Environ. Sci. Technol. 2003, 37, 5389-5397
Constrained Optimization for Green Engineering Decision-Making DEBORAH L. THURSTON* AND SURESH SRINIVASAN University of Illinois at Urbana-Champaign, 104 South Mathews, Urbana, Illinois 61801
Green engineering requires the designer to consider a very extensive set of environmental impacts. To minimize these impacts, the designer must significantly expand his or her “toolset” of product design concepts, alternative materials, manufacturing systems, and analytic methods for addressing life cycle impacts. This can overwhelm a designer, who then resorts to overly simplistic rules or checklists out of necessity. The central issue is how to identify all “pollution prevention pays” opportunities and then how to deal with the unavoidable tradeoffs that arise after all these opportunities have been exhausted. This paper presents a framework for employing mathematical decision modeling toward this end. A domain-independent constrained optimization formulation is presented. A multiattribute utility function reflects the willingness to pay for environmental improvement and is the basis of the objective function. The feasibility constraints reflect the unavoidable tradeoffs. Several case studies are presented, including power systems, floor tile manufacturing, and computer systems.
Introduction Green engineering is extremely complex. Designers are advised to consider many different impacts that occur over a wide range of time, media, and stakeholders. Multiple product life cycles should be considered, including material and energy inputs and outputs. This is a daunting task. Some useful tools have been developed, such as streamlined life cycle analysis and design for disassembly checklists. Anastas and Zimmerman (1) provide an overview of developments in green engineering, organized into 12 principles. These principles cover all four design scales: molecular, process, product, and system. They stress the need for inherency and are domain-independent in nature. For example, Principle 11 is “design for commercial afterlife”, but it is up to the designer to identify which product components might be worth salvaging and to develop techniques to refurbish them in such a way that the customer remains satisfied. Furthermore, designers are under tremendous pressure from stockholders to design for the environment while simultaneously decreasing cost. Once all synergistic “pollution prevention pays” approaches have been exhausted, the designer must evaluate tradeoffs between cost and environmental impact or even tradeoffs between environmental impacts across media. Where should we “draw the box” around life cycle impacts for this particular product? What tradeoffs are unavoidable? What tradeoffs are we willing to make? * Corresponding author phone: (217)333-6456; e-mail: thurston@ uiuc.edu. 10.1021/es0344359 CCC: $25.00 Published on Web 11/01/2003
2003 American Chemical Society
In dealing with these complexities, even well-meaning designers can fall prey to several decision traps that lead them away from their intended goal of environmental protection. These traps include a myopic focus on end-ofpipe treatment, unwitting transfer of pollution across media and life cycle stages, isolation of objectives due to institutional barriers, and “anchoring” on the status quo. In short, a holistic view is not taken. This paper presents a systematic method for integrating environmental issues directly into the analytic toolset traditionally employed by engineers during product development. In contrast to ad hoc methods in which a large number of assumptions might be buried, mathematical modeling provides a transparent framework for green engineering. In this way, environmental impacts are treated not as an afterthought but rather as a worthy component of the focused analytic effort that has served industry so well.
Constrained Optimization Design Tradeoffs. Figure 1 illustrates in general the tradeoff issue in green design decision-making. The axes indicate the cost and environmental impact of design alternatives. Alternatives that are nearest the origin are preferred since they have both low cost and low environmental impact, but the region below the Pareto optimal (PO) frontier (2) is infeasible. Alternatives that lie directly on the PO frontier are those where it is not possible to improve one attribute (such as environmental impact) without worsening another (such as cost). The upper right quadrant shows alternatives that are inferior to those on the PO frontier. Design changes that result in simultaneous improvements in environmental impact and cost savings help manufacturers move from the inferior region to the PO frontier. These changes should of course be made. The decision problem we address is 2-fold. First, what specific design modifications will help the designer move to the PO frontier? Second, what specific location on the frontier provides the best combination of cost and environmental impact? Figure 1 indicates that the optimal solution lies on the iso-utility curve where U ) 0.5. This design is superior to all others since its utility is greatest. To reach the PO frontier, designers typically employ a “house of quality” (HOQ) matrix approach to clarify the cause and effect relationships between product quality, cost, and engineering decisions (3). The matrix rows list all attributes of product performance xi, such as cost, quality, weight, environmental