Dealing with High-Dimensionality of Criteria in ... - ACS Publications

Article pubs.acs.org/IECR

Dealing with High-Dimensionality of Criteria in Multiobjective Optimization of Biomass Energy Supply Network Lidija Č uček,† Jiří Jaromír Klemeš,‡ Petar Sabev Varbanov,‡ and Zdravko Kravanja*,† †

Faculty of Chemistry and Chemical Engineering, University of Maribor, Smetanova ulica 17, 2000 Maribor, Slovenia Centre for Process Integration and Intensification−CPI2, Research Institute of Chemical and Process Engineering−MŰ KKI, Faculty of Information Technology, University of Pannonia, Egyetem utca 10, 8200 Veszprém, Hungary

‡

ABSTRACT: This contribution presents a novel dimensionality reduction methoda Representative Objectives Method (ROM)applied to environmental footprints, by which the number of environmental footprints within the multiobjective optimization (MOO) is reduced to a minimum number of representative ones. The number of footprints is reduced according to similarities among those footprints showing similar behavior. The proposed method consists of three steps: (i) generation of solution points for analyzing similarities among footprints, (ii) identification of similarities among footprints, and the selection of representative footprints (those footprints that show similar behavior are grouped into subsets, each subset’s representative footprint is then selected), and (iii) the performing of MOO for maximizing profit with respect to the representative footprints. In this way, the dimensionality of the criteria within the MOO can be significantly reduced. Rather than obtaining the remaining footprints through correlations among the representative “independent” footprints, they are read directly from Pareto solutions. The presented dimensionality reduction method is applicable in cases when the model is known. The presented approach is illustrated using a demonstration case study of different biomass energy supply chains. The similarities among carbon (CF), energy (EF), water (WF), water pollution (WPF), and land footprints (LF) were investigated, from which only two representative footprints, CF and WF, were selected. This case study indicated that using this novel approach makes MOO more practical for real life problems. (SEPI),8 the sustainable process index (SPI),12 and the waste reduction (WAR) algorithm,13 to name but a few. However, all approaches, even these, have some drawbacks. One of them is that the weighted sum methods are based on subjective weighting and possible difficulty when selecting the best solution. The SPI and SEPI have difficulties, as ecological footprints, when converting them to be expressed as area units. For simplicity, just one objective is often considered in many studies, and evaluated besides an economic criterion, such as greenhouse gas (GHG) emissions,14 environmental sustainability index,10 reliability,15 and responsiveness of the system,16 among others. However, more realistic solutions can be obtained if more impacts are considered, and therefore, in this respect, MOO is becoming increasingly important.17 An important limitation in this case is that the computational burden increases rapidly in size with the number of objectives.18 Other limitations are that MOO can be time-consuming, and there are often difficulties when visualizing and interpreting the objective spaces.19 It also prevents the carrying-out of an exact optimization, resulting in only two, three, or, at most, fourdimensional (4D) Pareto projections, thus providing only a narrow view producing underestimated environmental metric estimates.20 Reduction of the MOO dimensionality is thus

1. INTRODUCTION Economic, environmental, and social challenges are becoming increasingly important within numerous areas of everyday life, due to less sustainable human lifestyles over recent decades. The solution to these various challenges is “to pave a pathway” toward more sustainable development that should encompass the integration of economic, environmental, and social components at all levels, thus requiring complex multiobjective optimization (MOO).1 MOO would even deal with scarce and uncertain data.1 In order to progress toward sustainability and sustainable development, appropriate methods should be used for measuring sustainability and sustainable development.2 Yet the defining and usage of a suitable sustainability metric for support objective sustainability assessments is still an open issue in the literature.3 Plenty of indicators have been developed for measuring economic, environmental, and social issues,4 and among which footprints of various types play an important role.5 Footprints are usually measured in units of area, such as, e.g., an ecological footprint.6 However, data expressed in units of area display high variability and highly inherited inaccuracy since they could be based on a variety of different assumptions.7 Converting some of the footprints into area units can thus prove to be problematic, especially for processes that are not primarily area-based.8 One of the common ways of presenting the indicator data is by plotting them on a spider diagram.9 Often, the number of different objectives is reduced by aggregating them within an aggregated single sustainability indicator, using the weighted sum method,10 the geometric mean of the applied indicators’ ratios,11 the sustainable environmental performance indicator © 2013 American Chemical Society

Special Issue: PSE-2012 Received: Revised: Accepted: Published: 7223

September 24, 2012 February 5, 2013 February 11, 2013 February 11, 2013 dx.doi.org/10.1021/ie302599c | Ind. Eng. Chem. Res. 2013, 52, 7223−7239

Industrial & Engineering Chemistry Research

Article

energy supply chains27 extended for simultaneous assessment of footprints.28

required, without compromising the qualities of the Pareto solutions. The reductions and aggregations of different objectives have so far mostly relied on the decision-makers’ preferences.18 Yet, it should be based on a systematic mathematical approach. Various methods have been developed for this purpose that are generally categorized into linear and nonlinear methods.21 The more widely used linear methods are principal component analysis (PCA) and factor analysis, which are second-order methods, i.e. they rely only on information contained within the covariance matrix.21 Among the higher-order linear methods are projection pursuit and independent component analysis. Different methods are also used in the case of nonlinearity, such as nonlinear PCA, nonlinear principal curves, multi-D scaling, and topologically continuous map and vector quantization. A review of the different methods was conducted by Lygoe.21 Several research papers have addressed the reduction of dimensionality regarding MOO problems. A few of them are analyzed here. Deb and Saxena19 proposed an evolutionary MOO procedure, and a PCA-based evolutionary MOO procedure. Saxena and Deb22 further proposed two nonlinear dimensionality reduction algorithms for evolutionary MOO, one based on correntropy (technique based on the generalized correlation function) and one based on the maximum variance unfolding principle (a method based on proximity matrices). Brockhoff and Zitzler23 determined the minimum objective subset and calculated an approximation error for quantifying to what extent the dominance structure of the problem changes when omitting objectives. Pozo et al.3 used a method based on PCA for identifying the redundant environmental metrics within MOO. Guillén-Gosálbez18 developed a mixed-integer linear programming (MILP) based method, which minimizes the errors when omitting objectives. He demonstrated that some of objectives behave in a nonconflicting manner, thus reducing the dimension of the MOO problem. The emphasis of his work was on environmental problems and on minimizing a set of life cycle assessment (LCA) metrics. Vaskan et al.24 applied the MILP method for the optimal design of heat exchanger networks by considering environmental impacts. Gutiérrez et al.25 used PCA and multi-D scaling methodology in order to reduce the dimension of the problem. Lygoe et al.26 proposed a systematic and modular MOO decision making process; including clustering, PCA, and an efficient search method for dimension reduction. This contribution presents a MOO approach based on the novel Representative Objectives Method (ROM), by which those footprints that show similarities are grouped into subsets, each subset being then represented within the MOO by its corresponding representative footprint. The proposed method can be applied for problems when the model and matrix coefficients for evaluating footprints from optimization variables are known. Different environmental footprints are considered, and different measurements are proposed for determining the similarities among the footprints and selecting the representative ones: (i) normalized ratios between pairs of footprints, (ii) overlapping of footprint pairs regarding process variables, and (iii) average absolute normalized deviation between pairs of footprints. Optimistic and pessimistic scenarios are then considered with respect to the profit and environmental burdens when handling the remaining nonrepresentative footprints. This novel approach is illustrated using a demonstration case study by synthesizing biomass

2. DESCRIPTION OF THE REPRESENTATIVE OBJECTIVES METHOD The dimensionality reduction by the proposed ROM for objectives o ∈ O, comprised of annual profit P and different environmental footprints Ff,k(x), f ∈ FP, is only performed for footprints, and is solved over three main steps. First, the points are generated for when considering similarities among the footprints. Then, the subsets of similar footprints are identified after calculating the above-mentioned measurements using the generated points, and the representative footprints are selected, one from each subset. In the third step, the maximization of profit in MOO is performed versus the selected representative footprints using the ε-constraint method and the values of the remaining footprints are read directly from the obtained multiD Pareto solutions. The proposed approach for reducing the dimensionality of the criteria and the corresponding MOO are described below. 2.1. Generation of Points Used for Analyzing Similarities among Footprints. The environmental footprints Ff,k(x), f ∈ FP are obtained using the following equation: Ff , k(x) =

