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Process Systems Engineering

Mixed-Integer Programming Approach for Dimensionality Reduction in Data Envelopment Analysis: Application to the Sustainability Assessment of Technologies and Solvents Phantisa Limleamthong, and Gonzalo Guillen Gosalbez Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b05284 • Publication Date (Web): 15 Mar 2018 Downloaded from http://pubs.acs.org on March 16, 2018

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Industrial & Engineering Chemistry Research

Mixed-Integer Programming Approach for Dimensionality Reduction in Data Envelopment Analysis: Application to the Sustainability Assessment of Technologies and Solvents Phantisa Limleamthonga, Gonzalo Guillén-Gosálbeza,* a

Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, South Kensington Campus, London, SW7 2AZ

(United Kingdom). *Corresponding author. Tel.: +44 (0)20 7594 1478 E-mail addresses: [email protected] (P. Limleamthong), [email protected] (G. Guillén-Gosálbez).

ABSTRACT Data Envelopment Analysis (DEA) has recently emerged as an effective method for the sustainability assessment of industrial systems. Unfortunately, sustainability studies require the evaluation of a wide range of indicators (i.e. inputs and outputs in DEA notation), which can weaken the discriminatory power of DEA and ultimately lead to results that are less meaningful and hard to interpret. Here we develop a systematic MILP-DEA approach that identifies redundant metrics that can be omitted in DEA models with minimum information loss. Our approach is based on a bi-level programming model where binary variables denote the selection of the metrics and the objective functions are constraints are formulated according to the DEA models. The capabilities of this method are illustrated through the assessment of several industrial systems evaluated according to multiple criteria, some of which are based on life cycle metrics. Our results show that our systematic approach can effectively reduce the number of variables in DEA studies. This method can also be used to enhance the discriminatory power of DEA by diminishing the number of units deemed efficient considering a maximum allowable error.

KEYWORDS Sustainability assessment; Data Envelopment Analysis; Dimensionality reduction; Variables reduction; Life cycle impacts.

1. Introduction Sustainability principles have gained wider importance in our society. The chemical industry is not an exception to this trend, as it is at present striving to incorporate environmental and social aspects in process design and operations so as to become more sustainable whilst keeping profits high. With 1 ACS Paragon Plus Environment

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advances in technology, a wide variety of chemical products have been developed and launched at a very fast pace. In this context, there is a clear need for tools capable of assessing technologies in terms of some predefined sustainability criteria. This task can be posed as a multi-criteria decision-making problem requiring the simultaneous consideration of several goals. Among them, there are often inherent trade-offs that prevent the existence of one single process performing best in all of the objectives. Systematic mathematical techniques can assist decision-making in these sustainability studies, shedding light on inherent trade-offs and helping to articulate the decision-makers’ preferences.1–5 Among the available multi-criteria decision-support techniques, we focus here on Data Envelopment Analysis (DEA), a method originally developed by Charnes et al.6 and widely applied in economics and operations research to measure the comparative efficiency of a set of homogeneous units considering specific criteria. In the context of DEA, a set of entities referred to as Decision Making Units (DMUs) is considered, each of which converts multiple inputs into multiple outputs. The assessment criteria are therefore classified in DEA as either inputs (i.e. to be minimized) or outputs (i.e. to be maximized), which in sustainability studies reflect economic, environmental and social indicators.7 The main advantage of DEA is that it allows evaluating systems without the need to define subjective quantitative weights. DEA can also identify sources of inefficiencies and provide efficiency targets for the poorly-performed entities that if achieved would make them efficient. This technique, used for many years in economics, has recently found applications in engineering problems, including energy and environmental studies.8,9 DEA has been also coupled with Life Cycle Assessment (LCA) principles to evaluate the eco-efficiency of systems,10–12 and more recently the sustainability efficiency of several technologies.7,13–16 While applications in the chemical industry have been quite scarce, it is very likely that interest in this method will substantially grow as sustainability gain even more importance in the near future. One shortcoming of DEA is that its discriminatory power worsens with the number of inputs and outputs. This limitation reflects the curse of dimensionality, namely that as the number of inputs and outputs increases, the DMUs are projected onto a growing number of orthogonal directions. This results in a significant number of DMUs being deemed efficient, thereby making the analysis less meaningful.17 Several strategies have been introduced to tackle this issue. The most common ones are based on: (i) weight restrictions,18,19 where upper and lower bounds are assigned to inputs and outputs; and (ii) on assurance regions,20 where bounds are imposed on the ratio of weights. These strategies require additional preferential information and value judgement, both highly subjective and dependent on the preferences of the users.

