Full article: Sustainability assessment of plastic circular economy: transitional probabilities with innovative separation

Formulae display: $MathJax Logo$ ?Mathematical formulae have been encoded as MathML and are displayed in this HTML version using MathJax in order to improve their display. Uncheck the box to turn MathJax off. This feature requires Javascript. Click on a formula to zoom.

ABSTRACT

The significant effect of plastic waste separation on quality enhancement of recycled plastics comes with a cost of reduced quantity with its negative implications of higher prices. This paper aims to examine the effect of innovation on the sustainability of plastic circular economy (PCE) using a two-state cyclical dynamic closed model of plastic waste management based on ordinary differential equations. The performances of plastic waste separation models with and without innovation were compared under four pragmatic scenarios; the full force of plastic waste discard and incineration, the single force of plastic waste discard, the single force of plastic waste incineration and the complete riddance of plastic waste discard and incineration. In general, the simulated results evince that PCE cannot be sustained subject to the first three scenarios; it is however, sustainable only under the fourth scenario which endorses the complete prohibition of plastic waste discard and incineration. Under all the scenarios, the innovation-driven separation model outperformed the model without innovation in both the forward and reverse transitional phases of PCE. The paper therefore has policy directives for sustainable: PCE, environment, public health, water resources, climate through reduced greenhouse emission, employment and poverty reduction, which constitute strategic goals of the SDGs.

KEYWORDS:

1. Introduction

The transition from the wasteful plastic linear economy to plastic circular economy has become a cutting-edge technique to achieve sustainable plastic waste management and resource conservation (Addor et al., Citation2022a, Citation2022b; Bucknall, Citation2020; Ellen MacArthur Foundation EMF, Citation2013; Global Environment Community GEF, Citation2018; Grigore, Citation2017; Ledsham, Citation2023; Nunez, Citation2021; Wiah et al., Citation2022). It has however been argued that plastic circular economy cannot be sustained in the face of rapid production of virgin plastics (Addor et al., Citation2022b; Global Environment Community GEF, Citation2018). Sustainable plastic waste management can therefore be achieved within a closed system of plastic waste management which prohibits plastic production via virgin resources (Addor et al., Citation2022a, Citation2022b; Wiah et al., Citation2022). Typical replica of such a system is reported in (Addor et al., Citation2022a, Citation2022b; Wiah et al., Citation2022). Although plastic recycling constitutes a cure to plastic waste management, its sustainability within a pure closed system of plastic waste management remains a great concern. The issue of critical concern relates to the adequacy of recycling feedstock or resources, which significantly, is plastic waste. A critical interrogation settles on if there exists enough plastic waste to produce to meet the global plastic demand. To answer this question, Wiah et al. (Citation2022) proposed and derived transitional probabilities as a quantitative technique to assess the sustainability of plastic circular economy. The idea of transition defines movement in and out of a state (Addor et al., Citation2015; Shimer, Citation2012). The objective was to ascertain if plastic recycling or plastic circular economy can be sustained by depending on only plastic waste. It was established that the volume of plastic waste generated since 1950 can be recycled to meet the demand for plastic by the global economy. This is only possible subject to a holistic riddance of plastic waste incineration and discard. A critical review reveals that the derived recycling and waste generation transitional probabilities are less reflective of the complete dynamics of plastic waste management since it did not account for the role of either separation or innovative-driven separation in sustainable plastic circular economy. Contentiously, the separation target and the same driven by innovation are significant in determining both quantity and quality of recycled plastics (Addor et al., Citation2022b; Eriksen et al., Citation2018; Moroni & Mei, Citation2020; Serranti & Bonifazi, Citation2019; Zhao et al., Citation2018) due to the huge level of impurities in the plastic waste stream (Tretsiakova McNally et al., Citation2023; Wiah et al., Citation2022). Hence ignoring their roles in plastic waste generation and recycling models fosters sustainability analysis with unsettled information on quantity and quality. To summarize, the derived transitional probabilities which were applied as a quantitative measure of sustainable plastic circular economy does not reflect the role of separation and innovation. This has therefore, necessitated the need to integrate the separation target and innovation rate into the cyclical dynamic models to reflect these parameters on the transitional probabilities, and account for the role of innovation in the sustainability of PCE. In this respect, sustainability analysis of PCE will realistically encapsulate both quantity and quality.

The main objective of this study is to examine the impact of innovation on the sustainability of PCE. Specifically, the paper intends to: modify the linear cyclical dynamic closed (CDC) (Wiah et al., Citation2022) by integrating innovation and separation parameters, derive the plastic waste generation and recycling transitional probabilities, conduct the sustainability analysis and compare them in both the innovation and non-innovation models using numerical simulations.

The paper makes crucial contribution by integrating innovation into the plastic separation process, which enhances the real-world representation of the CDC model. By integrating these parameters, it is possible to examine and compare the performances of the innovative and non-innovative separation models of plastic waste management and their implications on the sustainability (or otherwise) of PCE. As a result of the innovation parameter integrated, the degree of sustainable plastic waste generation and recycling improved progressively right from the worse-case to the best-case scenario. Thus, innovation reduces the rate of contamination or impurities and diverts recyclable plastic wastes from landfills to recycling. This can ensure sustainable plastic waste management via sustainable PCE with outright dependence on plastic waste without resorting to plastic alternative materials.

2. Materials and methods

Transitional probabilities are basically derived from the CDC model of plastic waste management and applied quantitatively to assess the sustainability status of PCE. The referred CDC model is a model that reflects both the forward and reverse transitional phases of a closed-loop plastic waste management. By making use of a two-state system of ordinary differential equations (ODEs), the CDC model reflects the time heterogeneous property of the plastic life cycle. The underlying basic assumption is the model operates within a closed system, that ignores the production of plastics using virgin resources so that the concept of industrial ecology is intrinsically enforced. Considering also as assumptions in this paper; the plastic recycling and waste generation rates follow a Poisson process to characterize them with randomness to derive the transitional probabilities. In addition, the value of innovation is set based on the value of the separation parameter.