∑ a f ,v·xv ,k

∀f ∈ FP , ∀ k ∈ K (1)

v∈V

where af,v are the matrix coefficients and xv,k are the corresponding process variables at iteration k ∈ K. Note that the matrix coefficients correspond to specific environmental footprints. Both, the coefficients and values of the process variables are used for the calculating of different measurements when identifying similarities among footprints. As the measurements should reflect the obtaining of maximal profits at various values of footprints, the multicriteria approach should be applied along the whole range of footprints. The mixed-integer (non)linear programming (MI(N)LP) synthesis model from previous work,27 including different environmental footprints,28 is solved within MOO by maximizing profit as the main objective, while the footprints are constrained by ε when applying the ε-constraint method.29 The footprints at iteration k are normalized in order to adjust their values to a common scale. Relative footprints (Frf,k(x)) are obtained in this way and are defined as footprints obtained by optimization (Ff,k(x)) divided by the footprints obtained at maximum profit solution, where the footprints are relaxed (F0f (x)):

F rf , k(x) =

Ff , k(x) F 0f (x)

(2)

In order to keep the problem at a manageable level, 2D MOO is applied and the following (2D MI(N)LP)k problem defined at iteration k: max Pk = c Ty + f (x) − x ,y

7224

∑

w·Ff (x)

f ∈ FP dx.doi.org/10.1021/ie302599c | Ind. Eng. Chem. Res. 2013, 52, 7223−7239


Article

Note, that geometric means are applied in order that smaller normalized ratios contribute more than larger values. For perfect similarity, the normalized ratio between pairs of footprints is 1. (ii) Overlapping Pairs of Footprints30 ( f and f f) within the Process Variables for the Selected Optimal Point xk. a f , v·xv , k Of , ff , k = ∑ Ff , k v ∈ V ,a f , v ≠ 0 ∧ a ff , v ≠ 0

s.t. Ay + h(x , y) = 0 By + g (x , y) ≤ 0

(2D MI(N)LP)k

F rf , k(x) ≤ εk ,

∀ f ∈ FP

(x LO ≤ x ≤ x UP) ∈ X ⊂ Rn , y ∈ Y = {0, 1}m

where, during the sequence of MI(N)LPs, all the footprints are simultaneously forced to decrease sequentially from their maximal values (εk = 1) by a suitable step-size Δε; until there is no feasible solution: εk − 1 = εk − Δε ,

Δε =

1 , |K | − 1

k∈K

∀ (f ∈ FP , ff ∈ FP , k ∈ K )

Because Of,ff,k can be different from Off,f,k, geometric mean is calculated as GOf , ff , k =

(3)

Note that x denotes a vector of continuous variables (flowrates, design variables, etc.), while y is the vector of binary decision variables or discrete decisions (existence of a particular stream, process unit, technology, supply chain, etc.). Function f(x) is a continuous (non)linear function involved in objective function, and h(x,y) and g(x,y) are continuous equality (mass and energy balances, design equations, etc.) and inequality (product specifications, environmental constraints, etc.) constraint functions. Note also that all footprints within the objective function are simultaneously minimized to provide solutions with the least values for those footprints in cases where multiple footprint solutions exist. The weight w should be very small in order not to interfere with the maximization of profit, e.g., 10−6. In this way |K| Pareto optimal solutions xk are obtained in terms of the overall environmental burden. As all the footprints are simultaneously constrained, the solutions are located below all footprint-individual Pareto curves. As profit decreases linearly with footprints in most cases, the number of selected Pareto optimal solutions can be kept considerably low, e.g. 10−20. 2.2. Identifications of Similarities among Footprints. Three partitioning criteria for identifying similarities among footprints are proposed. (i) Normalized Ratios Between Pairs of Footprints30 (f and ff) for Selected Optimal Point xk. R f , ff , k

Om f , ff =

R f , ff , k ·R ff , f , k

GR m f , ff =

|K | ∑k ∈ K GR f , ff , k |K |

∀ (f ∈ FP , ff ∈ FP)

∀ (f ∈ FP , ff ∈ FP)

|K | ∑k ∈ K GOf , ff , k

∀ (f ∈ FP , ff ∈ FP)

|K |

⎛ ∑v ∈ V ,a ≠ 0 ⎜ ⎜ ff , v ⎝

Df , ff , k =

(∑v ∈ V ,a

1−

a f ,v a ff , v

Ff , k Fff , k

ff , v ≠ 0

·

a f , v·xv , k

(10)

(11)

Ff , k

⎞ ⎟ ⎟ ⎠

1) − 1

∀ (f ∈ FP , ff ∈ FP , k ∈ K )

(12)

Small values of Df,ff indicate good agreement between pairs of footprints. Now, smaller values should contribute as equally as larger values. The arithmetic mean is thus calculated between Df,f f and Dff,f: ADf , ff , k = (Df , ff , k + Dff , f , k )/2 ∀ (f ∈ FP , ff ∈ FP , k ∈ K )

(13)

Finally, their mean values for the generated points are (4)

Dm f , ff =

∑k ∈ K Df , ff , k

ADm f , ff =

∀ (f ∈ FP , ff ∈ FP , k ∈ K )

∀ (f ∈ FP , ff ∈ FP)

(9)

If footprint f is defined by the same process variables as footprint ff, then the overlap coefficient is 1. (iii) Average Absolute Normalized Deviation between Pairs of Footprints30 ( f and ff) for Selected Optimal Point xk

|K | ∑k ∈ K ADf , ff , k |K |

∀ (f ∈ FP , ff ∈ FP)

∀ (f ∈ FP , ff ∈ FP)

(14)

(15)

Based on the values of above partitioning criteria, footprints f ∈ FP are partitioned into NS subsets s ∈ S of similar footprints fss ∈ FSs, each subset being composed of one representative footprint f rs ∈ FSfrs and remaining unrepresentative footprints fr fu f us ∈ FSfu s ; FSs ∪ FSs = FSs. Thus, NS representative footprints f rs, s ∈ S are identified (ideally two or three). In a given subsets, those footprints are selected, that have the normalized ratios close to one. Overlap values and average absolute normalized deviations have to be checked if they are close to 1 and 0. One representative footprint is selected from among footprints within a given subset, either based on their priority or their

and then the mean values for the generated points are finally obtained, which are calculated as the ratios between pairs of footprints (and its geometric mean) divided by the number of iterations k (cardinality of a set K): ∑k ∈ K R f , ff , k

∑k ∈ K Of , ff , k

GOm f , ff =

(5)

Rm f , ff =

∀ (f ∈ FP , ff ∈ FP)

and

Because Rf,f f,k can differ from Rf f,f,k, geometric mean is calculated: GR f , ff , k =

Of , ff , k ·Off , f , k

The mean values then are

⎛ a f ,v ⎞ ⎜ a ff ,v a f , v·xv , k ⎟ = ∑ ⎜F ⎟ f ,k Ff , k ⎟ v ∈ V ,a ff , v ≠ 0 ⎜ ⎝ Fff ,k ⎠

∀ (f ∈ FP , ff ∈ FP , k ∈ K )

(8)

(6)

(7) 7225

dx.doi.org/10.1021/ie302599c | Ind. Eng. Chem. Res. 2013, 52, 7223−7239


Article

overlap values. In a situation where the above criteria do not yield the sufficient and desired similarities among the footprints, more subsets with more representative footprints should be selected, up to the number of evaluated footprints. 2.3. Multiobjective Optimization. In the last step, a MOO is performed for NS selected representative footprints, f rs, s ∈ S, where the main criterion is the maximization of the profit. The following (MI(N)LP)k ,..., k problem is defined max Pk1,..., k N = c y + f (x) − x ,y

S

∑

w·Ff , k(x)

f ∈ FP

s.t. Ay + h(x , y) = 0 By + g (x , y) ≤ 0 F rfrs , ks(x)

(MI(N)LP)k1,..., k N

s

≤ εs , ks ,

∀ frs ∈ FSsfr , s ∈ S (x LO ≤ x ≤ x UP) ∈ X ⊂ Rn , y ∈ Y = {0, 1}m

The ε-constraint method is applied to this MOO problem, with representative footprints being the varying parameters within the loops. The sequences of single-objective (MI(N)LP)k ,..., k 1

δoR, k =

NS

δoC, k =

μoR =

μoC =

okR

∀ (k ∈ K , o ∈ O , okR ≠ 0, okC ≠ 0)

,

∑k ∈ K , o R ≠ 0, o C ≠ 0 δoR, k k

k

NK

∑k ∈ K , o R ≠ 0, o C ≠ 0 δoC, k k

k

NK

∀o∈O

(21)

∀o∈O

(22)

where NK is the total number of obtained feasible solutions from iterations K (NK = Σk∈K,ORk ≠0,OCk ≠0 1). Standard deviation is calculated using eq 23 for relaxed and eq 24 for constrained solutions regarding each feasible objective: σoR =