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Other alternative approaches to reduce inputs/outputs in DEA are based on statistical criteria that require no additional subjective information. These include principal component analysis (PCA-DEA)21–24 and variable reduction (VR)25–30. These two methods aim to reduce the dimensionality of the problem by identifying redundant variables, according to some statistical criteria, that can be excluded from the analysis. They often exploit correlations between input and output variables, some of which are omitted to simplify the analysis without losing information. PCA is a statistical procedure whereby uncorrelated linear combinations of original variables are generated for dimensionality reduction. More precisely, PCA is designed to reduce the number of variables in a data set via an orthogonal transformation of a larger dimensional space of original correlated variables into a reduced dimensional space of linearly uncorrelated variables (called principal components, PCs). Adler and Golany22,23 proposed to incorporate PCA into DEA to improve its discrimination capabilities whilst minimizing information loss. They showed that the omission of principal components with little explanatory power (small variance) could reduce the number of efficient units in the DEA results. PCA was also applied to identify redundant environmental metrics in the multiobjective optimization of industrial systems.31 Nevertheless, the combined approach DEA-PCA shows some drawbacks. First, the results provided are hard to interpret, since the dimensionality reduction is performed in the coordinate system of PCs. Furthermore, there are no clear guidelines on how to specify the minimum number of PCs needed. Lastly, the translation of the improvement targets defined for the PCs into their corresponding counterparts in the space of the original variables is not unique, which complicates the interpretation of the DEA results.21 The VR method was firstly introduced by Jenkins and Anderson.25 They proposed a multivariate statistical approach to identify a set of original correlated variables that can be eliminated from the DEA analysis with minimum information loss. Their method employs partial correlation as a measure of the extent to which the original information is retained. They concluded that omitting even highly correlated variables could significantly affect the efficiency scores. Therefore, the choice of variables to be omitted should be performed cautiously. Amirteimoori et al.26 proposed an alternative variables reduction procedure that aggregates pairs of the most highly correlated inputs and outputs. This step was repeated iteratively, eliminating inputs and outputs until satisfying the following rule of thumb: ≥ × , 3 +

(1)

where is the number of DMUs, and and are the number of inputs and outputs, respectively.32 At each iteration, the variables to be omitted were chosen according to a correlation analysis. This approach may affect strongly the DEA results, as argued by Dyson et al.27 More precisely, the existence of high



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correlation between variables does not necessarily mean that one of them could be removed without affecting the efficiency score. Therefore, selecting the variables to omit based exclusively on the correlation analysis should be discouraged. Other VR methods for DEA that are based on a correlation analysis were reported in the literature.28–30 Wagner and Shimshak33 proposed a stepwise procedure for variable selection that measures the effect of changes in the efficiency scores as variables are included/excluded from the analysis one at a time. Their main contribution is the backwards approach, which starts by considering a full set of input and output variables to then eliminate some of them sequentially. Unfortunately, this approach offers no guarantee of convergence to the selection of the best reduced combination of variables. Simplifying the DEA application is crucial in sustainability studies because many metrics need to be considered simultaneously to cover its three pillars (economic, environmental and social).34 However, some of these metrics – or input and output variables in DEA terminology – might be considered redundant in the sense that they can be excluded from the analysis without modifying the outcome of the DEA models. This offers the opportunity to reduce the number of variables in DEA studies, thereby enabling a better visualization and interpretation of the results. Furthermore, by removing the redundant variables from the analysis, the model will contain only those variables that are key contributors to the systems’ performance. Consequently, by focusing efforts on improving these indicators, we will very likely improve as well the performance in the redundant variables, as the latter are strongly connected with the former. Hence, our main goal in this study is to introduce an approach to reduce the number of inputs and outputs in DEA studies whilst minimizing information loss. Compared to other approaches in the literature, our method is unique, since it selects redundant attributes using bi-level programming and an MIP model that minimizes an approximation error (hence, no statistical criteria are employed in the calculations). This method can identify redundancies in variables, thereby providing valuable guidelines on how to judiciously simplify the DEA application. Furthermore, this approach can be used in turn to improve the discriminatory power of DEA by diminishing the number of units deemed efficient, thereby further simplifying the problem and enabling a better visualization. The paper is organized as follows. A general background on DEA is introduced in Section 2. Section 3 describes the problem addressed in this study, followed by the methodology proposed to solve it, which is delineated in detail in Section 4. Four case studies are presented and comprehensively discussed in Section 5. Finally, the conclusions are drawn at the end of the paper.


