On Carbon Tax Effectiveness in Inducing a Clean Technology Transition: An Evaluation Based on Optimal Strategic Capacity Planning

Abstract

This paper studies carbon tax effectiveness in inducing a transition to cleaner production when a firm faces different technologies and demands over a planning horizon. To determine carbon tax effectiveness, we propose a model based on strategic capacity production planning under carbon taxes that considers proper performance measures. The model, which is formulated as a mixed integer linear problem (MILP), considers issues that previous works have not studied jointly, and that are relevant in a technological transition, such as machine replacement, workforce planning, and maintenance. The effectiveness measures consider levels of clean production and periods to reach a technological transition. Our computational experiments, based on a real case, have shown that in the absence of carbon taxes, a firm has no incentive to transition to clean technology. Still, the effectiveness of carbon taxes depends on the characteristics of the technology available for the production process and the magnitude of the demand. We include managerial insights aimed at both companies and the environmental authority.

1. Introduction

The balance of gases in the atmosphere creates what is known as the greenhouse effect, which traps the sun’s heat and, so far, has maintained the planet’s temperature at suitable levels for the development of life. However, over the last few centuries, human activity has disrupted the balance of greenhouse gases [1,2]. Since the Industrial Revolution, thousands of tons of carbon dioxide have been released into the atmosphere each year. Carbon dioxide is one of the primary greenhouse gases, and once its concentration in the atmosphere increases, more heat is trapped on the planet’s surface, raising temperatures beyond normal levels. The consequences of this disruption to the greenhouse effect, labeled as climate change, are already perceived through the ecosystem, and predictions state that there are less than two decades to reduce carbon emissions before reaching a no-return point [3]. Therefore, the sustained increase in carbon emissions is of global concern [4].

In response to the threat of climate change, 195 countries signed the Paris Agreement in 2016, with the main goal of holding the increase in global temperatures 2 °C below pre-industrial levels. However, efforts to accomplish this goal have proven to be insufficient; even more, it was estimated that by 2030, global emissions will double what should be emitted to fulfill the controlled rise of temperatures [5].

To control emissions, several policies and mechanisms have been designed to promote renewable energy and cleaner technologies [6]. Carbon pricing is the primary strategy used by environmental authorities [7], as it has been shown to reduce national emissions [8]. According to the World Bank [9], more than seventy-three carbon pricing mechanisms have been implemented. These pricing mechanisms are divided into two groups. The first one is carbon taxes, which set a specific value per ton of carbon emissions, providing certainty in the cost faced by emitters. Carbon taxes are often increased periodically, as are French, Swedish, and Canadian taxes [10]. The second group is emissions trading systems (ETSs), which set a limit on the total tonnes of carbon emissions. ETSs distribute the carbon emissions set by the environmental authority among emitters, who can then sell or buy them from other emitters [11]. Consequently, the value of carbon emissions is subject to the supply and demand law. ETSs should be reviewed periodically in terms of cap setting, covering sectors, and permit allocation [12]. Both mechanisms aim to reduce carbon emissions [13]. However, it is not clear which is more effective to reduce emissions. On the one hand, carbon taxes generate certainty in the clean technologies investment, but compared to ETS, it could be not clear if they induce a technological change that allows for meeting the goals set in the Paris Agreement. On the other hand, ETSs are aligned with the targets that each country sets to comply with the Paris Agreement, but a market mechanism to set the value per ton of carbon emissions could generate uncertainty in clean technologies investment. Despite the above, carbon taxes seems to be more effective than ETSs [14,15], since ETSs allow for emissions prevented in one region to be released in the same quantity elsewhere [16,17]. Furthermore, carbon taxes are easier to implement than ETSs, making them the preferred option among environmental authorities [18]. Thus, in this paper we focus on carbon taxes as a mechanism to reduce emissions and their effectiveness in inducing a clean technology change.

Carbon taxes are public policies to reduce emissions. Their effectiveness in inducing a technological transition to cleaner production has been studied mainly at the global [9], national [19,20], or large industrial sectors such as power generation [15,21], mining [22], and transportation [23]. Being a public policy, its implementation affects individual firms in all productive industries. From the literature review, we observe that the effect of carbon taxes on operational decisions and supply chain design has been extensively studied. Nevertheless, operational modeling and supply chain design approaches, both under carbon taxes, focus on analyzing how the production systems operation and the network distribution design are affected by carbon taxes. Thus, the effectiveness of carbon taxes to induce a technological transition to cleaner production in individual firms is still scarce [24].