σoC

unconstrained during the MOO, thus giving rise to more relaxed Pareto solutions in terms of environmental burdens and profit (optimistic solutions). In the second constrained scenario, the whole subset of footprints FSsC is now simultaneously constrained to the same εs (eq 17) and to the corresponding values obtained at the relaxed optimistic scenario, FSRs (eq 18):

)

okC − okR

Finally, the means of errors are calculated using eq 21 for optimistic relaxed solutions and using eq 22 for pessimistic constrained solutions for each feasible objective:

NS

(

∀ (k ∈ K , o ∈ O , okR ≠ 0, okC ≠ 0)

,

(20)

(16)

∀ fss ∈ FSs , s ∈ S

okC

The deviations associated with pessimistic constrained solutions with respect to the corresponding optimistic relaxed solutions are similarly defined as

A finer Pareto front is obtained when Δεs values are smaller, e.g. 1%. An NSD, mostly 3D graph of Pareto optimal solutions is thus obtained. The number of optimization runs depends on the number of representative footprints and step-sizes Δεs, and is equal to Πs∈S(|Ks|); e.g., for two representative footprints and a selected Δεs of 1%, the number of required optimization runs k ∈ K being about 10 000. The remaining footprints can be read directly from the MOO solutions and can be presented together with the selected representative footprints on the NSD plots. In this way, the exact solutions are obtained, and any errors associated with the calculations of “dependent” footprints from their correlations with the “independent” ones, are thus avoided. Two scenarios are then considered with respect to handling the remaining nonrepresentative footprints. In the first one, as in problem (MI(N)LP)k ,..., k the remaining footprints are

F rfs,C (x) ≤ εs , ks , s , ks

okR − okC

(19)

ε1, ks − 1 = ε1, ks − Δε1 , ..., (εNS , ks − 1 = εNS , ks − ΔεNS ,

1

(18)

δCo,k

the sequence of MI(N)LPs each representative footprint f rs, s ∈ S is forced to decrease sequentially from its maximal value (εs = 1) by its suitable step-size Δεs until there is no feasible solution:

k NS ∈ K Ns), ..., ks ∈ Ks

)

NS

problems are thus solved for representative footprints as the maximization of the profit subjected to relative representative footprints. NS -embedded loop statements are used to repeatedly solve the (MI(N)LP)k ,..., k problem, where during 1

(

∀ fss ∈ FSs , s ∈ S

In this way, footprints of the second scenario never exceed those of the first scenario. More rigid Pareto solutions in terms of profit and environmental burdens are thus obtained (pessimistic solutions). Which option to select depends on the decisionmakers, either moderated or restricted toward the environment. Both optimistic and pessimistic scenarios were applied within the demonstrated case study. Note that when footprints grouped within the same subsets are completely correlated, the optimistic and pessimistic solutions are the same, and the error of the dimensionality reduction is zero. If, however, they differ, their difference can be used as a measure of the error of this MOO approach. When presenting optimistic solutions, a percentage deviation δRo,k at iteration k ∈ K for the optimistic value of objective oRk obtained for the remaining relaxed footprints, with respect to the corresponding pessimistic value oCk obtained for constrained remaining footprints, is defined for nonzero feasible objectives as follows:

NS

1

T

F rfs,C (x) ≤ F rfs,R , s , ks s , ks

=

∑k ∈ K , o R ≠ 0, o C ≠ 0 (δoR, k − μoR )2 k

k

NK − 1 ∑k ∈ K , o R ≠ 0, o C ≠ 0 (δoC, k − μoC )2 k

k

NK − 1

∀o∈O (23)

∀o∈O (24)

3. DEMONSTRATION CASE STUDY The simple methodology described above was applied within the case study of regional biomass and bioenergy supply chains,27 extended for the simultaneous assessment of footprints.28 Biomass and bioenergy supply chains follows a four layer structure, which consists of harvesting (agricultural

(17) 7226



Article

Table 1. Matrix Coefficients (Specific Environmental Footprints) for the Demonstration Case Study footprint WPF (kg/(t·km2))

LF (1/t)

1.726

0.032

1.37 × 10−6

1.726

0.032

0

0

0

0

0.75

0

0

0

0

0

0.5

0

0

0.5

1.251

0

0

0.00262

0.005

0.01504

0

0

ei L2 f ,timber

0.00078

0.004

0.0108

0

0

ei L3 f ,corngrain,DGP

0.147

1.3

2.5

0

0

ei L3 f ,woodchips,incineration

0

0

0

0

0

ei L3 f ,cornstover,incineration

0

0

0

0

0

ei L3 f ,MSW,MSWincineration

0.415

0.31

0

0.0016

0

ei L3 f ,woodchips,MSWincineration

0

0

0

0

0

ei L3 f ,cornstover,MSWincineration

0

0

0

0

0

ei L3 f ,cornstover,AD

0

0.091

0

0

0

ei L3 f ,manure,AD

0

0.091

0

0

0

ei L3 f ,timber,timbersawing

0.00125

0.036

0

0

ei L4 f ,corngrain

0

0

0

0

0

ei L4 f ,heat

0

0

0

0

0

ei L4 f ,electricity

0

0

0

0

0

ei L4 f ,ethanol

0

0

0

0

0

ei L4 f ,board

0

0

0

0

0

ei L4 f ,digestate

0.017

0

0

0.0201

0

ei L4 f ,DDGS

0

0

0

0

0

matrix coefficient (af,v)

2

CF (kg/(t·km ))

2

WF (kg/(t·km ))

ei L1 f ,corngrain

0.154

900

ei L1 f ,cornstover

0.154

900

ei L1 f ,manure

0

ei L1 f ,woodchips

0.066

ei L1 f ,MSW

0

ei L1 f ,timber

0.044

ei L2 f ,corngrain

0.09

ei L2 f ,cornstover

0.75 2,500 0.229 1,500

10.6

EF (MJ/(t·km2))

footprint matrix coefficient (af,v)

CFa (kg/(t·km3))

WF (kg/(t·km3))

EF (MJ/(t·km3))

WPF (kg/(t·km3))

LF (1/(t·km))

,L1,L2 ei trf ,corngrain

0.000053

0.000136

0.000389

0

0

,L1,L2 ei trf ,woodchips

0.00024

0.00049

0.0014

0

0

,L1,L2 ei trf ,MSW

0.000113

0.00056

0.0016

0

0

,L1,L2 ei trf ,cornstover

0.0011

0.00233

0.00667

0

0

,L1,L2 ei trf ,manure

0.000053

0.0001

0.00028

0

0

,L1,L2 ei trf ,timber

0.000053

0.000245

0.0007

0

0


0.000008

0.000073

0.000208

0

0

,L2,L3 ei trf ,woodchips

0.000008

0.000263

0.00075

0

0

,L2,L3 ei trf ,MSW

0.00013

0.00056

0.0016

0

0

,L2,L3 ei trf ,cornstover

0.000008

0.00125

0.00357

0

0

,L2,L3 ei trf ,manure

0.000053

0.0001

0.00028

0

0

7227



Article

Table 1. continued footprint a

CF (kg/(t·km ))

WF (kg/(t·km ))

EF (MJ/(t·km3))

WPF (kg/(t·km3))

LF (1/t·km)

,L2,L3 ei trf ,timber

0.000008

0.000131

0.000375

0

0


0.00001

0.00008

0.000264

0

0

,L3,L4 ei trf ,heat

0

0

0

0

0

,L3,L4 ei trf ,electricity

0

0

0

0

0

,L3,L4 ei trf ,ethanol

0.000027

0.000124

0.00035

0

0

,L3,L4 ei trf ,DDGS

0.000053

0.0001

0.00028

0

0

,L3,L4 ei trf ,board

0.000008

0.000131

0.000375

0

0

,L3,L4 ei trf ,digestate

0.000053

0.0001

0.00028

0

0

matrix coefficient (af,v)

