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Abstract

To achieve the carbon neutrality goal, enterprises should consider not only the development of new low-carbon emission projects but also the adjustment of the existing high-carbon emission projects. This paper discusses a multi-period project adjustment and selection (MPPAS) problem under the carbon tax and carbon quota policies. First, we propose an uncertain mean-chance MPPAS model for maximizing the profit of the project portfolio under the carbon tax and carbon quota policies. Then, we provide the deterministic equivalent of the proposed model and conduct the theoretical analysis of the impact of carbon tax and carbon quota policies. Next, we propose an improved adaptive genetic algorithm to solve the proposed model. Finally, we give numerical experiments to verify the proposed algorithm’s performance and show the proposed model’s applicability. Research has shown that the government can achieve the carbon neutrality goal by determining reasonable carbon tax and carbon quota policies, and companies can make the optimal investment decisions for the project portfolio by the proposed model. In addition, the proposed algorithm has good performances in robustness, convergence speed, and global convergence.

Keywords

Project portfolio uncertainty theory carbon emission reduction adaptive genetic algorithm

1 Introduction

The project portfolio refers to projects or project sets combined to facilitate effective management and achieve strategic objectives. The low-carbonization of industrial structures is one of the important tasks to reduce carbon emissions and is a long-term strategy that cannot be achieved in the short term. To achieve the low-carbonization of industrial structures, the government should formulate reasonable carbon emission policies and enterprises should adjust existing high-carbon emission projects while developing new low-carbon emission projects. Therefore, the project portfolio with the strategic objective of carbon emissions reduction is described as a multi-period project adjustment and selection (MPPAS) problem. At present, many governments are formulating various policies to reduce carbon emissions. Among them, carbon cap-and-trade and carbon tax policies are the most widely used policies [30]. The carbon cap-and-trade and carbon tax aim to promote enterprises’ carbon emissions reduction through economic means [6]. Recently, many scholars have studied the mixed low-carbon policy combining various carbon emission reduction policies. Shen and Zhao [28] showed that the “carbon trading-carbon tax" mixed policy is more conducive to improving the macro-economy and welfare than a single carbon emission reduction policy. Under the background of the mixed low-carbon policy (carbon tax and carbon trading), Zhao et al. [38] constructed a duopoly game model between supply chains and analyzed the complexity of competition decisions. Shen et al. [29] constructed a supply chain game model under a mixed low-carbon policy that embraced carbon cap-and-trade and carbon tax and discussed the possibility of low-carbon transformation for high-carbon manufacturers. Han et al. [7] explored the incentive effect of revenue-sharing and cost-sharing with subsidy offered by a retailer in the low-carbon supply chain under the carbon tax policy. Theoretical analysis and numerical experiments show that the government should impose the highest-possible carbon tax under the revenue-sharing and the medium-level carbon tax under the cost-sharing with subsidy. Zhu et al. [39] took the automobile manufacturing industry as the research object and identified the effective green manufacturing mode under the carbon tax and carbon quota policies.

Most existing research on the project portfolio problem considers the selection of new projects and ignores the adjustment of existing projects [1, 4, 8, 16, 24, 26, 32]. To maintain competitiveness, enterprises should consider not only the development of new projects but also the adjustment of the existing projects. Some scholars have studied the project portfolio problem considering the selection and adjustment of projects. Huang and Zhao [13] proposed a project selection and adjustment model with consideration of the cost overrun risk. Li et al. [17] proposed an uncertain mean-variance model for project selection and adjustment problems with divisibility. However, the MPPAS problem considering carbon emission reduction has not yet been studied.

In the research of the project portfolio, how to deal with indeterminate parameters is a significant problem. Traditionally, scholars have used probability theory to deal with indeterminate parameters. Probability theory is very effective for handling indeterminate parameters. The premise of using probability theory is that the probability distribution estimated from historical data should be very close to the actual frequency. However, some parameters sometimes have nothing or insufficient available data. For example, unexpected events (such as the COVID-19 outbreak or the outbreak of a trade war) invalidate historical data. Yang et al. [33] and Zhang et al. [36] revealed that using people’s belief levels as probabilities to make investment decisions can lead to counterintuitive results when the economic and social environments change significantly. In addition to probability theory, various theories are proposed to deal with indeterminate parameters, such as fuzzy theory, grey theory, and uncertainty theory. Uncertainty theory proposed by Liu [19] is a new tool for modeling indeterminate parameters. No paradox arises when dealing with people’s inaccurate estimates by uncertainty theory [21]. To deal with people’s inaccurate estimates, many scholars have applied uncertainty theory to study decision-making problems [3, 5, 11, 12, 18, 22, 25, 34, 37]. Huang studied the application of uncertainty theory in the project portfolio problem [10, 14]. This paper discusses the MPPAS problem under the carbon tax and carbon quota policies in the framework of uncertainty theory.

Optimization algorithm is essential to find the exact solution of the proposed model in logical time. In the research of project portfolio problems, heuristic algorithms have become a powerful tool for finding optimal solutions to large-scale problems, and there are many types, such as Genetic Algorithm [23], Ant Colony Optimization Algorithm [2], Gravitational Search Algorithm [27], and Hybrid TLBO-TS Algorithm [15]. Genetic Algorithm (GA) is the most widely used heuristic algorithm. In GA, the proper selection of parameters has a decisive effect on the performance of GA. For example, if the crossover probability is large, the genetic structure of the individual with high fitness is easily damaged. If the crossover probability is small, the convergence speed to the optimal individual is slow. Therefore, many scholars discuss the adaptive genetic algorithm (AGA) to preserve the highly adaptable individuals and population’s diversity [35]. Hu et al. [9] discussed the adaptive crossover probability and adaptive mutation probability to preserve the diversity of the population and highly adaptable individuals. Tang et al. [31] presented an AGA using adaptive crossover and mutation probability to optimize the train timetable, showing that the proposed algorithm quickly yields the good quality solution. This paper proposes an improved AGA. In the proposed algorithm, we present a feasibility repair operation to improve the search efficiency of the feasible solutions.

Table 1 lists the recent work related to our study. From Table 1, it can be seen that most research on reducing carbon emissions is related to the impact of low-carbon policies. Moreover, most existing research on the project portfolio problem focuses on the selection of new projects and ignores the adjustment of existing projects. This paper studies the uncertain multi-period project portfolio problem considering the adjustment of existing projects and the impact of low-carbon policies. In addition, this paper proposes an improved AGA to effectively solve the proposed model.

Table 1
Recent work related to our study

Researcher (year) [Ref.] Low-carbon policy Modeling Theory Algorithm

Shen and Zhao (2022), [28] Carbon tax and carbon trading A computable general equilibrium-based study of low-carbon policy Probability

Shen et al. (2022), [29] Carbon tax and cap-and-trade Low-carbon transition models of high carbon supply chains Probability

Zhao et al. (2022), [38] Carbon tax and carbon trading Complex dynamics and chaos control of the duopoly supply chain

Han et al. (2023), [7] Carbon tax and carbon subsidy Retailer-driven incentive contracts for carbon emission reduction Probability

Zhu et al. (2023), [39] Carbon tax and carbon quota Green manufacturing models under the low-carbon policy

Kosztyan (2020), [16] A matrix-based multilevel multimode project scheduling model Exact algorithm

Dursun et al. (2020), [4] A cognitive map integrated intuitionistic decision-making procedure Fuzzy

Beseiso and Kumar (2021), [1] Selecting interdependent projects using prioritized criteria Fuzzy Genetic algorithm

Huang et al. (2022), [10] Mean-variance model considering synergy between projects Uncertainty TLBO algorithm

Ma et al. (2022), [23] Proactive project scheduling with resource transfer times Uncertainty Genetic algorithm

Hong et al. (2023), [8] Mean-semivariance model with reinvestment and synergy between projects Uncertainty Binary meta-heuristic algorithms

This paper Carbon tax and carbon quota Multi-period project adjustment and selection model Uncertainty Improved adaptive genetic algorithm

Researcher (year) [Ref.]	Low-carbon policy	Modeling	Theory	Algorithm
Shen and Zhao (2022), [28]	Carbon tax and carbon trading	A computable general equilibrium-based study of low-carbon policy	Probability
Shen et al. (2022), [29]	Carbon tax and cap-and-trade	Low-carbon transition models of high carbon supply chains	Probability
Zhao et al. (2022), [38]	Carbon tax and carbon trading	Complex dynamics and chaos control of the duopoly supply chain
Han et al. (2023), [7]	Carbon tax and carbon subsidy	Retailer-driven incentive contracts for carbon emission reduction	Probability
Zhu et al. (2023), [39]	Carbon tax and carbon quota	Green manufacturing models under the low-carbon policy
Kosztyan (2020), [16]		A matrix-based multilevel multimode project scheduling model		Exact algorithm
Dursun et al. (2020), [4]		A cognitive map integrated intuitionistic decision-making procedure	Fuzzy
Beseiso and Kumar (2021), [1]		Selecting interdependent projects using prioritized criteria	Fuzzy	Genetic algorithm
Huang et al. (2022), [10]		Mean-variance model considering synergy between projects	Uncertainty	TLBO algorithm
Ma et al. (2022), [23]		Proactive project scheduling with resource transfer times	Uncertainty	Genetic algorithm
Hong et al. (2023), [8]		Mean-semivariance model with reinvestment and synergy between projects	Uncertainty	Binary meta-heuristic algorithms
This paper	Carbon tax and carbon quota	Multi-period project adjustment and selection model	Uncertainty	Improved adaptive genetic algorithm

The structure of the paper is as follows: Section 2 proposes the uncertain mean-chance MPPAS model under the carbon tax and carbon quota policies. Section 3 provides the deterministic equivalent and the analytical results of the proposed model, and Section 4 proposes an improved AGA to solve the proposed model. Section 5 presents the numerical experiments to illustrate the proposed algorithm’s efficiency and the proposed model’s applicability. Section 6 concludes the paper.