In this paper, innovation defines any form of technology incorporated into the process of separating plastic waste. Innovative separation may come in one of the following forms: gravity separation, electrostatic separation, magnetic density separation, floatation separation and sensor-based separation. Regardless of the form, it is critical to simulate a value based on the value of the separation parameter to ensure sufficient waste recovery.

Given the variables $x (t)$ , $y (t)$ , $μ$ , $ψ$ , and the parameters $w = w_{i} + w_{d}$ ; where $x (t)$ and $y (t)$ respectively define the volume of recycled plastic and plastic waste at time period $t$ . The parameters $ψ$ and $μ$ are the plastic: recycling and waste generation rates respectively; $w$ is the combined rates of plastic waste discard ( $w_{d}$ ) and incineration ( $w_{i}$ ). Given a separation rate (target) $β$ with a rate of innovation $γ$ , the proposed innovation-driven separation model is

(1)

\{\begin{matrix} \frac{dy}{dt} = μ (1 - γβ) x - (1 - γ) (ψ + ω) y \\ \frac{dx}{dt} = (1 + γ) ψy - μ (1 - γβ) x \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}

(1)

The system model (EquationEquation 1(1) $\{\begin{matrix} \frac{dy}{dt} = μ (1 - γβ) x - (1 - γ) (ψ + ω) y \\ \frac{dx}{dt} = (1 + γ) ψy - μ (1 - γβ) x \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (1) ) is the innovative form of the non-innovative separation model given by

(2)

\{\begin{matrix} \frac{dy}{dt} = μ (1 - β) x - (ψ + ω) y \\ \frac{dx}{dt} = ψy - μx \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}

(2)

The significant modification made to the system model (EquationEquation 2(2) $\{\begin{matrix} \frac{dy}{dt} = μ (1 - β) x - (ψ + ω) y \\ \frac{dx}{dt} = ψy - μx \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (2) ) is the introduction of the separation and innovation parameters, $β$ and $γ$ respectively. In this paper, all the assumptions underpinning the system model (EquationEquation 2(2) $\{\begin{matrix} \frac{dy}{dt} = μ (1 - β) x - (ψ + ω) y \\ \frac{dx}{dt} = ψy - μx \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (2) ) as held in Wiah et al. (Citation2022) are in play. In addition, a given level of separation target is set and innovation is integrated at a level to balance the effect of separation.

The process of plastic waste generation and product creation through recycling is defined within a discrete time space $t = 0, 1, \dots, n$ . Short-lived plastic waste and products are denoted by $y_{s}$ and $x_{s}$ respectively. Detailed definitions of $y_{s}$ and $x_{s}$ can be found in Wiah et al. (Citation2022). With an assumption that plastic waste generation and recycling within a closed system of plastic waste management commence at time period $t$ , and persist for time $τ$ , both plastic: recycling and waste generation persist for $t + τ$ period. The volumes of plastic: waste generated and products recycled at time period $t + τ$ are $y (t + τ)$ and $x (t + τ)$ respectively. The following variables are further defined: $p_{yx}$ denotes the transitional probability for plastic recycling; $p_{xy}$ is the transitional probability for plastic waste generation; $y_{s} (t, τ)$ is the volume of short-lived (uncensored) plastic waste and; $x_{s} (t, τ)$ represents the volume of short-lived (uncensored) recycled plastics. By definition, $y_{s} (t, τ)$ and $x_{s} (t, τ)$ are, respectively, plastic waste and recycled plastics at time period $t + τ$ , which were neither plastic waste nor recycled plastics somewhere within the period [t, $t + τ$ ].

Presented below, are the sets of linear systems of ODEs in the innovative and non-innovative models of PCE.

(3)

\{\begin{matrix} \frac{dy (t + τ)}{dt} = μ (1 - γβ) x (t + τ) - (1 - γ) (ψ + ω) y (t + τ) \\ \frac{d y_{s} (t, τ)}{dt} = μ (1 - γβ) x (t + τ) - (1 - γ) (ψ + ω) y_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}

(3)

(4)

\{\begin{matrix} \frac{dx (t + τ)}{dt} = (1 + γ) ψy (t + τ) - μ (1 - γβ) x (t + τ) \\ \frac{d x_{s} (t, τ)}{dt} = (1 + γ) ψy (t + τ) - μ (1 - γβ) x_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}

(4)

In the above systems of ODEs, (EquationEquation 3(3) $\{\begin{matrix} \frac{dy (t + τ)}{dt} = μ (1 - γβ) x (t + τ) - (1 - γ) (ψ + ω) y (t + τ) \\ \frac{d y_{s} (t, τ)}{dt} = μ (1 - γβ) x (t + τ) - (1 - γ) (ψ + ω) y_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (3) ) and (EquationEquation 4(4) $\{\begin{matrix} \frac{dx (t + τ)}{dt} = (1 + γ) ψy (t + τ) - μ (1 - γβ) x (t + τ) \\ \frac{d x_{s} (t, τ)}{dt} = (1 + γ) ψy (t + τ) - μ (1 - γβ) x_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (4) ) represent the probabilistic characterization of (Equation 1) since the terms $y_{s} (t, τ)$ and $x_{s} (t, τ)$ introduce some forms of randomness into the system. The system (EquationEquation 3(3) $\{\begin{matrix} \frac{dy (t + τ)}{dt} = μ (1 - γβ) x (t + τ) - (1 - γ) (ψ + ω) y (t + τ) \\ \frac{d y_{s} (t, τ)}{dt} = μ (1 - γβ) x (t + τ) - (1 - γ) (ψ + ω) y_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (3) ) is the random form of plastic waste generation from which the plastic recycling transitional probability can be derived. Similarly, (EquationEquation 4)(4) $\{\begin{matrix} \frac{dx (t + τ)}{dt} = (1 + γ) ψy (t + τ) - μ (1 - γβ) x (t + τ) \\ \frac{d x_{s} (t, τ)}{dt} = (1 + γ) ψy (t + τ) - μ (1 - γβ) x_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (4) presents the random version of plastic recycling, which is useful for the derivation of the plastic waste generation transitional probability. Innovation is integrated into the system to ensure that the effect of separation on quantity of recycled plastics is mitigated by the existing innovative processes. Thus, the total volume of waste generated $μx$ is supposed to reduce by $μβx$ due to separation at a target $β$ in (EquationEquation 3(3) $\{\begin{matrix} \frac{dy (t + τ)}{dt} = μ (1 - γβ) x (t + τ) - (1 - γ) (ψ + ω) y (t + τ) \\ \frac{d y_{s} (t, τ)}{dt} = μ (1 - γβ) x (t + τ) - (1 - γ) (ψ + ω) y_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (3) ); however, innovation guarantees that the loss is minimized to a volume of $μγβx$ and the same volume is regained as indicated in the system (EquationEquation 4(4) $\{\begin{matrix} \frac{dx (t + τ)}{dt} = (1 + γ) ψy (t + τ) - μ (1 - γβ) x (t + τ) \\ \frac{d x_{s} (t, τ)}{dt} = (1 + γ) ψy (t + τ) - μ (1 - γβ) x_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (4) ). Therefore, the possibility to increase both quality and quantity of recycled plastics exists. In the non-innovative system model, the probability model is given by

