Measuring incineration plants’ performance using combined data envelopment analysis, goal programming and mixed integer linear programming

Research on incineration addresses various issues such as how to improve energy recovery performance and control pollution (Pai et al. 2008), how to select most appropriate location for incineration plants (Alçada-Almeida et al. 2009; Eiselt and Marianov 2015) and derive compensation package for effected people (Chang et al. 2002), and what extent the recycling volume would change following the introduction of an incineration tax (Sahlin et al. 2007). These facilitate analysing feasibility of incineration plants. Due to the variability of quality and quantity of waste the output of incineration plants (energy and heat, quantities of ferrous and non-ferrous metals and emissions) vary. Also, based on the main ingredients of the waste incinerated, the energy conversion technology is selected (Di Gregorio and Zaccariello 2012). Additionally, quality of waste governs the plant maintenance (predictive, preventive and shut down) requirements. The following paragraphs demonstrate what has been contributed by prior researches on incineration plant performance measurement and the knowledge gaps that need still addressing.

Data envelopment analysis (DEA) is one of the most widely known techniques for measuring units’ performance based on a set of inputs that are consumed in order to produce outputs. In their work, Chen and Chen (2012) propose a DEA model to assess the efficiency of Municipal Solid Waste (MSW) incineration services. Application of DEA technique has been demonstrated in 29 MSW incineration plants in Portugal (Simões et al. 2010; Marques and Simões 2009). The environmental performance of large MSW incineration plants has been proposed by Chen et al. (2010). Using power capacity, operation cost, operation time and power consumption as inputs and a plethora of undesirable outputs—such as hazardous emissions, municipal waste suspended, bottom ash, and opacity, environmental efficiency of MSW incineration plants was derived.

Besides assessing MSW incineration plants, DEA has been applied to measure the performance of household refuse collection systems and refuse collection services (Benito-Lopez et al. 2011). Super efficiency and cross efficiency DEA models has also been applied for assessing environmental performance of MSW treatment facilities (Sarkis and Weinrach 2001). The efficiency of waste management systems (generation, sorting of recyclables and collection) has been also examined with a DEA/Analytic Hierarchy Process (AHP) application (Chen 2010). The efficiency of each has been extracted using a conventional input oriented DEA model while weighting factors for each waste management system have been extracted from a panel of experts using the AHP. An aggregated efficiency index is constructed from the two methodologies. In a similar context, managerial preferences are tackled with ANP providing input to DEA for the selection of the best location of solid waste facilities in Iran (Khadivi and Ghomi 2012). The productivity of waste management in Taiwan has been also examined through a non-radial network DEA approach considering different stages and layers (Huang et al. 2014). The packaging waste management system has been examined with Benefit of Doubt (BoD), a DEA like methodology in Belgium; the cost efficiency of 35 Belgian municipal waste joint ventures in 2010 is assessed (De Jaeger and Rogge 2014). Quality of 293 Belgian municipality solid waste collection and processing facilities has been examined with a shared DEA model; the waste cost is treated as a shared input among treatment efforts of multiple municipal solid waste fractions (Rogge and De Jaeger 2012, 2013).The water treatment is closely related with waste management while reuse of wastewater is achieved through plants. The efficiency of 338 plants has been investigated based on cost factors (inputs) and reused water quality (output) in Valencia, Spain (Hernández-Sancho and Sala-Garrido 2009).

In their work Costi et al. (2004) proposed a Mixed Integer Non-Linear Programming (MINLP) model for the selection of plant waste treatment type and its capacity. A mixed integer linear programming (MILP) model that examines cost minimization has been proposed for landfill location (Fiorucci et al. 2003). Similar mathematical models have been used as decision support system (DSS) for selecting optimal waste treatment facility (Fiorucci et al. 2003; Chang and Chang 1998). Models for treatment type, flows and technology have been proposed using multi-objective programming analysis (Galante et al. 2010; Alumur and Kara 2007). Applications of Goal Programming (GP) in solid waste management (SWM) have been also suggested in the relevant literature for assessing the environmental impact while minimizing operational costs (Galante et al. 2010). The transportation of hazardous waste and location of disposal and recycling centers is important as well to environment and to total population. Samanlioglu (2013) proposed a GP formulation for location-routing of hazardous waste under economic (cost minimization) and risk (total transportation risk related to the population, total risk for the population around treatment and disposal centers) objectives.

