Hierarchical Pareto Optimization for the ... - ACS Publications

Ind. Eng. Chem. Res. 2006, 45, 3265-3279

3265

GENERAL RESEARCH Hierarchical Pareto Optimization for the Sustainable Development of Industrial Ecosystems Aditi Singh and Helen H. Lou* Department of Chemical Engineering, Lamar UniVersity, Beaumont, Texas 77710

Establishing industrial ecosystems (IEs) is a viable approach for the sustainable development of industries. The ultimate objective of an IE is to minimize the material and energy consumption and reduce or even eliminate waste disposal through various recycle and reuse mechanisms. Yet, a rigorous scientific approach for the analysis and design of sustainable IE is lacking. The major obstacles include the challenge of evaluating each dimension of the “triple bottom line” of sustainability, handling the complex interdependency among different members, and the need of a holistic decision-making/decision-analysis method across different scales, from the individual production unit up to the ecosystem. To tackle this difficulty, the improvement of economic and environmental sustainability of IEs is defined as a multilayer multi-objective optimization problem in this paper. A novel Hierarchical Pareto Optimization Methodology is introduced to achieve the most sustainable solution. This methodology provides a systematic and flexible framework to solve multiscale, multidimensional problems, and it provides clear guidance for improving sustainability. The effectiveness of this approach is demonstrated by a case study. Introduction Sustainability is a vital issue for the long-term development of human society and the ecosystem. The principal aim of sustainable development is to achieve economic prosperity, environmental cleanness, and societal well-being simultaneously.1,2 To satisfy this “triple bottom line”, sustainable development becomes multidimensional in nature. The establishment of industrial ecosystems (IEs) is a viable approach for the sustainable development of industries.3-5 An IE is a symbiosis of many industries, where the product/ byproduct/waste generated by one industry can be used as a resource by another industry. These mass/energy exchange activities not only help in improving the economic status of the industry, by generating additional revenue from the waste and byproducts, but also reduce environmental stress, by reducing the consumption of fresh resource and minimizing waste disposal. The intensive mass and energy exchange activities in an IE result in strong interdependence among the member entities. Therefore, the economic as well as environmental performance of each entity in an IE becomes strongly dependent on the other members of the IE. A small change in the production of one member may have large impact on the performance of another member and, eventually, the sustainability of the entire IE through “chain reactions”. Hence, to establish an efficient IE, the complex economic, environmental, and societal issues must be considered simultaneously from the process/plant level up to the ecosystem level. However, so far, a scientific approach for the analysis and design of sustainable IE is lacking. The major obstacles include * To whom correspondence should be addressed. Tel.: 409-8808207. Fax: 409-880-2197. E-mail: [email protected].

the challenge of evaluating each dimension of the “triple bottom line” of sustainability, handling the complex interdependency among different members and various uncertainties, and the need of a holistic decision-making method across different scales, from the individual production unit up to the ecosystem.6 Considering these difficulties, in this paper, a hierarchical Pareto optimization methodology has been developed to identify the optimal configuration of an IE for sustainability. Hierarchical Pareto optimization methodology decomposes a complicated multi-objective optimization problem into multiple levels. The upper level constitutes the overall objectives for the entire system, and the lower level resembles individual subsystems that are interconnected with interaction variables. A recursive Pareto optimization algorithm is deployed to identify the optimum solution for the overall system. This methodology provides a systematic and flexible framework to solve multiscale, multidimensional problems. The effectiveness of this approach is demonstrated by a case study. Because of the difficulty in encompassing societal performance, in the current case study, two dimensions of sustainability are included, i.e., to maximize the economic benefits and minimize the environmental pressure. Hierarchical Pareto Optimization Methodology It is extremely difficult to analyze and optimize large-scale complex systems that consist of different members with multiple objectives. The computational difficulty that occurs is due to the large number of variables in the problem, high nonlinearity in coupling, and the interactions among variables. An effective and efficient decision-making method is needed to solve these problems. Pareto Optimization Methodology. The multi-objective decision-making/decision-analysis methodologies can be clas-

10.1021/ie050487q CCC: $33.50 © 2006 American Chemical Society Published on Web 03/17/2006

3266

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006

Figure 2. Pareto optimum (objective space).12

Figure 1. Illustration of Pareto optimum points (design space).13

sified into four different categories: no preference methods, a priori methods, interactive methods, and a posteriori methods. The Pareto optimization methodology belongs to the fourth category, i.e., a posteriori methods, in which the user can specify his/her preference after generating a set of solutions for the multi-objective optimization problem.7-11 Pareto optimization is used to calculate the equilibrium solution in a cooperative scenario. A set of solutions is Pareto optimum if, by moving from that solution to another in the feasible solution space, any improvement in the value of one of the objective functions results in the deterioration of at least one of the remaining objective functions.12 Mathematically, if the goal is to minimize the objectives, the set of Pareto optimal solutions (P) is such that

P ) {x* ∈ X: ∀x ∈ X, where Fi(x) g Fi(x*) ∀i, ∨Fi(x) > Fi(x*) for at least one i }12 (1) where F is the vector of objective functions, i the index of different objective functions, X the feasible region, x the vector of decision variables (which represents a point in the feasible region),5 and x* the Pareto optimal solution. Consider a bi-objective maximization problem with f1 and f2 as their objective functions with two decision variables x1 and x2. These two objective functions are interdependent. If each objective function is optimized independently, the result will be its global maxima, namely, the utopia point, irrespective of the other objective function. The utopia points for f1 and f2 are A and B, respectively, as shown in Figure 1. The curves encircling these utopia points are each objective function’s indifference curves, which are also known as isovalue curves. For the maximization problem, the value of each function increases as it approaches the utopia point from the periphery. Hence, each objective function prefers the points on the inside of the indifference curve to the points outside the curve, because they are closer to its utopia point.13 Consider the point X in Figure 1: this point does not qualify to be the Pareto optima, because both objectives can improve their values by transcending to their inner indifference curves without degrading the value of the other objective function. In contrast, both objective functions prefer point Y, because the value of f2 can be improved further by transcending from point X to point Y without reducing the value of f1. Similarly, the value of f1 can be

improved by transcending from point X to point Z, while the value of f2 remains the same, because it remains on the same isovalue curve. Thus, both point Z and Y are more desirable, compared to point X. If we transcend away from these points, any further improvement in the value of one objective function happens only by deteriorating the value of the other objective function. Therefore, the Pareto optimal solution exists either at the point where the indifference curves are tangent to each other (points Z and Y) or at the boundary of the feasible region. Common tangency means that the marginal rate of change of the objective functions, with respect to each decision variable, is the same for both objective functions at this point. However, not every point of tangency of the indifference curves is Pareto optimal. Consider point P in Figure 1: at this point, the indifference curves are tangent to each other but the value of both functions can be improved further by moving to an inner point. Hence, this point does not satisfy the basic definition of Pareto optimality. At point Y, a higher value of x1 and a lower value of x2 are required to improve the value of f1, because these conditions lead to values close to its utopia point. Similarly, at point Y, a higher value of x2 and a lower value of x1 is required to improve the value of f2, because because these conditions lead to points closer to its utopia point. Conversely, at point P, both functions prefer to move toward a higher value of x2 and a lower value of x1 to increase their value. Therefore, the necessary condition for Pareto optimality is that the marginal rate of substitution, with respect to various decision variables, should be equal and in opposite directions. This constitutes “good tangency”, similar to the tangency at points Z and Y. Note that one problem may have multiple Pareto solutions. All the Pareto optimal solutions in the entire objective space together form the Pareto frontier,13 as shown in Figure 2. Depending on the preference of the decisionmaker, a specific Pareto optimum solution can be selected from the Pareto frontier. Many algorithms are available to generate the Pareto frontier for multi-objective optimization problems. Some commonly used methods include the Constraint Proposal Method, Normal Constraint Method, and Linear Weight Method.7,14,15 Currently, genetic and evolutionary algorithms are being applied to solve complex multi-objective problems, because of their capability of being able to find solutions in a complex solution space quickly. They provide a framework for effectively sampling large search spaces, using an appropriate encoding scheme and target function. In this paper, the Linear Weight Method is used to identify Pareto optimal solutions for the case study.