2 Uncertain multi-period project adjustment and selection (MPPAS) model under the carbon tax and carbon quota policies

2.1 Notations

To discuss the MPPAS problem under the carbon tax and carbon quota policies, we use the following notations:

1) Parameters

n : the number of existing projects;

k : the number of new candidate projects;

$T_{i}, T_{j}^{'} :$ the lifetime period of existing project i and new project j;

$S_{i}, S_{j}^{'} :$ the upgrade duration of existing project i and the investment duration of new project j;

Q_it : the output of existing project i in t-th period. The t-th period indicates from time t to t + 1;

$Q_{jt}^{'} :$ the output of new project j in t-th period after new project j is selected;

IC_it : the income per unit output of existing project i in t-th period;

${\tilde{IC}}_{jt}^{'} :$ the uncertain income per unit output of new project j in t-th period after new project j is selected;

${\tilde{IO}}_{it} :$ the uncertain upgrading outlay of existing project i in upgrading t-th period;

${\tilde{IO}}_{jt}^{'} :$ the uncertain investment outlay of new project j in t-th period after new project j is selected;

$F_{it}, F_{jt}^{'} :$ the carbon emission of existing project i and new project j in t-th period;

F_b : after carbon neutralization, the allowable carbon emission per period for project portfolio;

F_t : the allowable carbon emission of project portfolio in t-th period;

B_t : the investment budget in t-th period;

q : the carbon tax per unit carbon emission;

M_it : the income from selling production lines or facilities of existing project i when existing project i is abandoned at time point t;

$w_{i}, w_{j}^{'} :$ the carbon emission per unit output of existing project i and new project j;

r : the discount rate.

2) Decision variables

x_i : the abandonment decision variable for existing project i,

$x_{i} = {\begin{matrix} 1, & if the existing project i is abandoned, \\ 0, & otherwise, \end{matrix}$ y_i : the upgrade decision variable for existing project i,

$y_{i} = {\begin{matrix} 1, & if the existing project i is upgraded, \\ 0, & otherwise, \end{matrix}$ The decisions to keep, abandon, and upgrade are mutually exclusive for the existing project i. Therefore, we know that it is {x_i = 0, y_i = 1} if the existing project i is upgraded, {x_i = 0, y_i = 0} if the existing project i is kept, and {x_i = 1, y_i = 0} if the existing project i is abandoned.

z_j : the selection decision variable of the new project j,

$z_{j} = {\begin{matrix} 1, & if the new project j is selected, \\ 0, & otherwise, \end{matrix}$ b_i : the abandonment or upgrade time point of existing project i;

$b_{j}^{'} :$ the selection time point of new project j;

2.2 Uncertain mean-chance multi-period project adjustment and selection (MPPAS) model with carbon emissions constraint

The enterprise aims to maximize profit while satisfying the government’s carbon emission requirements by adjusting existing projects and selecting new projects. To simplify things, we assume that: each project has no salvage value at the end of its lifetime. When the existing project i is abandoned at the time point t, the enterprise obtains a net income of $M_{it} = \frac{T_{i} - t + 1}{T_{i}} M_{i 1}$ by selling the production lines or facilities of the existing project i. If the existing project i is upgraded, there is no income during the upgrade. However, after upgrading the existing project i, its output increases by c_i, and the carbon emission per unit output decreases by h_i.

Throughout this paper, the profit of projects expresses as the net present value (NPV), considering the time value of money. If the existing project i is kept, the profit of existing project i is

$\sum_{t = 1}^{T_{i}} \frac{Q_{it} {IC}_{it}}{{(1 + r)}^{t}} .$ If the existing project i is abandoned at time point b_i, the profit of existing project i is

$\frac{T_{i} - b_{i} + 1}{T_{i} (1 + r)^{b_{i} - 1}} M_{i 1} + \sum_{t = 1}^{b_{i} - 1} \frac{Q_{it} {IC}_{it}}{(1 + r)^{t}} .$ If the existing project i is upgraded at time point b_i (see Fig. 1), the profit of existing project i is

Fig. 1

Cash flows when the existing project i is upgraded.

$\begin{matrix} \sum_{t = 1}^{b_{i} - 1} \frac{Q_{it} {IC}_{it}}{(1 + r)^{t}} + \sum_{t = b_{i} + S_{i}}^{T_{i}} \frac{(1 + c_{i}) Q_{it} {IC}_{it}}{(1 + r)^{t}} \\ - \sum_{t = b_{i}}^{b_{i} + S_{i} - 1} \frac{{\tilde{IO}}_{i, t - b_{i} + 1}}{(1 + r)^{t - 1}} . \end{matrix}$

For the existing project i, since the decisions to keep, abandon, and upgrade are mutually exclusive, the total profit of the existing project portfolio is

$\begin{matrix} {\tilde{NPV}}_{1} & = & \sum_{i = 1}^{n} ((\frac{T_{i} - b_{i} + 1}{T_{i} (1 + r)^{(b_{i} - 1)}} M_{i 1} + \sum_{t = 1}^{b_{i} - 1} \frac{Q_{it} {IC}_{it}}{(1 + r)^{t}}) x_{i} + \sum_{t = 1}^{T_{i}} \frac{Q_{it} {IC}_{it}}{(1 + r)^{t}} (1 - x_{i} - y_{i}) \\ + & (\sum_{t = 1}^{b_{i} - 1} \frac{Q_{it} {IC}_{it}}{(1 + r)^{t}} + \sum_{t = b_{i} + S_{i}}^{T_{i}} \frac{(1 + c_{i}) Q_{it} {IC}_{it}}{(1 + r)^{t}} - \sum_{t = b_{i}}^{b_{i} + S_{i} - 1} \frac{{\tilde{IO}}_{i, t - b_{i} + 1}}{(1 + r)^{t - 1}}) y_{i}) \\ = & \sum_{i = 1}^{n} (\sum_{t = 1}^{T_{i}} \frac{Q_{it} {IC}_{it}}{(1 + r)^{t}} + (\frac{T_{i} - b_{i} + 1}{T_{i} (1 + r)^{(b_{i} - 1)}} M_{i 1} - \sum_{t = b_{i}}^{T_{i}} \frac{Q_{it} {IC}_{it}}{(1 + r)^{t}}) x_{i} \\ + & (\sum_{t = b_{i} + S_{i}}^{T_{i}} \frac{c_{i} Q_{it} {IC}_{it}}{(1 + r)^{t}} - \sum_{t = b_{i}}^{b_{i} + S_{i} - 1} (\frac{Q_{it} {IC}_{it}}{(1 + r)^{t}} + \frac{{\tilde{IO}}_{i, t - b_{i} + 1}}{(1 + r)^{t - 1}})) y_{i}) . \end{matrix}$

If the new project j is selected at time point $b_{j}^{'}$ (see Fig. 2), the profit of the new project j is

Fig. 2

Cash flows when the new project j is selected.

$\sum_{t = 1}^{T_{j}^{'}} \frac{Q_{jt}^{'} {\tilde{IC}}_{jt}^{'}}{(1 + r)^{t + b_{i} + S_{i} - 1}} - \sum_{t = b_{j}^{'}}^{b_{j}^{'} + S_{j}^{'} - 1} \frac{{\tilde{IO}}_{j, t - b_{i} + 1}^{'}}{(1 + r)^{t - 1}} .$

Thus, the profit of the new project portfolio is

$\begin{matrix} {\tilde{NPV}}_{2} = \sum_{j = 1}^{k} \\ (\sum_{t = 1}^{T_{j}^{'}} \frac{Q_{jt}^{'} {\tilde{IC}}_{jt}^{'}}{(1 + r)^{t + b_{j}^{'} + S_{j}^{'} - 1}} - \sum_{t = b_{j}^{'}}^{b_{j}^{'} + S_{j}^{'} - 1} \frac{{\tilde{IO}}_{j, t - b_{j}^{'} + 1}^{'}}{(1 + r)^{t - 1}}) z_{j} . \end{matrix}$

When the existing projects are not adjusted, the future net cash flow of existing projects is the opportunity cost, and it is as follows (see [17]).