As stated above, majority of the DEA models help assess the operational performance of incineration plants. DEA technique also measures efficiency of incineration plants based on inputs and outputs through benchmarking. An efficient unit would be the one that uses fewer resources to produce energy with fewer emissions. While examining the performance of an energy production unit, additional constraints could be introduced—such as social constraints, or analyzing the demand and supply of each plant. Moreover, as incineration plants contribute to overall power and heat requirements from renewable sources, and also reduces the environmental impact by consuming waste (reducing the landfill) and enhances harmful emissions, one has to study the overall impact of incineration plants in a region so as to maximize the benefits and minimize the emission impact. Chen et al. (2012) propose a network DEA model for incineration plants’ efficiency measurement from waste treatment to electricity generation. A directional distance function is used to construct a modified DEA model in accordance with the production characteristics of incineration plants, while allowing for differentiation between desirable (waste disposal and energy production) and undesirable outputs (bottom ashes and pollutants). This helps both policymakers and plant operators to enhance overall performance of incineration in a region. However, as the model doesn’t set a goals/targets for desirable and undesirable outputs as per the policymakers’ and plant owners’ objectives, the outcomes of the model may not very realistic and decision-making on the basis of proposed model’s results may not be accurate. Ideally incineration plants consume huge amount of MSW, produce maximum power and heat, and discharge as little emission as possible. However, in reality due to varied quantity and quality of waste within a specific region one has to optimize the outputs through setting up the targets/goals. Additionally, there must be provision of withdrawing some facilities if their undesirable outputs are beyond certain limit. This would warrant policy makers and plant owners to run the units optimally and shutting down the suboptimal units for undertaking maintenance and other desirable activities to bring them in full operations in the due course. To authors’ knowledge there is no work that has modeled the performance of incineration plants in a region for power and heat production, reduction of waste and harmful emissions with specific targets for both desirable and undesirable outputs. This study proposes a hybrid data envelopment analysis (DEA), Goal Programming (GP) and mixed integer linear programming (MILP) model to assess the performance of incineration plants in a specific region to enhance overall power production, reduction of waste and emissions. This model provides a flexible framework that utilizes the strengths of DEA and GP. Additionally, it enables to make decision on shutting down low performer plants that could be undergone desired maintenance to improve their performance and incorporated into the system again. The proposed model in one hand facilitates policy makers to enhance overall performance of incineration plants within a specific region to achieve desired targets and on the other hand helps individual plant operators to address their issues and challenges related to input and output variables in order to achieve the desired throughput (energy production and waste handling) and undesirable pollutants (reduce bottom ash and emissions)

This research uses secondary data from literature to formulate the proposed performance measurement model. Information of 22 incineration plants in the UK has been used to demonstrate the effectiveness of the model. The following paragraphs formulate the proposed performance measurement model.

3.1 Model formulation

Incineration plants’ performance assessment needs consideration of multiple criteria. DEA derives efficiency scores of decision making units (DMUs) based on pre-determined inputs and outputs. Integrating GP with DEA allows to set goals for multiple objectives (e.g. Operational, economic, environmental, social performance) (Zografidou et al. 2016) In this paper, a hybrid GP/DEA model is presented to measure incineration plants’ performance. The model integrates binary variables (0, 1) in order to segregate plants that are performing in line with the desired environmental and operational targets. The proposed model is based on an existing GP/DEA model (Izadikhah et al. 2014) and extends its features by adding binary variables and various scenarios to form a mixed integer linear programming (MILP) model.

Data envelopment analysis (DEA) helps deriving efficiencies of DMUs by comparing the inputs and outputs of each unit against each other. Goal Programming helps to set goals for each criterion and derives decision variables through minimizing the deviations for each goal. Combining DEA and GP into a single framework, helps derive efficiency score of each unit within the desired goals. The use of binary variables facilitates selection of efficient units on the basis of the achievement of the desired goals.