(5)

\{\begin{matrix} \frac{dy (t + τ)}{dt} = μ (1 - β) x (t + τ) - (ψ + ω) y (t + τ) \\ \frac{d y_{s} (t, τ)}{dt} = μ (1 - β) x (t + τ) - (ψ + ω) y_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}

(5)

(6)

\{\begin{matrix} \frac{dx (t + τ)}{dt} = ψy (t + τ) - μx (t + τ) \\ \frac{d x_{s} (t, τ)}{dt} = ψy (t + τ) - μ x_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}

(6)

The system (EquationEquation 5(5) $\{\begin{matrix} \frac{dy (t + τ)}{dt} = μ (1 - β) x (t + τ) - (ψ + ω) y (t + τ) \\ \frac{d y_{s} (t, τ)}{dt} = μ (1 - β) x (t + τ) - (ψ + ω) y_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (5) ) proxies the probabilistic form of plastic waste generation in the non-innovative model, which will be used to derive the plastic recycling transitional probability. Also, the probabilistic representation of plastic recycling in the non-innovative model is given by (EquationEquation 6(6) $\{\begin{matrix} \frac{dx (t + τ)}{dt} = ψy (t + τ) - μx (t + τ) \\ \frac{d x_{s} (t, τ)}{dt} = ψy (t + τ) - μ x_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (6) ). Total volume of plastic waste generated will reduce by $μβx$ due to separation, which further translates into huge quantity loss of recycled plastics. Thus, in this non-innovative model, the process of separation will enhance quality of recycled plastics, but at a cost of a decreased quantity.

2.1. Derivation of the transitional probabilities

To solving for the recycling transitional probability $p_{yx}$ , eliminate $μx (t + τ)$ from (EquationEquation 3(3) $\{\begin{matrix} \frac{dy (t + τ)}{dt} = μ (1 - γβ) x (t + τ) - (1 - γ) (ψ + ω) y (t + τ) \\ \frac{d y_{s} (t, τ)}{dt} = μ (1 - γβ) x (t + τ) - (1 - γ) (ψ + ω) y_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (3) ) and apply the Laplace transform to obtain.

(7)

sY (s + τ) - y (0 + τ) - [s Y_{s} (s, τ) - y_{s} (0, τ)] = (1 - γ) (ψ + w) Y_{s} (s, τ) - (1 - γ) (ψ + w) Y (s + τ)

(7)

Within the discrete time frame $τ \in [0, 1]$ , and $y_{s} (0, τ) = y_{s 0} (τ) = 0$ . Using $y (0) = y_{0}$ , (6) reduces to

sY (s + 1) - y (t) - s Y_{s} (s, 1) = (1 - γ) (ψ + w) Y_{s} (s, 1) - (1 - γ) (ψ + w) Y (s + 1)

(8)

\Rightarrow Y (s + 1) = Y_{s} (s, τ) + \frac{y (τ)}{s + (1 - γ) (ψ + w)}

(8)

Define $y (t, τ) = y (t + 1)$ , then, transforming (EquationEquation 8(8) $\Rightarrow Y (s + 1) = Y_{s} (s, τ) + \frac{y (τ)}{s + (1 - γ) (ψ + w)}$ (8) ) via the inverse Laplace transforms gives

(9)

y (t + 1) = y_{s} (t + 1) + y (t) exp [- (1 - γ) (ψ + w) t]

(9)

A Poisson process is assumed for the recycling rate $ψ$ , which is subject to an arrival time $ψt$ given by

(10)

exp (- ψt) = 1 - p_{yx}

(10)

Simplifying (EquationEquation 9(9) $y (t + 1) = y_{s} (t + 1) + y (t) exp [- (1 - γ) (ψ + w) t]$ (9) ) using (EquationEquation 10(10) $exp (- ψt) = 1 - p_{yx}$ (10) ) yields

y (t + 1) = y_{s} (t + 1) + y (t) {(1 - p_{yx})}^{1 - γ} exp {(- wt)}^{1 - γ}

{(1 - p_{yx})}^{1 - γ} = \frac{y (t + 1) - y_{s} (t + 1)}{y (t) exp [(1 - γ) wt]}

(11)

p_{yx} = 1 - {\{\frac{y (t + 1) - y_{s} (t + 1)}{y (t) exp [(1 - γ) wt]}\}}^{\frac{1}{1 - γ}}

(11)