impact, etc. The columns list all the engineering decision variables yj that the designer can directly control in order to improve each attribute xi. Typical decision variables yj include material choice, manufacturing process, assembly technology, etc. The 12 principles of green engineering (1) can help guide the designer toward including decision variables that might otherwise remain unexplored. For example, Principle 6: “Conserve Complexity” could direct the automotive design engineer to consider the possibility of using remanufactured engine blocks. Next, expert judgment is employed in order to fill in the matrix with symbols that indicate the degree to which each attribute is influenced by each design-specific decision variable. The subjective nature of the expert judgments necessary to place the symbols makes the assessment difficult. After the possible existence of all influences is indicated, the relative strength of each influence is assessed. For design problems of any realistic size, the matrix grows quite large and unwieldy. The central problem is how to identify the design decision variable values that result in the optimal combination of attributes. For example, what materials, manufacturing processes, assembly VOL. 37, NO. 23, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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(y1, y2, y3, ..., ym) as a vector of functions h ) (h1, h2, h3, ..., hn) or x ) h(y):
FIGURE 1. Design alternative space. methods, and end-of-life strategies result in the best combination of cost, quality, and environmental impact? Constrained Optimization Modeling. Engineering design is typically an iterative configure-evaluate-reconfigure process. So, too, is the process of defining the elements in a normative constrained multiattribute optimization. The definition of each element influences the others, which in turn might lead to redefining the first element. The four major elements in constrained optimization modeling are as follows: an objective function which one is attempting to maximize or minimize; a vector of product performance attributes x ) (x1, x2, x3, ..., xn) that contribute to the objective function and often conflict with one another (such as cost, quality, and environmental impact); a vector of decision variables y ) (y1, y2, y3, ..., ym) over which the designer has direct control (such as material choice, assembly process, etc.); and a vector of constraint functions h(y) ) x that model the cause and effect relationships between design decision variables y and the product performance vector x. Table 1 summarizes model elements. The first step is to reduce the very large number of product attributes x and engineering parameters y listed in the matrix to the subset upon which experimental and product development time are best spent (4, 5). A preliminary conception of the objective function can be used to guide this process, but its explicit definition is not necessary at this stage. The performance attribute vector x ) (x1, x2, x3, ..., xn) is then defined in terms of the design decision variable vector y )
These constraint equations are most often determined using traditional engineering analytic techniques including activity-based cost estimation (where x1 is cost) and statistical process control (where x2 is scrap rate, finite element analysis, etc). The cause and effect relationships between design decisions y and environmental impacts are much more difficult to assess, but life cycle analysis methods are continually improving, including commercially available software [PreConsultants (6)]. Several methods are available for expressing environmental benefits in terms of externality costs (7). These approaches are most commonly employed in evaluating regulatory programs but could be used by manufacturers as well. The constraints shown in eq 1 must be satisfied simultaneously, so only certain combinations of attributes x ) (x1, x2, x3, ..., xn) are technically feasible. For example, it is not possible to improve x1 (say, stiffness) by increasing y1 (thickness of a component) without simultaneously worsening x2 (weight), all else being held the same. Now the central problem is how to identify which feasible combination of attributes x is best. Traditional design objective functions minimize a single attribute, such as weight or perhaps cost. However, our central problem is how to decide which combination of several attributes x ) (x1, x2, x3, ..., xn) best suits our purpose. Since x ) h(y), our objective function is of the form:
f (x1, x2, x3, ..., xn) ) f [(h1(y), h2(y), h3(y), ..., hn(y)]
(2)
) g(y) The form of g(y) should reflect preferences for the conflicting attributes x. The simplest form is a linear weighted average shown in eq 3, but this approach has been demonstrated to be highly unreliable for design problems. The reason is that the arbitrary assessment of “relative importance” employed
TABLE 1. Elements of Mathematical Modeling for Green Engineering Decision-Making model element
expression
cost
x ) (x1, x2, x3, ..., xn), for example: x1 ) cost, x2 ) quality, x3 ) environmental impact, etc. y ) (y1, y2, y3, ..., ym), for example: y1 ) material choice, y2 ) material production process, y3 ) manufacturing settings, etc. hi(y1, y2, ..., ym) ) xi for i ) 1, 2, ..., n x1,l e h1(y1, y2, y3, ..., ym) e x1,u
quality
x2,l e h2(y1, y2, y3, ..., ym) e x2,u
environmental impact
x3,l e h3(y1, y2, y3, ..., ym) e x3,u