The proposed model has been formulated following a few steps. Figure 1 depicts the flow chart of proposed performance measurement model. Firstly, based on DEA formulation, the units (incineration plants) are identified and the inputs/outputs of each unit are derived from the available data. The GP formulation is used to model operational goals, environmental targets, and capital constraints. The combined DEA and GP model helps derive efficiency of each unit under assessment. In order to identify the plants that need to be closed on the basis of their overall performance in relation to the overall targets, binary variables (0, 1) are introduced within the combined GP/DEA model along with the linearisation constraints. The overall outcome is a MILP model, which identifies the inefficient plants that must be closed in order to keep the overall targets (e.g. MSW consumption, energy and heat production, emissions reduction) of specific zone through analyzing various scenarios. The following paragraphs explained the proposed model mathematically.

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3.3 Model description

Economic systems entail production processes; assuming that \( i = 1, \ldots ,m \) inputs are consumed to produce \( r = 1, \ldots ,s \) outputs, then the following DEA models are used in order to assess the efficiency of the entities that are examined (DMUs). Focusing only on envelopment models, the following Linear Programming (LP) formulations are input or output oriented DEA models (Charnes et al. 1981, 1984).

Models presented in Table 3 (a) and (b) are the conventional DEA models. The outputs are treated all as desirable. However, every economic entity may produce undesirable outputs by consuming inputs. In the present study, incineration plants produce, among desirable outputs, undesirable outputs as well. During the incineration procedure, municipal waste is consumed producing power but also harmful gas emissions. The DEA model that assesses a production procedure based on desirable and undesirable outputs is presented with the following LP formulation (Sueyoshi and Goto 2011a, b; Sueyoshi et al. 2010).

$$ \begin{aligned} &\hbox{max} \,\,\beta \hfill \\ &s.t. \hfill \\ &\quad \sum\limits_{j = 1}^{n} {\lambda_{j} \cdot x_{j,i} } \le x_{o,j} ,\quad i = 1, \ldots ,m \hfill \\ &\quad \sum\limits_{j = 1}^{n} {\lambda_{j} \cdot y_{{j,r_{1} }} } \ge y_{{o,r_{1} }}^{des} \cdot \left( {1 + \beta } \right),\quad r_{1} = 1, \ldots ,s_{1} \hfill \\ &\quad \sum\limits_{j = 1}^{n} {\lambda_{j} \cdot y_{{j,r_{2} }} } = y_{{o,r_{2} }}^{undes} \cdot \left( {1 - \beta } \right),\quad r_{2} = 1, \ldots ,s_{2} \hfill \\ &\quad \lambda_{j} \ge 0, \, j = 1, \ldots ,n \hfill \\ &\quad \beta \, free \hfill \\ \end{aligned} $$

(1)

Table 3

Input (a) and output (b) oriented DEA models

\( \begin{aligned} &\hbox{min} \, \theta \\ & s.t. \hfill \\& \sum\limits_{j = 1}^{n} {x_{ij} \cdot \lambda_{j} } \le x_{io} \cdot \theta , \, i = 1,\ldots,m \hfill \\ & \sum\limits_{j = 1}^{n} {y_{rj} \cdot \lambda_{j} } \ge y_{ro} , \, r = 1,\ldots,s \hfill \\ & \lambda_{j} \ge 0, \, j = 1,\ldots,n \hfill \\& \theta \, free \hfill \\ \end{aligned} \)	\( \begin{aligned} & \hbox{max} \, \varphi \\ & s.t. \hfill \\ & \sum\limits_{j = 1}^{n} {x_{ij} \cdot \lambda_{j} } \le x_{io} \, i = 1,\ldots,n \hfill \\ & \sum\limits_{j = 1}^{n} {y_{rj} \cdot \lambda_{j} } \ge y_{ro} \cdot \varphi , \, r = 1,\ldots,s \hfill \\ & \lambda_{j} \ge 0, \, j = 1,\ldots,n \hfill \\ & \varphi \, free \hfill \\ \end{aligned} \)
(a)	(b)

In DEA model (1), a DMU is considered efficient if \( \beta^{*} = 0 \). However, DEA models assess the production process based on a number of given datasets for inputs and outputs.