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3267

In the case where the maxima or minima solution does not exist or is too expensive to calculate, it is required to choose some approximate solutions for normalization. Consider the following optimization problem:

upper level to arrive at an optimal solution for the entire system. To achieve subsystem independence, some of the necessary conditions for optimality can be relaxed for generating optimal solutions at the lower level for individual subsystems and then applied again at the upper level. The advantage of hierarchical optimization method is that it conceptually simplifies complex systems by reducing the dimensionality and modularizes each individual system. Hence, a flexible and efficient mathematical framework can be developed. Depending on the nature of the optimization problem at each level, different optimization algorithms can be applied at different levels of hierarchy to achieve the optimal solution. Hierarchical Pareto Optimization Methodology. Consider a multi-objective optimization problem for a large-scale complex system with m subsystems:

min F(F1,F2, ..., Fn)

min F(F1,F2,...,Fj,..Fn)

The Linear Weight Method replaces the multicriteria problem with a parametrized scalar problem. Each parameter combination corresponds to one Pareto optimal solution. By varying the values of these parameters, it is possible to generate a part or all of the Pareto frontier. While using the Linear Weight Method, it is necessary to normalize the objective functions a priori using the following formula:

F h (x) )

F(x) - Fmin Fmax - Fmin

(2)

x

u

w.r.t.

w.r.t.

G(X) e 0

H(u,D,y) ) 0

H(X) ) 0

G(u,D,y) e 0 (3)

u l e u e uu

In the Linear Weight Method, each objective function (Fi) is assigned a weight factor (wi), such that

Dl e D e Du

Xl e X e X u

n

wi ) 1 ∑ i)1

(4)

A scalar optimization problem then is defined as n

min

wiFi(X) ∑ i)1

w.r.t.

G(X) e 0 H(X) ) 0 Xl e X e Xu

(5)

By varying the weights wi and solving this scalar problem, Pareto optimal solutions are generated for the multi-objective optimization problems. It is guaranteed to generate the entire Pareto solution for convex problems.14 In the case of nonconvex problems, a piece-wise analysis of the objective function might be required. Hierarchical Optimization Hierarchical optimization is a proven competent optimization technique that is based on the decomposition of large-scale complex systems and subsequent modeling of the systems into “independent subsystems”.16-21 This provides an allowance to study the behavior of the subsystems at a lower level and to transmit the information obtained to fewer subsystems at a higher level. Each subsystem can be optimized separately and independently, using the appropriate optimization technique, depending on the nature of the problem, to identify a lowerlevel solution. These subsystems are joined together by coupling variables or interaction variables, which are manipulated at the

yl e y e yu

(6)

where u is the vector of the manipulated variable, D the disturbance variable, and y the vector of the output variable. The overall system and each subsystem i has n objectives. The subscripts “u” and “l” respectively denote the upper bound and lower bound of the variables. The performance of the overall system is determined by the actions of each individual subsystem. Each individual subsystem concerns its own objectives, whereas the performance of its own objectives is influenced by the system’s outcome as well. Thus, an efficient algorithm should be utilized to communicate the decisionmaking at different levels toward various objectives. As stated previously, the hierarchical approach has proven power in tackling complex, multiscale problems, and Pareto optimization is a popular decision-making/decision-analysis method for multi-objective problems. In this regard, the hierarchical optimization method can be coupled with a Pareto optimization technique to identify the Pareto optimal solution of large-scale systems with multiple objectives at different hierarchies via recursive calculation. This new methodology is called Hierarchical Pareto Optimization Methodology. In the Hierarchical Pareto Optimization approach, first, the Pareto frontier is generated for all the subsystems concerning their own objectives. After the Pareto optimum solutions are selected from the Pareto frontier for all the subsystems at a lower level, based on some criteria, the optimal value of their local variables is passed to the upper level as parameters. After that, the Pareto frontier is generated for the overall system at the upper level with these new parameters by varying the interaction variables. The optimal value of these interaction variables then is passed back to the lower level as parameters and a Pareto optimum solution is generated for each subsystem with these new parameters. This iterative process continues until the optimum solution for the overall system is converged. The

3268


Figure 3. Scheme for hierarchical Pareto optimization.

solution strategy of hierarchical Pareto optimization methodology is illustrated in Figure 3. Depending on the complexity of the system, multiple levels can be adopted. In the following application, a bilevel approach is utilized to maximize the sustainability of an IE through the coordination of different plants. Mathematically, hierarchical Pareto optimization can be formulated as follows. At the lower level:

min Fi(Fi1,Fi2,...,Fij,...,Fin,λ) ui

w.r.t.

Hi(ui,Di,yi,λ) ) 0 Gi(ui,Di,yi,λ) e 0 ui,l e ui e ui,u Di,l e Di e Di,u yi,1 e yi e yi,u 1eiem 1ejen

(7)

where i denotes the index of subsystems, while j denotes the index of different objectives. At the upper level:

min F(F1,F2,...,Fn) λ

Using this formulation, a Pareto frontier can be generated for each subsystem, as well as for the overall system. The selection criteria to identify the best solution from a pool of optimal solutions (the Pareto frontier) can be influenced by various environmental, economic, and societal factors. MultiCriteria Analysis (MCA) helps in comparing as well as ranking different outcomes.22,23 MCA is a decision-making tool, which allows decision makers to consider a full range of related factors.24,25 It is a structured approach used to choose the best solution from a pool of alternatives available. In this technique, specific indicators will be defined and associated with different objectives. In many existing approaches, these indicators are the weights associated with each objective and these weights are defined subjectively. In this proposed approach, after the Pareto frontier has been generated for a given multi-objective optimization problem, a selection criteria must be defined to select the most suitable solution from a pool of solutions. To define a homogeneous selection criterion, first, the objective functions are normalized, so that all the functions are evaluated on the same scale. If the multiple objectives involve both maximization and minimization functions, then let F′(X) denote the normalized form of the objectives that needs to be maximized and G′(X) denote the normalized form of the objectives that needs to be minimized. These functions are normalized using eq 2. This is to make sure that the changes of different objectives are evaluated fairly. To select the best solution from a set of w Pareto optimal solutions residing on the Pareto frontier, calculate the value of Ck at each Pareto optimal point as

w.r.t.

p

Ck )

H(u,D,y,λ) ) 0

q

F′i(X)

∏ ∏ i)1 j)1 G′(X) j

G(u,D,y,λ) e 0 where

y l e y e yu λl e λ e λu where λ is the vector of interaction variable.