design attributes
decision variables
constraints
objective function
max ) U(y)
5390
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1 K
[[∏ n
meaning elements of product performance that are both relevant and negotiable over a limited range
engineering parameters the designer can directly control in order to improve attributes (x1, x2, x3, ..., xm)
] ]
(KkiUi[hi(y1, y2, y3, ..., ym)] + 1) - 1
i)1
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relationships between engineering parameters and product performance cost (x1) is a function of decision variables (y1, y2, y3, ..., ym); typical units are total amortized cost per unit product quality (x2) is a function of decision variables (y1, y2, y3, ..., ym) typical units are percent scrap rate, reliability, etc. environmental impact (x3) is a function of decision variables (y1, y2, y3, ..., ym); typical units are pounds of waste, ecopoints, etc. utility maximization is determined through identification of the optimal set of design decision variables y
to assign the weighting factors wi can be systematically biased by irrelevant factors. The perceived relative importance (and thus the willingness to make tradeoffs) often does not remain static throughout the real design space (4). Thus, relative weighting factor approaches can lead to designs that do not best satisfy the decision maker’s preferences: n
g(y) )
∑w [h (y)] i
(3)
i
i)1
Instead, we recommend a multiattribute utility function formulation as shown below in eq 4. Multiattribute utility analysis is a rigorously normative methodology based on the axioms of utility theory (8). The scaling factor ki more accurately reflects the designer’s willingness to make tradeoffs among the attributes over the entire range of feasibility. The n single attribute utility functions Ui[hi(y)] can reflect the decision-maker’s nonlinear valuation of each attribute i and their attitude toward risk and are assessed using the lottery method (8). With careful problem definition, they can often take the form of a well-behaved exponential function Ui(xi) ) bi + ai exp(-cixi). Coefficient ci reflects the degree of risk aversion, and ai and bi are calculated such that Ui(xi) is scaled from 0 to 1. The multiplicative form in eq 4 can be employed after conditions of preferential and utility independence are verified. The goal is to determine the set of design decision variables y that maximize overall utility. It is very important to note that the attribute scaling constants ki should not be viewed as “weights” reflecting the “relative importance” of each attribute. In contrast, they help provide a more accurate assessment of the decision maker’s willingness to make tradeoffs as it changes throughout the design space (4). In addition, inequality constraint eqs 6 and 7 define the range over which the designer is both able and willing to make tradeoffs among the attributes x. Where less is preferred to more (such as cost or environmental impact), the upper bound is defined as the “worst” that the decision maker is willing to tolerate (not the worst possible). The lower bound is defined as an optimistic, yet realistic, “best” from the viewpoint of technical feasibility. This is the range of technical and preferential negotiability. Additional equality or inequality constraints eq 8 and 9 may be necessary, and the specific form of eqs 4-9 will depend on the problem:
max ) U(y)
1
[[
K
n
∏(Kk U [h (y , y , y , ..., y i
i
i
1
2
3
m)]
i)1
] ]
+ 1) - 1
(4)
where
hi(y1, y2, y3, ..., ym) ) xi
for i ) 1, 2, ..., n
(5)
hi(y1, y2, y3, ..., ym) g xi,l
for i ) 1, 2, ..., n
(6)
hi(y1, y2, y3, ..., ym) e xi,u
for i ) 1, 2, ..., n
(7)
for k ) 1, 2, ..., p
(8)
for j ) 1, 2, ..., r
(9)
subject to
and perhaps
qk(y1, y2, y3, ..., ym) ) 0 hn+j(y1, y2, y3, ..., ym) g 0
where K is the normalizing parameter, calculated from n
1+K)
∏(1 + Kk ) i
i)1
(10)
Computational Issues. While the nonlinear objective function in eq 4 appears to present some computational complexity, several factors facilitate its solution. First, many design problems distill to a small number of incommensurate, conflicting objectives after elements of the attribute set (x1, x2, x3, ..., xn), and their ranges have been defined in such a way that conditions of preferential and utility independence are satisfied. If satisfied, these conditions facilitate straightforward identification of single attribute utility functions Ui(xi) and indicate the multiplicative form shown in eq 4. For example, ergonomics might appear in the HOQ, but demand functions (if known) can directly convert ergonomic performance to expected profits. In fact, some design theory researchers argue that expected profit alone is the only accurate metric for design evaluation. Second, with careful definition of attributes and their ranges, single attribute utility functions can often take the form of the well-behaved exponential function Ui(xi) ) bi + ai exp(-cixi).The third factor that reduces computational complexity is that several elements of the design decision variable set (y1, y2, y3, ..., ym) are often discrete or even binary variables, such as material choice, enabling exhaustive enumeration techniques. All of the optimization problems presented in this paper were readily solved on a personal computer using a spreadsheet add-in solver that employs simplex and branch and bound for linear and integer problems and a generalized reduced gradient algorithm for nonlinear problems.