3

3

a

The CF of rail transport varies significantly depending on the traction, diesel, electric, or diesel−electric traction. The CF from electricity production varies considerably between countries according to the share of power plant technologies used. For this reason, and because in the previous reference by authors27 all biomass and bioproducts had the same values for CF, also here the same values for CF have been assumed. More or less European average conditions are considered.33

layerL1), storage and pretreatment (preprocessing layer L2), conversion steps (processing layerL3), and usage of products (use layerL4), including transportation flows within and between the layers. The total area of the region was assumed to be 1,000 km2. Several biomass types, technology options, and bioproducts were considered within the synthesis. Different biomass types were utilized: corn grain, corn stover, wood chips, municipal solid waste (MSW), manure, and timber. The dry-grind process (DGP), anaerobic digestion (AD), MSW incineration, timber sawing, and incineration converted biomass into valuable products, electricity, heat, ethanol, boards, organic fertilizer, and distiller dried grains with solubles (DDGS). Besides the processed products, food could also be produced. Supply chains incorporated different environmental pressures and considered different footprints, carbon (CF), energy (EF), water (WF), water pollution (WPF), and land footprints (LF). The model formulated as an MILP within the General Algebraic Modeling System (GAMS)31 consisted of 632 equations, 1,064 single variables, and 21 binary variables. Its single-objective problem was solvable in a fraction of a second by MILP solver CPLEX.32 It should be noted that, as this model was defined as an MILP and was applied without any optimality gap, the obtained solutions are globally optimal. 3.1. Selection of Points Used for Obtaining Similarities among the Footprints. Identifications of similarities among footprints were performed directly from the matrix of process variables and footprints. The matrix coefficients are shown in Table 1. The specific environmental footprints in Table 1 represent

ei L3 f , pi , t , (pi , t ) ∈ PT , PT ⎧(corn grain, DGP), (wood chips, incineration), ⎫ ⎪ ⎪ ⎪ (corn stover, incineration), (MSW, ⎪ ⎪ MSW incineration), (wood chips, ⎪ ⎬ =⎨ ⎪ MSW incineration), (corn stover, ⎪ ⎪ MSW incineration), (corn stover, AD), ⎪ ⎪ ⎪ ⎩ (manure, AD), (timber, sawing) ⎭

footprint f of intermediate product pi and the selected technology t at L3 in kg/(t·km2), MJ/(t·km2) or 1/t; ei L4 f , p , p ∈ P , P = {corn grain, heat, electricity, ethanol, board, digestate, DDGS}

footprint f of product p at L4 in kg/(t·km2), MJ/(t·km2) or 1/t; ei trf , pi,L1,L2 , pi ∈ PI , PI = {corn grain, corn stover, wood chips, MSW, manure, timber}

footprint f from transportation of intermediate products pi from L1 to L2 in kg/(t·km3), MJ/(t·km3) or 1/(t·km); ei trf , pi,L2,L3 , pi ∈ PI , PI = {corn grain, wood chips, MSW, corn stover, manure, timber}

footprint f from transportation of intermediate products pi from L2 to L3 in kg/(t·km3), MJ/(t·km3) or 1/(t·km);

ei L1 f , pi , pi ∈ PI , PI = {corn grain, corn stover, manure,

,L2,L4 ei trf , pd , pd ∈ PD , PD = {corn grain}

wood chips, MSW, timber}

footprint f from transportation of direct products pd from L2 to L3 in kg/(t·km3), MJ/(t·km3) or 1/(t·km);

footprint f of intermediate product pi at L1 in kg/(t·km2), MJ/ (t·km2) or 1/t;

,L3,L4 ei trf , pp , pp ∈ PP , PP = {heat, electricity, ethanol,

ei L2 f , pi , pi ∈ PI , PI = {corn grain, corn stover, timber}

board, digestate, DDGS}

footprint f of intermediate product pi at L2 kg/(t·km2), MJ/ (t·km2) or 1/t;

footprint f from transportation of produced products pp from L3 to L4 in kg/(t·km3), MJ/(t·km3) or 1/(t·km). 7228



Article

Figure 1. Profit versus direct specific relative footprints.

420.11, while Rm WF,CF = 2.47. The selection into subsets could be achieved either from normalized average ratios (Table 4a) or from their geometric means (Table 4b). From both tables it can be seen that two or three groups could be selected. With a higher tolerances for deviations from 1 (e.g., ± 0.25 for geometric means in Table 4b), two groups were chosen, the first group included CF and EF (highlighted by the light gray color), and the second group WF, WPF, and LF (highlighted by both the medium and darker gray colors). However, with a smaller tolerances in deviations three groups were chosen, the first group included CF and EF; the second group, WF and WPF (highlighted by the darker gray color); and the third group, LF only. Table 5 presents (a) the average results obtained from the second criteria, overlap of pairs of footprints in process variables, and (b) their geometric means. If the overlaps of pairs of footprints f and ff in the process variables was equal to 1, then footprint f was defined by the same process variables as footprint ff. Again, overlap values Om f,f f m could differ significantly from Om f f,f , e.g. OCF,LF = 0.31, while Om LF,CF = 1.00. From Table 5, it can be seen that in the case of smaller tolerances and three subsets, the overlap values for the similar footprints were equal or very close to 1.00. In the case of larger tolerances and two subsets, the overlap values for other similar footprints and LF in the last subset were quite large (Table 5a), and in within ±0.25 range in the case of geometric mean values (Table 5b). From Table 5b, it can be seen that two groups can be selected in two possible ways: (i) the first group can contain CF and EF and the second group can contain WF, WPF, and LP and (ii) the first group can contain CF, WF, and EF, and the second group contains WPF and LF. In the first, representative footprints cannot be chosen in the first group, since in this group there are only two footprints, which have the same overlap coefficients. However, from the second group, WPF or WF could be chosen as representative footprints with the same interpretation applied as in Table 4. In the second, WF was selected as a representative footprint in the first group, since all coefficients equaled 1. From the second group, a representative footprint could not be chosen since in this group there were only two footprints that had the same overlap coefficients. Table 6 presents (a) the average absolute normalized deviations between pairs of footprints and (b) their arithmetic means.

Also, it should be noted that the data in the literature vary significantly, since footprints are specific for each system. Footprints differ for the same system regarding location, time, technology, composition etc. Proper evaluation of footprints is therefore impossible without measuring them. This case is just a demonstration case study. However, the methodology can be applied for any regional biomass energy supply chain by only changing the data, since the model for synthesizing regional biomass supply chains and the ROM for dealing with the highdimensionality of criteria in MOO is data-independent. Maximization of profit with relaxed footprint constraints was carried out first. The solution obtained corresponded to the one with maximal footprints. Those maximal values were set as reference footprints F0f and when normalized, relative footprints Frf,k were obtained and set to 1. The problem (2D MI(N)LP)k, introduced in section 2.1, was then applied in order to generate those Pareto solutions (green curve in Figure 1) used for analyzing similarities among the footprints. Then, the corresponding variables xv,k were selected at the optimal values of the footprints when maximizing the profit. In order to present the solution more easily at a smaller size, only several points along the Pareto curves were selected, namely at Frf,k = 1, 0.8, 0.6, 0.4, and 0.2. Note that all the values at Frf,k = 0 were 0. The selected Pareto solutions are presented in Table 2. Individual footprints’ Pareto curves are also presented in Figure 1. As expected, they are all located above the green curve. The environmental footprints could be calculated using eq 1 by applying data from Tables 1 and 2. Environmental footprints obtained at the selected Pareto points are presented in Table 3. Note that the values obtained at Frf,k = 1 were the reference footprints F0f . 3.2. Identifications of Similarities among the Footprints. Applying proposed measurements (i)−(iii) for determining the similarities among the footprints (eqs 4−15) at five different points, at Frf,k = 1, 0.8, 0.6, 0.4, and 0.2, their arithmetic mean values were calculated and are presented in Tables 4−6. Table 4 presents (a) the average results obtained from the first criteria, comparisons between normalized ratios regarding pairs of footprints, and (b) their geometric means. The ratio between footprints f and f f was equal to 1 for perfect similarity. In Table 4a, it can be seen that when the values for normalized ratios between footprints were not close m m to 1, Rm f,ff could differ significantly from Rff,f , e.g., RCF,WF = 7229



Article

Table 2. Process Variables at Selected Pareto Solutions footprints at Frf,k process variable (xv,k)