Similarly, to derive the plastic waste generation probability $p_{xy}$ , the term $ψy (t + τ)$ is eliminated from both equations in (EquationEquation 4(4) $\{\begin{matrix} \frac{dx (t + τ)}{dt} = (1 + γ) ψy (t + τ) - μ (1 - γβ) x (t + τ) \\ \frac{d x_{s} (t, τ)}{dt} = (1 + γ) ψy (t + τ) - μ (1 - γβ) x_{s} (t, τ) \\ x (0) = x_{0} \geq 0 \\ y (0) = y_{0} \geq 0 \end{matrix}$ (4) ); applying the Laplace transform to the results of the elimination gives

sX (s + τ) - x (0 + τ) - [s X_{s} (s, τ) - x_{s} (0, τ)] = μ (1 - γβ) X_{s} (s, τ) - μ (1 - γβ) X (s + τ)

sX (s + τ) - x (t) - s X_{s} (s, τ) = μ (1 - γβ) X_{s} (s, τ) - μ (1 - γβ) X (s + τ)

(12)

\Rightarrow X (s + 1) = X_{s} (s, 1) + \frac{x (t)}{s + μ (1 - γβ)}

(12)

The inverse Laplace transform converts (EquationEquation 12(12) $\Rightarrow X (s + 1) = X_{s} (s, 1) + \frac{x (t)}{s + μ (1 - γβ)}$ (12) ) to

(13)

x (t + 1) = x_{s} (t + 1) + x (t) exp [- μ (1 - γβ) t]

(13)

The rate plastic waste generation rate $μ$ follows a Poisson process with an arrival time $μt$ , and it is defined by

(14)

exp (- μt) = 1 - p_{xy}

(14)

The expression (EquationEquation 13(13) $x (t + 1) = x_{s} (t + 1) + x (t) exp [- μ (1 - γβ) t]$ (13) ) is simplified by substituting (EquationEquation 14(14) $exp (- μt) = 1 - p_{xy}$ (14) ) to give

{(1 - p_{xy})}^{1 - γβ} = \frac{x (t + 1) - x_{s} (t + 1)}{x (t)}

(15)

p_{xy} = 1 - {[\frac{x (t + 1) - x_{s} (t + 1)}{x (t)}]}^{\frac{1}{1 - γβ}}

(15)

Equations (EquationEquation 11(11) $p_{yx} = 1 - {\{\frac{y (t + 1) - y_{s} (t + 1)}{y (t) exp [(1 - γ) wt]}\}}^{\frac{1}{1 - γ}}$ (11) ) and (EquationEquation 15(15) $p_{xy} = 1 - {[\frac{x (t + 1) - x_{s} (t + 1)}{x (t)}]}^{\frac{1}{1 - γβ}}$ (15) ) are the innovative-based transitional probabilities for plastic recycling and plastic waste generation respectively.

Following similar approach, the non-innovative-driven transitional probabilities for plastic recycling and plastic waste generation are, respectively, represented by

(16)

p_{y x}^{n o n_i} = 1 - \frac{y (t + 1) - y_{s} (t + 1)}{y (t) exp [- w t]}

(16)

(17)

p_{x y}^{n o n_i} = 1 - \frac{x (t + 1) - x_{s} (t + 1)}{x (t)}

(17)

A summary of the transitional probabilities based on four practical scenarios is presented in .

Table 1. Summary of plastic recycling and waste generation based on four scenarios

Display Table

2.2. Variables and parameter values

The values of $x$ and $y$ can be obtain in (Addor et al., Citation2022a, Citation2022b), while that of the parameters and their sources are presented (). Further, details on the computation of the values of $x_{s}$ and $y_{s}$ which were computed from the values of $x$ and $y$ , respectively, can be obtained in Wiah et al. (Citation2022). The data span the period 1988 ( $t = 0$ ) to 2021 ( $t = 34$ ).

Table

Display Table

3. Results and discussions

Based on the above parameter values including that of $x_{s}$ and $y_{s}$ , the transitional probabilities ( $p_{yx}$ , $p_{xy}$ ) were computed under the given scenarios ( $S 1, S 2, S 3, S 4$ ). The sustainability analysis of PCE will be performed using the transitional probabilities, this will be followed by sensitivity analysis.

Results

Figure presents the transitional probabilities for plastic waste recycling subject to the role of the rate of innovation ( $γ$ ) in a plastic waste separation model at a separation rate or target ( $β$ ), under $S 1$ (a), $S 2$ (b), $S 3$ (c) and $S 4$ (d). Under the full force of waste plastic incineration and discard ( $S 1$ ), the transitional probability for plastic recycling starts from zero in 1988 and attains positive in 1989, then declines continuously thereafter to attain negative for the rest of the 32 years (1990–2021). Under the operative force of only plastic waste discard, the plastic recycling transitional probability begins from zero in the initial period (1988), then gains positivity over a two-year period (1989–1990), but attains negative for the rest of the 31-year period (1991–2021). It is also clear that from the initial period under the single force of plastic waste incineration, the plastic recycling probability starts from zero, then attains positive for 7 years (1989–1995); however, it attains negative over a 26-year period (1996–2021). Finally, under the complete riddance of plastic waste discard and incineration, the plastic recycling transitional probability persistently attains positive values from 1989 through to 2021 after rising from zero in 1988. These results imply that in the innovation-driven separation model, plastic recycling cannot be sustained under $S 1$ , $S 2$ and $S 3$ , although sustainability persists for seven out of 34 years. It is sustainable only under $S 4$ .

Figure 1. Plastic recycling transitional probabilities with innovation-driven separation.

The transitional probability for plastic waste recycling in the non-innovative plastic waste separation model, under $S 1$ (e), $S 2$ (f), $S 3$ (g) and $S 4$ (h) are depicted (). It is obvious that under the combined operative forces of plastic waste discard and incineration (e) and the single operative force of plastic waste discard (f), the plastic recycling transitional probabilities consistently assume negative values for the rest of the period (1988–2021) after attaining zeros in 1988, with unprecedented negative values under $S 1$ relative to $S 2$ . As an improvement over the situations under $S 1$ and $S 2$ , the transitional probability for plastic recycling under $S 3$ rises from zero in 1988 and assumes a positive value in 1989, but attains negative values from 1990 to 2021.

Figure 2. Plastic recycling transitional probability with non-innovation-driven separation.