Constrained Optimization for Power Systems Electricity generation is one of the primary causes of environmental pollution. In the United States alone, it is estimated that 63% of sulfur dioxide emissions, 22% of NOx emissions, 40% of carbon emissions, and 37% of mercury emissions are caused by the production of electricity (9). Conventional power generation technologies such as coalbased systems and nuclear reactors further damage the environment via particulates, ozone accumulation, mercury emissions, land impacts, and radioactive wastes. At least 72% of the nation’s energy demand is served by a network of such coal and nuclear facilities (10). As our reliance on these technologies continues, irreparable damage is being inflicted upon the environment. Although cleaner technologies such as wind, solar, and biomass exist, they have not yet been utilized to their fullest potential. Reasons include high frontend costs, intermittent supply characteristics, social barriers, etc. The extent to which these renewables can be included in the energy mix in a cost-effective fashion needs to be explored. Mathematical programming techniques have proved useful for considering cost and efficiency of an energy mix (11). The case study below considers both the cost and the environmental impacts of including alternative energy sources in the energy mix. Estimating Environmental Impacts. For this case study, a tool for estimating the environmental impacts of energy sources, Power Scorecard (12) has been employed. It was developed as a part of the Pace Law School Energy Project by a group of six nationally recognized environmental organizations consisting of the Environmental Defense, the Izaak Walton League, the Natural Resources Defense Council, the NW Energy Coalition, the Pace Law School, and the Union of Concerned Scientists. It was developed as an educational tool to aid consumers in choosing their energy mix depending upon their expressed level of environmental concern. It provides an overall rating and individual environmental impact ratings for various energy sources. The scoring scale resulted from a collective exercise of expert judgment by these participating organizations and is based upon available data and application of state-ofthe-art techniques for quantifying environmental impacts of power generation from various sources. This methodology VOL. 37, NO. 23, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 2. Default Impact Scores for Common Power-Generating Technologies air impact rating
water
land use
technology
overall score
CO2
SO2
NOx
mercury
use
quality
on-site
off-site
wind; low land impact Solar Central Station PV solar distributed PV biomass (wood fuel, high NOx, has waste) hydro plant, default low impact hydro gas-fired steam, electric oil-fired steam, electric coal-fired steam electric nuclear
0.1 2.6 0 5.4
0 0 0 10
0 0 0 1
0 0 0 6
0 0 0 6
0 1 0 1
0 6 0 6
1 14 0 5
0 0 0 4
5.6 1.8 4.4 6.2
0 0 6 8
0 0 1 6
0 0 6 7
0 0 1 4
10 4 5 6
10 4 6 6
15 4 4 4
15 4 5 7
8.8 11.8
10 0
10 0
10 0
10 0
9 10
6 6
5 55
9 34
TABLE 3. Levelized Cost and Energy Profile for State of Illinois levelized cost generated power percentage share energy type (in cents/kW-h) in MW-h in Illinois in Illinois coal nuclear hydro gas petroleum biomass wind
4.2 2.5 2.4 3.8 3.6 7.0 4.3
74 155 385 81 737 134 142 094 5 667 530 764 789 1 133 709
45.32 49.96 0.08 3.46 0.46 0.69
has been an important stimulus to the CRS’s (Center for Resource Solutions) Green-e certification program and has also been used for rating electricity products offered in Pennsylvania, New Jersey, and California markets. For each technology, Power Scorecard provides a rating in eight categories: climate change, acid rain, ozone (smog) and fine particulates, air toxics (mercury), water resource consumption, pollution of water bodies, on-site land impacts (permanent plant footprint), and off-site land impacts (solid waste disposal and fuel processing). Impacts (such as SOx emissions) per MW-h are estimated for each technology and are then used to assign scores (most often in the range of 0-10) in each category. A score of one corresponds to minimal environmental impact while a score of 10 indicates extensive environmental damage. Scores below 0 and above 10 are also possible (for example, the land use impacts due to longterm storage of radioactive wastes). Table 2 shows the default values and overall impact scores for several technologies in each of the eight categories. The overall environmental impact score is obtained by taking the weighted average of scores for each impact where global climate change impact is weighted twice indicating its greater significance. These default scores are representative of typical power plants in the United States and are used when no information is available about specific facilities. The score for a particular plant may differ. Environmental Impact Estimation for Illinois. For illustrative purposes, we consider the power profile of the State of Illinois as of the year 1999. The second column of Table 3 shows levelized costs estimates (the equivalent annual cost of constructing and operating the plant over its economic life) for producing electricity from various energy sources. These data have been collected from various resources (1316). Columns 3 and 4 show that the primary energy sources for Illinois are nuclear power and coal. Together they meet approximately 95% of the energy demand. On the basis of emissions data gathered from ref 10, scores are given for the various environmental impacts. Default scores for the State of Illinois similar to those in Table 2 have been used appropriately when enough site-specific information was not available. 5392