$ONPV = \sum_{i = 1}^{n} \sum_{t = 1}^{T_{i}} \frac{Q_{it} {IC}_{it}}{{(1 + r)}^{t}} .$

Therefore, the total profit of the project portfolio considering the opportunity cost is

$\begin{matrix} \tilde{NPV} & = & {\tilde{NPV}}_{1} + {\tilde{NPV}}_{2} - ONPV = \sum_{i = 1}^{n} ((\frac{T_{i} - b_{i} + 1}{T_{i} (1 + r)^{(b_{i} - 1)}} M_{i 1} - \sum_{t = b_{i}}^{T_{i}} \frac{Q_{it} {IC}_{it}}{(1 + r)^{t}}) x_{i} \\ + & (\sum_{t = b_{i} + S_{i}}^{T_{i}} \frac{c_{i} Q_{it} {IC}_{it}}{(1 + r)^{t}} - \sum_{t = b_{i}}^{b_{i} + S_{i} - 1} (\frac{Q_{it} {IC}_{it}}{(1 + r)^{t}} + \frac{{\tilde{IO}}_{i, t - b_{i} + 1}}{(1 + r)^{t - 1}})) y_{i}) \\ + & \sum_{j = 1}^{k} (\sum_{t = 1}^{T_{j}^{'}} \frac{Q_{jt}^{'} {\tilde{IC}}_{jt}^{'}}{(1 + r)^{t + b_{j}^{'} + S_{j}^{'} - 1}} - \sum_{t = b_{j}^{'}}^{b_{j}^{'} + S_{j}^{'} - 1} \frac{{\tilde{IO}}_{j, t - b_{j}^{'} + 1}^{'}}{(1 + r)^{t - 1}}) z_{j} . \end{matrix}$

Because the total profit of the project portfolio contains uncertain variables, the enterprise aims to maximize the expected $\tilde{NPV}$ , i.e.,

$max E [\tilde{NPV}] .$

The enterprise has a limited budget. In the t-th period, the outlay ${\tilde{O}}_{it}$ by the upgrade of the existing project i is

${\tilde{O}}_{it} = {\begin{matrix} {\tilde{IO}}_{i, t - b_{i} + 1}, & b_{i} \leq t \leq b_{i} + S_{i} - 1, \\ 0, & otherwise, \end{matrix}$ and the outlay ${\tilde{O}}_{jt}^{'}$ by the selection of the new project j is

${\tilde{O}}_{jt}^{'} = {\begin{matrix} {\tilde{IO}}_{j, t - b_{j}^{'} + 1}^{'}, & b_{j}^{'} \leq t \leq b_{j}^{'} + S_{j}^{'} - 1, \\ 0, & otherwise . \end{matrix}$ To prepare for the lack of capital, the enterprise can require that the chance of the investment capital exceeding the budget in each investment period should not be greater than a preset enough tolerance level α, i.e.,

$\begin{matrix} M {\sum_{i = 1}^{n} {\tilde{O}}_{it} y_{i} + \sum_{j = 1}^{k} {\tilde{O}}_{jt}^{'} z_{j} \geq B_{t}} \\ \leq α, t = 1, 2, \dots, S, \end{matrix}$ (1) where $M$ represents the uncertain measure, and S represents the maximum investment period.

The enterprise also needs to consider the carbon emission of the project portfolio.

If the existing project i is kept, i.e., {x_i = 0, y_i = 0}, the carbon emission of the existing project i in the t-th period is

${\begin{matrix} w_{i} Q_{it}, & t \leq T_{i}, \\ 0, & otherwise . \end{matrix}$ If the existing project i is abandoned at time point b_i, i.e., {x_i = 1, y_i = 0}, the carbon emission of the existing project i in the t-th period is

${\begin{matrix} w_{i} Q_{it}, & 1 \leq t \leq b_{i} - 1, \\ 0, & otherwise . \end{matrix}$

If the existing project i is upgraded at time point b_i, i.e., {x_i = 0, y_i = 1}, the carbon emission of existing project i in the t-th period is

${\begin{matrix} w_{i} Q_{it}, & 1 \leq t \leq b_{i} - 1, \\ 0, & b_{i} \leq t < b_{i} + S_{i}, \\ (1 - h_{i}) w_{i} (1 + c_{i}) Q_{it}, & b_{i} + S_{i} \leq t \leq T_{i} . \\ 0, & T_{i} < t . \end{matrix}$ Therefore, the carbon emission of existing project i in the t-th period is

$F_{it} = {\begin{matrix} w_{i} Q_{it}, & 1 \leq t \leq b_{i} - 1, \\ w_{i} Q_{it} ((1 - x_{i} - y_{i}) & b_{i} \leq t < b_{i} + S_{i}, \\ w_{i} Q_{it} (1 - x_{i} - (h_{i} - c_{i} + h_{i} c_{i}) y_{i}), & b_{i} + S_{i} \leq t \leq T_{i}, \\ 0, & T_{i} < t . \end{matrix}$

If the new project j is selected at time point $b_{j}^{'}$ , the carbon emission of the new project j in the t-th period is

$F_{jt}^{'} = {\begin{matrix} 0, & 1 \leq t \leq b_{j}^{'} + S_{j}^{'} - 1, \\ w_{j}^{'} Q_{j, t - b_{j}^{'} - S_{j}^{'} + 1}^{'} z_{j}, & b_{j}^{'} + S_{j}^{'} \leq t < b_{j}^{'} + S_{j}^{'} + T_{j}^{'}, \\ 0, & b_{j}^{'} + S_{j}^{'} + T_{j}^{'} \leq t . \end{matrix}$

Thus, the carbon emission of the project portfolio in the t-th period is

$\sum_{i = 1}^{n} F_{it} + \sum_{j = 1}^{k} F_{jt}^{'} .$ The carbon emission of the project portfolio in the t-th period must not exceed the allowable carbon emission F_t. Therefore, the carbon emission constraint is

$\sum_{i = 1}^{n} F_{it} + \sum_{j = 1}^{k} F_{jt}^{'} \leq F_{t}, t = 1, 2, \dots, T .$ (2) Finally, let’s consider the relationship between decision variables. As previously mentioned, since the decision variables x_i and y_i are mutually exclusive, we have

$x_{i} + y_{i} \leq 1, i = 1, 2, \dots, n .$ (3) In addition, the relationship between decision variables x_i, y_i, and b_i for the existing project i is as follows:

$\begin{matrix} b_{i} = d_{i} (x_{i} + y_{i}), i = 1, 2, \dots, n . \\ \begin{matrix} d_{i} \in {1, 2, \dots, T}, & if x_{i} = 1, \\ d_{i} \in {1, 2, \dots, T - S_{i}}, & if y_{i} = 1 . \end{matrix} \end{matrix}$ (4) The relationship between decision variables z_j and $b_{j}^{'}$ for the new project j is as follows:

$\begin{matrix} b_{j}^{'} = d_{j}^{'} z_{j}, j = 1, 2, \dots, k . \\ d_{j}^{'} \in {1, 2, \dots, T - S_{j}^{'}}, \end{matrix}$ (5)

Therefore, the uncertain mean-chance MPPAS model with carbon emissions constraint is as follows:

${\begin{matrix} max E [\tilde{NPV}], \\ s . t . \\ Equations (1), (2), (3), (4), (5), \\ x_{i}, y_{i} \in {0, 1}, i = 1, 2, \dots, n, \\ z_{j} \in {0, 1}, j = 1, 2, \dots, k . \end{matrix}$ (6)

2.3 Uncertain mean-chance multi-period project adjustment and selection (MPPAS) model without carbon emissions constraint

If the carbon emissions of the project portfolio in the t-th period exceed the allowable carbon emissions F_t, the enterprise must pay the carbon tax according to the excess carbon emission. The excess carbon emissions EF_t are

$\begin{matrix} {EF}_{t} = {\begin{matrix} \sum_{i = 1}^{n} F_{it} + \sum_{j = 1}^{k} F_{jt}^{'} - F_{t}, & \sum_{i = 1}^{n} F_{it} + \sum_{j = 1}^{k} F_{jt}^{'} > F_{t}, \\ 0, & otherwise, \end{matrix} t = 1, 2, \dots, T_{e}, \end{matrix}$ where T_e is the end period of the project portfolio lifetime. Then, the profit of the project portfolio is

$\tilde{NPV} - q \sum_{t = 1}^{T_{e}} \frac{{EF}_{t}}{(1 + r)^{t}} .$

Therefore, the uncertain mean-chance MPPAS model without carbon emissions constraint is as follows:

${\begin{matrix} max E [\tilde{NPV}] - q \sum_{t = 1}^{T_{e}} \frac{{EF}_{t}}{(1 + r)^{t}}, \\ s . t . \\ Equations (1), (3), (4), (5), \\ x_{i}, y_{i} \in {0, 1}, i = 1, 2, \dots, n, \\ z_{j} \in {0, 1}, j = 1, 2, \dots, k . \end{matrix}$ (7)

3 Deterministic equivalent and analysis of the proposed model

3.1 Deterministic equivalent of the proposed model

Since models (6) and (7) contains uncertain variables, it cannot be directly calculated. To solve models (6) and (7), we use the knowledge of uncertainty theory. Firstly, we introduce some definitions and theorems of uncertainty theory.