Comparing the negative values in the last years shows that $S 3$ (g) is better than $S 2$ , which agrees with the results in under the respective scenarios. Finally, under $S 4$ (total riddance of plastic waste discard and incineration), the plastic recycling transitional probability rises from zero and assumes positive values for the rest of the period. Therefore, as it was in the case of the innovation-driven separation model, plastic recycling is sustainable only under $S 4$ . These results are consistent with the results in Wiah et al. (Citation2022).

The comparative performances of the plastic recycling transitional probabilities under $S 1$ , $S 2$ , $S 3$ and $S 4$ in the separation with innovation and without innovation are summarized in . It is evidently clear that under $S 1$ (j), $S 2$ (k), $S 3$ (l) and $S 4$ (m), the plastic recycling transitional probabilities associated with the innovation-characterized separation model lie above the plastic recycling transitional probabilities for the non-innovation separation model. It is important to note that although the plastic recycling transitional probabilities are positive for both innovative and non-innovative separation models under $S 4$ (m), the values are unprecedentedly higher in the model that reflects the integrated role of innovation than they are in the separation model without innovation.

Figure 3. Innovative versus non-innovative separation-based recycling.

The simulated results of the transitional probabilities for plastic waste generation in the models characterizing the integrated role of innovation and non-innovation are presented under $S 1$ , $S 2$ , $S 3$ and $S 4$ in . It is crucial to note that since the plastic waste generation transitional probabilities are independent on $w = w_{i} + w_{d}$ , the values are the same under the four given scenarios. Throughout the entire period, from 1988 ( $t = 0$ ) to 2021 ( $t = 34$ ), the values of the plastic waste generation transitional probabilities non-negatives (positives), indicating that plastic waste generation continues unabated. As illustrated, the values of the transitional probabilities for plastic waste generation in the integrated model of separation and innovation are higher than the values in the separation without innovation model.

Figure 4. Plastic waste generation transitional probabilities with the integrated roles of separation with and without innovation.

To further illustrate the effect of innovation on plastic waste generation and recycling, a sensitivity analysis is conducted () to determine the degree of responsiveness of plastic recycling and waste generation to variations in the innovation parameter ( $γ$ ).

Figure 5. Sensitivity of plastic waste recycling to variations in innovation.

Figure 6. Sensitivity of the plastic waste generation probability subject to variation in innovation.

presents 10% and 20% variations in the rate of innovation under $S 3$ (p) and $S 4$ (q). Focusing the analysis under $S 3$ indicates that a 10% increase in innovation induces about 186% increase in the plastic recycling probability; while a 10% reduction in the rate of innovation induces approximately 259% decreases in the recycling probability. Increasing the innovation rate further to 20%, yields positive probabilities (close to unity) over the entire period.

Expectedly under $S 4$ , the same 10% upward variation in innovation accounts for a more than proportionate upward variation (21% increase) in the recycling probabilities; however, a 10% decrease in the rate of innovation explains a more than proportionate decrease (29% approximately) in the downward trend.

Finally, shows the effect of a 10% upward variation in innovation on the transitional probability for plastic waste generation. As a result of a 10% upward variation in the rate of innovation, the plastic waste generation transitional probability increases by 31.4%, while the same probability decreases by 31.9% due to a 10% downward variation in the rate of innovation.

Discussions

It is eminent to emphasize that negative values indicate undefined probabilities and are useful in defining unsustainable plastic recycling or waste generation as the case may apply. To make meaning of the concept of sustainable plastic circular economy, both the plastic recycling and waste generation transitional probabilities play critical roles. In the foregoing analysis, while the plastic waste recycling probabilities are completely positive only under $S 3$ , the plastic waste generation probabilities are positives in all the specified scenarios. In all scenarios where the positive plastic waste generation transitional probabilities correspond with negative plastic recycling transitional probabilities, the plastic-waste-material loop opens at one end (reverse end), which defines unsustainable plastic recycling. This explains the fact that the amount of waste generated cannot be recycled to perpetuate the cycle of plastic: waste generation (forward transition) and product creation (reverse transition). Implacably, in all scenarios with corresponding positive values of plastic waste generation and recycling probabilities, plastic circular economy is sustainable as the cycle that defines both forward transition to reverse transition continuously remains unbroken.

Summarizing from the results, it can be established that plastic recycling cannot be sustained under the following events: the combined forces of plastic waste discard and incineration ( $S 1$ ), the single force of plastic waste discard ( $S 2$ ) and the single force of plastic waste incineration ( $S 3$ ). Plastic recycling can only be sustained in the event of complete riddance of plastic waste discard and wasteful incineration. Thus, plastic circular economy can only be sustained when plastic waste discard and wasteful incineration are completely avoided.

Reflecting on the results underscores the fact that plastic circular economy is unsustainable in both the innovation-drivenand non-innovative separation models under $S 1$ , $S 2$ and $S 3$ . Also, in the same vein, plastic circular economy is sustainable in both models under $S 4$ . This implies that innovation is necessary but not sufficient to engineer sustainable plastic circular economy. A sufficient condition is needed to check human knowledge and attitude in line with indiscriminate plastic waste discard. This probably constitutes a key reason for achieving sustainable plastic circular economy in the event of complete prohibition of plastic waste discard and unproductive incineration. That notwithstanding, the role of innovation in sustainable plastic circular economy cannot be downplayed as the innovation-driven separation model outperformed the non-innovation separation model in plastic recycling. Reflecting more, the innovation-driven separation produced higher values for plastic recycling under $S 4$ than the separation model without innovation. Specifically, under $S 3$ , while sustainable plastic recycling is impossible over the entire period in the separation model without innovation, this is achievable over 7 years in the separation model with innovation. Elucidating further, while sustainability is attainable under $S 4$ in both models, the plastic recycling values are unprecedentedly higher in the integrated model of separation with innovation than they are in the separation model without innovation. This shows that while a lot of plastic waste are recovered for recycling in the innovative model, a lot of plastic waste are lost in the non-innovative model during the separation process (Dahlbo et al., Citation2017; Eriksen et al., Citation2018; Heinzel et al., Citation2015; Huysman et al., Citation2017; LACW, Citation2017; Petersen et al., Citation2015; Ragaert et al., Citation2017; Vadenbo et al., Citation2016; van der Harst et al., Citation2016) due to lack of innovation to treat impurities.