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2. Background 2.1 Motivating example In this section, we introduce the fundamentals of DEA. Consider the example given in Figure 1, where we wish to assess five different flowsheets that produce one kg of a given compound. Each such alternative is evaluated in terms of environmental and economic performance. More precisely, we aim to answer the following two questions: which flowsheets perform best? And, by how much the suboptimal flowsheets should be improved to become optimal? The solution of the DEA models described in detail later in the article would identify flowsheets A, B and C (displayed in green circles) as optimal/efficient (efficiency score of one, as they cannot be improved simultaneously in both objectives); meanwhile, flowsheets D and E (displayed in white circles) would emerge as suboptimal/inefficient (efficiency score strictly less than one, as for each of them it is always possible to find another unit or linear combination of units with better performance in both indicators). The latter could be improved by projecting them on the Pareto front (displayed in a blue solid line), which would allow to define improvement targets (displayed in red circles) that could make them optimal.

Figure 1. DEA applied to the flowsheets example with two criteria. The solution to this motivating example is relatively straightforward since the problem is based upon two indicators. However, sustainability problems typically require more than two objectives, which 5 ACS Paragon Plus Environment


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leads to more challenging assessments. As an example, consider the case of four indicators, like in Figure 2, where every polyline represents a flowsheet and we are interested in answering the questions posed above while at the same time elucidating whether all the metrics are needed in the analysis. The method introduced next tackles this problem in a systematic manner.

Figure 2. Flowsheets example with four criteria. Every polyline represents a different design connecting its performance in every indicator.

2.2 Fundamentals of DEA DEA is a non-parametric mathematical programming technique that carries out a multi-criteria evaluation of a set of decision making units (DMUs), each of them consuming a given amount of inputs to produce a certain level of outputs.6 In general, DMUs represent alternatives whose performance needs to be evaluated (e.g. process technologies, chemical plants, solvents, etc.). DEA was originally develop to assess the efficiency of economic entities; however, it can be effectively used as a multi-criteria decisionmaking tool (MDCM) where inputs correspond to performance metrics to be minimized and outputs to indicators to be maximized.35,36 The concept of DEA was originally developed from the traditional “output-to-input ratio”, a definition of efficiency commonly adopted in engineering and science.32 In mathematical terms, the relative efficiency of a given DMUo is calculated by solving an optimization problem aiming at maximizing the ratio between the weighted-sum of inputs and the weighted-sum of outputs. This maximization problem is subjected to the condition that the efficiency of the other DMUs, for which an


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analogous ratio is defined, needs to be less than or equal to one.6 For the DMU being studied (DMUo), the original fractional problem formulated in DEA can be mathematically expressed as follows: max ,

s. t.

=

∑ ∑" !

∑ ' ∑" '

≤ 1; ∀, ∈ . ; , ≠ 0

12 , 34 ≥ 0.