Definition 1. Let Γ be a nonempty set, and $L$ be a σ-algebra over Γ. Each element $Λ \in L$ is called an event. A set function $M {Λ}$ is called an uncertain measure if it satisfies the following three axioms:

(i) (Normality axiom) $M {Γ} = 1 .$

(ii) (Duality axiom) $M {Λ} + M {Λ^{c}} = 1 .$

(iii) (Subadditivity axiom) For every countable sequence of events {Λ_i}, we have

$M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} .$ The triplet $(Γ, L, M)$ is called an uncertainty space.

Definition 2. [19] An uncertain variable ξ is called a linear uncertain variable if it has a linear uncertainty distribution function

$Φ (t) = {\begin{matrix} 0, & t \leq a, \\ \frac{t - a}{b - a}, & a \leq t \leq b, \\ 1, & otherwise, \end{matrix}$ we denote it by ξ ∼ L (a, b) , where a and b are real numbers. The inverse uncertain function of Φ (t) is

$Φ^{- 1} (α) = a + (b - a) α, 0 \leq α \leq 1 .$

Theorem 1. [20] Let ξ₁ and ξ₂ be independent uncertain variables with finite expected values. Then for any real numbers a₁ and a₂, we have

$E [a_{1} ξ_{1} + a_{2} ξ_{2}] = a_{1} E [ξ_{1}] + a_{2} E [ξ_{2}] .$

When the uncertain variables are represented by uncertainty distributions, the operational law is given by [19] as follows:

Theorem 2. [19] Let uncertain variables ξ₁, ξ₂, ⋯ , ξ_n be independent and their uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n be regular. If f (ξ₁, ξ₂, ⋯ , ξ_n) increases monotonically with respect to ξ₁, ξ₂, ⋯ , ξ_m and decreases monotonically with respect to ξ_m+1, ξ_m+2, ⋯ , ξ_n, then the inverse uncertainty distribution of

$ξ = f (ξ_{1}, ξ_{2}, \dots, ξ_{n})$ is expressed as

$Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (α), \dots,$

$Φ_{m}^{- 1} (α), Φ_{m + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)),$

$0 < α < 1 .$

Let the uncertainty distribution of uncertain variables ${\tilde{IO}}_{it}$ , ${\tilde{IO}}_{jt}^{'}$ and ${\tilde{IC}}_{jt}^{'}$ are Ψ_it, $Ψ_{jt}^{'}$ and $Φ_{jt}^{'}$ , respectively. According to Theorem 1, the objective function of model (6) is

$\begin{matrix} {NPV}^{'} & = & \sum_{i = 1}^{n} ((\frac{T_{i} - b_{i} + 1}{T_{i} (1 + r)^{(} b_{i} - 1)} M_{i 1} - \sum_{t = b_{i}}^{T_{i}} \frac{Q_{it} {IC}_{it}}{(1 + r)^{t}}) x_{i} \\ + & (\sum_{t = b_{i}}^{b_{i} + S_{i} - 1} (\frac{Q_{it} {IC}_{it}}{(1 + r)^{t}} - \frac{μ_{i, t - b_{i} + 1}}{(1 + r)^{t - 1}}) + \sum_{t = b_{i} + S_{i}}^{T_{i}} \frac{c_{i} Q_{it} {IC}_{it})}{(1 + r)^{t}}) y_{i}) \\ + & \sum_{j = 1}^{k} (\sum_{t = 1}^{T_{j}^{'}} \frac{Q_{jt}^{'} e_{jt}^{'}}{(1 + r)^{t + b_{i} + S_{i} - 1}} - \sum_{t = b_{j}^{'}}^{b_{j}^{'} + S_{j}^{'} - 1} \frac{μ_{j, t - b_{i} + 1}^{'}}{(1 + r)^{t - 1}}) z_{j}, \end{matrix}$ (8) and the objective function of model (7) is

$\begin{matrix} {NPV}^{″} & = & \sum_{i = 1}^{n} ((\frac{T_{i} - b_{i} + 1}{T_{i} (1 + r)^{(} b_{i} - 1)} M_{i 1} - \sum_{t = b_{i}}^{T_{i}} \frac{Q_{it} {IC}_{it}}{(1 + r)^{t}}) x_{i} \\ + & (\sum_{t = b_{i}}^{b_{i} + S_{i} - 1} (\frac{Q_{it} {IC}_{it}}{(1 + r)^{t}} - \frac{μ_{i, t - b_{i} + 1}}{(1 + r)^{t - 1}}) + \sum_{t = b_{i} + S_{i}}^{T_{i}} \frac{c_{i} Q_{it} {IC}_{it})}{(1 + r)^{t}}) y_{i}) \\ + & \sum_{j = 1}^{k} (\sum_{t = 1}^{T_{j}^{'}} \frac{Q_{jt}^{'} e_{jt}^{'}}{(1 + r)^{t + b_{i} + S_{i} - 1}} - \sum_{t = b_{j}^{'}}^{b_{j}^{'} + S_{j}^{'} - 1} \frac{μ_{j, t - b_{i} + 1}^{'}}{(1 + r)^{t - 1}}) z_{j} - q \sum_{t = 1}^{T_{e}} \frac{{EF}_{t}}{(1 + r)^{t}}, \end{matrix}$ (9) where μ_it, $μ_{jt}^{'}$ , and $e_{jt}^{'}$ are the expected values of ${\tilde{IO}}_{it}$ , ${\tilde{IO}}_{jt}^{'}$ , and ${\tilde{IC}}_{jt}^{'}$ , respectively.

According to the duality property of uncertainty measure (Definition 1 (ii)), the budget risk constraint Equation (1) transforms as

$M {\sum_{i = 1}^{n} {\tilde{O}}_{it} y_{i} + \sum_{j = 1}^{k} {\tilde{O}}_{jt}^{'} z_{j} \leq B_{t}}$

$\geq 1 - α, t = 1, 2, \dots, S .$

According to the monotonicity of the uncertain measure and the operational law (Theorem 2), we have

$\begin{matrix} \sum_{i = 1}^{n} Θ_{it}^{- 1} (1 - α) y_{i} + \sum_{j = 1}^{k} {Θ^{'}}_{jt}^{- 1} (1 - α) z_{j} \leq B_{t}, \\ t = 1, 2, \dots, S, \end{matrix}$ (10)

where

$\begin{matrix} Θ_{it}^{- 1} (1 - α) \\ = {\begin{matrix} Ψ_{i, t - b_{i} + 1}^{- 1} (1 - α), & b_{i} \leq t \leq b_{i} + S_{i} - 1, \\ 0, & otherwise, \end{matrix} \end{matrix}$

$\begin{matrix} Θ_{jt}^{' - 1} (1 - α) \\ = {\begin{matrix} Ψ_{j, t - b_{j}^{'} + 1}^{' - 1} (1 - α), & b_{j}^{'} \leq t \leq b_{j}^{'} + S_{j}^{'} - 1, \\ 0, & otherwise . \end{matrix} \end{matrix}$ Therefore, the deterministic equivalent of model (6) is

${\begin{matrix} max Equation (8) \\ s . t . \\ Equations (2), (3), (4), (5), (10), \\ x_{i}, y_{i} \in {0, 1}, i = 1, 2, \dots, n, \\ z_{j} \in {0, 1}, j = 1, 2, \dots, k, \end{matrix}$ (11) and the deterministic equivalent of model (7) is

${\begin{matrix} max Equation (9) \\ s . t . \\ Equations (3), (4), (5), (10),, \\ x_{i}, y_{i} \in {0, 1}, i = 1, 2, \dots, n, \\ z_{j} \in {0, 1}, j = 1, 2, \dots, k . \end{matrix}$ (12)

3.2 Analysis of the proposed model

Proposition 1. The optimal solution X of model (11) is a feasible solution of model (12).

Proof. Let Ω₁ and Ω₂ be the feasible sets of models (11) and (12), respectively. Comparing the constraints of models (11) and (12) makes it easy to know Ω₁ ⊆ Ω₂. Therefore, the optimal solution of model (11) belongs to Ω₂, i.e., X ∈ Ω₁ ⊆ Ω₂. Thus, Proposition 1 is proved.

Proposition 2. When the optimal solution X of model (12) satisfies the carbon emission constraint Equation (2) of model (11), X is the optimal solution of model (11), and the objective function values of models (11) and (12) at X are the same, i.e.,

$\begin{matrix} {NPV}^{'} (X) = {NPV}^{″} (X), if X \\ satisfies the carbon emission constraint Eq . (2) . \end{matrix}$

Proof. When the optimal solution X of model (12) satisfies the carbon emission constraint Equation (2), the objective function values of models (11) and (12) at X are the same because EF_t = 0 in Equation (9), and X is a feasible solution of model (11), i.e., X ∈ Ω₁. Since X ∈ Ω₁ ⊆ Ω₂, the optimal solution X on Ω₂ is the optimal solution on Ω₁. Thus, Proposition 2 is proved.