Real world problems have more than one objective, based on which criteria are optimized (maximized or minimized). Goal Programming (GP) models are generally used in order to handle such problems. A typical GP model is formulated as follows:

$$ \begin{aligned}& \hbox{min} \, w^{1} \cdot \delta^{ - } + w^{2} \cdot \delta^{ + } + w^{3} \cdot \left( {\delta^{ - } + \delta^{ + } } \right) \hfill \\& s.t. \hfill \\ &\quad \quad \sum\limits_{k = 1}^{K} {a_{k} \cdot x_{k} } { + }\delta^{ - } - \delta^{ + } = G \hfill \\& \quad \quad \sum\limits_{k = 1}^{K} {a_{k} \cdot x_{k} } \le b \hfill \\ &\quad \quad x_{k} \ge 0, \, \forall k \hfill \\ \end{aligned} $$

(2)

In formulation (2), slack variables are minimized in the objective function according to the direction of each goal. Extending model (2), then the following model (3) is formulated with the integration of DEA technique (Izadikhah et al. 2014).

$$ \begin{aligned} &\hbox{min} \, \sum\limits_{r = 1}^{{s_{1} }} {\delta_{r}^{ - } } + \sum\limits_{r = 1}^{{s_{2} }} {\delta_{r}^{ + } } + \sum\limits_{r = 1}^{{s_{3} }} {\left( {\delta_{r}^{ - } + \delta_{r}^{ + } } \right)} \hfill \\ &s.t. \hfill \\ &\quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \cdot y_{rj} } { + }\delta^{ - } - \delta^{ + } = G_{r}^{1} ,\quad r = 1, \ldots ,s_{1} \hfill \\ &\quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \cdot y_{rj} } { + }\delta^{ - } - \delta^{ + } = G_{r}^{2} ,\quad r = 1, \ldots ,s_{2} \hfill \\ &\quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \cdot y_{rj} } { + }\delta^{ - } - \delta^{ + } = G_{r}^{3} ,\quad r = 1, \ldots ,s_{3} \hfill \\ &\quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \cdot x_{ij} } \le x_{io} ,\quad i = 1, \ldots ,m \hfill \\ &\quad \quad \lambda_{j} \ge 0,\quad \, j = 1, \ldots ,n \hfill \\ \end{aligned} $$

(3)

The methodology used in this paper, is a hybrid DEA/GP model. With this combination, the inputs and outputs are utilized, while each combination of incineration plants is examined with the GP approach. The resulting model is a MILP mathematical programming model and is presented in formulation (4). In this model, the aim is to minimize the slack variables that overestimate the goals of undesirable outputs which concern harmful gas emissions. The first set of constraints is introduced to model the goals for undesirable outputs; the second set of constraints, concern the inputs. In both cases, a binary variable is introduced in the model, examining only those DMUs which satisfy constraints third or fourth set of constraints. Binary variable \( \eta \) is introduced to model the disjunction \( \sum\nolimits_{j = 1}^{n} {PE_{j} \cdot \xi_{j} \ge PE^{U} } \, \vee \sum\nolimits_{j = 1}^{n} {AV_{j} \cdot \xi_{j} \ge AV^{U} } \)(either power exported or annual availability of each incineration plant should be over a specific threshold). Parameters \( M^{1} \) and \( M^{2} \) are sufficient upper bounds of each constraint. With disjunctive formulation, incineration plants are selected based on the thresholds set to power exported or annual availability. The power generation of each incineration is more or less steady; however the waste incinerated may vary in type, consistency and power generation value. Thus, the aim is to set constraints on power generation and annual availability. Annual availability could have not been introduced in classical DEA models, as is not part of the production process, however, when considering a centralized model with decisions that concern the selection of a plant, similar to the one proposed in this paper, then this type of information regarding exogenous operations of each plant can be utilized.

As it can be seen from Fig. 2, parameters for power exported and annual availability have very low correlation (\( p = 0.10 \)); thus, DMUs should be selected on one of the two parameters as the more the annual availability does not imply more power exported to the grid.