(8)

1ekew and


p+q)n

(9)

where p is the total number of objective functions that must be maximized, q the total number of objective functions that must be minimized, n the total number of objective functions, and w the total number of Pareto optimal solutions available in the Pareto frontier. The intention of this criteria is to determine an optimal solution where F′(X) attains its maximum possible value and G′(X) attains its minimum possible value simultaneously. Finally, to get the best solution for the system, select the solution set with the highest value of Ck, i.e.,

max {C1,C2,...,Cw}

(10)

This criterion allows us to quantitatively identify the best solution from a pool of solutions. It needs to be pointed out that this criterion is developed specifically for the sustainable development of industrial systems, i.e., the simultaneous improvement of economic as well as environmental performance to the maximum extent. The decision-makers may define their own criterion in order to suit the specific requirements of the system under consideration. Emergy-Based Sustainability Indices for Industrial Systems At the ecological level, the contributions of the ecosystem to industrial activities, and the effect of industrial activities on the ecosystem, should both be considered. It will be nice if all these can be quantified on a common ground. System ecologists initiated an environmental accounting method through the Emergy Analysis (EMA) method to assess and analyze the sustainability of ecosystems. In EMA, the values of nonmoneyed and moneyed resources, services, and commodities are quantified based on the evaluation of the energy used for making products or services. This “used” energy is called emergy.26,27 In other words, emergy is the available energy of one type previously required, directly and indirectly, to make a specific resource, product, or service. The unit for emergy is the emjoule (ej). To calculate the emergy of a particular substance or service, transformity of that substance or service will be needed. Transformity is defined as the emergy of one type of available energy required to make 1 J of energy of another type. The unit for transformity is the solar emjoule per joule (Sej), when emergy is calculated in terms of solar energy. In case the transformity is unknown, the emergy/money ratio can be used to calculate the emergy of the substances or the services. Emdollars, i.e., the ratio of emergy to money, is calculated by dividing the total emergy use of a country by its gross economic product.27 Emergy analysis provides a common platform to quantitatively express the economic values, as well as the environmental factors. It facilitates the comparison of the economic and environmental status of different entities on a common ground. Hence, the sustainability performance of the ecosystem can be exploited conveniently using a set of indices.26 However, traditional emergy-based sustainability indices were derived based on the study of agricultural systems or natural ecological systems, and they are not very effective in assessing industrial systems. By addressing the unique features of industrial systems (i.e., waste treatment, recovery, reuse, and

Figure 4. Emergy flow diagram for industrial systems.28

recycle), and considering all the material/energy flows and investments in industrial systems, Lou et al.28 introduced a comprehensive emergy flow diagram (Figure 4) for industrial systems and a set of new emergy-based sustainability indices for industrial systems. Such a scenario, where the waste generated by one plant can be reused internally or can even be used as a resource by another plant, is common in an IE. For example, the wastewater generated by one plant may be sent to another plant to be used for steam generation. Similarly, excessive carbon dioxide that has been generated by one plant as waste can be sent to another nearby plant that uses carbon dioxide as a raw material. As depicted in this diagram, an industrial process may consume various renewable (R) and nonrenewable resources (N) to produce yield (YP). Waste (W) is also generated during production as an undesired byproduct. Some of the waste, which is not hazardous to the environment, can be disposed without treatment (WU), whereas some must be treated before it is discharged into the environment (WT). The WU materials can have one or more of the following fates: i.e., directly discharged into the environment (WUW), recycled as a nonrenewable resource (WURN) or renewable resource (WURR), or sold in the market as yield (YUW). Similarly, WT may also have one or more of the following fates: discharged to the environment after waste treatment (WTW), recycled as a nonrenewable resource (WTRN) or renewable resource (WTRR), or sold in the market as yield (YTW). WTRN and WURN together constitute the total amount of waste recycled as nonrenewable resource (WRN). Similarly, WTRR and WURR together constitute the total amount of waste recycled as renewable resource (WRR). The summation of WUW and WTW gives the total waste discharge (Ww). YUW and YTW together constitute the total amount of waste that can be sold as a useful product (YW). The investment of each industry can be classified as the investment on production (Fp), waste treatment (Fw), waste recycle (FR), and waste disposal (FD). FR consists of the recycling cost of nonrenewable resources from waste (FRN) and the recycling cost of renewable resources from waste (FRR) including the hand-

3270


ling and transportation cost for each of them. FD also consists of two parts: the cost of disposing treated waste (FTD) and the cost of disposing untreated waste (FUD) into the environment. The emergy-based indices28 for the sustainability assessment of industrial systems are briefly reviewed as follows, where all the emergy flows have common units of solar emjoules per year (Sej/yr). 1. Index of Economic Performance. The index of economic performance (IEcP) is the ratio of total yield to the total investment:

YP + YW IEcP ) FP + F W + I

(11)

The numerator is the sum of the emergy of the main product and byproduct (YP) and the yield generated from waste (YW). The denominator is the emergy of the total investment to obtain the required quantity and quality of the product, while satisfying the environmental regulations. It includes the production cost (FP), the waste treatment and disposal cost (FW), and the total resource (I) intake for that process. It will be economically beneficial to have a high IEcP value. A process can be made profitable by maximizing the production while minimizing the total investment. 2. Index of Environmental Performance. The index of environmental performance (IEvP) quantifies the environmental performance of a process or service:

IEvP )

N + WUW + WTW R + WRR + WRN

(12)

The numerator of the IEvP is the sum of the emergy of nonrenewable resources consumed (N) and the waste disposed into the environment with or without treatment (WUW + WTW) during the production process. The denominator of the IEvP is the total emergy of the renewable resources (R) used in the process and the internal recycle streams for renewable and nonrenewable resources generated from treated as well as untreated waste (WRR + WRN). A low value of IEvP is always desirable, because it indicates lesser pressure on the environment. The value of IEvP can be improved by replacing nonrenewable resources with appropriate renewable resources. Implementing internal recycle of waste as renewable or nonrenewable resource can also largely reduce the value of the IEvP. 3. Index of Sustainable Performance. The index of sustainable performance (ISP) is the ratio of IEcP and IEvP:

ISP ) )

IEcP IEvP (YP + YW)/(FP + FW + I) (N + WUW + WTW)/(R + WRR + WRN)

to the increase in IEcP and decrease in IEvP. ISP can have a high value, even if the IEvP remains the same and IEcP increases many-fold or if IEcP remains the same and IEvP decreases many-fold. Therefore, the use of only the ISP to evaluate the environmental performance may not be able to reach a balance between the environmental and economic concerns. Because sustainability must consider the economic performance and environmental stress simultaneously, it is best to use both the IEcP as well as the IEvP explicitly to evaluate the overall performance. Hierarchical Pareto Optimization for Sustainable Industrial Ecosystems Based on the emergy-based indices for industrial systems, the following objective function can be used to optimize an IE that is economically as well as environmentally sustainable:

F ) (max IEcP,min IEvP)

(14)