Proposition 3. Let X′ and X″ be the optimal solutions of models (11) and (12), respectively. When X″ does not satisfy carbon emission constraint Equation (2), there exists carbon tax q^* ⩾ 0 satisfying

${NPV}^{″} (X^{″}, q^{*}) = {NPV}^{'} (X^{'}) .$

Proof. Since Ω₁ ⊆ Ω₂, when q = 0, the optimal value of model (12) is equal to or greater than the optimal value of model (11), i.e.,

${NPV}^{″} (X^{″}, 0) ⩾ {NPV}^{'} (X^{'}) .$ When the optimal solution X″ of model (12) does not satisfy the carbon emission constraint Equation (2), NPV″ (X″, q) is strictly decreasing function with respect to q, because $\sum_{t = 1}^{T_{e}} {EF}_{t} > 0$ . Therefore, there exists q^* ⩾ 0 satisfying

${NPV}^{″} (X^{″}, q^{*}) - {NPV}^{'} (X^{'}) = 0 .$ Thus, Proposition 3 is proved.

Proposition 4. If the carbon tax is greater than q^*, the optimal value of model (12) is less than the optimal value of model (11). If the carbon tax is less than q^*, the optimal value of model (12) is greater than the optimal value of model (11).

According to the objective function of model (12) is a strictly decreasing function with respect to q and Proposition 3, Proposition 4 is easy to prove.

Feasibility repair operation I:
01: For i = 1 to n
02: If x_i + y_i = 0, then b_i = 0.
03: If x_i + y_i = 2, then set either x_i or y_i to 0.
04: If x_i = 1, then b_i is randomly generated from the set {1, 2, ⋯ , T}.
05: If y_i = 1, then b_i is randomly generated from the set {1, 2, ⋯ , T - S_i}.
06: End For.
07: For j = 1 to k
08: If z_j = 0, then $b_{j}^{'} = 0$ .
09: If z_j = 1, then $b_{j}^{'}$ is randomly generated from the set ${1, 2, \dots, T - S_{j}^{'}}$ .
10: End For.

Feasibility repair operation II:
01: For i = 1 to n
02: If x_i + y_i = 0, then b_i = 0.
03: If x_i + y_i = 2, then set either x_i or y_i to 0.
04: If x_i = 1, then
05: If b_i ≠ 0, then b_i is kept.
06: If b_i = 0, then b_i is randomly generated from the set {1, 2, ⋯ , T}.
07: If y_i = 1, then
08: If b_i ≠ 0, then b_i is kept.
09: If b_i = 0, then b_i is randomly generated from the set {1, 2, ⋯ , T - S_i}.
10: End For.
11: For j = 1 to k
12: If z_j = 0, then $b_{j}^{'} = 0$ .
13: If z_j = 1, then
14: If $b_{j}^{'} \neq 0$ , then $b_{j}^{'}$ is kept.
15: If $b_{j}^{'} = 0$ , then $b_{j}^{'}$ is randomly generated from the set ${1, 2, \dots, T - S_{j}^{'}}$ .
16: End For.

Proposition 3 shows that there is a carbon tax q^* at which the optimal profits of the project portfolio with and without carbon emission constraints are equal. Proposition 4 shows that if the carbon tax is less than q^*, the enterprise does not consider the government’s carbon emission reduction requirement when making the investment decision. It is because the purpose of investment is to maximize profits. Therefore, if the carbon tax is less than q^*, the carbon neutrality goal cannot achieve, i.e., the carbon tax policy is invalid. If the carbon tax is equal to or higher than q^*, the enterprise does consider the government’s carbon emission reduction requirement when making the investment decision and ultimately achieves the carbon neutrality goal. However, the higher the carbon tax, the lower the profit of the enterprise. It will reduce the investment desire of the enterprise. Therefore, when the carbon tax is q^*, it can minimize the losses of the enterprise and achieve the carbon neutrality goal simultaneously.

4 Improved adaptive genetic algorithm

4.1 Representation structure

The proposed model has 3n + 2k decision variables. To represent the solution of the proposed model, we use the integer vector G = (g₁, g₂, ⋯ , g_3n+2k) as the individual chromosome, where g_i (i = 1, 2, ⋯ , 2n + k) are 0 or 1 and g_i (i = 2n + k + 1, ⋯ , 3n + 2k) are the non-negative integer numbers. Table 2 shows the corresponding representation structure between the individual and the decision variables.

Table 2
Corresponding representation structure between the individual chromosome and the decision variables

variable x ₁ ⋯ x _n y ₁ ⋯ y _n z ₁ ⋯ z _k b ₁ ⋯ b _n $b_{1}^{'}$ ⋯ $b_{k}^{'}$

individual g ₁ ⋯ g _n g _n+1 ⋯ g _2n g _2n+1 ⋯ g _2n+k g _2n+k+1 ⋯ g _3n+k g _3n+k+1 ⋯ g _3n+2k

variable	x ₁	⋯	x _n	y ₁	⋯	y _n	z ₁	⋯	z _k	b ₁	⋯	b _n	$b_{1}^{'}$	⋯	$b_{k}^{'}$
individual	g ₁	⋯	g _n	g _n+1	⋯	g _2n	g _2n+1	⋯	g _2n+k	g _2n+k+1	⋯	g _3n+k	g _3n+k+1	⋯	g _3n+2k

We perform the genetic operation about the decision variables x_i, y_i, and z_j. The decision variables b_i and $b_{j}^{'}$ are determined by the feasibility repair operation according to x_i, y_i, and z_j.

4.2 Feasibility repair operation

To improve the efficiency of obtaining the feasible solution of the proposed model, we introduce the feasibility repair operation. In model (11), we refer to constraints Equations (3)-(5) as “Constraint $A_{1}^{″}$ and the other constraints Equations (1) and (2) as “Constraint $A_{2}^{″}$ . In the feasibility repair operation, we determine b_i and $b_{j}^{'}$ to satisfy “Constraint $A_{1}^{″}$ according to {x_i, y_i, z_j}. There are two algorithms to determine b_i and $b_{j}^{'}$ , i.e., feasibility repair operation I and feasibility repair operation II. The feasibility repair operation I determines b_i and $b_{j}^{'}$ to satisfy “Constraint $A_{1}^{″}$ according to x_i, y_i, and z_j generated randomly in the generation process of the new individual. After performing the genetic operation, the feasibility repair operation II determines b_i and b_j to satisfy the “Constraint $A_{1}^{″}$ according to the changed values of x_i, y_i, and z_j.

4.3 Generation of the initial population

The generation process of the initial population is as follows.

Step 1. Set i = 1.

Step 2. The genes {g₁, g₂, ⋯ , g_2n+k} in the individual G_i = {g₁, g₂, ⋯ , g_3n+2k} are randomly generated from the set {0, 1}.

Step 3. The genes {g_2n+k+1, g_2n+k+2, ⋯ , g_3n+2k} are generated by feasibility repair operation I.

Step 4. Check the feasibility of the individual G_i. If the individual G_i satisfies “Constraint $A_{2}^{″}$ , then go to Step 5, else go to Step 2.

Step 5. If i < m, then i = i + 1, go to Step 2, where m is the population size.

Step 6. Return the generated initial population.

4.4 Evaluation of individual

The individual G_i generated by the feasibility repair operation satisfies “Constraint $A_{1}^{″}$ . If the individual G_i does not satisfy “Constraint $A_{2}^{″}$ , the fitness of the individual G_i evaluates as 0. If the individual G_i satisfies “Constraint $A_{2}^{″}$ , the fitness of the individual G_i evaluates as Fitness (G_i) = f (G_i) - f_min, nonumber where f (G_i) is the objective function value of model (11) for the individual G_i and f_min is the minimum of the objective function values for the individuals satisfying the “Constraint $A_{2}^{″}$ .

4.5 Selection operation

The selection operation is performed by the roulette wheel. The specific process is as follows:

Step 1. Based on the fitness values of each individual G_i (i = 1, 2, ⋯ , m), the selection probability P_s and the cumulative selection probability P_cs are calculated by

$P_{s} (G_{i}) = \frac{Fitness (G_{j})}{\sum_{j = 1}^{m} Fitness (G_{j})}, i = 1, 2, \dots, m,$

$P_{cs} (G_{i}) = \sum_{j = 1}^{i} P_{s} (G_{j}), i = 1, 2, \dots, m .$ Step 2. Set j = 1.

Step 3. Generate a uniformly distributed random number R in (0, 1).

Step 4. Select the G_i satisfying P_cs (G_i-1) < R ≤ P_cs (G_i) as the j-th individual of the new population.

Step 5. If j < m, let j = j + 1, and go to Step 3.

Step 6. Return the new population.

4.6 Adaptive crossover operation

We use the adaptive crossover probability to improve the performance of GA. Based on the fitness values of individuals G_j and G_j+1, the adaptive crossover probability is determined by

$P_{ac} (G_{j}, G_{j + 1}) = {\begin{matrix} \frac{F_{\max} - f}{F_{\max} - F_{mean}} P_{c}, & f \geq F_{mean}, \\ P_{c}, & otherwise, \end{matrix}$ (13) where f = min {Fitness (G_j) , Fitness (G_j+1)} and P_c is the basic crossover probability, and F_mean and F_max are the population’s average fitness value and max fitness value, respectively. The specific process is as follows:

Step 1. Set a basic crossover probability P_c and j = 1.

Step 2. Generate a uniformly distributed random number R_c in (0, 1), and calculate the adaptive crossover probability P_ac (G_j, G_j+1) for individuals G_j and G_j+1 by Equation (13).

Step 3. If R_c > P_ac (G_j, G_j+1), keep the individuals G_j and G_j+1 and go to Step 5.