1.0

0.8

0.6

0.4

0.2

2.34 × 105

1.87 × 105

1.40 × 105

9.34 × 104

4.67 × 104

1.41 × 105

1.13 × 105

8.44 × 104

5.63 × 104

2.82 × 104

7.30 × 103

0

0

7.30 × 103

7.30 × 103

2.21 × 103

2.21 × 103

2.21 × 103

2.21 × 103

2.21 × 103

2.12 × 104

2.12 × 104

1.65 × 104

0

0

2.21 × 104

0

0

6.64 × 103

0

qim, m,L1,L2 , t/y ,corngrain

2.34 × 105

1.87 × 105

1.40 × 105

9.34 × 104

4.67 × 104

qim, m,L1,L2 , t/y ,cornstover

1.41 × 105

1.13 × 105

8.45 × 104

5.63 × 104

2.82 × 104

qim, m,L1,L2 , t/y ,timber

2.21 × 104

0

0

6.64 × 103

0

1.92 ·105

1.49 × 105

1.10 × 105

7.70 × 104

3.85 × 104

2.65 × 103

2.65 × 103

2.65 × 103

2.65 × 103

2.65 × 103

1.39 × 105

1.13 × 105

8.44 × 104

5.45 × 104

2.63 × 104

2.12 × 104

2.12 × 104

1.65 × 104

0

0

,T,L2,L3 qnm,woodchips,MSWincineration , t/y

0

0

0

0

0

,T,L2,L3 qnm,cornstover,MSWincineration , t/y

0

0

0

0

0

1.83 × 103

0

0

1.83 × 103

1.83 × 103

7.30 × 103

0

0

7.30 × 103

7.30 × 103

1.77 × 104

0

0

5.31 × 103

0

0

5.52 × 103

5.80 × 103

0

0

1.25 × 109

1.02 × 109

7.71 × 108

4.79 × 108

2.54 × 108

2.43 × 105

1.97 × 105

1.50 × 105

9.29 × 104

4.92 × 104

6.22 × 104

4.79 × 104

3.54 × 104

2.48 × 104

1.24 × 104

9.50 × 103

0

0

2.86 × 103

0

3.65 × 103

0

0

3.65 × 103

3.65 × 103

4.81 × 104

3.72 × 104

2.74 × 104

1.92 × 104

9.62 × 103

1.50 × 106

1.10 × 106

7.58 × 105

5.16 × 105

2.21 × 105

,L1 , t/y ∑ qim,corngrain i∈I ,L1 , t/y ∑ qim,cornstover i∈I ,L1 , t/y ∑ qim,manure i∈I ,L1 , t/y ∑ qim,woodchips i∈I ,L1 , t/y ∑ qim,MSW i∈I ,L1 , t/y ∑ qim,timber i∈I

∑∑ i∈I m∈M

∑∑ i∈I m∈M

∑∑ i∈I m∈M

∑

,T,L2,L3 qnm,corngrain,DGP , t/y

∑

n ∈ N (corngrain,DGP) ∈ PT

∑

,T,L2,L3 qnm,woodchips,incineration , t/y

∑

n ∈ N (woodchips,incineration) ∈ PT

∑

,T,L2,L3 qnm,cornstover,incineration , t/y

∑

n ∈ N (cornstover,incineration) ∈ PT

∑

,T,L2,L3 qnm,MSW,MSWincineration , t/y

∑

n ∈ N (MSW,MSWincineration) ∈ PT

∑

∑

n ∈ N (woodchips,MSWincineration) ∈ PT

∑

∑

n ∈ N (cornstover,MSWincineration) ∈ PT

∑

,T,L2,L3 qnm,cornstover,AD , t/y

∑

n ∈ N (cornstover,AD) ∈ PT

∑

∑

,T,L2,L3 qnm,manure,AD , t/y

n ∈ N (manure,AD) ∈ PT

∑

∑

,T,L2,L3 qnm,timbersawing , t/y

n ∈ N (timbersawing) ∈ PT

, t/y ∑ ∑ qnm,j,L2,L4 ,corngrain m∈M j∈J

, MJ/y ∑ ∑ qnm,j,L3,L4 ,heat n∈N j∈J

, MWh/y ∑ ∑ qnm,j,L3,L4 ,electricity n∈N j∈J

, t/y ∑ ∑ qnm,j,L3,L4 ,ethanol n∈N j∈J

, t/y ∑ ∑ qnm,j,L3,L4 ,board n∈N j∈J

, t/y ∑ ∑ qnm,j,L3,L4 ,digestate n∈N j∈J

, t/y ∑ ∑ qnm,j,L3,L4 ,DDGS n∈N j∈J

∑∑

road,L1,L2 m ,L1,L2 DiL1,L2 ·qi , m ,corngrain , (t ·km)/y , m ·f i , m

i∈I m∈M

7230



Article

Table 2. continued footprints at Frf,k process variable (xv,k)

∑∑

road,L1,L2 m ,L1,L2 DiL1,L2 ·qi , m ,woodchips , (t ·km)/y , m ·f i , m

1.0

0.8

0.6

0.4

0.2

2.14 × 104

2.14 × 104

3.03 × 104

2.14 × 104

3.54 × 104

3.67 × 105

3.67 × 105

2.95 × 105

0

0

9.19 × 105

6.67 × 105

4.57 × 105

3.00 × 105

1.33 × 105

1.99 × 105

0

0

1.99 × 105

1.82 × 105

2.17 × 105

0

0

6.12 × 104

0

2.02 × 106

2.08 × 106

1.54 × 106

1.24 × 106

1.90 × 105

1.04 × 104

2.95 × 104

6.34 × 103

3.69 × 104

5.30 × 103

1.11 × 105

1.11 × 105

3.31 × 104

0

0

6.30 × 105

6.73 × 105

3.73 × 105

3.13 × 105

6.00 × 104

1.46 × 104

0

0

1.46 × 104

2.92 × 104

1.42 × 105

0

0

1.06 × 104

0

0

1.56 × 104

1.64 × 104

0

0

1.82 × 1010

1.33 × 1010

8.05 × 109

1.30 × 109

6.66 × 108

5.17 × 106

3.72 × 106

2.21 × 106

4.51 × 105

1.29 × 105

1.93 × 106

1.92 × 106

1.40 × 106

8.53 × 105

4.01 × 105

2.47 × 104

0

0

7.43 × 103

0

9.49 × 103

0

0

9.49 × 103

1.03 × 104

1.02 × 106

1.05 × 105

7.75 × 104

5.00 × 104

2.50 × 104

i∈I m∈M

∑∑

road,L1,L2 m ,L1,L2 DiL1,L2 ·qi , m ,MSW , (t ·km)/y , m ·f i , m

i∈I m∈M

∑∑

road,L1,L2 m ,L1,L2 DiL1,L2 ·qi , m ,corn stover , (t ·km)/y , m ·f i , m

i∈I m∈M

∑∑

road,L1,L2 m ,L1,L2 DiL1,L2 ·qi , m ,manure , (t ·km)/y , m ·f i , m

i∈I m∈M

∑∑

road,L1,L2 m ,L1,L2 DiL1,L2 ·qi , m ,timber , (t ·km)/y , m ·f i , m

i∈I m∈M road,L2,L3 m ,L2,L3 ·qm , n ,corngrain , (t ·km)/y ∑ ∑ DmL2,L3 , n ·f m, n m∈M n∈N road,L2,L3 m ,L2,L3 ·qm , n ,woodchips , (t ·km)/y ∑ ∑ DmL2,L3 , n ·f m, n m∈M n∈N road,L2,L3 m ,L2,L3 ·qm , n ,MSW , (t ·km)/y ∑ ∑ DmL2,L3 , n ·f m, n m∈M n∈N road,L2,L3 m ,L2,L3 ·qm , n ,cornstover , (t ·km)/y ∑ ∑ DmL2,L3 , n ·f m, n m∈M n∈N road,L2,L3 m ,L2,L3 ·qm , n ,manure , (t ·km)/y ∑ ∑ DmL2,L3 , n ·f m, n m∈M n∈N road,L2,L3 m ,L2,L3 ·qm , n ,timber , (t ·km)/y ∑ ∑ DmL2,L3 , n ·f m, n m∈M n∈N

road,L2,L4 m ,L2,L4 ·qm , j ,corngrain , (t ·km)/y ∑ ∑ DmL2,L4 , j ·f m, j m∈M j∈J

·f nroad,L3,L4 ·qnm, j,L3,L4 , (MJ·km)/y ∑ ∑ DnL3,L4 ,j ,j ,heat n∈N j∈J

·f nroad,L3,L4 ·qnm, j,L3,L4 , (MWh·km)/y ∑ ∑ DnL3,L4 ,j ,j ,electricity n∈N j∈J

·f nroad,L3,L4 ·qnm, j,L3,L4 , (t ·km)/y ∑ ∑ DnL3,L4 ,j ,j ,ethanol n∈N j∈J

·f nroad,L3,L4 ·qnm, j,L3,L4 , (t ·km)/y ∑ ∑ DnL3,L4 ,j ,j ,board n∈N j∈J

·f nroad,L3,L4 ·qnm, j,L3,L4 , (t ·km)/y ∑ ∑ DnL3,L4 ,j ,j ,digestate n∈N j∈J

·f nroad,L3,L4 ·qnm, j,L3,L4 , (t ·km)/y ∑ ∑ DnL3,L4 ,j ,j ,DDGS n∈N j∈J

when the smaller tolerances and three subsets were selected. When the larger tolerances were considered and two subsets were selected, the values for the absolute normalized deviations and their arithmetic means, were again close to 0, except for WPF/LF (ADm WPF,LF = 0.15 in Table 6b), which was still close to 0. From Tables 4−6, it could be concluded that with a smaller tolerance three subsets of footprints could be selected (first group CF and EF; second group WF and WPF; and third group LP), and with larger tolerances only two subsets of footprints could be chosen (first group CF and EF; second group WF, WPF, and LF). In the first case, three representative footprints and, in the second case, two representative footprints had to be selected, one from each subset. Let us first consider the two subsets. The selections of representative footprints could be carried out either by considering quantitative criteria based on normalized ratios and overlap values or qualitative criteria. When an overlap value