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When the Power Scorecard methodology is applied to the current energy profile of Illinois, the scores shown in column 4 (Table 4) are obtained. The resulting weighted average overall impact score is 9.72, which makes this energy mix very undesirable. Since the environmental impact is high, the potential of alternative energy sources should be considered. A recent report (17) indicates that Illinois has the potential to generate all of its electricity demand from its renewable energy sources. Table 5 shows that there is a significant untapped potential from less-polluting energy sources, including wind and biomass (16-19). Yet, these sources have not yet been utilized. Their current contribution to the State’s needs is less than 1%. Results for Power Systems. Two constrained optimization models are developed to analyze tradeoffs between cost and environmental impacts of energy mix alternatives. The decision variables are defined as y ) (y1, y2, y3, ..., y7) where yj is the amount of electricity generated from the jth energy source (in MW-h). First, a cost minimization model is employed in order to locate the PO frontier. Then, a multiattribute utility maximization model is formulated to determine the optimal location on the PO frontier. This location corresponds to the energy mix that represents the best combination of cost and environmental impact. Model 1: Cost Minimization Subject to Environmental Impact Constraint. The traditional formulation shown in eq 11 substitutes a cost function h1(y) into the objective function to minimize total costs. Constraint eq 12 limits the total environmental impact score from all sources to some maximum level E as specified by the user. The environmental impact score Sj for each source j is determined using Power Scorecard. Equation 13 shows that demand must be satisfied. Equation 14 indicates that the amount of electricity from each source cannot exceed its resource potential Pj, which is the maximum amount of electricity that can be generated from a particular energy source j, including both tapped and untapped potential: 7
minimize
∑C y
(11)
j j
j)1
subject to 7
h2(y1, y2, y3, ..., ym) )
∑S (y /D) e E j
j
(12)
j)1 7
h3(y1, y2, y3, ..., ym) )
∑y - D ) 0 j
(13)
j)1
h4(y1, y2, y3, ..., ym) ) yj - Pj e 0
for j ) 1, 2, 3, ..., m (14)
TABLE 4. Environmental Impact Ratings for Current Energy Mix power score card parameter
type of effect on environment
wt
scores
carbon dioxide emissions sulfur dioxide emissions nitrogen oxides emissions mercury levels water usage water quality on-site land impact off-site land impact
global climate change acid rain ozone and particulate impacts air toxics water usage water quality on-site land impact off-site land impact
2 1 1 1 1 1 1 1
4.40 2.76 4.83 4.17 9.28 6.00 29.94 21.75
TABLE 5. Potential of Renewable Energy Technologies in Illinois type of energy source wind biomass hydro solar geothermal
untapped potential in MW-h/yr 15.57 × 106 70.00 × 106 1.53 × 106 technology improvements necessary for large-scale production not feasible due to low-to-moderate temperature resources
where Cj is the levelized cost (in $) of producing electricity from the jth energy source, yj is the amount of electricity produced from jth energy source (MW-h), Sj is the overall environmental score rating of the jth type of energy source, D is the demand (MW-h), E is the maximum allowable environmental impact score specified by the user, Pj is the resource potential of the jth type of energy source (MW-h), and m is the number of energy sources ) 7. The optimal solution is the energy mix y* that minimizes cost, while simultaneously satisfying environmental impact score, demand, and resource potential constraints. To determine the PO frontier with respect to cost and environmental impact, the model is solved repeatedly for values of E ranging from 5.44 to 10.06. The purple curve in Figure 2 shows the resulting PO frontier. It should be noted that the same frontier would be defined if the model were reformulated to minimize environmental impact subject to a userspecified cost constraint.
Defining the PO frontier reveals that the current energy mix appears to be an inferior solution and that it is possible to decrease cost without worsening environmental impact. The current mix is shown as point n, with a cost of $5483 M and environmental impact score of 9.72. Table 6 and Figures 2 and 3 show details of the possible solutions discussed next. By modifying the energy mix as shown in Table 6 and Figure 3, one can move from point n to point a, and costs could be reduced to $5287M with no change in environmental impact score. This analysis is intended primarily to demonstrate the implementation of constrained optimization modeling, provide a first-order assessment of the current situation, and identify sources of uncertainty that could affect the outcome. For example, the model indicates that a 3.6% cost decrease could be achieved with no adverse environmental impact by moving from point n to point a by decreasing reliance on coal from 45% to 38% and increasing the use of both nuclear (from 51% to 55%) and wind sources (from 0 to 1.7%). The resulting 3.6% cost savings are within the same range of uncertainty as the cost estimates. So before this step is undertaken, it would be prudent to validate the relevant cost and environmental impact estimation parameters for nuclear and wind sources. Furthermore, the slope of the PO frontier in this region is relatively flat, so improving the environmental score would be relatively inexpensive. For example, moving from point a to point b improves the impact score from 9.72 to 9.2 at a marginal cost of only $11M or $21M per unit improvement in environmental impact score. Table 6 and Figure 3 indicate that this move requires further decreasing reliance on coal from 38% to 31% and increasing reliance on wind to 8.3%. Table 7 shows the individual impact ratings obtained for
FIGURE 2. Iso-utility curves for kE ) 0.4 and kC ) 0.7. VOL. 37, NO. 23, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 6. Cost and Environmental Impact of Alternative Energy Mix Scenarios decision criterion current cost & energy mix
minimize cost
cost-efficient solution
maximize multiattribute utility (kC ) 0.4, kE ) 0.7)
minimize environmental impact
on Figure 2 energy mix
point n 45.3% coal 50% nuclear 3.5% gas 1.2% other
point a 38% coal 55% nuclear 3.8% gas 1.7% wind 1.5% other
point b 31.5% coal 55% nuclear 3.8% gas 8.3% wind 1.4% other
cost (in $ million) environmental score
$5,483 9.72
$5,287 9.72
$5,298 9.2
point c 50% coal 33.2% nuclear 3.8% gas 9.5% wind 2.1% biomass 1.4% other $5,998 8.25
42.5% coal 0% nuclear 3.8% gas 9.5% wind 42.8% biomass 1.4% other $8,787 5.44
feature
FIGURE 3. Alternative energy mix scenarios.