Step 4. If R_c ≤ P_ac (G_j, G_j+1), then the offspring individuals $G_{j}^{'}$ and $G_{j + 1}^{'}$ are generated from the two parent individuals G_j and G_j+1. We use the two-point crossover to generate the offspring individuals. Assume that the two selected parent individuals are $G_{j} = {g_{1}^{j}, g_{2}^{j}, \dots, g_{3 n + 2 k}^{j}}$ and $G_{j + 1} = {g_{1}^{j + 1}, g_{2}^{j + 1}, \dots, g_{3 n + 2 k}^{j + 1}}$ . Using the randomly generated crossover points i and l (1 ≤ i < l ≤ 2n + k), the offspring individuals are generated by

$\begin{matrix} G_{j}^{'} = {g_{1}^{j}, g_{2}^{j}, \dots, g_{i - 1}^{j}, g_{i}^{j + 1}, \underset{︸}{g_{i + 1}^{j + 1}, \dots, g_{l - 1}^{j + 1}, g_{l}^{j + 1}}, g_{l + 1}^{j}, \dots, g_{2 n + k}^{j}, g_{2 n + k + 1}^{j}, \dots, g_{3 n + 2 k}^{j}}, \end{matrix}$

$\begin{matrix} G_{j + 1}^{'} = {g_{1}^{j + 1}, g_{2}^{j + 1}, \dots, g_{i - 1}^{j + 1}, \underset{︸}{g_{i}^{j}, g_{i + 1}^{j}, \dots, g_{l - 1}^{j}, g_{l}^{j}}, g_{l + 1}^{j + 1}, \dots, g_{2 n + k}^{j + 1}, g_{2 n + k + 1}^{j + 1}, \dots, g_{3 n + 2 k}^{j + 1}} . \end{matrix}$ Step 5. If j < m - 2, set j = j + 2 and go to Step 2.

Step 6. Return the new population.

4.7 Mutation operation

The specific process of the mutation operation is as follows:

Step 1. Set j = 1 and the mutation probability P_m.

Step 2. Generate a uniformly distributed random number R_m in (0, 1).

Step 3. If R_m > P_m, keep the individual G_j and go to Step 5.

Step 4. If R_m ≤ P_m, randomly select a mutation position i (1 ≤ i ≤ 2n + k). If the i-th gene of the individual G_j is 0, it converts to 1. Otherwise, it converts to 0.

Step 5. If j < m, set j = j + 1 and go to Step 2.

Step 6. Return a new population.

4.8 Algorithm procedure

Figure 3 shows the procedure of the proposed algorithm. In Figure 3, “Set Parameters" includes population size (m), basic crossover probability (P_c), mutation probability (P_m), and termination generation number. According to Figure 3, the specific step of the proposed algorithm is as follows:

Fig. 3

The procedure of the proposed algorithm.

Step 1. Set parameters.

Step 2. Generation of the initial population (please see Section 4.3 for the specific process).

Step 3. Evaluate the fitness of each individual (please see Section 4.4 for the specific process).

Step 4. Record the best individual according to the elitism preservation strategy.

Step 5. Update the population through selection, adaptive crossover, and mutation operations (please see Sections 4.5–4.7 for the specific process).

Step 6. Repeat Steps 3-5 until satisfying the termination condition.

Step 7. Report the best individual as the optimal solution for the model.

5 Numerical experiments

The steel industry is a typical high-carbon emission industry, and steel enterprises’ crude steel production mainly depends on blast furnace steelmaking. These steelmaking projects have high energy consumption and carbon emission intensity, which is not suitable for the sustainable development strategy of the steel industry under the background of the low-carbon emission policy. The upgrading of the steelmaking process via technologies such as energy efficiency improvement and CCUS (carbon capture, utilization and storage) is a method to reduce carbon emissions in the steel industry. However, the efficiency of reducing carbon emissions via these technologies is limited under the process structure and technology of current steel production. Ultra-low CO₂ steelmaking (ULCOS) is a new project for carbon neutralization in the steel industry.

This section discusses the technological improvement of existing blast furnace steelmaking (adjustment of existing high-carbon projects) and the establishment of new ULCOS (selection of new low-carbon projects) for a steel enterprise under the context of carbon tax and carbon quota policies. The data used in the numerical simulation is based on information from China Metallurgical News (https://www.csteelnews.com/).

5.1 Problem description

Consider a steel enterprise that produces crude steel based on five steelmaking processes. Under the background of the carbon tax and carbon quota policies, the enterprise plans to adjust existing projects and establish new ULCOS projects. The enterprise is considering the typical non-focusing smelting reduction steelmaking HIsmelt process as a new ULCOS project. For the convenience of calculation, the outputs per period of each project are all Q=100,000 (ton), the available investment budget for each period is B=250 million dollars, and the discount rate is r=0.08. The enterprise requires that the chance of the investment capital exceeding the budget in each investment period should be less than 5%. According to the expert evaluation, the upgrade cost of the existing blast furnace process, the investment cost and the income per unit output of the HIsmelt projects estimate as uncertain variables with the linear distribution. The information on the existing blast furnace projects and the HIsmelt projects are shown in Tables 3 and 4, respectively.

Table 3
Parameters of the existing blast furnace projects

Existing Project IC_i, ($) ${\tilde{IO}}_{i} \sim L (u_{i}, v_{i})$ , (×10⁶$) w_i, (CO₂t/t) c _i h _i M_i1, (×10⁶$) T _i S _i

P₁ (Blast furnace 1) 168 L(16.8625, 18.6375) 2.2 0.30 0.450 75.5000 20 1

P₂ (Blast furnace 2) 156 L(41.8000, 46.2000) 2.1 0.35 0.675 63.6000 20 1

P₃ (Blast furnace 3) 138 L(18.4300, 20.3700) 2.0 0.30 0.600 62.5000 20 1

P₄ (Blast furnace 4) 108 L(51.6800, 57.1200) 1.9 0.35 0.450 73.4000 20 1

P₅ (Blast furnace 5) 90 L(30.8750. 34.1250) 1.8 0.35 0.525 66.6000 20 1

Existing Project	IC_i, ($)	${\tilde{IO}}_{i} \sim L (u_{i}, v_{i})$ , (×10⁶$)	w_i, (CO₂t/t)	c _i	h _i	M_i1, (×10⁶$)	T _i	S _i
P₁ (Blast furnace 1)	168	L(16.8625, 18.6375)	2.2	0.30	0.450	75.5000	20	1
P₂ (Blast furnace 2)	156	L(41.8000, 46.2000)	2.1	0.35	0.675	63.6000	20	1
P₃ (Blast furnace 3)	138	L(18.4300, 20.3700)	2.0	0.30	0.600	62.5000	20	1
P₄ (Blast furnace 4)	108	L(51.6800, 57.1200)	1.9	0.35	0.450	73.4000	20	1
P₅ (Blast furnace 5)	90	L(30.8750. 34.1250)	1.8	0.35	0.525	66.6000	20	1

Table 4

Parameters of the HIsmelt projects

New Project	${\tilde{IC}}_{j}^{'} \sim L (a_{j}^{'}, d_{j}^{'})$ , ($)	${\tilde{IO}}_{j}^{'} \sim L (u_{j}^{'}, v_{j}^{'})$ , (×10⁶$)	$w_{j}^{'}$ , (CO₂t/t)	$T_{j}^{'}$	$S_{j}^{'}$
$P_{1}^{'}$ (HIsmelt 1)	L(137.750, 152.250)	L(147.25, 162.75)	0.80	30	1
$P_{2}^{'}$ (HIsmelt 2)	L(161.975, 179.025)	L(142.50, 157.50)	0.40	30	1
$P_{3}^{'}$ (HIsmelt 3)	L(149.150, 164.850)	L(171.00, 189.00)	0.60	30	1
$P_{4}^{'}$ (HIsmelt 4)	L(194.275, 214.725)	L(228.00, 252.00)	0.55	30	1
$P_{5}^{'}$ (HIsmelt 5)	L(199.025, 219.975)	L(190.00, 210.00)	0.70	30	1

In this case, the objective function of model (11) is

$\begin{matrix} \sum_{i = 1}^{5} ((\frac{15 - b_{i} + 1}{15 (1 + r)^{b_{i} - 1}} M_{i 1} - \sum_{t = b_{i}}^{20} \frac{{IC}_{i} Q}{(1 + r)^{t}}) x_{i} + (\frac{{IC}_{i} Q}{(1 + r)^{b_{i}}} - \frac{0.5 (u_{i} + v_{i})}{(1 + r)^{b_{i} - 1}} + \sum_{t = b_{i} + 1}^{20} \frac{c_{i} {IC}_{i} Q}{(1 + r)^{t}}) y_{i}) \\ + \sum_{j = 1}^{5} (\sum_{t = 1}^{30} \frac{(a_{j}^{'} + d_{j}^{'}) Q}{2 (1 + r)^{t + b_{i}}} - \frac{u_{j}^{'} + v_{j}^{'}}{2 (1 + r)^{b_{j}^{'} - 1}}) z_{j} . \end{matrix}$ (14)