To achieve high sustainability, a high value of Profit (high IEcP) and lesser environmental stress (low IEvP) is desired. As indicated in eq 13, it is clear that, with the maximal IEcP and minimal IEvP, maximal sustainability can be achieved. Because of the multiscale and multidimensional nature of sustainability, hierarchical Pareto optimization methodology should be a valuable approach for optimizing the sustainability of an IE. At the lower level, this methodology is used to identify the individual economic and environmental objectives of each member industry, independent of the objectives of the other members. At the upper level, the optimization algorithm will calculate the overall objectives of the entire IE, i.e., to maximize the overall economic performance and minimize the environmental pressure and eventually maximize the overall sustainability of the entire ecosystem. Mathematically, at the upper level, a bi-objective optimization problem can be formulated for the entire IE as

F ) max (IEcP, -IEvP)

(15)

Consider an IE with m member industries at the lower layer: the objective of each industry can be formulized as

Fi ) max (IEcPi, -IEvPi)

(∀ i, 1 ei em)

(16)

Using the previously discussed formulation, a Pareto frontier can be generated for each plant, as well as the IE between the two objectives. In this work, it was found that the trend followed by the IEcP was similar to that of Profit but has slight variations at a few points. We report the frontier between Profit and IEvP, because it is easier to calculate and use Profit in decisionmaking. Equation 9 is tailored for this case study to calculate the value of Ck for the points on the Pareto frontier.

(13)

It is a measure of the overall sustainability of a process, because it combines the economic as well as the environmental performance of a process. It seeks a balance between the economic and environmental aspects of a process by providing appropriate consideration to all of the inflows and outflows, and various cost factors. A large ISP value indicates high overall sustainability. However, the ISP may not be equally sensitive

Ck )

F′(X) G′(X)

where

F′(X) ) Profit G′(X) ) IEvP p ) 1, q ) 1, n ) 2

(17)


Figure 5. Schematic diagram of an industrial ecosystem (IE) with four plants.

The number of points on the Pareto frontier (w) of each plant may be different, because of some overlaps in the optimal solution points that are generated. After checking the value of Ck for all the Pareto optimal points in the Pareto frontier, the best solution from the Pareto frontier was selected using eq 10. In this paper, the Branch and Reduce Optimization Navigator (BARON) solver in the GAMS software has been used to calculate the global optimal solution. In the BARON program, the global optimization strategy integrates conventional branch and bound procedures with a variety of range reduction tests that are applied to every subproblem of the search tree. This helps in contracting the search space and reduce the relaxation gap. It implements heuristic techniques for generating solutions and is especially useful for highly nonlinear problems.29 This bilevel, bi-objective optimization problem of an IE is solved using the Linear Weight Method in the case study discussed in the next section. Case StudysAn Industrial Ecosystem with Four Members The emergy flow diagram of an IE with four memberssPlant 1, Plant 2, Plant 3, and Plant 4sis depicted in Figure 5. In Plant 1, some of the waste (W1) that is generated during the manufacture of yield (YP1) is treated (WT1), because of environmental concerns. Some of this treated waste can be recycled internally as a nonrenewable resource (WTRN1), and the remainder can be sold to the market (YTW1). Some of the untreated (WU1) waste can be recycled internally as a renewable

Table 1. Price and Cost Details for the Case Study

R N YP YUW YTW FUD FTD

Plant 1 ($/ton)

Plant 2 ($/ton)

Plant 3 ($/ton)

Plant 4 ($/ton)

35 70 105 20 (to P2,R) 55 80 65

30 80 150 30 (to P4,R) 50 75 60

40 90 130 20 (to P1,R) 45 60 50

45 95 170 25 (to P3,R) 55 80 65

resource (WURR1), while the remainder can be sold to Plant 2 to be used as a renewable resource (YUW1). In Plant 2, the yield generated from the product (YP2) is sold in the market, along with the yield generated from treated waste (YTW2). The remaining amount of treated waste can be recycled internally as a nonrenewable resource (WTRN2). Some of the untreated waste is sold to Plant 4 (YUW2), to be used as a renewable resource, and the remainder is recycled internally as a renewable resource (WURR2). Plant 3 recycles a fraction of its treated waste internally to substitute fresh nonrenewable resource (WTRN3) and sells the remaining in the market as yield (YTW3). Some of its untreated waste is recycled internally as a renewable resource (WURR3) and remaining is sold to Plant 1 as renewable resource (YUW3). In Plant 4, some of the treated waste is recycled internally as a nonrenewable resource (WTRN4) and the remainder is sold in the market as yield (YTW4). Some of the untreated waste is sold to Plant 3 as a renewable resource (YUW4), and the remainder is recycled internally as a renewable resource (WURR4). A mathematical model is developed to depict the production and waste transfer. In this model, at every node

3272


Figure 6. Pareto frontier for Plant 1 after the first iteration.


where a stream divides into two or more streams, a coefficient is defined to indicate the fraction of primary stream going into secondary streams. Complete details of the model for this IE are given in the Appendix. The details of the price and cost information for this IE are given in Table 1; the cost of handling and transporting the marketable waste (YUW or YTW) is included in those prices. In this case study, most of the material recycled internally between the members of the IE is a renewable resource extracted from the untreated waste. This is frequently encountered in practice. The sustainability of this IE is optimized using the hierarchical Pareto optimization methodology. The lower level is comprised of four subsystems: Plant 1, Plant 2, Plant 3, and Plant 4. Each plant’s objective is to maximize its IEcP value and minimize its IEvP value, to maximize its economic performance while causing minimal or no harm to the environment (hence, maximize its overall sustainability).

In this model, the interaction variables are the fraction of mass/energy streams going from one plant to another, the fraction of waste discharged into the environment that are influenced by the fraction of reuse stream, and the amount of total resource consumption by these plants (for Plant 1: a31, a32, a33, X1; for Plant 2: b21, b22, b23, X2; for Plant 3: c21, c22, c23, X3; and Plant 4: d21, d22, d23, X4). On the other hand, the local variables are the variables that have a critical role in determining the economic and environmental performance of each plant but do not have any direct impact on any other member of the ecosystem (for example, the fraction of internal recycle, product sale to the market, and waste discharge that will not be affected by the mass and energy reuse between plants (for Plant 1: a21, a22, a23; for Plant 2: b31, b32, b33; for Plant 3: c31, c32, c33; and for Plant 4: d31, d32, d33)). This model can also be developed using mass-flow rates instead of split fractions, based on the user’s preference. The results achieved




in both cases are identical. After defining the objective functions at both levels, the Pareto optimal solutions are generated for each subsystem at the lower level using the Linear Weight Method. The Pareto optimum solution for each plant then is selected from the frontier based on the criterion explained in eq 17. For illustration, the Pareto frontier for Plant 1, Plant 2, Plant 3, and Plant 4, after the first iteration, is presented in Figures 6-9, respectively. The selected Pareto optimal point has been circled in each figure. The Pareto optimal solution then is allocated from this frontier. The corresponding value of Profit and all the other indices at each plant’s local Pareto optima has been listed in Tables 2-5 for Plant 1, Plant 2, Plant 3, and Plant 4, respectively. The results demonstrated that the local Pareto optimum attained in the initial iteration for one plant may not be the best strategy for the other plants’ specific individual interests and it may not be the best strategy for the overall IE. For example, as shown in Table 2, at the Pareto optima of Plant 1, the profit of

Table 2. Performance Indices at Local Pareto Optima of Plant 1 after the First Iteration Profit (M$/yr)a IEvP IEcP ISP a

Plant 1

Plant 2

Plant 3

Plant 4

overall IE

19510 0.872 1.935 2.220

27913 2.012 2.572 1.278

18170 2.127 1.694 0.796

31900 1.584 1.818 1.148

97371 1.562 1.989 1.273

Millions of dollars per year.