One approach to determine the effectiveness of carbon taxes to induce a technological transition to cleaner production in individual firms is to use a strategic capacity plan that considers carbon taxes and cleaner production technology alternatives. Such a plan can define a firm’s optimal transition to clean technology, from which the effectiveness of carbon taxes can be measured in terms of the level of clean production achieved at the end of the planning horizon and/or the periods it takes for the firm to reach different levels of clean production. To our knowledge, no work has considered strategic capacity planning to analyze the optimal technological transition induced by carbon taxes in an individual firm.

This paper addresses the effectiveness of carbon taxes to induce a transition to clean production technologies in a firm, using a strategic capacity planning model under carbon taxes. The strategic capacity planning model is formulated as a MILP and incorporates decisions that previous work has not considered jointly. Such decisions are capacity expansion through investment and replacement of technology, expansion, or reduction through operating costs (workforce planning), demand satisfaction through production and inventory, allocation, maintenance, sale of discarded machines, and carbon taxes. We define effectiveness measures to determine the effectiveness of carbon taxes to induce a transition to clean production technologies. These effectiveness measures are functions of the optimal variables of the strategic capacity planning model and are based on transition levels and periods. The transition level is the proportion of the total production allocated to low-emission technologies in each period of the planning horizon. The transition period is the time to reach a given clean production rate. Thus, the carbon tax is effective if a technological change is induced in the firm at the end of the planning horizon; the greater the transition to clean production and the earlier it occurs, the greater the effectiveness of the carbon tax. In this sense, this paper aims to contribute to eco-efficiency and green manufacturing [25,26] by highlighting the effectiveness of clean technology deployment mechanisms and their relationship with a firm’s long-term capacity decisions. To show the applicability of the strategic capacity planning model and illustrate the transition to clean technologies in terms of the level and periods of transition, we present an illustrative example based on a real case. The illustrative example is then perturbed to explore the effect of demand and technologies on the effectiveness of carbon taxes.

The main contributions of this study are summarized as follows: (1) We address for the first time the effectiveness of carbon taxes to induce a transition to clean production technologies from the perspective of an individual manufacturing firm, using a strategic capacity planning model. Thus, we expanded the alternatives to analyze the effectiveness of carbon taxes as a mechanism to reduce carbon emissions. (2) We show how to formulate a strategic capacity planning model under carbon taxes as a MILP. Unlike previous capacity planning models, we consider decisions that have not been addressed jointly. (3) We propose new measures to evaluate the effectiveness of carbon taxes to induce a transition to cleaner production technologies in a firm. (4) We determine how demand and the relationship between different technologies available in the market, in terms of emissions and investment costs, affect the effectiveness of carbon taxes to induce a transition to cleaner production technologies in a firm.

The remainder of this paper is structured as follows. A review of related work is discussed in Section 2. Section 3 presents the strategic capacity planning model under carbon taxes. In Section 4, we propose measures to evaluate the effectiveness of carbon taxes. Numerical results to analyze the effectiveness of carbon taxes are reported in Section 5. Section 6 shows the main implications of the results for companies and environmental authorities. Finally, Section 7 presents our conclusions and future lines of research.

2. Literature Review

The literature review is divided into two parts. The first one explores how the operational problems of a firm have considered carbon prices. That is the implication of carbon prices in models of inventory management, routing, and supply chain design. The second one explores the formulation of strategic capacity planning models, specifically those that consider technology replacement, as well as workforce planning.

A comprehensive review of operations models under carbon pricing can be found in Zhou and Wen [27]. De-la Cruz-Márquez et al. [28] and Yadav and Khanna [29] model inventory decisions under carbon pricing. De-la Cruz-Márquez et al. [28] propose an inventory model for items with imperfect quality with sensitive demand under carbon taxes. They determine the optimal selling policy, order quantity, and backorder quantity. Yadav and Khanna [29] develop an inventory model for perishable products with the objective of determining the optimal cycle length and selling price to maximize profit based on the imposed carbon taxes. Qin et al. [30] and Pu et al. [31] model vehicle routing problems and their optimal path under carbon pricing. Qin et al. [30] model the simultaneous pickup and delivery vehicle routing problem under carbon taxes. They found that under carbon taxes, vehicle speed becomes more important to consider as the price of the tax increases. Pu et al. [31] model a multi-depot vehicle routing problem (MDVRP) of same-city delivery under carbon taxes. They found that the cost of emissions can be reduced by driving smaller distances. Zhou and Wen [27] identified that supply chain design network models are the more studied in conjunction with carbon prices [32,33,34,35,36]. For example, Wang et al. [37] study the relationship between supply chains and the environmental authority that set a carbon tax. They use a Stackelberg game approach and consider two different types of supply chains. Some works on supply chain study the technological transition induced by carbon pricing using supply chain models. Turken et al. [38], Li et al. [39], and Saberi et al. [40] study the technology investment decisions in the supply chain under carbon pricing. Turken et al. [38] model a multi-facility firm that can invest in green technology. Li et al. [39] examine government subsidy impacts on green technology investment in a supply chain under ETS. Saberi et al. [40] study supply chain firms’ investment decisions in green technologies under uncertainties in ETS prices and demand. However, having a scope encompassing the entire operational chain does not allow for discerning the effects of carbon taxes at the more particular level of the production process. To the best of our knowledge, only Drake et al. [41] use a capacity planning model to study how carbon taxes affect the technology investment decisions from an individual firm perspective. However, they use a Stackelberg game approach, which only allows analyzing the capacity expansion in the short term. Song et al. [42] expand the Drake model but do not focus on the impact of carbon taxes on the firm’s technology. In summary, no work has explored the effectiveness of carbon taxes to induce a transition toward cleaner technologies of a firm over the long term.