According to Definition 2, the budget risk constraint Equation (10) is

$\begin{matrix} \sum_{i = 1}^{n} Θ_{it}^{- 1} (1 - α) y_{i} + \sum_{j = 1}^{k} {Θ^{'}}_{jt}^{- 1} (1 - α) z_{j} \leq B, \\ t = 1, 2, \dots, S, \end{matrix}$ (15) where

$\begin{matrix} Θ_{it}^{- 1} (1 - α) \\ = {\begin{matrix} v_{i} + 0.05 (u_{i} - v_{i}), & b_{i} \leq t \leq b_{i} + S_{i} - 1, \\ 0, & otherwise, \end{matrix} \\ Θ_{jt}^{' - 1} (1 - α) \\ = {\begin{matrix} v_{j}^{'} + 0.05 (u_{j}^{'} - v_{j}^{'}), & b_{j}^{'} \leq t \leq b_{j}^{'} + S_{j}^{'} - 1, \\ 0, & otherwise . \end{matrix} \end{matrix}$

The government’s carbon emission requirements include the following: Before the T′=4th period, the carbon emissions per period of the enterprise shall not exceed the carbon emission of the current period. After the 4th period, the carbon emissions of the enterprise linearly reduce in each period, and the carbon emission before the T=15th period (carbon neutrality) must be less than F_b = 5 ×10⁵ (ton). From Table 3, the current carbon emission of the existing project portfolio is as follows:

$F_{0} = \sum_{i = 1}^{5} w_{i} Q = 10^{6} (ton) .$ According to the government requirement, the carbon emission constraint in the t-th period is

$\sum_{i = 1}^{n} F_{it} + \sum_{j = 1}^{k} F_{jt}^{'} \leq F_{t},$ (16) where

$F_{t} = {\begin{matrix} F_{0}, & 1 \leq t \leq T^{'}, \\ \frac{(T - t) F_{0} + (t - T^{'}) F_{b}}{T - T^{'}}, & T^{'} + 1 \leq t \leq T, \\ F_{b}, & T + 1 \leq t . \end{matrix}$

Thus, model (11) is transformed as

${\begin{matrix} max Equation (14) \\ s . t . \\ Equations (3), (4), (5), (15), (16), \\ x_{i}, y_{i} \in {0, 1}, i = 1, 2, \dots, n, \\ z_{j} \in {0, 1}, j = 1, 2, \dots, k . \end{matrix}$ (17)

5.2 Evaluation of the proposed algorithm

This section provides the results of numerical experiments to evaluate the performance of the proposed algorithm. In the numerical experiments, we use the PC with the Intel Core i3-5005U CPU 2.00GHz and 8GB RAM.

To verify the robustness of our proposed algorithm, we use the data from Section 5.1 and change the parameter values, such as population size, P_c, P_m, and maximum generation number. We perform the proposed algorithm 20 times for each set of parameter settings and report the best results. Table 5 shows the computation results on model (17). In Table 5, Relative error is calculated as

Table 5
Computation results on model (17)

Pop _ size P _c P _m Optimal value (×10⁶) Relative error (%) GN ST (s) ST^* (s)

50 0.7 0.05 39.4309 5.4147 656 110.13 72.2453

50 0.9 0.10 41.6882 0.0000 636 111.95 71.2002

50 0.7 0.15 40.3781 3.1426 587 112.27 65.9025

100 0.7 0.10 41.6882 0.0000 274 185.89 50.9339

100 0.9 0.10 41.6882 0.0000 232 188.38 43.7042

100 0.9 0.15 41.6882 0.0000 237 186.82 44.2763

150 0.7 0.10 41.6882 0.0000 204 245.58 50.0983

150 0.7 0.15 41.6882 0.0000 212 248.36 52.6523

150 0.9 0.10 41.6882 0.0000 193 246.72 47.6170

Enumeration method 41.6882 - - 0.1651 ×10⁶ -

Pop _ size	P _c	P _m	Optimal value (×10⁶)	Relative error (%)	GN	ST (s)	ST^* (s)
50	0.7	0.05	39.4309	5.4147	656	110.13	72.2453
50	0.9	0.10	41.6882	0.0000	636	111.95	71.2002
50	0.7	0.15	40.3781	3.1426	587	112.27	65.9025
100	0.7	0.10	41.6882	0.0000	274	185.89	50.9339
100	0.9	0.10	41.6882	0.0000	232	188.38	43.7042
100	0.9	0.15	41.6882	0.0000	237	186.82	44.2763
150	0.7	0.10	41.6882	0.0000	204	245.58	50.0983
150	0.7	0.15	41.6882	0.0000	212	248.36	52.6523
150	0.9	0.10	41.6882	0.0000	193	246.72	47.6170
Enumeration method	41.6882	-	-	0.1651 ×10⁶	-

$\begin{matrix} Relative error \\ = \frac{actual optimal value - optimal value}{actual optimal value}, \end{matrix}$ where actual optimal value is the optimal value obtained by the enumeration method. GN is the generation number when the optimal solution first appears and ST indicates the spent time of the proposed algorithm when the maximum generation number is 1000, and ST^* indicates the spent time of the proposed algorithm when the optimal solution first appears.

From Tables 5, we obtain the following:

For each set of parameter settings, the generation numbers to the first appearance of the optimal solution are different, but the optimal values are the same as the optimal value obtained by the enumeration method when the population size is equal to or greater than 100.

The enumeration method takes a long time to find the optimal solution, but our algorithm can find the optimal solution of the proposed model in a short time.

The crossover and mutation probabilities of the algorithm have little effect on the proposed algorithm’s spent time. However, the population size affects the spent time of the proposed algorithm. From Tables 5, it can be seen that the reasonable parameter values to solve the proposed model are Pop _ size = 100, P_c = 0.9, and P_m = 0.1.

Next, to evaluate the performance of our proposed algorithm for the large-scale problem, we obtain the optimal solutions of model (17) when the number of the existing and new projects is 5, 10, and 20, respectively. Then we compare them with the calculation results by the proposed algorithm (Standard genetic algorithm: SGA) in Ref. [13] and the proposed algorithm (adaptive genetic algorithm: AGA) in Ref. [31]. All algorithms are coded in MATLAB R2019a. The used parameters are as follows: The maximum generation number is 1000, the population size is m = 100, the crossover probability is P_c = 0.9, and the mutation probability is P_m = 0.1. For each problem, the algorithms repeat twenty times, and compare the performances of each algorithm by the average value.

We evaluate the algorithm’s performance by using the following three metrics.

1) Convergence Speed metric (CS) : CS is a metric to evaluate the convergence speed of the given algorithm. CS is measured by

$CS = \frac{1}{num} \sum_{i = 1}^{num} {ST}_{i} \times {GN}_{i},$ where num is the repeated execution number of the algorithm, ST_i is the average spent time per generation in the i-th repeat execution of the algorithm, and GN_i is the generation number when the optimal individual first appears in the i-th repeat execution of the algorithm. The lower the CS, the better the algorithm’s convergence performance.

2) Optimal Value (OV) : OV is a metric to evaluate the quality of the optimal solution obtained by the given algorithm. OV is the maximum of the optimal values obtained by repeating twenty times for the given algorithm. The higher the OV, the better the optimal solution quality for the given algorithm.

3) Solution Uniformity metric (SU) : SU is a metric to evaluate the uniformity of the optimal solution obtained by the given algorithm. SU is measured by

$SU = \frac{1}{num} \sum_{i = 1}^{num} \frac{OV - {FV}_{i}}{OV},$ where FV_i is the optimal value in the i-th repeat execution of the algorithm. The lower the SU, the better the uniformity of the optimal solution obtained by the algorithm.

Table 6 shows the performance metric values of each algorithm, and Table 7 shows the average spent time ( $\bar{ST}$ ) per generation, the average generation number ( $\bar{GN}$ ) and spent time (ST^*) when the optimal individual first appears. From Table 6, we obtain the following results:

Comparing the CS values in Table 6, our algorithm gives better results than the other algorithms. As shown in Table 7, our algorithm’s spent time per generation is longer than the SGA and the AGA because of the feasibility repair operation in our algorithm. However, the GN values of our algorithm are smaller than the SGA and the AGA. Comprehensively considering the ST and GN values, we know that the convergence speed of our algorithm is faster than the SGA and the AGA.

Comparing the OV values in Table 6, all algorithms provide the same optimal solution when the problem size is small (n = 5, k = 5). However, when the problem size is large, our algorithm provides the optimal solution with better quality than SGA and AGA. This shows that our algorithm overcomes premature convergence and has good global convergence.

Comparing the SU values in Table 6, we know that the SU value of our algorithm is smaller than the SGA and AGA. This means that our algorithm can obtain a more uniform optimal solution.