Table 3. Calculated Environmental Footprints at Selected Pareto Points footprint Frf,k

CF (t/ (km2·y))

WF (t/ (km2·y))

EF (GJ/ (km2·y))

WPF (t/ (km2·y))

LF (km2/ (km2·y))

1.0 0.8 0.6 0.4 0.2

118.66 94.93 71.20 43.87 21.87

376,176.78 275,363.97 207,871.48 150,470.71 72,992.38

1,446.78 1,134.39 846.10 578.01 287.50

12.02 9.62 7.21 4.80 2.40

0.32 0.26 0.19 0.13 0.06

When checking absolute normalized deviation, Table 6 had to be considered. The smaller values indicated good agreement between pairs of footprints. Note that deviations of footprints in pairs with LF were 0, since LF had defined just one variable, the LF of corn grain. As can be seen from the table, the deviations for similar footprints in the subsets were negligible 7231



Article

Table 4. (a) Normalized Average Ratios between Pairs of Footprints and (b) Their Geometric Means

Table 5. (a) Average Overlaps of Pairs of Footprints in Process Variables and (b) Their Geometric Means

Table 6. (a) Average Absolute Normalized Deviations between Pairs of Footprints and (b) Their Arithmetic Means

conclusion could be obtained by considering qualitative judgment. As CF in biomass supply chains is probably the most important footprint, WF is currently more important34 than WPF, and LF is less important, so for these qualitative reasons CF and WF could be selected as representative footprints. It can be concluded that with a larger tolerance and two subsets of similar footprints CF could be selected as a representative footprint in the first subset (CF and EF), and WF in the second subset (WP, WPF, and LF). In the case of a smaller tolerance, LF could also be selected as a representative footprint for the third subset (LF only). 3.3. Multiobjective Optimization. In this study, MOO was only performed for two representative footprints, CF and WF, because it was more transparent, and the solution could be

was considered, smaller values meant that the similarities of the process variables between the first and the second footprints were smaller, while the similarities between the second and first were higher. This implied that the first footprints were defined with a larger number of variables to which the variables of the second footprints were just subsets. The footprint with a smaller overlap value was more suitable for selecting as a representative footprint. In the first subset of similar footprints, CF was selected as the representative footprint as Om CF,EF = 0.95 = 1.00 (Table 5a). Similarly, in the second subsets while Om EF,CF m = 0.93 and even O = 0.58, WF was selected as Om WF,WPF WF,LF m while both Om WPF,WF and OLF,WF were 1.00 (Table 5a). Note also that the normalized average ratios between representative footprints, and other footprints within the same footprint subsets, should differ at least from 1 (Table 4a). The same 7232



Article

Figure 2. Profit vs representative footprints in the optimistic scenario.

Removing outliers35 from the set of nonzero deviations, the percentage deviations of the MOO approach are shown on smaller figures as another dimension presented by color scaling. Note again that those deviations present a measure for the range of possible solutions between optimistic and pessimistic scenarios. Figure 2 presents deviation as percent profit (on the left), in CF (in the middle), and WF (on the right), while Figure 3a presents deviation in EF, Figure 3b presents deviation in WPF, and Figure 3c presents deviation in LF. As can be seen, the smallest deviations are associated with WP (μRWF = 0.72%), the medium ones are associated with profit (μRP = 3.30%) and CF (μRCF = 4.58%), and the largest are associated with WPF (μRWPF = 6.28%), LF (μRLF = 6.57%), and EF (μREF = 7.60%). From Figures 2 and 3, the whole range of feasible Pareto solution space and the deviation of optimistic vs pessimistic solution for the representative footprints can be seen. The best compromise profit−footprints solution could be selected. As the remaining footprints were not constrained, the solution space showed somewhat optimistic values for both the profit and the footprints. (b) Pessimistic Scenario. Figure 4 also shows the results for the profit vs the representative footprints obtained by MOO, where the remaining footprints were as constrained as their corresponding representative footprints; EF was constrained as CF, and WPF and LF constrained as WF, and additionally those footprints had to be either equal or lower than those obtained by the optimistic scenario. 3D projections of the

better graphically represented. A 3D problem was thus obtained; where the profit was the main criterion, and the representative footprints were constrained by ε. As a step-size Δε of 1% was selected, the number of optimization runs for the two footprints, and also the number of points within the plots was equal to 10,000. Two scenarios of MOO were performed (i) profit versus representative footprints where all the remaining footprints were relaxed (optimistic scenario) and (ii) profit versus representative footprints where the remaining footprints were constrained as their corresponding representative footprints (pessimistic scenario); EF was constrained as CF, and WPF and LF as WF. In addition, the footprints were restricted to being equal to or lower than their corresponding footprints when relaxed. The means of errors and standard deviations for optimistic (μRo and σRo ) and pessimistic scenario (μCo and σCo ) for each objective o ∈ O are calculated from eqs 21−24 and shown in 3D plots. (a) Optimistic Scenario. Figure 2 shows a 3D projection of profit vs the two representative footprints, and Figure 3 shows 3D projections of the remaining footprints as read from the MOO solutions; 3D projection of EF from the first subset vs the representative footprints is represented in Figure 3a and 3D projections of WPF and LF from the second subset are given in Figure 3b and c. 7233



Article

Figure 3. 3D projections of the remaining footprint vs WF and CF in the optimistic scenario: (a) for EF from the first subset, (b) for WPF, and (c) LF from the second subset.

between pessimistic and optimistic scenarios. The figures show the percentage deviations as compared with the optimistic scenario, as another dimension presented by color scaling. Figure 4 presents deviation in percent for profit (on the left), CF (in the middle), and WF (on the right), while Figure 5a

remaining footprints are plotted in Figure 5a for EF from the first subset and Figure 5b and c for WPF and LF from second subset. The smaller Figures 4 and 5 again show the error of the MOO approach and present the range of possible solutions 7234



Article

Figure 4. Profit vs representative footprints in the pessimistic scenario.

are shown in Figure 6 for both the optimistic and pessimistic scenarios. The main difference between employing 2D vs 3D projections is that 2D projections are subject to some uncertainty regarding the relationship between EF and CF e.g. due to the varying of WP. Ranges of up to 135 GJ/(km2·y) in EF were now obtained at certain values of CF. Note that in the case of 3D projections, unique EF values vs CF and WF are shown. In respect to the profit, the same tendencies toward lower profit values could also be seen in the 2D projections when employing the pessimistic scenario. In order to summarize the presented novel ROM method, by which the number of footprints in MOO is reduced to a minimum number of representative footprints, a procedure is briefly outlined in the following: (i) Definition of the MOO model and identification of matrix coefficients. (ii) Generation of appropriate points for determining similarities among footprints performing 2D MOO by maximizing profit vs equally constrained all footprints. (iii) Identifications of similarities among footprints using the criteria as in eqs 4−15. (iv) Groupings of the representative and remaining footprints into subsets containing footprints with similar behavior. Applying normalized average ratios between footprints as the main grouping criteria. (v) Selection of representative footprints. Identification is performed by quantitative (normalized overlaps in process variables and normalized average ratios) and qualitative judgment.

presents the deviation in EF, Figure 5b present deviation in WPF, and Figure 5c presents deviation in LF. As can be seen again, the smallest deviations are associated with WF (μCWF = −0.69%), the medium ones are associated with profit (μCP = C −3.14%) and CF (μCF = −4.24%), and the largest are associated with WPF (μCWPF = −5.52%), LF (μCLF = −5.68%), and EF (μCEF = −6.78%). The negative values indicate that objectives obtained from the pessimistic solutions are smaller than those from the optimistic solutions. As all the remaining footprints were now constrained the same way as their representative footprints, and restricted to being equal or lower than those obtained at optimistic scenario, the Pareto solution space was more constrained and showed a tendency toward lower values for profit. It can be seen from Figures 4 and 5 that the range of feasible solution space for footprints at given profits has been enlarged when compared to the optimistic scenario. As to which option to select, optimistic or pessimistic, depends on decision makers. It was suggested in favor of the optimistic scenario. Even if the remaining footprints were relaxed, giving rise to somewhat overestimated profit solutions, rather good environmental solutions could still be obtained. The remaining footprints actually had similar behavior as their corresponding representative footprints anyway, and therefore, their solutions could not be far from the optimistic solution, and they also had higher profits. Furthermore, for comparison, the results could also be presented on 2D Pareto projections. For illustration, the similarities among the footprints in the first group (EF vs CF) 7235



Article

Figure 5. 3D projections of the remaining footprints vs WF and CF in the pessimistic scenario: (a) for EF from the first subset, (b) for WPF, and (c) LF from the second subset. 7236



Article

The methodology should be further upgraded to those cases where the model is unknown. The presented approach was namely based on the matrix coefficients from the model, as well as the calculated process variables at their selected Pareto optimal values. By upgrading this methodology to unknown models, the methodology would become universally applicable.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +386 2 22 94 481. Fax: +386 2 25 27 774. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors are grateful for the financial support from the EC FP7 project “Design Technologies for Multi-scale Innovation and Integration in Post-Combustion CO2 Capture: From Molecules to Unit Operations and Integrated Plants” CAPSOL, Grant No. 282789, and from the Slovenian Research Agency (Program No. P2-0032 and PhD research fellowship contract No. 1000-08-310074).