Plant 1 is close to its maxima (Profit1 ) 19,510 M$/yr, Profit1max ) 20,050 M$/yr) and the IEvP value also is very similar to its minima (IEvP1 ) 0.872, IEVP1min ) 0.850). (The term “M$/ yr” denotes millions of dollars per year.) On the other hand, the Profit values for Plant 2 (Profit2 ) 27,913 M$/yr, Profit2max ) 54,417 M$/yr), Plant 3 (Profit3 ) 18,170 M$/yr, Profit3max ) 47,903 M$/yr), and Plant 4 (Profit4 ) 31,900 M$/yr, Profit4max ) 51,190 M$/yr) are very low, compared to its individual maximum possible profit. As far as the IEvP value is concerned, at the Pareto optima of Plant 1, the IEvP value for Plant 2 (IEvP2

3274


Table 3. Performance Indices at Local Pareto Optima of Plant 2 after the First Iteration Profit (M$/yr)a IEvP IEcP ISP a

Plant 1

Plant 2

Plant 3

Plant 4

overall IE

3317 2.359 1.266 0.537

53719 1.554 1.62 1.042

35710 1.742 1.687 0.968

28940 1.989 1.471 0.740

121684 1.821 1.888 1.037

Table 4. Performance Indices at Local Pareto Optima of Plant 3 after the First Iteration Plant 1 Profit IEvP IEcP ISP a

1363.45 3.957 1.165 0.294

Plant 2 9699 2.796 1.668 0.597

Plant 3 44920 1.23 2 1.626

Plant 4 21470 2.385 1.316 0.552

77372 1.956 1.583 0.809

a

Plant 1

Plant 2

Plant 3

Plant 4

overall IE

28.64 2.262 1.0036 0.444

9701 2.795 1.668 0.597

34960 2.059 1.654 0.803

48590 1.18 2.089 1.770

93138 1.717 1.808 1.053


) 2.012, IEvP2min ) 1.532), Plant 3 (IEvP3 ) 2.127, IEvP3min ) 1.226), and Plant 4 (IEvP4 ) 1.584, IEvP4min ) 1.173) is fairly high, compared to their individual minima. Moreover, at this point, the Profit value for the overall IE is much lower than its maximum possible profit (Profitie ) 97,371, ProfitIEmax ) 172,259). The IEvP value that IE achieves at this point is a moderately higher than the minimum possible IEvP value (IEvPIE ) 1.562, versus IEvPIEmin ) 1.315). Hence, it is evident that the Pareto optimal solution for one plant regarding its maximum sustainability may not give the best solution for the profit or environmental cleanliness of the other members. Also, the Pareto solution for one plant may not give the maximum Profit or minimum IEvP value for the entire IE. This statement also holds true for the local Pareto optima for Plant 2, Plant 3, and Plant 4. After the local Pareto optima for each plant has been identified, the Pareto optimal values of the local variables for each subsystem are passed as parameters to the upper level. At the upper level, the Pareto optimal solution is generated for the entire IE, using the local variables generated at the lower level as parameters while adjusting the interaction variables. The values of interaction variables then are sent to the lower level as parameters and Pareto solutions are generated for each subsystem again. The Pareto optimal solution at the upper level converges after a few iterations. The number of Pareto optimum solutions that are generated to make the Pareto frontier in the later iterations may be limited to one or two points. Because of this observation, it may not be possible to generate a Pareto frontier at a latter stage. The final values of Profit and IEvP for Plant 1, Plant 2, Plant 3, and Plant 4, at their respective local optima, are listed in Tables 6-9, respectively. In contrast with the values for local Pareto optima that each plant attained in the first iteration (see Tables 2-5), the Profit values for all the plants improves along the iterations while the IEvP value deteriorates for some of the

Plant 2

Plant 3

Plant 4

overall IE

20050 0.919 1.986 2.162

45796 2.416 2.127 0.881

41350 1.960 1.865 0.952

49750 1.253 2.140 1.708

156735 1.632 2.092 1.282

Table 7. Performance Indices at Final Local Pareto Optima of Plant 2

Profit IEvP IEcP ISP a

Table 5. Performance Indices at Local Pareto Optima of Plant 4 after the First Iteration

Plant 1


overall IE


Profit (M$/yr)a IEvP IEcP ISP

Profit (M$/yr)a IEvP IEcP ISP a


(M$/yr)a

Table 6. Performance Indices at Final Local Pareto Optima of Plant 1

(M$/yr)a

Plant 1

Plant 2

Plant 3

Plant 4

overall IE

18600 1.266 1.787 1.412

54417 1.620 2.415 1.491

46310 1.687 2.057 1.219

47520 1.471 2.004 1.362

166658 1.527 2.207 1.445


Table 8. Performance Indices at Final Local Pareto Optima of Plant 3 Profit (M$/yr)a IEvP IEcP ISP a

Plant 1

Plant 2

Plant 3

Plant 4

overall

20050 0.919 1.986 2.162

45796 2.416 2.127 0.881

47900 1.517 2.162 1.425

49750 1.253 2.140 1.708

163302 1.507 2.192 1.455


Table 9. Performance Indices at Final Local Pareto Optima of Plant 4 Profit (M$/yr)a IEvP IEcP ISP a

Plant 1

Plant 2

Plant 3

Plant 4

overall

20050 0.919 1.986 2.162

45796 2.416 2.127 0.881

41340 1.960 1.865 0.952

49750 1.253 2.140 1.708

156735 1.632 2.092 1.282


Table 10. Values of Local Variables at the Industrial Ecosystem’s Pareto Optima Plant 1

Plant 2

Plant 3

Plant 4

variable

value

variable

value

variable

value

variable

value

a21 a22 a23

1 0 0

b31 b32 b33

1 0 0

c31 c32 c33

1 0 0

d31 d32 d33

1 0 0

plants. For example, at the final local Pareto optima of Plant 1, the Profit value for Plant 1 increases from 19,510 M$/yr after the first iteration, to 20,050 M$/yr, but the IEvP1 value increases from 0.872 to 0.919. On the other hand, at the same solution point, the Profit value for Plant 3 increases from 18,170 M$/yr to 41,350 M$/yr and, simultaneously, the IEvP3 value decreases from 2.127 to 1.960. The Profit of IE increases from 97,371 M$/yr to 156,735 M$/yr and the IEvP value for IE increases from 1.562 to 1.632. The reason for this is that, to take care of each other’s interests, sometimes, the plants must compromise some of their own benefits in a cooperative scenario, which may help improve the benefits for the entire system. In the next step, the value of all the performance indices is calculated at the upper level to identify the global optimal solution for the overall IE using the final optimal values of local as well as global variables. The optimum values of the local variables and the interaction variables at the system’s Pareto point are listed in Tables 10 and 11, respectively. As can be seen from Table 12, when compared to the previous iterations, a better value of Profit (ProfitIE ) 171,924 M$/yr, ProfitIEmax

