal value 0.58 $/lb
Variation range (0.56 - 0.60) $/lb
z2Zn
0.55 $/lb
(0.53 - 0.57) $/lb
P5Zn
5.93 $/lb
(5.75 - 6.11) $/lb
P6Zn
2.93 $/lb
(2.84 - 3.02) $/lb
u1max t0 u2max t0 u3max t0 u4max t0
5
4×10 $
(3.5 – 4.5) ×105 $
3×105 $
(2.5 – 3.5) ×105 $
2×105 $
(1.5 – 2.5)×105 $
3×105 $
(2.5 – 3.5) ×105 $
26
Table 5. Sustainability assessment and investment strategy ranking after one Monte Carlo simulation Strategy No. 1 2 3 4 5 6 7 8 9 10
Budget distribution (×103 $) u1* t0 u*2 t0 u*3 t0 u*4 t0 161 125 89 125 178 79 86 156 178 85 120 118 209 132 93 67 190 91 122 97 173 106 172 48 208 143 83 65 127 129 140 104 191 108 126 75 171 133 116 80
27
S Rt1
Rank
0.8337 0.8344 0.8350 0.8351 0.8355 0.8325 0.8348 0.8312 0.8350 0.8339
8 6 4 2 1 9 5 10 3 7
Table 6. Sustainability assessment and investment strategy ranking after 1,000 Monte Carlo simulation runs. Strategy No. 1 2 3 4 5 6 7 8 9 10
Budget distribution (×103 $) u1* t0 u*2 t0 u*3 t0 u*4 t0 161 125 89 125 178 79 86 156 178 85 120 118 209 132 93 67 190 91 122 97 173 106 172 48 208 143 83 65 127 129 140 104 191 108 126 75 171 133 116 80
28
S * Rt1
Score
0.8315 0.8321 0.8326 0.8330 0.8331 0.8302 0.8327 0.8290 0.8327 0.8317
0 0 88 6992 9004 0 944 0 972 0
Table 7. Comparison of the best sustainability improvement strategies with different emphases on categorized sustainability Coeff’s in Eq. 12 (α, β, ) (1, 1, 1) (5, 2, 1)
Budget distribution (×103 $)
Sustainability after improvement
u t0
u t0
u t0
u t0
E * Rt1
V * Rt1
L* Rt1
S * Rt1
190 255
91 68
122 51
97 91
0.9231 0.9250
0.8011 0.8032
0.7670 0.7561
0.8331 0.8760
* 1
* 2
* 3
* 4
29
zi
Pi
wi
Hi fi , j
f k,i (a) From
z1
...
z2
z1 z2
zn
0
M
... H n
H1 H 2
0
p1
w1
p2
w2
...
0
zn
To
H1
z1
0
...
0
f1,1
H2
0
z2
...
0
f 2,1 f 2,2
M
M
M
O ...
M
M
M
zn
f n ,1
f n ,2
p1
p1
0
w1
w1
p2
0
0 ... p2 ... w2 ...
Hn
0
w2
0
0
0
f1,2
M
M
M
pn
0 0
0
wn
...
f1,n
... f 2 ,n
0
O M ... f n ,n
0
...
0 0
0
0
M
O
pn
... 0 ...
wn
(b) Fig. 1.
Structure of an e-EIOA model: (a) a basic element in a system, and (b) a system model that consists of n sub-systems.
30
pn
wn
(1,1,1)
Social (L)
1
0.5 S(t) S
L(t) V(t) (0,0,0)
E(t)
Fig. 2. Sustainability status representation scheme: the Sustainability Cube.
31
Input system information, budget data, uncertainty data, sustainability goal, etc. Generate an e-EOIA model for the system under study
Generate a parameterized system optimization model Solve the optimization model to identify alternative strategies
Conduct Monte Carlo simulation by incorporating uncertain information Generate sustainability assessment result for each strategy No All scenarios evaluated?
Yes Rank sustainability improvement strategies and identify the best one Output the best strategies with complete sustainability and cost information
Fig. 3. Flowchart of the procedure for uncertainty incorporation and optimal strategy development.
32
Suppliers (Chemicals) z 1Zn
H1 (Chemical Supplier # 1) f Zn 3,1
Tier I Manufacturing (Metal Plating) H3 (Plating Shop # 1) f Zn 3,3
f Zn 3,2 z Zn 2
OEM (Automotive Assembly) H5 (Automotive OEM # 1)
Zn f 5,3
Zn f 4,2
Product
Zn f 3,5
Zn f 5,4
H4 (Plating Shop # 2)
H2 (Chemical Supplier # 2)
pZn 5
H6 (Automotive OEM # 2)
Zn f 4,4 Zn f 4,6 Zn f 4,5
p Zn 6
Zn f 6,4 w6Zn w5Zn w4Zn w3Zn w2Zn w1Zn
Fig. 4. Sketch of a multi-sector involved industrial system.
33
Waste
(1,1,1) 1
Social
0.8
S(R(t1 )
0.6 S(R(t0) 0.4 0.2
(0,0,0)
Fig. 5. Sustainability performance comparison of the system before and after optimal strategy implementation.
34