(M.1a) (M.1b) (M.1c)

Where the notation used is as follows. Indices: 0

Index for the DMU that is being assessed for 0 = 1, … ,

7

Index for the criteria classified as inputs for 7 = 1, … ,

,

8

Index for the other DMUs for , = 1, … , and , ≠ 0

Index for the criteria classified as outputs for 8 = 1, … ,

Sets: . 9

:

Set of DMUs Set of inputs Set of outputs

Parameters (data given by users):

Number of DMUs Number of inputs consumed by DMUj

Number of outputs generated by DMUj

? 12 ? 34 4@ = 1

(M.2b)

∑=2>? 12 ? 34 4; ≤ 0; ∀, ∈ . ; , ≠ 0

(M.2c)

12 , 34 ≥ 0.

(M.2d)

,

Model M.2 is then computed repetitively for all DMUs to obtain their individual relative efficiency. If the calculated efficiency score has a value of one, then such a DMUo is regarded as efficient. Otherwise, i.e. the efficiency score is strictly less than one, then DMUo is deemed inefficient. Model M.2, referred to as the primal DEA model, focuses on efficiency assessment (i.e. it provides the relative efficiency score for each DMU). In addition, there is a dual DEA model that focuses on the inefficiency assessment of inefficient DMUs. For the latter DMUs, the dual model calculates improvement targets and identifies source of inefficiency. Note that here we focus on the constant returns to scale (CRS) model, yet the approach presented next can be easily adapted to the variable returns to scale case. This implies that the output to input ratio is assumed to be independent of the magnitude of the inputs.

3. Problem statement Having introduced the fundamentals of DEA, we next formally state the problem of interest. In essence, we aim to develop a method based on DEA to evaluate the sustainability performance of a set of entities using as little information as possible. More precisely, given a set of DMUs .: 1, … , , and a set

of criteria classified as either inputs 9: 1, … , , or outputs :: 1, … , , we aim to determine the minimum 8 ACS Paragon Plus Environment

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number of them such that their efficiency can be assessed with minimum information loss. This information loss is quantified via the difference in efficiency scores between the case that includes all the inputs and outputs and the one solved in a reduced domain of them. In mathematical terms, the problem is aimed at determining the smallest inputs subset 9′ ⊆ 9 with given size F (i.e., |9′| = ′) and outputs subset :′ ⊆ : with given size F (i.e., |:′| = ′) such that the difference in efficiency scores for each , ∈ .

(between the base case with all the inputs/outputs 9 and :, respectively and the ones calculated for the

reduced problem with 9′ and :′) is minimum. Furthermore, we are also interested in enhancing the

discriminatory power of DEA by understanding how the reduction in the number of inputs and outputs can affect the DEA results.

4. Proposed methodology To solve the problem mentioned above, we derive an approach based on bi-level optimization, in which binary variables model the selection of inputs and outputs in the master outer problem, while the continuous ones represent the DEA weights in the inner slave problem. The outer problem seeks to minimize the difference between the efficiency scores calculated considering all the inputs and outputs and those obtained in a reduced subset of them. On the other hand, the inner problem provides the efficiency scores that would be obtained for any potential combination of inputs and outputs proposed by the outer problem. The bi-level program can be expressed in compact form as follows: min ∗

∗ L ∑M ;>?K; − ; K

s. t.

(M.3a)

max

= ∑=2>? 12 ? 34 4@ = 1

(M.3c)

∑=2>? 12 ? 34 4; ≤ 0; ∀, ∈ . ; , ≠ 0

(M.3d)

12 , 34 ≥ 0.

(M.3e)

,

To solve this model, we replace the primal DEA problem by its Karush-Kuhn-Tucker (KKT) conditions. As will be later discussed in more detail, this approach allows us to calculate the efficiency scores of all the DMUs by solving a single optimization problem rather than a set of LPs. The general form of the KKT conditions are as follows: Stationarity for minimizing N 9 ACS Paragon Plus Environment


∇N ∗ − P

R∈T

QR ∇SR ∗ − P

V∈X

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UV ∇ℎV ∗ = 0

Primal feasibility SR ∗ ≤ 0

∀Y ∈ Z

ℎV ∗ = 0

∀ ∈ [

Dual feasibility QR ≥ 0

∀Y ∈ Z

Complementary slackness QR SR ∗ = 0

∀Y ∈ Z

where N is the objective function; Z is the set of inequality constraints; [ is the set of equality

constraints; and are decision variables. Considering the primal DEA model defined in M.2, we have that: Z = ., 9, :, = 0, ∗ = \34∗ , 12∗ , U@ , Q; , Q4 , Q2 ]. Where the notation is as follows. 34∗