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3275 Table 11. Values of Interaction Variables at the Industrial Ecosystem’s Pareto Optima Plant 1

Plant 2

Plant 3

Plant 4

variable

value

variable

value

variable

value

variable

value

X1 a31 a32 a33

500 0 1 0

X2 b21 b22 b23

700 0 1 0

X3 c21 c22 c23

800 0.5 0.5 0

X4 d21 d22 d23

750 0 1 0

Table 12. Performance Indices at the Final Global Pareto Optima of the Industrial Ecosystem Profit (M$/yr)a IEvP IEcP ISP a

Plant 1

Plant 2

Plant 3

Plant 4

overall IE

20051 0.919 1.987 2.162

54221 1.621 2.690 1.659

47902 1.517 2.162 1.425

49748 1.253 2.145 1.712

171924 1.345 2.339 1.739


) 172,259 M$/yr) and IEvP (IEvPIE ) 1.345, IEvPIEmin ) 1.315) for the entire IE can be achieved at the Pareto optimal solution for the entire IE. At this point, the values of Profit and IEvP are not their best possible values, which is due to the compromise of two objectives. As is obvious from the coefficients listed in Tables 10 and 11, the best strategy for the entire IE, as well as each plant, is to opt for zero discharge. For example, in the case of Plant 1, the best strategy is to internally recycle all the treated waste as a nonrenewable resource (a21 ) 1) and all the untreated waste as a renewable resource (a31 ) 1). This not only minimizes the overall fresh raw material requirement but also minimizes the waste discharge in the environment, thus improving the economic as well as environmental performance simultaneously. Note that, in this case study, the objective functions are highly nonlinear. Because of the constraints applied and nonlinear nature of objective functions (see Appendix), the feasible region becomes very small. This is the reason the system converges to the final solution after only a few iterations. The small feasible solution space is also the reason the number of feasible Pareto optimum solutions is limited in some iterations. In the initial iterations, there are multiple Pareto solutions and the Pareto frontier can be generated; the best solution for each subsystem, as well as for the entire IE, can be chosen from the Pareto frontier, according to the requirements under various scenarios. As the system converges toward the final solution, the number of Pareto solutions reduces drastically. It also needs to be explained that these results are highly dependent on how the cost functions for each member is defined. If the costs are changed, then entirely different results may be generated for the same IE. Conclusion and Discussion In conclusion, the hierarchical Pareto optimization methodology provides an efficient and flexible approach for multiobjective decisionmaking of complex systems that consist of several subsystems. This methodology provides a modular structure, where each member industry is treated as a separate module at the lower level. Therefore, it is easy to add a new member, remove one member, or change one specific member, while maintaining the integrity of the algorithm. It provides a comprehensive analysis and multidimensional decision-making/ decision-analysis power for the entire system as well as

individual members. Thus, hierarchical Pareto optimization methodology is very suitable for optimizing the sustainable development of industrial ecosystems (IEs). The case study results show that, to maximize the sustainability of the entire IE, at the upper level, the Pareto optimum solution regarding the economic and environmental objectives of the IE must be identified. At the lower level, the Pareto optimum solution of each individual member also must be considered. Sometimes, compromise might be necessary to ensure maximum benefits for the overall system. Acknowledgment This work is in-part supported by the National Science Foundation under Grant CTS-0407494, the Texas Advanced Technology Program under Grant No. 003581-0044-2003, and the Gulf Coast Hazardous Substance Research Center. Appendix Mathematical Model for Case Study. The modeling details for the industrial ecosystem (IE) used in the current case study with four member industriessPlant 1, Plant 2, Plant 3, and Plant 4sare provided here. A1. Model for Plant 1. The production and waste streams for the Plant 1 model are as follows:

MYP1 ) 0.75X1 MW1 ) 0.25X1 MW1 ) MWT1 + MWU1 MWT1 ) MWTRN1 + MWTW1 + MYTW1 MWU1 ) MWURR1 + MWUW1 + MYUW1 MWU1 ) a11MW1 MWT1 ) a12MW1 MWTRN1 ) a21MWT1 MYTW1 ) a22MWT1 MWTW1 ) a23MWT1 MWURR1 ) a31MWU1 MYUW1 ) a32MWU1 MWUW1 ) a33MWU1 The constraints for the Plant 1 model are as follows:

X1 ) R1 + MYUW3 + MWURR1 + N1 + MWTRN1 N1 + MWTRN1 ) 0.7(R1 + MYUW3 + MWURR1) a21 + a22 + a23 ) 1 a31 + a32 + a33 ) 1 a21 g a22 a32 g a31 100 e X1 e 500 The Plant 1 model has the following parameters:

a11 ) 0.4 a12 ) 0.6

3276


The Plant 1 model has the following cost functions:

CFP1 ) 1000 + X10.7

X2 ) R2 + MWURR2 + MYUW1 + N2 + MWTRN2 N2 + MWTRN2 ) 0.75(R2 + MWURR2 + MYUW1) b21 + b22 + b23 ) 1

CFW1 ) 100 + MWT10.6

b31 + b32 + b33 ) 1

CFRR1 ) 10MWURR10.5

b22 g b21

CFRN1 ) 25MWTRN1

0.5

b31 g b32

CFUD1 ) 80MWUW1 CFTD1 ) 65MWTW1

200 e X2 e 700 The Plant 2 model has the following parameters:

The Profit, IEcP, IEvP, and ISP values for the Plant 1 model are as follows:

b11 ) 0.55 b12 ) 0.45

Profit1 ) 105MYP1 + 55MYTW1 + 20MYUW1 - 1100 X10.7 - MWT10.6 - 65MWTW1 - 80MWUW1 - 10MWURR10.5 -

The Plant 2 model has the following cost functions:

25MWTRN10.5 - 35R1 - 70N1 - 20MYUW3

CFP2 ) 1200 + X20.7

IEcP1 ) (105MYP1 + 55MYTW1 + 20MYUW1) ÷

CFW2 ) 120 + MWT20.6

(1100 + X10.7 + MWT10.6 + 65MWTW1 + 80MWUW1 +

CFRR2 ) 12MWURR20.5

10MWURR10.5 + 25MWTRN10.5 + 35R1 + 70N1 + 20MYUW3) IEvP1 )

70N1 + 65MWTW1 + 80MWUW1

CFRN2 ) 25MWTRN20.5

35R1 + 10MWURR10.5 + 25MWTRN10.5 + 20MYUW3

CFUD2 ) 75MWUW2

IEcP1 IEvP1

CFTD2 ) 60MWTW2

ISP1 )

A2. Model for Plant 2. The production and waste streams for the Plant 2 model are as follows:

MYP2 ) 0.8X2 MW2 ) 0.2X2 MW2 ) MWT2 + MWU2

The Profit, IEcP, IEvP, and ISP values for the Plant 2 model are as follows:

Profit2 ) (150MYP2 + 50MYTW2 + 30MYUW2) (1320 + X20.7 + MWT20.6 + 60MWTW2 + 75MWUW2 + 12MWURR2 + 25MWTRN2 + 30R2 + 80N2 + 20MYUW1)