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Fifth constraint guarantees Variable Returns to Scale (VRS) technology for the selected DMUs while sixth and seventh constraints are introduced to bound the number of selected DMUs in the range of \( \left[ {\mu ,K} \right] \).

$$ \begin{aligned} & \hbox{min} \, \sum\limits_{r = 1}^{s} {\frac{{d_{r}^{ + } }}{{G_{r}^{und} }}} \hfill \\ & s.t. \hfill \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \cdot \xi_{j} \cdot } y_{r,j}^{und} + d_{r}^{ - } - d_{r}^{ + } = G_{r}^{und} , \, r = 1, \ldots ,s \hfill \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \cdot \xi_{j} \cdot } x_{i,j} \le x_{{i,j_{0} }} , \, i = 1, \ldots ,m \hfill \\ & \quad \quad \sum\limits_{j = 1}^{n} {PE_{j} \cdot \xi_{j} \ge PE^{U} } + M^{1} \cdot \eta \hfill \\ & \quad \quad \sum\limits_{j = 1}^{n} {AV_{j} \cdot \xi_{j} \ge AV^{U} } + M^{2} \cdot \left( {1 - \eta } \right) \hfill \\ & \quad \quad \sum\limits_{j = 1}^{n} {\lambda_{j} \cdot \xi_{j} = 1} \hfill \\ & \quad \quad \sum\limits_{j = 1}^{n} {\xi_{j} } \le K \hfill \\ & \quad \quad \sum\limits_{j = 1}^{n} {\xi_{j} } \ge \mu \hfill \\& \quad \quad \xi_{j} \in \left\{ {0,1} \right\}, \, \hfill \\& \quad \quad \eta \in \left\{ {0,1} \right\}, \hfill \\& \quad \quad \lambda_{j} \ge 0, \, j = 1, \ldots ,n \hfill \\ \end{aligned} $$

(4)

Due to the existence of bilinear terms (product of binary and continuous variable), model (3) is a Mixed Integer Non – Linear Programming (MINLP) model. In order to avoid any local optima (due to existence of a nonlinear model), the following triplets of constraints (5)–(7) are introduced to linearize bilinear term (\( \lambda \cdot \xi \)).

$$ \hat{\lambda }_{j} - M_{j} \cdot \xi_{j} \le 0, \, j = 1,\ldots,n $$

(5)

$$ \hat{\lambda }_{j} - \lambda_{j} \le 0, \, j = 1,\ldots,n $$

(6)

$$ \lambda_{j} - \hat{\lambda }_{j} + M_{j} \cdot \xi_{j} \le M_{j} , \, j = 1,..,n $$

(7)

In linearization constraints (4), (6) the upper bound of λ value (\( M_{j} \)) is 1.

Model (3) is formulated, based on (4)–(6) as follows:

$$ \begin{aligned}& \hbox{min} \, \sum\limits_{r = 1}^{s} {\frac{{d_{r}^{ + } }}{{G_{r}^{und} }}} \hfill \\& s.t. \hfill \\& \, \sum\limits_{j = 1}^{n} {\hat{\lambda }_{j} \cdot } y_{r,j}^{und} + d_{r}^{ - } - d_{r}^{ + } = G_{r}^{und} , \, r = 1,\ldots,s \hfill \\& \, \sum\limits_{j = 1}^{n} {\hat{\lambda }_{j} \cdot } x_{i,j} \le x_{{i,j_{0} }} , \, i = 1,\ldots,m \hfill \\& \, \sum\limits_{j = 1}^{n} {PE_{j} \cdot \xi_{j} \ge PE^{U} } + M^{1} \cdot \eta \hfill \\& \, \sum\limits_{j = 1}^{n} {AV_{j} \cdot \xi_{j} \ge AV^{U} } + M^{2} \cdot \left( {1 - \eta } \right) \hfill \\& \, \sum\limits_{j = 1}^{n} {\hat{\lambda }_{j} = 1} \hfill \\& \, \sum\limits_{j = 1}^{n} {\xi_{j} } \le K \hfill \\& \, \sum\limits_{j = 1}^{n} {\xi_{j} } \ge \mu \hfill \\& \, \hat{\lambda }_{j} - \xi_{j} \le 0, \, j = 1,\ldots,n \hfill \\& \, \hat{\lambda }_{j} - \lambda_{j} \le 0, \, j = 1,\ldots,n \hfill \\& \, \lambda_{j} - \hat{\lambda }_{j} + \xi_{j} \le 1, \, j = 1,..,n \hfill \\& \, \eta ,\xi_{j} \in \left\{ {0,1} \right\}, \, j = 1,..,n \, \hfill \\& \, \lambda_{j} ,\hat{\lambda }_{j} \ge 0, \, j = 1,\ldots,n \hfill \\ \end{aligned} $$

(8)