TABLE 7. Environmental Impact Ratings for Point b
max )
effect on environment
scores
effect on environment
scores
global CC acid rain ozone and FP air toxics
3.15 1.93 3.42 2.88
water usage water quality on-site LI off-site LI
8.58 5.48 32.09 22.09
U(y)
1 K
[[∏
] ]
2
(KkiUi[hi(y1, y2, y3, ..., y7)] + 1) - 1
i)1
(15)
where 7
∑C y
(16)
∑S (y /D)
(17)
h1(y1, y2, y3, ..., ym) )
j j
j)1
point b. Comparing Tables 7 and 4 (impact ratings for the current energy mix) shows a decrease in individual scores in all categories except land use, which increases reflecting the shift from coal to nuclear. The effect of these changes can be quantified. For example, the improvement in the global climate change score from 4.4 to 3.15 translates to a reduction in CO2 emissions of about 19 632 thousand ton/yr (based upon 1999 electricity demand). As we move further left on the PO frontier, the slope begins to increase, reflecting increased marginal costs for further improvements in environmental impact. The frontier identifies many other possible solutions, ranging from one that would minimize the environmental impact score (5.44) at a cost of $8,787M to one that would minimize cost ($5,287M) with an environmental impact score of 9.72. The question is which combination of cost and environmental impact is best. Model 2: Maximization of Multiattribute Utility. To determine the optimal location on the PO frontier, the constrained optimization formulation shown in eqs 15-20 is employed, with cost given by h1(y) and environmental impact score given by h2(y): 5394
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7
h2(y1, y2, y3, ..., ym) )
j
j
j)1
Ui(xi) ) bi + ai exp(-cixi)
for i ) 1, 2
(18)
subject to 7
h3(y1, y2, y3, ..., ym) )
∑y - D ) 0 j
(19)
j)1
h4(y1, y2, y3, ..., ym) ) yi - Pi e 0
for j ) 1, 2, 3, ..., m (20)
The instantaneous slopes of the iso-utility curves shown in Figure 2 reflect the willingness to make tradeoffs between cost and environmental impact for the utility function shown in eq 15 assuming kC ) 0.7, kE ) 0.4, and c1 ) c2 ) -0.8. At point a, the utility function reflects a willingness to pay up to $4520M per unit improvement in environmental impact score (within a small range). Since this is greater than the
FIGURE 4. Product deployment through market segments after take-back. tradeoff necessary for moving from point a to point b ($21M/ unit), utility improves and the designer is better off at point b than at point a. At point b (with utility of approximately 0.85), the willingness to pay decreases to $3239M/unit but remains greater than the tradeoff necessary ($51M/unit) to continue along the PO frontier. Note that as one moves from right to left on the PO frontier, the willingness to pay to improve the environmental impact score decreases, while the tradeoff necessary to achieve further improvement increases. The optimal solution is the point where the PO frontier is tangent to the greatest iso-utility curve, point c where utility achieves its maximum possible value of 0.9. Here, the slope of the utility function indicates that the willingness to pay has decreased to $974M/unit, while the necessary tradeoff has increased significantly to $974M/unit. As one moves further left on the PO frontier, the willingness to pay further decreases, becoming less than the necessary tradeoff, so point c is the optimal solution. Table 6 and Figure 3 indicate that, as compared with the current energy mix, the optimal solution calls for an increase in the use of coal sources from 45% to 50%, a reduction in nuclear from 51% to 33%, increasing wind sources to 10%, and increasing biomass sources to 2%.