Figure 6. EF versus CF by relaxed and constrained remaining footprints.

(vi) Performing MOO with representative footprints, by relaxing the remaining footprints. (vii) Representation of multi-D projections of profit vs representative footprints and remaining footprints vs representative footprints.

■

4. CONCLUSIONS AND FUTURE WORK Presented contribution introduced a methodology (principle and procedure) based on the novel representative footprint method for identifying similarities among different objectives (footprints) within MOO. The number of footprints was reduced through similarities among those footprints that showed similar behavior. Following the presented procedure, the dimensionality of the criteria set can be reduced significantly to a minimum of representative footprints. Different footprints were investigated, such as CF, EF, WF, WPF, and LF. The methodology offers several advantages, such as results are presented in multi-D projections for footprints directly as obtained from optimization solutions. Any errors, usually associated with correlation calculations, are thus circumvented; however, there was deviation in results from optimistic and pessimistic scenarios. Since this method deals with footprints directly, the subjective weighting of environmental footprints is thus avoided, such as for the environmental index. The method is simple and can be easily implemented. The methodology was successfully applied to a demonstration case study of biomass energy supply chains where the dimensionalities of the footprints were reduced from five to two. This case study indicated that CF and WF are probably the more generally representative footprints for biomass supply chains and resulted in 3D projections of profit, and the remaining footprints with respect to the representative ones were transparent and can be well-presented graphically. Any additional representative footprints would contribute to 4D projections that are harder to represent and comprehend. In regard to future work, the similarities among footprints should also be investigated regarding several other footprints such as nitrogen and phosphorus footprints, and the issue of biodiversity as measured by biodiversity footprints. It should be noted that only direct environmental footprints were considered during this contribution. In order to achieve more realistic solutions, the indirect (unburdening) effects should be included with the direct effects, therefore considering the total effects (burdening and unburdening).28

NOMENCLATURE

Superscripts

C = constrained remaining footprints L1 = harvesting and supply layer L2 = collection and preprocessing layer L3 = main processing layer L4 = use layer LO = lower bound r = relative R = relaxed remaining footprints road = road conditions tr = transport UP = upper bound Sets

FP = set of footprints FR = set of representative footprints FS = set of similar footprints FU = set of unrepresentative remaining footprints I = set of supply zones J = set of demand locations K = set of iterations M = set of collection and intermediate process centers N = set of process plants O = set of objectives P = set of products S = set of groups (subsets) of similar footprints T = set of technology options V = set of variables Subsets

PD = set of directly used products (subset of P) PI = set of intermediate products (subset of P) PP = set of produced products from plants (subset of P) PT = PI × T = set of pairs of intermediate products and the applicable process technology for it Indexes

f = index for footprints f f = index for footprints, other than footprint f f r = index for representative footprints 7237



Article

GRf,ff,k = geometric mean of normalized ratio between pairs of footprints f and f f for selected optimal point xk GRm f,ff = mean value of the geometric mean of the normalized ratio between pairs of footprints f and f f OCk = objective by constrained remaining footprints at iteration K ORk = objective by relaxed remaining footprints at iteration K Of,ff,k = overlap of footprints pairs f and f f in process variables for selected optimal point xk Om f,ff = mean value of the overlap re footprints pairs f and f f in process variables Rf,f f,k = normalized ratio between pairs of footprints f and ff for selected optimal point xk Rm f,f f = mean value of the normalized ratio between pairs of footprints f and f f δCo,k = delta percentage for each feasible objective o ∈ O when the remaining footprints are constrained, at each iteration k ∈K δRo,k = delta percentage for each feasible objective o ∈ O when the remaining footprints are relaxed, at each iteration k ∈ K εk = ε-constraint depending on iteration k εs,k = ε-constraint for each representative footprint depending on iteration k μCo = means of errors for constrained solutions for each feasible objective o ∈ O μRo = mean of errors for relaxed solutions for each feasible objective o ∈ O σCo = standard deviation for constrained solutions for each feasible objective o ∈ O σRo = standard deviation for relaxed solutions for each feasible objective o ∈ O

fs = index for similar footprints f u = index for unrepresentative remaining footprints i = index for supply zones j = index for demand locations k = index for iterations m = index for collection and intermediate process centers n = index for process plants o = index for objectives p = index for products pd = index for directly used products pi = index of intermediate products pp = index for produced products s = index for similar footprints t = index for technology options v = index for variables Scalars

|K| = cardinality of a set K (the number of iterations k) NK = number of obtained feasible solutions from K iterations NS = number of subsets (groups) of similar footprints (and embedded loop statements) w = small weight in objective function to simultaneously minimize footprint values ε = ε-constraint Parameters

af,v = matrix coefficient (specific environmental footprint) ADf,ff,k = arithmetic mean of average absolute normalized deviation between pair of footprints f and f f for selected optimal point xk ADm f,ff = mean value of the arithmetic mean of average absolute normalized deviation between pair of footprints f and f f Df,ff,k = average absolute normalized deviation between pair of footprints f and f f for selected optimal point xk Dm f,ff = mean value of average absolute normalized deviation between pair of footprints f and f f La,Lb Dx,y = distance between object x in layer a and object y in layer b, km eiLa f,pi = specific environmental footprint f of intermediate product pi at layer a, a = {1, 2}, kg/(t·km2), MJ/(t·km2), or 1/t eiL3 f,pi,t = specific environmental footprint f of intermediate product pi and the selected technology t at processing layer, kg/(t·km2), MJ/(t·km2), or 1/t eiL4 f,p = specific environmental footprint f of product p at use layer, kg/(t·km2), MJ/(t·km2), or 1/t eitr,La,Lb = transport environmental footprint f from layer a to f,p the layer b, kg/(t·km3), MJ/(t·km3), or 1/(t·km) Ff,k(x) = environmental footprint obtained by optimization depending on iteration k, t/(km2·y), GJ/(km2·y), or km2/ (km2·y) F0f (x) = footprint obtained at maximum profit solution, where footprints are relaxed, t/(km2·y), GJ/(km2·y), or km2/ (km2·y) r Ff,k (x) = relative footprint obtained by optimization depending on iteration k, t/(km2·y), GJ/(km2·y), or km2/ (km2·y) Froad,La,Lb (x) = road condition factor between object x in x,y layer a and object y in layer b GOf,ff,k = geometric mean of overlap re footprints pairs f and f f in process variables for selected optimal point xk GOm f,ff = mean value of the geometric mean of overlap re footprints pairs f and f f in process variables

Variables

f(x) = continuous function involved in objective function g(x,y) = continuous inequality constraints function h(x,y) = continuous equality constraints function Pk = profit obtained by iteration k, M€/y qm,L1 i,pi = production rate of intermediate product pi at supply zone i, t/y qm,T,L2,L3 = mass flow of intermediate product pi to the n,pi,t selected technology t at the process plant n, t/y qm,La,Lb = mass flow of product p from object x in layer a to x,y,p object y in layer b, t/y m,T,L2,L3 qn,pi,pp,t = mass flow of produced products pp from intermediate product pi with the selected technology t at the process plant n, t/y x = vector of continuous variables xk = Pareto optimal solution xv,k = optimal values of process variables at iteration k y = vector of binary variables Abbreviations

AD = anaerobic digestion CF = carbon footprint (t/(km2·y)) D = dimensional DDGS = distiller’s dried grains with solubles DGP = dry-grind process EF = energy footprint (GJ/(km2·y)) GAMS = General Algebraic Modeling System GHG = greenhouse gas LCA = life cycle assessment, also life cycle analysis LF = agricultural land footprint (km2/(km2·y)) MILP = mixed-integer linear programming MINLP = mixed-integer nonlinear programming 7238