MWT2 ) MWTRN2 + MWTW2 + MYTW2 MWU2 ) MWURR2 + MWUW2 + MYUW2 MWT2 ) b11MW2 MWU2 ) b12MW2

IEcP2 ) (150MYP2 + 50MYTW2 + 30MYUW2) ÷ (1320 + X20.7 + MWT20.6 + 60MWTW2 + 75MWUW2 + 12MWURR2 + 25MWTRN2 + 30R2 + 80N2 + 20MYUW1)

MWURR2 ) b21MWU2 MYUW2 ) b22MWU2 MWUW2 ) b23MWU2

IEvP2 )

80N2 + 60MWTW2 + 75MWUW2 30R2 + 12MWURR20.5 + 25MWTRN20.5 + 20MYUW1

MWTRN2 ) b31MWT2 MYTW2 ) b32MWT2 MWTW2 ) b33MWT2 The constraints for the Plant 2 model are as follows:

ISP2 )

IEcP2 IEvP2



MYP3 ) 0.85X3 MW3 ) 0.15X3

Profit3 ) 130MYP3 + 20MYUW3 + 45MYTW3 - 1210 X30.7 - MWT30.6 - 60MWUW3 - 50MWTW3 - 17MWURR30.5 22MWTRN30.5 - 40R3 - 90N3 - 25MYUW4

MW3 ) MWT3 + MWU3 MWT3 ) MWTRN3 + MWTW3 + MYTW3 MWU3 ) MWURR3 + MWUW3 + MYUW3 MWU3 ) c11MW3 MWT3 ) c12MW3 MWURR3 ) c21MWU3 MYUW3 ) c22MWU3 MWUW3 ) c23MWU3 MWTRN3 ) c31MWT3

IEcP3 ) (130MYP3 + 20MYUW3 + 45MYTW3) ÷ (1210 + X30.7 + MWT30.6 + 60MWUW3 + 50MWTW3 + 17MWURR30.5 + 22MWTRN30.5 + 40R3 + 90N3 + 25MYUW4) IEvP3 )

90N3 + 50MWTW3 + 60MWUW3 40R3 + 25MYUW4 + 17MWURR30.5 + 22MWTRN30.5 ISP3 )

IEcP3 IEvP3


MYTW3 ) c32MWT3

MYP4 ) 0.7X4

MWTW3 ) c33MWT3

MW4 ) 0.3X4 MW4 ) MWT4 + MWU4

The constraints for the Plant 3 model are as follows:

X3 ) R3 + MYUW4 + MWURR3 + N3 + MWTRN3 N3 + MWTRN3 ) 0.65(R3 + MYUW4 + MWURR3) c21 + c22 + c23 ) 1

MWT4 ) MWTRN4 + MWTW4 + MYTW4 MWU4 ) MWURR4 + MWUW4 + MYUW4 MWU4 ) d11MW4 MWT4 ) d12MW4

c31 + c32 + c33 ) 1

MWURR4 ) d21MWU4

c22 g c21

MYUW4 ) d22MWU4

c31 g c32

MWUW4 ) d23MWU4

400 e X3 e 800

MWTRN4 ) d31MWT4

The Plant 3 model has the following parameters:

c11 ) 0.6 c12 ) 0.4

MYTW4 ) d32MWT4 MWTW4 ) d33MWT4 The constraints for the Plant 4 model are as follows:

X4 ) R4 + MYUW4 + MWURR4 + N4 + MWTRN4 The Plant 3 model has the following cost functions:

N4 + MWTRN4 ) 0.72(R4 + MYUW4 + MWURR4)

CFP3 ) 1100 + X30.7

d21 + d22 + d23 ) 1

CFW3 ) 110 + MWT30.6

d31 + d32 + d33 ) 1

CFRR3 ) 17MWURR30.5 CFRN3 ) 22MWTRN30.5 CFUD3 ) 60MWUW3 CFTD3 ) 50MWTW3 The Profit, IEcP, IEvP, and ISP values for the Plant 3 model are as follows:

d22 g d21 d31 g d32 300 e X4 e 750 The Plant 4 model has the following parameters:

d11 ) 0.7 d12 ) 0.3 The Plant 4 model has the following cost functions:

3278


CFP4 ) 1115 + X40.7 CFW4 ) 115 + MWT4

0.6

CFRR4 ) 15MWURR40.5 CFRN4 ) 26MWTNR4

0.5

CFUD4 ) 80MWUW4 CFTD4 ) 65MWTW4 The Profit, IEcP, IEvP, and ISP values for the Plant 4 model are as follows:

Profit4 ) 170MYP4 + 25MYUW4 + 55MYTW4 - 1230 -

IEcP ) (105MYP1 + 55MYTW1 + 20MYUW1 + 150MYP2 + 50MYTW2 + 30MYUW2 + 130MYP3 + 20MYUW3 + 45MYTW3 + 170MYP4 + 25MYUW4 + 55MYTW4) ÷ (1100 + X10.7 + MWT10.6 + 65MWTW1 + 80MWUW1 + 10MWURR10.5 + 25MWTRN10.5 + 35R1 + 70N1 + 20MYUW3 + 1320 + X20.7 + MWT20.6 + 60MWTW2 + 75MWUW2 + 12MWURR20.5 + 25MWTRN20.5 + 30R2 + 80N2 + 20MYUW1 + 1210 + X30.7 + MWT30.6 + 60MWUW3 + 50MWTW3 + 17MWURR30.5 + 22MWTRN30.5 + 40R3 + 90N3 + 25MYUW4 + 1230 + X40.7 + MWT40.6 + 65MWTW4 + 80MWUW4 + 15MWURR40.5 + 26MWTRN40.5 + 45R4 + 95N4 + 30MYUW2)

X40.7 - MWT40.6 - 65MWTW4 - 80MWUW4 - 15MWURR40.5 26MWTRN40.5 - 45R4 - 95N4 - 30MYUW2 IEcP4 ) (170MYP4 + 25MYUW4 + 55MYTW4) ÷ (1230 + X40.7 + MWT40.6 + 65MWTW4 + 80MWUW4 + 15MWURR40.5 + 26MWTRN40.5 + 45R4 + 95N4 + 30MYUW2)

IEvP4 )

95N4 + 65MWTW4 + 80MWUW4

IEvP ) (70N1 + 65MWTW1 + 80MWUW1 + 80N2 + 60MWTW2 + 75MWUW2 + 90N3 + 50MWTW3 + 60MWUW3 + 95N4 + 65MWTW4 + 80MWUW4) ÷ (35R1 + 10MWURR10.5 + 25MWTRN10.5 + 20MYUW3 + 12MWURR20.5 + 25MWTRN20.5 + 20MYUW1 + 40R3 + 25MYUW4 + 17MWURR30.5 + 22MWTRN30.5 + 45R4 + 15MWURR40.5 + 26MWTRN40.5 +

45N4 + 15MWURR40.5 + 26MWTRN40.5 + 30MYUW2 ISP4 )

30MYUW2)

IEcP4 IEvP4

ISP )

IEcP IEvP

A5. Model for the Entire Industrial Ecosystem. For the entire ecosystem, the definitions of all the streams, constraints, and cost functions remain the same as those for the member plants. While calculating the Profit and performance indices (IEcP, IEvP, and ISP) for the entire IE, the mass and energy exchanges between the member industries are counted as internal recycle within the IE. Furthermore, the cost of purchasing lessexpensive resources from other members is not included. The Profit function and the performance indices for the entire IE are defined below:

In these models, Xi is the amount of resources being processed in each plant, where the subscript i represents the plant number (in millions of tons/year), M is the flow rate of each stream (in millions of tons/year), and C is the money value of the streams or investment ($/ton). In calculating the index values, the emergy value of each term is calculated based on its money value divided by the money-to-emergy ratio, ξ ) 1.75 × 1012 Sej/$.1 Because on the fact that ξ appears in both the numerator as well as the denominator, eventually ξ is eliminated from the indices.