Results for Statistical Manufacturing Process Control This section presents an overview of ref 20, an example of Principle 1: Material and Energy Inputs Should Be Inherently Nonhazardous, and an example of Principle 2: Prevention Instead of Treatment (1). A floor tile manufacturer sought assistance in preventing the formation of hazardous air pollutant (HAP) emissions. It was first observed that emission rates varied considerably over short periods of time in ways that could not be explained by unavoidable sampling error alone. The stochastic, or random, nature of the manufacturing process that caused these variations was viewed as an opportunity for pollution prevention. The hypothesis was
that opportunities existed for reducing emissions by modifying the product (for example, raw materials) and/or modifying the manufacturing system. First, brainstorming sessions resulted in a list of 30 product and process control settings y that were suspected of influencing the production of emissions, either in type or quantity. These included 13 different raw material ingredients (8 new and 5 recycled), 13 line settings (temperature, speed, thickness, etc. at various points), and 4 different mixer settings. Three attributes were deemed to be relevant: x1 ) hazardous air pollutants in ln[HAP in ng/L], x2 ) quality in % scrap rate, and x3 ) marginal cost in dollars. Extensive data from a large number of production runs was recorded and analyzed. Statistical analysis revealed that within the original set of 30 possible decision variables y, two raw materials and two manufacturing process settings could in fact be controlled in such a way as to reduce emissions, scrap rate, and/or cost. So, the decision variable set y was defined as y1 ) raw material A (lb), y2 ) mass flow rate, y3 ) mixer orifice diameter, and y4 ) raw material B (lb). A set of linear constraints of the form h(y) ) x shown in eqs 21-24 was obtained from a multivariate regression, where a and b are the slope and intercept, respectively: n
hi(y1, y2, y3, y4) )
∑a (y ) + b ) x i
i
i
i
for i ) 1, 2, 3 (21)
i)1
or
-0.0066y1 + 0.183y2 - 0.734y3 + 0.057y4 + 6.005 ) x1 ln [HAP, ng/L] (22) -0.0867y1 + 0y2 + 6.59y3 + 0y4 - 0.134 ) x2 % scrap rate (23) 0.1y1 + 0y2 + 0y3 + 0y4 + 27.68 ) x3 marginal cost, $ (24) VOL. 37, NO. 23, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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A multiattribute utility assessment revealed nonlinear single attribute utility functions of the form Ui(xi) ) bi + ai exp(-cixi) and a multiplicative multiattribute utility function as shown in eq 4. Constrained optimization was then employed to identify the best set of tradeoffs. A 21% reduction in HAP emissions and a 21% reduction in scrap rate were achieved at a 13% increase in the marginal cost. An additional, ironic result was the discovery that one of the raw materials that were contributing significantly to hazardous air pollutant production was a scrap material that the manufacturer was attempting to recycle in order to decrease solid waste.
Results for Computer Marketing, Design, and Product Take-Back Aspects of Principles 6-8 (1) are exhibited in ref 21, a market portfolio approach to product take-back legislation for personal computers. Three market segments are identified; “green” consumers, utilitarians, and technophiles. The willingness to make tradeoffs among cost, reliability, and environmental impact varies between each group. Green customers are the most willing to pay to reduce environmental impact, while technophiles are the least. After product take-back, the designer must decide how to best recover the economic value-added by the manufacturing process. An optimization model determines which of 88 computer components should be reused (cleaned, inspected, and reassembled), remanufactured (cleaned, refurbished, and reassembled), recycled (reprocessed back to raw material form), or disposed and replaced with new components. Figure 4 illustrates the basic concept of utilizing product components from one market segment in a cascading fashion in other market segments during subsequent life cycles. For example, when products in the technophile market are taken-back, their components can be employed for a second life cycle in the utilitarian market segment, then in the green market segment for a third life cycle. For illustrative purposes, the figure shows entire computer systems, but the model deals with 88 separate components and 8 manufacturing/remanufacturing operations. The model is shown in eqs 25-30: r
max ) U(X)
z
1
[[
3
∑∑K ∏(K k l)1 p)1 p
p j,pU(Xj,p,l)
j)1
] ]
+ 1) - 1
for p ) (1, ..., z); l ) (1, ..., r) (25)
where s
∑
5
[Y1,i,p,l(C8i +
i)1
∑
6
Cni) + Y2,i,p,l(C4i) + Y3,i,p,l(
n)1
∑C
ni)
+
n)3 5
Y4,i,p,l(C7i +
∑C
ni)]
) X1,p,l
n)2
for p ) (1, ..., z); l ) (1, ..., r) (26) s
∏[(Y
1,i,p,l
+ Y4,i,p,l) exp{-(hdyp,l/θi)bi} + (Y2,i,p,l +
i)1
c-1
∑(D
Y3,i,p,l) exp(-hd
i,p-f,l-fNp-f,l-f)/θi)]
) X2,p,l
f)0
for p ) (1, ..., z); l ) (1, ..., r) (27)
s
∑
5
[Y1,i,p,l(E8i +
i)1
∑
6
Emi) + Y2,i,p,l(E4i) + Y3,i,p,l(
m)1
∑E
mi)
+
m)3 5
Y4,i,p,l(E7i +
∑E
mi)]
) X3,p,l
m)2
for p ) (1, ..., z); l ) (1, ..., r) (28) 5396