Shifting the focus to the forward link that completes the loop (cycle) of plastic circular economy, the positive values of plastic waste generation are higher in the innovation-driven separation model than they are in the separation model without innovation. This is consistent with the fact that huge volume of plastic waste is lost to impurities (Dahlbo et al., Citation2017; Eriksen et al., Citation2018; Heinzel et al., Citation2015; Huysman et al., Citation2017; LACW, Citation2017; Petersen et al., Citation2015; Ragaert et al., Citation2017; Vadenbo et al., Citation2016; van der Harst et al., Citation2016) due to lack of innovation leading to lower rate of recycling in the non-innovative separation model. On the other hand, huge volume of high-quality plastic can be recovered through innovative separation (Bonifazi et al., Citation2019; Wu et al., Citation2020) to engineer higher rate of plastic recycling. This explains why recycling is unprecedentedly high in the integrated model of separation and innovation. The results are finally buttressed by: Yu et al. (Citation2022), who established that achieving circular economy with sustainable resource recycling can be engineered with innovative sorting devices which provide potentials for accurately analyzing the chemical composition of plastic waste; Tao et al. (Citation2023), who attributed enhanced accuracy and speed of waste separation processes to hyperspectral imaging and machine learning models, which has transformed plastic recycling; Choi et al. (Citation2023), who proposed integrated image sensors and deep learning algorithms to advance plastic waste classification and recycling efficiency; Zheng et al. (Citation2018), who ascribed higher rates of plastic waste sorting and recycling to a novel discriminating model using near-infrared hyperspectral imaging; Kara et al. (Citation2022), who alluded the development of sustainable environmental practices to innovative focus on circular economy from technical and economical to political and socio-cultural dimensions.

As policy prioritization is informed by the sensitivities of responds variables with respect to variations in key parameters, the degree of responsiveness of plastic recycling and waste generation to variations in the innovation parameter was conducted. Reflecting on the results of the sensitivity analysis, the recycling probability is more sensitive to variation in innovation; however, it is more sensitive in the downward trend than it is in the upward trend. In general, the rate of innovation is a highly sensitive correlate of plastic recycling with a higher rate of sensitivity in downward variations than in upward variation. Also, evidenced from the sensitivity analysis is the fact that the plastic waste generation transitional probability correlates directly with the rate of innovation with almost the same degree of sensitivity in both upward and downward trends. Indicatively, innovation is a positive determinant of recyclable plastic waste. Based on the findings of this paper, it is necessary to integrate innovation in the plastic separation process to reduce impurities or maximize recovery of recyclable plastic wastes to optimize plastic recycling.

The paper derives its strength from an inherent non-conservative property of the innovation-driven separation model which makes it possible for it to thrive in the event of an emerging powerful innovation. This is evinced by the role of innovation as a positive predictor of both plastic recycling and recyclable waste. Hence, innovation can be enhanced to increase the volume of recyclable waste and recycled plastics in the event of technology obsolescence. The limitation of the paper is that the model does not reflect the role of population characteristics such as growth rate, knowledge and attitude. It is thus, not easy to draw conclusion on sustainability status of plastic circular economy in the event of these neglected variables. Albeit this neglection, the study has been able to assess the role of innovation in achieving sustainable plastic circular economy, all other things being equal. Another limitation arises from the fact that the value of the innovation parameter was a simulated value set based on the value of the separation target, which can raise concern about the generalizability of results. That notwithstanding, the other parameters were based on real plastic data spanning the period 1988–2021. This has the tendency to increase generalizability of the study above 80%.

It is therefore, an earnest recommendation that population characteristics be integrated in future studies on sustainable plastic circular economy models. In addition, a reasonable method should be applied to estimate the innovative separation parameter in future studies; attempts should be made to compare the impact of specific innovative separations on sustainable PCE.

4. Conclusion

The main aim of the paper is to examine the effect of innovation on the sustainability of plastic circular economy in a waste separation CDC model of plastic waste management. This was achieved by comparing the comparative performances of plastic waste separation models with and without innovation under four pragmatic scenarios; the full force of plastic waste discard and incineration ( $S_{1}$ ), the single force of plastic waste discard ( $S_{2}$ ), the single force of plastic waste incineration ( $S_{3}$ ) and the complete riddance of plastic waste discard and incineration ( $S_{4}$ ). The simulated results evince that in both models (with and without innovation) in general, plastic circular economy cannot be sustained subject to the first three scenarios; but it can be sustained only under the fourth scenario which emphasizes the complete riddance of plastic waste discard and unproductive incineration. This underscores the fact that innovation is a necessary but not a sufficient condition for the attainment of sustainable plastic circular economy. There is the need for such an intrinsic driver as positive human attitude towards plastic waste discard together with productive incineration of plastic waste. However, under all the scenarios, the innovation-driven separation model outperformed the model without innovation in both the forward and reverse transitional phases of plastic circular economy. This therefore, establishes the fact that innovation enhances the degree to which plastic circular economy can be sustained by reducing the rate of contamination or impurities in the stream of plastic waste. This helps in recovering or diverting plastic wastes away from landfills to recycling. In general, the rate of innovation is a highly positive sensitive determinant of plastic recycling with a higher rate of sensitivity in the downward than it is in upward trend. Also, innovation is a positive determinant of recyclable plastic waste with somewhat equal rates of sensitivities in both directional variations.