12∗

Linear weights assigned to input 7

Linear weights assigned to output 8

U@

Lagrange multiplier for the equality constraint ℎ@ ∗

Q4

Lagrange multiplier for the inequality constraint S4 ∗

Q;

Q2

Lagrange multiplier for the inequality constraint S; ∗ Lagrange multiplier for the inequality constraint S2 ∗

Hence, the KKT conditions applied to the primal DEA take the following form: N ∗ = − ∑=2>? 12∗ ? 34∗ 4@

(3) A

S; ∗ = ∑=2>? 12∗ ? 34∗ 4; ≤ 0

(4)

S4 ∗ = −34∗ ≤ 0

(5)


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S2 ∗ = −12∗ ≤ 0

(6)

And the Lagrangian function will be delineated as follows: ^ ∑=2>? 12∗ ? 12∗ ? 34∗ 4; b A

− ∑4>? 34∗ 4@ b = 0

(7)

It is important to note that the last condition of complementary slackness can be reformulated by introducing binary and slack variables.37 Adopting this approach, binary variables e; , e4 and e2 are assigned for all inequality constraints to model whether a constraint is active or not in the optimal solution. If it is active, then the slack is set to zero and the multiplier can take any positive value. If it is inactive, then the slack can take any value and the multiplier must be zero. Furthermore, we introduce binary variables f4 and ? Q; 4; + Q4 = g 1 − f4

∀7 ∈ 9

(M.4a)

? Q; ? 12∗ ? 34∗ 4; + h; = 0

∀, ∈ .; , ≠ 0

(M.4e)

−34∗ + h4 = 0

∀7 ∈ 9

(M.4f)

−12∗ + h2 = 0

∀8 ∈ :

(M.4g)

h; ≤ ga1 − e; b

∀, ∈ .; , ≠ 0

(M.4h)

h4 ≤ g 1 − e4

∀7 ∈ 9

(M.4i)

h2 ≤ g 1 − e2

∀8 ∈ :

(M.4j)

Q; ≤ ge;

∀, ∈ .; , ≠ 0

(M.4k) 11

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Q4 ≤ ge4

∀7 ∈ 9

(M.4l)

Q2 ≤ ge2

∀8 ∈ :

(M.4m)

34∗ ≤ gf4

∀7 ∈ 9

(M.4n)

12∗ ≤ g? f4 = i

(M.4p)

∑=4>? ?K; − ; K

(M.4r)

Note that alternative error metrics could be used, like the sum of square errors between the original efficiencies and the ones calculated after removing inputs/outputs. Using the L1-norm or the absolute-value norm in the model has the advantage of yielding a linear MILP formulation that can be efficiently solved by branch and bound methods. The L2-norm or the Euclidean norm lead to convex MINLPs that require initialization strategies and result in larger CPU times. The L1-norm can be reformulated into linear constraints as follows: k; ≥ ;∗ − ;L

(8)

k; ≥ ;L − ;∗

(9)

min ∗

∑M ;>? k;

(10)

Where the notation used is as follows: Parameters: g

Large parameter, assumed as 103

j

The total number of outputs for j = 1, … ,

i

The total number of inputs for i = 1, … ,


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Positive variables: ;∗

;L k;

Relative efficiency score of DMUj of the reduced case Relative efficiency score of DMUj of the base case Difference or error in efficiency score of DMUj between the original problem and the reduced domain problem

h;

Slack variable for the inequality constraint S; ∗

h2

Slack variable for the inequality constraint S2 ∗

h4

Slack variable for the inequality constraint S4 ∗

Binary variables: f4

Binary variable to select input 7

e;

Binary variable to denote whether the constraint S; ∗ is active or not

e2

Binary variable to denote whether the constraint S2 ∗ is active or not

Mixed-Integer Programming Approach for ... - ACS Publications

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