Profit ) 105MYP1 + 55MYTW1 + 20MYUW1 - 1100 - X10.7 -

Literature Cited

MWT10.6 - 65MWTW1 - 80MWUW1 - 10MWURR10.5 25MWTRN10.5 - 35R1 - 70N1 - 20MYUW3 + 150MYP2 + 50MYTW2 + 30MYUW2 - 1320 - X20.7 + MWT20.6 60MWTW2 - 75MWUW2 - 12MWURR20.5 - 25MWTRN20.5 30R2 - 80N2 - 20MYUW1 + 130MYP3 + 20MYUW3 + 45MYTW3 - 1210 - X30.7 - MWT30.6 - 60MWUW3 50MWTW3 - 17MWURR30.5 - 22MWTRN30.5 - 40R3 - 90N3 25MYUW4 + 170MYP4 + 25MYUW4 + 55MYTW4 - 1230 X40.7 - MWT40.6 - 65MWTW4 - 80MWUW4 - 15MWURR40.5 26MWTRN40.5 - 45R4 - 95N4 - 30MYUW2

(1) Sikdar, S. Sustainable Development and Sustainability Metrics. AIChE J. 2003, 49, 1928. (2) AIChE Center for Waste Reduction and Technology (CWRT). CollaboratiVe ProjectssFocus Area: Sustainable DeVelopment. AIChE: New York, 2000. (3) Garner, A.; Keoleian, G. A. Industrial Ecology: An Introduction; National Pollution Prevention Center for Higher Education, University of Michigan: Ann Arbor, MI, 1995. (4) Lange, J. P. Sustainable Development: Efficiency and Recycling in Chemicals Manufacturing. Green Chem. 2002, 4, 546. (5) Leigh, N. G.; Realff, M. J. A Framework for Geographically Sensitive and Efficient Recycling Networks. J. EnViron. Plann. Manage. 2003, 46, 147. (6) Bakshi, B. R.; Fiksel, J. The Quest for Sustainability: Challenges for Process Systems Engineering. AIChE J. 2003, 49, 1350. (7) Miettinen, K. Nonlinear MultiobjectiVe Optimization; Kluwer Academic Publishers: Boston, MA, 1999. (8) Coello, C.; Coello, A. EVolutionary Algorithms for SolVing MultiObjectiVe Problems; Kluwer Academic Publishers: Boston, MA, 2002.

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3279 (9) Hwang, C.; Paidy, S.; Yoon, K. Mathematical Programming with Multiple Objectives: A Tutorial. Comput. Operat. Res. 1980, 7, 5. (10) Bhaskar, V.; Gupta, S. K.; Ray, A. K. Applications of Multiobjecitve Optimization in Chemical Engineering. ReV. Chem. Eng. 2000, 16 (1), 1. (11) Xu, A. Chemical Production: Complex Optimization, Pollution Reduction and Sustainable Development, Ph.D. Dissertation, Louisiana State University, Baton Rouge, LA, December 2004. (12) Messac, A.; Ismail-Yahaya, A.; Mattson, C. A. The Normalized Normal Constraint Method for Generating the Pareto Frontier. Struct. Multidisciplinary Optimiz. 2003, 25, 86. (13) Bergstrom, T. A Primitive Public Economy; University of California: Santa Barbara, CA (www.econ.ucsb.edu/∼tedb/Lec4.ps). (14) Fescher, M. The Measure of Pareto Optima: Applications to MultiObjectiVe Metaheuristics; The Institute for Systems Research, University of Maryland: College Park, MD, 2002. (15) Ehtamo, H.; Hamalainen, R. P.; Heiskanen, P.; Teich, J.; Verkama, M.; Zionts, S. Generating Pareto Solutions in a Two-Party Setting: Constraint Proposal Methods. Manage. Sci. 1999, 45, 1697. (16) Wismer, D. A. Optimization Methods for Large-Scale Systems with Applications; McGraw-Hill: Boston, MA, 1971. (17) Kravanja, Z.; Grossmann, I. E. Multilevel-Hierarchical MINLP Synthesis of Process Flowsheets. Comput. Chem. Eng. 1997, 21, S421. (18) Lee, M. H.; Lee, S. J.; Han, C.; Chang, K. S.; Kim, S. H.; You, S. H. Hierarchical On-Line Data Reconciliation and Optimization for an Industrial Utility Plant. Comput. Chem. Eng. 1998, 22, S247. (19) Stephanopoulos, G.; Ng, C. Perspectives on the Synthesis of PlantWide Control Structures. J. Process Control 2000, 10, 97. (20) Cheng, L.; Subramanian, E.; Westerberg, A. W. Design and Planning under Uncertainty: Issues on Problem Formulation and Solution. Comput. Chem. Eng. 2003, 27, 781.

(21) Levis, A. A.; Papageorgiou, L. G. A Hierarchical Solution Approach for Multi-Site Capacity Planning under Uncertainty in the Pharmaceutical Industry. Comput. Chem. Eng. 2004, 28, 707. (22) Triantaphyllou, E. Multi-Criteria Decision Making Methods: A ComparatiVe Study; Applied Optimization, Volume 44; Springer: Norwell, MA, 2000. (23) Deng, H.; Yeh, C. H.; Willis, R. J. Inter-company Comparison using Modified TOPSIS with Objective Weights. Comput. Operat. Res. 2000, 27, 963. (24) Geldermann, J.; Rentz, A. O. Multi-criteria Analysis for Technique Assessment: Case Study from Industrial Coating. J. Ind. Ecol. 2005, 9 (3), 127. (25) Ho, I.; Olson, D.; Rodgers, T. A Requirements Negotiations Model Based on Multi-Criteria Analysis. In Proceedings of the Fifth International Symposium on Requirements Engineering, IEEE; Institute for Electrical and Electronic Engineers (IEEE): New York, 2001. (26) Odum, H. T. EnVironmental Accounting; Wiley: New York, 1996. (27) Brown, M. T.; Herendeen, R. A. Embodied Energy Analysis and EMERGY Analysis: A Comparative View. Ecol. Econ. 1996, 19, 219. (28) Lou, H. H.; Kulkarni, M. A.; Singh, A.; Hopper, J. R. Sustainability Assessment for Industrial Systems. Ind. Eng. Chem. Res. 2004, 43, 4233. (29) Sahinidis, N.; Tawarmalani, M. BARONsSolVer Manual; University of Illinois at Urbana-Champaign: Urbana, IL (http://www.gams.com/dd/ docs/solvers/baron.pdf).

ReceiVed for reView April 25, 2005 ReVised manuscript receiVed February 7, 2006 Accepted February 15, 2006 IE050487Q

Hierarchical Pareto Optimization for the ... - ACS Publications

Recommend Documents