Considering the result of the closed model of plastic waste management, the direct impact of this research on preserving the environment, not only within Sub-Sahara Africa but within the global environment, can be achieved in the following ways: First, the closed model prohibits the production of plastics using virgin resources, which is critical for fossil fuel reserve conservation. Also, by prohibiting plastic production based on virgin resources, a waste-based plastic production is emphasized through recycling which has proven to be environmentally friendly due to its low emissions among the various plastic waste treatment techniques. Further, depending plastic waste as the only way to produce plastic practically enforces the concept of industrial ecology, this places monetary value on plastic waste since it now assumes the status of non-substitutable valuable resource which can discourage people from discarding as well as encourage others to recover them from landfills and the environment. Last but not least, with the role of innovation integrated into the separation process, high-quality plastics will be produced in higher quantities, and this is crucial for sustainable plastic recycling and waste generation. It is therefore expedient to integrate innovation in the plastic separation process to lessen impurities or maximize recovery of recyclable plastic wastes to enhance plastic recycling. By ensuring a closed plastic-waste-material loop, in the applied closed model, plastic circular economy can be sustained to achieve sustainable: environment, public health, water resources, climate through reduced greenhouse emission, employment and poverty reduction, which constitute strategic goals of the SDGs.

Public Interest Statement

The paper assessed the sustainability of plastic circular economy (PCE) using transitional probabilities by examining the role of innovation in a plastic waste separation model. The model was formulated using a two-state cyclical dynamic closed (CDC) model based on ordinary differential equations. The CDC model reflects both the forward and reverse transitions of plastic waste management to characterize a closed-loop plastic waste management with a zero tolerance for virgin plastic production. Sustainability of PCE was assessed using plastic waste separation models with and without innovation under four practical scenarios; the force(s) of plastic: waste discard and incineration, waste discard only, waste incineration only and the complete riddance of plastic waste discard and incineration. It was evidenced that PCE cannot be sustained subject to the first three scenarios; it is only sustainable under the fourth scenario, which confirms previous results. Generally, the separation model with innovation outperformed that without innovation.

Authors contributions

The original research idea was conceived by Authors John A. Addor (JAA) and Eric N. Wiah (ENW), while the need for improvement was discussed by Authors JAA and John Bentil (JB). The framework for the study including the mathematical formulation was designed by JAA and ENW. The improved version of the model was accomplished by JAA and JB, while derivations were executed by JAA. JAA and JB conducted the literature search and data collection was handled by JAA and ENW. The simulations were performed by JAA, JB and ENW. JAA and JB performed the analysis. Final editing was performed by JB and ENW. The original manuscript was prepared by JAA. All authors read through the work, agreed and submitted to the journal.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data will be made available upon request to the corresponding author.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Notes on contributors

Eric Neebo Wiah

John Awuah Addor is a Senior Lecturer at the Department of Mathematics, Statistics and Actuarial Science (MSA), Takoradi Technical University (TTU), Ghana. He holds a Doctor of Philosophy in Mathematics from the University of Mines and Technology, Tarkwa, Ghana, in 2022. His current research interest is in applied mathematics in the areas of mathematical modelling, dynamical systems, epidemiology, environment and sustainability, optimal control theory, circular and green economy. He has served MSA of TTU for the past 16 years. He is a member of the Technical University Teachers Association of Ghana (TUTAG) and Ghana Statistical Association (GSS).

The main research focus of the authors is in the area of Applied Mathematics and Engineering, with major emphasis on mathematical/engineering modelling and dynamical systems. Currently, the research focus of the authors is in the following areas: sustainable environment, plastic waste management, climate change, resource conservation and energy, social interaction dynamics and epidemiology. This paper falls within the context of plastic waste management, sustainable environment, resource conservation and energy, and climate change. Precisely, the paper is an aspect of a research project titled “Mathematical Models for the Cyclical Dynamics of Plastic Waste Management”.