Article

(18) Guillén-Gosálbez, G. A novel MILP-based objective reduction method for multi-objective optimization: Application to environmental problems. Comput. Chem. Eng. 2011, 35 (8), 1469−1477. (19) Deb, K.; Saxena, D. K. On finding pareto-optimal solutions through dimensionality reduction for certain large-dimensional multi-objective optimization problems; Kanpur Genetic Algorithms Laboratory (KanGAL), Indian Institute of Technology, 2005; Kangal report no. 2005011, Kanpur, India. (20) Kravanja, Z. Process systems engineering as an integral part of global systems engineering by virtue of its energy  environmental nexus. Curr. Opin. Chem. Eng. 2012, 1 (3), 231−237. (21) Lygoe, R. J. Complexity Reduction in High-Dimensional MultiObjective Optimisation. Ph.D. thesis, University of Sheffield: Sheffield, U.K.; 2010. (22) Saxena, D.; Deb, K. Non-linear Dimensionality Reduction Procedures for Certain Large-Dimensional Multi-objective Optimization Problems: Employing Correntropy and a Novel Maximum Variance Unfolding; In Evolutionary Multi-Criterion Optimization; Obayashi, S, Deb, K, Poloni, C, Hiroyasu, T, Murata, T, ed.: Springer: Berlin/ Heidelberg, Germany, 2007; pp 772−787. (23) Brockhoff, D.; Zitzler, E. Are All Objectives Necessary? On Dimensionality Reduction in Evolutionary Multiobjective Optimization. In Parallel Problem Solving from Nature - PPSN IX; Runarsson, T, Beyer, H-G, Burke, E., Merelo-Guervós, J, Whitley, L. D., Yao, X., Eds.; Springer: Berlin/Heidelberg, Germany, 2006; pp 533−542. (24) Vaskan, P.; Guillén-Gosálbez, G.; Jiménez, L. Multi-objective design of heat-exchanger networks considering several life cycle impacts using a rigorous MILP-based dimensionality reduction technique. Appl. Energy 2012, 98, 149−161. (25) Gutiérrez, E.; Lozano, S.; Moreira, M. T.; Feijoo, G. Assessing relationships among life-cycle environmental impacts with dimension reduction techniques. J. Environ. Manage. 2010, 91 (4), 1002−1011. (26) Lygoe, R.; Cary, M.; Fleming, P. A Many-Objective Optimisation Decision-Making Process Applied to Automotive Diesel Engine Calibration. In Simulated Evolution and Learning; Deb, K., Bhattacharya, A., Chakraborti, N., Chakroborty, P., Das, S., Dutta, J., et al., Eds.; Springer: Berlin/Heidelberg, Germany, 2010; pp 638−646. (27) Č uček, L.; Lam, H.; Klemeš, J.; Varbanov, P.; Kravanja, Z. Synthesis of regional networks for the supply of energy and bioproducts. Clean Technol. Environ. Pol. 2010, 12 (6), 635−645. (28) Č uček, L.; Varbanov, P. S.; Klemeš, J. J.; Kravanja, Z. Total footprints-based multi-criteria optimization of regional biomass energy supply chains. Energy 2012, 44 (1), 135−145. (29) Haimes, Y. Y.; Lasdon, L. S.; Wismer, D. A. On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization. IEEE Trans. Syst., Man. Cybern 1971, No. SMC1(3), 296−297. (30) Č uček, L.; Klemeš, J. J.; Varbanov, P. S.; Kravanja, Z. Reducing the dimensionality of criteria in multi-objective optimization of biomass energy supply chains. Chem. Eng. Trans. 2012, 29, 1231− 1236. (31) GAMSA User’s Guide; GAMS Development Corporation: Washington, DC, USA, 2010. (32) IBM. IBM ILOG CPLEX Optimizer. www-01.ibm.com/ software/integration/optimization/cplex-optimizer/ (accessed Jan 2013); 2012. (33) Spielmann, M.; Bauer, C.; Dones, R.; Tuchschmid, M. Transport Services; Swiss Centre for Life Cycle Inventories: Dübendorf, Switzerland, 2007; ecoinvent report No. 14. (34) Galli, A.; Wiedmann, T.; Ercin, E.; Knoblauch, D.; Ewing, B.; Giljum, S. Integrating Ecological, Carbon and Water footprint into a “Footprint Family” of indicators: Definition and role in tracking human pressure on the planet. Ecolog. Indicators 2012, 16, 100−112. (35) Kreyszig, E. Advanced Engineering Mathematics, 9th ed.; John Wiley & Sons, Inc.: Hoboken, NJ, 2006.

MOO = multiobjective optimization MSW = municipal solid waste PCA = principal component analysis ROM = representative objectives method SEPI = sustainable environmental performance indicator SPI = sustainable process index WAR = waste reduction algorithm WF = water footprint (t/(km2·y)) WPF = water pollution footprint (t/(km2·y))

■

REFERENCES

(1) Fu, Y.; Diwekar, U. M.; Young, D.; Cabezas, H. Process design for the environment: A multi-objective framework under uncertainty. Clean Prod. Processes 2000, 2 (2), 92−107. (2) Schlör, H.; Fischer, W.; Hake, J.-F. Methods of measuring sustainable development of the German energy sector. Appl. Energy 2013, 101, 172−181. (3) Pozo, C.; Ruíz-Femenia, R.; Caballero, J.; Guillén-Gosálbez, G.; Jiménez, L. On the use of Principal Component Analysis for reducing the number of environmental objectives in multi-objective optimization: Application to the design of chemical supply chains. Chem. Eng. Sci. 2012, 69 (1), 146−158. (4) Jeswani, H. K.; Azapagic, A.; Schepelmann, P.; Ritthoff, M. Options for broadening and deepening the LCA approaches. J. Cleaner Prod. 2010, 18 (2), 120−127. (5) Č uček, L.; Klemeš, J. J.; Kravanja, Z. A Review of Footprint analysis tools for monitoring impacts on sustainability. J. Cleaner Prod. 2012, 34, 9−20. (6) Rees, W. E. Ecological footprints and appropriated carrying capacity: what urban economics leaves out. Environ. Urban. 1992, 4 (2), 121−130. (7) Wiedmann, T.; Minx, J. A definition of “carbon footprint”. In Ecological Economics Research Trends; Pertsova, C. C., Ed.; Nova Science Publishers: New York, USA, 2008. (8) De Benedetto, L.; Klemeš, J. The Environmental Performance Strategy Map: an integrated LCA approach to support the strategic decision-making process. J. Cleaner Prod. 2009, 17 (10), 900−906. (9) Shonnard, D. R.; Kicherer, A.; Saling, P. Industrial Applications Using BASF Eco-Efficiency Analysis: Perspectives on Green Engineering Principles. Environ. Sci. Technol. 2003, 37 (23), 5340−5348. (10) Tokos, H.; Novak Pintarič, Z.; Yang, Y. Bi-objective optimization of a water network via benchmarking. J. Cleaner Prod. 2013, 39, 168−179. (11) Sikdar, S.; Sengupta, D.; Harten, P. More on aggregating multiple indicators into a single index for sustainability analyses. Clean Technol. Environ. Pol. 2012, 14, 765−773. (12) Krotscheck, C.; Narodoslawsky, M. The Sustainable Process Index a new dimension in ecological evaluation. Ecolog. Eng. 1996, 6 (4), 241−258. (13) Hilaly, A. K.; Sikdar, S. K. Pollution Balance: A New Methodology for Minimizing Waste Production in Manufacturing Processes. Air Waste 1994, 44 (11), 1303−1308. (14) Zamboni, A.; Bezzo, F.; Shah, N. Spatially Explicit Static Model for the Strategic Design of Future Bioethanol Production Systems. 2. Multi-Objective Environmental Optimization. Energy Fuels 2009, 23 (10), 5134−5143. (15) Busacca, P. G.; Marseguerra, M.; Zio, E. Multiobjective optimization by genetic algorithms: application to safety systems. Reliability Eng. Syst. Safety 2001, 72 (1), 59−74. (16) Zhong, Z.; You, F. Oil spill response planning with consideration of physicochemical evolution of the oil slick: A multiobjective optimization approach. Comput. Chem. Eng. 2011, 35 (8), 1614−1630. (17) Grossmann, I. E.; Guillén-Gosálbez, G. Scope for the application of mathematical programming techniques in the synthesis and planning of sustainable processes. Comput. Chem. Eng. 2010, 34 (9), 1365−1376. 7239


Dealing with High-Dimensionality of Criteria in ... - ACS Publications

Recommend Documents