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subject to
xi,p,l e Xi,p,l e xi,p,u for i ) (1, ..., s); p ) (1, ..., z); l ) (1, ..., r) (29) Y1,i,p,l + Y2,i,p,l + Y3,i,p,l + Y4,i,p,l ) 1 for i ) (1, ..., s); p ) (1, ..., z); l ) (1, ..., r) (30) where the following parameters apply: Decision Variables. Y1,i,p,l ) 1 if component i is new, 0 otherwise Y2,i,p,l ) 1 if component i is directly reused, 0 otherwise Y3,i,p,l ) 1 if component i is remanufactured, 0 otherwise Y4,i,p,l ) 1 if component i is recycled, 0 otherwise i ) (1, ..., s) ) components in product p, defined such that each is comprised of a single material j ) (1, ..., 3) ) attributes, j ) 1, cost; j ) 2, reliability; j ) 3, environmental impact p ) (1, ..., z) ) products in the portfolio l ) (1, ..., r) ) life cycles being modeled Di,p,l ) 1 if component i in product p is reused or remanufactured during life cycle l, 0 otherwise Np,l ) number of years until product p is taken back at the end of life cycle l Objective Function. X ) (X1,p,l, X2,p,l, X3,p,l) ) attributes X1,p,l ) total cost in dollars of product p summed over all components i for life cycle l X2,p,l ) total reliability of product p at the end of life cycle l in terms of the probability that a component will not fail before the expected lifetime X3,p,l ) environmental impact of product p for life cycle l summed over all components i expressed as a normalized value returned from SIMAPRO 4.0S U(X) ) multiattribute utility summed over all products p for all life cycles l U(Xj,p,l) ) utility of attribute j for product p for life cycle l Constraints. Cn,i ) cost associated with operation n for component i Em,i ) environmental impact resulting from operation m on component i n ) m ) (1, ..., 8) ) operations (material acquisition, manufacturing, assembly, take-back, disassembly, remanufacturing, recycling and disposal). Ri(t) ) reliability of component i t ) operating time (h) θi ) characteristic life of component i (h) h ) average hours of use per day product p d ) average number of days a year product p is used bi ) slope of the Weibull distribution curve (1 for useful life) for component i f ) (1, ..., c - 1) where c is the number of the current life cycle xi,p,l, xi,p,u ) lower and upper bounds, respectively, on each attribute i for each product Results indicate that even without market segmentation (where one product is intended for all three niche markets), when remanufacturing and reuse were considered over two 3-yr service lives, costs decreased by an average of 34% and environmental impacts decreased by 62%, with only a slightly lowering of reliability by 1%. When a portfolio approach and market segmentation are considered, costs decreased by 37% and environmental impacts decreased by 62%, with a
lowering of reliability by an average of 4%. More importantly, the utility of each market segment improved because each was offered the product that possessed the combination of cost, reliability, and environmental impact that best meets its needs. Further improvements were realized through a paradigm shift toward selling a service rather than selling a product, perhaps via leasing arrangements. If the leasing arrangement permits the manufacturer to control and tailor the take-back period for each market segment, the utility of each market niche was further improved.
Discussion This paper presented a constrained optimization approach to green engineering. This approach involves three basic elements. First, the large number of considerations must be reduced to a subset of elements with the greatest potential for improving the outcome along cost, quality and environmental metrics. Second, these elements are employed in the formulation of a constrained optimization model that clarifies the overall objective and unavoidable feasibility constraints, and identifies design parameters upon which analytic efforts should be focused. Last, normative decision theory is embedded in the model to explicitly reflect preferences for conflicting objectives. Several case studies illustrated the approach. Normative decision analysis can help model our preferences for conflicting objectives, such as minimizing cost and minimizing environmental impact. Scientific and engineering analysis can help model the unavoidable constraints on feasible combinations of those objectives. Constrained optimization can help identify the best design decisions, the ones that maximize utility while making tradeoffs that satisfy feasibility constraints. But none of these tools answer the really tough question: what should our preferences be for cost versus environmental impact tradeoffs? The constrained optimization model forces us to confront the environmental impacts of our current preferences, and can illuminate the need for their reevaluation.
Acknowledgments The authors are grateful for the support provided by the National Science Foundation Awards ECS-0224829 and DMI0217491.
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Received for review May 5, 2003. Revised manuscript received September 15, 2003. Accepted September 16, 2003. ES0344359
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