References

Addor, J. A., Wiah, E. N., & Alao, F. I. (2022a). An improved two-states cyclical dynamic model for plastic waste management. Asian Research Journal of Mathematics, 18(5), 52–13. https://doi.org/10.9734/arjom/2022/v18i530378
Google Scholar
Addor, J. A., Wiah, E. N., & Alao, F. I. (2022b). Mathematical model for the cyclical dynamics of plastic waste management: A two-state closed model. Journal of Materials Science Research and Reviews, 9(2), 15–36. https://journaljmsrr.com/index.php/JMSRR/article/view/30234
Google Scholar
Addor, J. A., Yeboah-Forson, D., & Padi, A. (2015). Robustness of the job-finding, job loss model in modelling the employment and unemployment rates of Ghana. Mathematical Theory and Modelling, 5(8), 114–133.
Google Scholar
Bonifazi, G., Capobianco, G., Palmieri, R., & Serranti, S. (2019). Hyperspectral imaging applied to the waste recycling sector. Spectrosc Europe, 31, 8–11. https://doi.org/10.1255/sew.2019.a3
Google Scholar
Bucknall, D. G. (2020). Plastics as a materials system in a circular economy. Philosophical Transactions of the Royal Society A, 378(2176), 2019026820190268. https://doi.org/10.1098/rsta.2019.0268
Google Scholar
Choi, J., Lim, B., & Yoo, Y. (2023). Advancing plastic waste classification and recycling efficiency: Integrating image sensors and deep learning algorithms. Applied Sciences, 13(18), 10224. https://doi.org/10.3390/app131810224
Google Scholar
Dahlbo, H., Poliakova, V., Myll¨ari, V., Sahimaa, O., & Anderson, R. (2017). Recycling potential of post-consumer plastic packaging waste in Finland. Waste Management, 71, 52–61. https://doi.org/10.1016/j.wasman.2017.10.033
PubMed Web of Science ®Google Scholar
Ellen MacArthur Foundation (EMF). (2013). Towards the circular economy: Economic and business rationale for an accelerated transition. online report. https://ellenmacarthurfoundation.org/towards-the-circular-economy-vol-1-an-economic-and-business-rationale-for-an
Google Scholar
Eriksen, M. K., Damgaard, A., Boldrin, A., & Astrup, T. F. (2018). Quality assessment and circularity potential of recovery systems for household plastic waste. Journal of Industrial Ecology, 23(1), 1–13. https://doi.org/10.1111/jiec.12822
Web of Science ®Google Scholar
Global Environment Community (GEF). (2018). Plastic and circular economy. 54th GEF councelling meeting, June 24-26, 2018, a STAP document. GEF/STAP/C.54/Inf.05.2018
Google Scholar
Grigore, M. E. (2017). Methods of recycling, properties and applications of recycled thermoplastic polymers. Recycling, 2(4), 24. https://doi.org/10.3390/recycling2040024
Google Scholar
Heinzel, J. V., Larsen, C. S., Tønning, K., Malmgren-Hansen, B., & Nilsson, N. H. (2015). Meka-nisk sortering af plastaffald fra husholdninger (mechanical sorting of plastic waste from households) [report]. Danish Environmental Protection Agency.
Google Scholar
Huysman, S., De Schaepmeester, J., Ragaert, K., Dewulf, J., & De Meester, S. (2017). Performance indicators for a circular economy: A case study on post-industrial plastic waste. Resources, Conservation & Recycling, 120, 46–54. https://doi.org/10.1016/j.resconrec.2017.01.013
Web of Science ®Google Scholar
Kara, S., Hauschild, M., Sutherland, J., & McAloone, T. (2022). Closed-loop systems to circular economy: A pathway to environmental sustainability? CIRP Annals, 71(2), 505–528. https://doi.org/10.1016/j.cirp.2022.05.008
Web of Science ®Google Scholar
LACW. (2017). Statistics on waste managed by local authorities in England in 2016/2017. Retrieved January 3, 2022, from. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/664594/LACW_mgt_annual_Stats_Notice_Dec_2017.pdf
Google Scholar
Ledsham, N. (2023). Creating circular economy for plastics. The sustainability institute by ERM, 2020-2023. https://www.sustainability.com/thinking/creating-a-circular-economy-for-plastics/
Google Scholar
Moroni, M., & Mei, A. (2020). Characterization and separation of traditional and bio-plastics by hyperspectral devices. Applied Sciences, 10(8), 2800. https://doi.org/10.3390/app10082800
Google Scholar
Nunez, C. (2021). Four key ideas to building a circular economy. National Geographics. https://www.nationalgeographic.com/science/article/paid-content-four-key-ideas-to-building-a-circular-economy-for-plastics
Google Scholar
Petersen, C., Mayland, C., & Manokaran, S. (2015). Plast fra restaffald—analyse af restaffald med fokus p°a plast (plastic from residual waste—analysis of the residual waste with focus on plastic). EcoNet and ARC.
Google Scholar
Ragaert, K., Delva, L., & Geem, K. (2017). Mechanical and chemical recycling of solid plastic waste. Waste Management, 69, 24–58. https://doi.org/10.1016/j.wasman.2017.07.044
PubMed Web of Science ®Google Scholar
Serranti, S., & Bonifazi, G. (2019). Techniques for separation of plastic wastes. In F. Pacheco-Torgal, J. Khatib, F. Colangelo, & R. Tuladhar (Eds.), Woodhead publishing series in civil and structural engineering (pp. 9–37). Woodhead Publishing. https://doi.org/10.1016/B978-0-08-102676-2.00002-5
Google Scholar
Shimer, R. (2012). Reassessing the ins and outs of unemployment. Review of Economic Dynamics, 15(2), 127–148. https://doi.org/10.1016/j.red.2012.02.001
Web of Science ®Google Scholar
Tao, J., Gu, Y., Hao, X., Liang, R., Wang, B., Cheng, Z., Yan, B., & Chen, G. (2023). Combination of hyperspectral imaging and machine learning models for fast characterization and classification of municipal solid waste. Resources, Conservation and Recycling, 188, 106731. https://doi.org/10.1016/j.resconrec.2022.106731
Web of Science ®Google Scholar
Tretsiakova McNally, S., Lubarsky, H., Vennard, A., Cairns, P., Farrell, C., Joseph, P., Arun, M., Harvey, I., Harrison, J., & Nadjai, A. (2023). Separation and characterization of plastic waste packaging contaminated with food residues. Polymers, 15(13), 2943. https://doi.org/10.3390/polym15132943
PubMed Web of Science ®Google Scholar
Vadenbo, C., Hellweg, S., & Astrup, T. F. (2016). Let’s be clear(er) about substitution—A reporting framework to account for product displacement in life cycle assessment. Journal of Industrial Ecology, 21(5), 1078–1089. https://doi.org/10.1111/jiec.12519
Web of Science ®Google Scholar
van der Harst, E., Potting, J., & Kroeze, C. (2016). Comparison of different methods to include recycling in Lcast of aluminium cans and disposable polystyrene cups. Waste Management, 48, 565–583. https://doi.org/10.1016/j.wasman.2015.09.027
PubMed Web of Science ®Google Scholar
Wiah, E. N., Addor, J. A., & Alao, F. I. (2022). Transitional probability for plastic waste management and implication on sustainability. Sustainable Environment, 8(1), 2118654. https://doi.org/10.1080/27658511.2022.2118654
Google Scholar
Wu, X., Li, J., Yao, L., & Xu, Z. (2020). Auto-sorting commonly recovered plastics from waste household appliances and electronics using near-infrared spectroscopy. Journal Cleaner Production, 246, 118732. https://doi.org/10.1016/j.jclepro.2019.118732
Web of Science ®Google Scholar
Yu, J., Liu, X., Manago, G., Tanabe, T., Osanai, S., & Okubo, K. (2022). New terahertz wave sorting technology to improve plastic containers and packaging waste recycling in Japan. Recycling, 7(5), 66. https://doi.org/10.3390/recycling7050066
Google Scholar
Zhao, Y. B., Lv, X. D., & Ni, H. G. (2018). Solvent-based separation and recycling of waste plastics: A review. Chemosphere, 209, 707–720. https://doi.org/10.1016/j.chemosphere.2018.06.095
PubMed Web of Science ®Google Scholar
Zheng, Y., Bai, J., Xu, J., Li, X., & Zhang, Y. (2018). A discrimination model in waste plastics sorting using NIR hyperspectral imaging system. Waste Manag, 72, 87–98. https://doi.org/10.1016/j.wasman.2017.10.015
PubMed Web of Science ®Google Scholar

Sustainability assessment of plastic circular economy: transitional probabilities with innovative separation

ABSTRACT

1. Introduction

2. Materials and methods

2.1. Derivation of the transitional probabilities