Table 6

Performance metrics of the algorithms

	n = 5, k = 5			n = 10, k = 10			n = 20, k = 20
Algorithm	CS	OV	SU	CS	OV	SU	CS	OV	SU
SGA	36.9454	41.6882	0.0025	101.6371	76.0637	0.0197	172.9125	146.0812	0.0471
AGA	34.8272	41.6882	0.0018	98.0356	79.0611	0.0156	163.3651	158.0798	0.0325
Our Algorithm	32.6503	41.6882	0.0010	89.4739	81.0898	0.0147	147.8569	165.0917	0.0213

Table 7

The average generation number ( $\bar{GN}$ ), the average spent time ( $\bar{ST}$ ) per generation and the spent time (ST^*) when the optimal individual first appears

	n = 5, k = 5			n = 10, k = 10			n = 20, k = 20
Algorithm	$\bar{GN}$	$\bar{ST}$	ST ^*	$\bar{GN}$	$\bar{ST}$	ST ^*	$\bar{GN}$	$\bar{ST}$	ST ^*
SGA	182.35	0.1964	35.8135	458.30	0.2158	98.9011	703.17	0.2468	173.5424
AGA	178.10	0.1985	35.3529	443.50	0.2177	96.5500	677.03	0.2445	165.5338
Our Algorithm	152.95	0.2040	31.2018	393.25	0.2252	88.5599	591.45	0.2521	149.1045

Overall, our algorithm is better than the proposed SGA and AGA in terms of robustness, convergence speed, and global convergence.

5.3 Illustration of the proposed model

To illustrate the application of the proposed model, we discuss the MPPAS problem of the iron and steel enterprise based on the data introduced in Section 5.1. Table 8 shows the optimal project portfolio of model (17).

Table 8
Optimal project portfolio of model (17)

Existing project New project Expected NPV(×10⁶$) Carbon emission in15th period (ton)

P ₁ P ₂ P ₃ P ₄ P ₅ $P_{1}^{'}$ $P_{2}^{'}$ $P_{3}^{'}$ $P_{4}^{'}$ $P_{5}^{'}$

y(1) y(5) y(1) x(12) x(7) – z(1) – – z(2) 41.6882 463,437

Existing project	New project	Expected NPV(×10⁶$)	Carbon emission in15th period (ton)
P ₁	P ₂	P ₃	P ₄	P ₅	$P_{1}^{'}$	$P_{2}^{'}$	$P_{3}^{'}$	$P_{4}^{'}$	$P_{5}^{'}$
y(1)	y(5)	y(1)	x(12)	x(7)	–	z(1)	–	–	z(2)	41.6882	463,437

In Table 8, x (12) means that it is abandoned at time point 12, y (1) means that it is upgraded at time point 1, and z (1) means that it is selected at time point 1. As shown in Table 8, to obtain maximum profits, the enterprise upgrade the existing blast furnace projects P₁, P₂, and P₃ at time points 1, 5, and 1, respectively. The existing blast furnace projects P₄ and P₅ abandon at time points 12 and 7, respectively. For the new HIsmelt projects, $P_{2}^{'}$ and $P_{5}^{'}$ select at time points 1 and 2, respectively. Then the expected NPV of the project portfolio is 41.6882 (×10⁶), and the carbon emission of the project portfolio from the 12th period is 463,437 (ton).

Figure 4 shows the carbon emissions of the optimal project portfolio and the allowable carbon emissions in each period.

Fig. 4

The carbon emissions of the project portfolio and the allowable carbon emissions in each period.

As shown in Figure 4, the carbon emission of the project portfolio in the 4th period is F₄=951,300 (ton). From the 12th period, the carbon emissions per period are 463,437 (ton), less than the allowable carbon emission F_b=500,000 (ton).

5.4 Comparison of models with and without carbon emissions constraint

The objective function of model (12) without the carbon emissions constraint is

${\begin{matrix} max Equation (18) \\ s . t . \\ Equations (3), (4), (5), (15), \\ x_{i}, y_{i} \in {0, 1}, i = 1, 2, \dots, n, \\ z_{j} \in {0, 1}, j = 1, 2, \dots, k . \end{matrix}$ (19) Table 9 shows the optimal project portfolio and the carbon emission in the 15th period of model (19) without carbon emissions constraint. As shown in Table 9, when the carbon emissions of the project portfolio are not limited, the enterprise upgrades the blast furnace projects P₁ and P₃ at time point 1 and selects the new HIsmelt projects $P_{2}^{'}$ and $P_{5}^{'}$ at time points 1 and 2, respectively. The existing blast furnace projects P₂, P₄, and P₅ keep their original state.

Table 9

The optimal project portfolio and the carbon emission in the 15th period by model (19)

Existing project					New project					Carbon emission in15th period (ton)
P ₁	P ₂	P ₃	P ₄	P ₅	$P_{1}^{'}$	$P_{2}^{'}$	$P_{3}^{'}$	$P_{4}^{'}$	$P_{5}^{'}$
y (1)	-	y (1)	-	-	z (3)	z (1)	z (4)	-	z (2)	1.0913 ×10⁶

Figure 5 shows the allowable carbon emissions and the carbon emissions of the optimal project portfolio in each period. The carbon emission of the project portfolio in the 15th period is 1.0913 ×10⁶ (ton), less than the allowable carbon emission F_b=500,000 (ton).

Fig. 5

The allowable carbon emissions and the carbon emissions of the optimal project portfolio by model (19).

Figure 6 shows the expected NPV of the optimal project portfolio with respect to the carbon tax by model (19). In Figure 6, the dashed line indicates the expected NPV of the optimal project portfolio of model (17), and q^* is the carbon tax at which the expected NPV of the optimal project portfolio of model (17) and model (19) are the same.

Fig. 6

The expected NPV of the optimal project portfolio with respect to the carbon tax.

As shown in Figure 6, the higher the carbon tax, the smaller the expected NPV of the project portfolio. When the carbon tax is smaller than q^*, the expected NPV of the optimal project portfolio of model (19) is larger than that of model (17). This show that the enterprise will make the investment decision without considering the government’s carbon emission requirements when the carbon tax is smaller than q^* because the enterprise’s purpose is to maximize profits. Then the government cannot achieve the carbon neutralization goal. In other words, the carbon tax policy is invalid when the carbon tax is smaller than q^*. When the carbon tax is larger than q^*, the expected NPV of the optimal project portfolio of model (19) is smaller than the expected NPV of the optimal project portfolio of model (17). This show that the enterprise will make an investment decision that satisfies the government’s carbon emission requirements when the carbon tax is larger than q^*. This is consistent with the result of Proposition 4.

6 Conclusion

This paper discusses the multi-period project adjustment and selection problem under the carbon tax and carbon quota policies. The main contributions of the paper are as follows. Firstly, this paper proposes the uncertain mean-chance model to solve the multi-period project adjustment and selection (MPPAS) problem under the background of the carbon tax and carbon quota policies. Secondly, this paper proposes the deterministic equivalent of the proposed model and conducts the theoretical analysis of the impact of carbon tax and carbon quota policies. Thirdly, this paper develops an improved adaptive genetic algorithm to solve the proposed model. Fourthly, through a case study of steel enterprises with high carbon emissions, the performance of the proposed algorithm and the applicability of the proposed model are proved.

This paper considers a project portfolio problem under the carbon tax and carbon quota policies but does not consider other low-carbon policies such as the carbon trading policy. In addition, this paper uses the uncertain mean-chance model as an evaluation model for project portfolios. Further research may focus on two aspects. One is to consider more real-life environments, such as various policies to reduce carbon emissions and the interdependence between projects, and how to use uncertain dominance criterion to select the portfolio. The other is developing effective programming algorithms to solve the proposed model.

Footnotes

Conflict of interest

No conflict of interest exists.

Acknowledgements

Our research have been supported by the National Social Science Foundation of China (17BGL052).

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Uncertain multi-period project adjustment and selection under the carbon tax and carbon quota policies

Abstract

Keywords

1 Introduction

2.1 Notations

2.2 Uncertain mean-chance multi-period project adjustment and selection (MPPAS) model with carbon emissions constraint

3.1 Deterministic equivalent of the proposed model

4 Improved adaptive genetic algorithm

4.1 Representation structure

Table 2 Corresponding representation structure between the individual chromosome and the decision variables variable x 1 ⋯ x n y 1 ⋯ y n z 1 ⋯ z k b 1 ⋯ b n b 1 ′ ⋯ b k ′ individual g 1 ⋯ g n g n+1 ⋯ g 2n g 2n+1 ⋯ g 2n+k g 2n+k+1 ⋯ g 3n+k g 3n+k+1 ⋯ g 3n+2k

4.3 Generation of the initial population

4.4 Evaluation of individual

4.5 Selection operation

4.6 Adaptive crossover operation

4.8 Algorithm procedure

5.1 Problem description

Table 8 Optimal project portfolio of model (17) Existing project New project Expected NPV(×106$) Carbon emission in15th period (ton) P 1 P 2 P 3 P 4 P 5 P 1 ′ P 2 ′ P 3 ′ P 4 ′ P 5 ′ y(1) y(5) y(1) x(12) x(7) – z(1) – – z(2) 41.6882 463,437

Footnotes

Conflict of interest

Acknowledgements

References

Table 8
Optimal project portfolio of model (17)

Existing project New project Expected NPV(×10⁶$) Carbon emission in15th period (ton)

P ₁ P ₂ P ₃ P ₄ P ₅ $P_{1}^{'}$ $P_{2}^{'}$ $P_{3}^{'}$ $P_{4}^{'}$ $P_{5}^{'}$

y(1) y(5) y(1) x(12) x(7) – z(1) – – z(2) 41.6882 463,437