In this paper, we address this problem by formulating a strategic capacity planning model under carbon taxes. A comprehensive review on strategic capacity planning can be found in Verter and Dincer [43], Van Mieghem [44], Wu et al. [45], Julka et al. [46], and the most recent in Martínez-Costa et al. [47]. Martínez-Costa et al. [47] classify the strategic capacity models according to the number of sites involved in the process (single-site or multi-site), the type of capacity considered (investing, outsourcing, reduction, and replacement), and the uncertainty considered in the formulation (deterministic or stochastic). Furthermore, they describe major decisions that a strategic capacity planning problem should address: capacity size, capacity location, allocation, technology selection, production and inventory, backorders, workforce planning, new product development, and financial planning. Following this classification, the capacity planning problem of Drake et al. [41] studies a single-site and single-item capacity model under uncertain demand. They consider capacity expansion through investment by acquiring units of the capacity of two different technologies (dirty or clean) and demand is satisfied through production. The model is formulated as a two-stage stochastic linear programming that maximizes the expected profit. They derived an optimal closed-form solution that depends on the problem parameters. Song et al. [42] extend the work of Drake et al. [41], considering existing capacity at the beginning of the planning horizon. However, the two related works of Drake et al. [41] and Song et al. [42] left out crucial decisions for the strategic capacity plan with carbon taxes; namely, machine replacement and workforce planning.

Carbon taxes promote a transition to clean technologies [48]. Therefore, it is essential to consider discarding old machinery when planning capacity under carbon taxes. A literature review of equipment replacement is presented by Hartman and Tan [49]. Chand et al. [50] analyze a single-site and single-item strategic capacity model under deterministic demand. They consider capacity expansion through investment and machine replacement, where maximum periods of use are allowed before a machine must be replaced. The model is formulated as a MILP and solved by a heuristic procedure. Mitra et al. [51] study a single-site and multi-item strategic capacity model under uncertain demand. They consider capacity expansion through investment and machine replacement, where an upgrade of equipment constitutes a machine replacement, and demand is satisfied by production and inventory. The model is formulated as a two-stage stochastic MILP and solved using a commercial solver. Benedito et al. [52] formulate a single-site and multi-item strategic capacity under deterministic demand. They consider expansion through investment and machine replacement, demand is satisfied through production and inventory, and the selling of machines associated with the salvage value of the equipment. The model is formulated as a MILP and solved using a commercial solver. Wang and Nguyen [53] analyze a single-site and multi-item strategic capacity under deterministic demand. They consider capacity expansion through investment and machine replacement, with technology selection, where demand is satisfied through production and inventory. The model is formulated as a MILP, using a stochastic dynamic approach to address technology uncertainty, and solved by a genetic algorithm. Regarding workforce planning, for capital-intensive firms, the implementation of extra work shifts is an additional tool to increment capacity and delay the costs of machine acquisition. Few papers consider the workforce in strategic capacity planning. Fleischmann et al. [54] study a multi-site and multi-item strategic capacity model under deterministic demand. They consider capacity expansion through investment and machine replacement, and through operational costs with overtime and change of work shifts. Additionally, they consider production planning with an allocation of machines to satisfy demand and included the location as a decision variable. The model is formulated as a MILP and solved using a commercial solver. Bihlmaier et al. [55] analyze a multi-site and multi-item strategic capacity model under uncertain demand. They consider expansion through investment by acquiring machines, and through operational costs by adding new work shifts, such as late, night, and Saturday work shifts. Additionally, they considered production planning with allocation and backorders to satisfy demand and included location as a decision variable. The model is presented as a two-stage stochastic MILP and solved using a commercial solver. Weston et al. [56] study a single-site and multi-item strategic capacity model under uncertain demand. They consider capacity expansion through investment and operational cost by deciding the number of work shifts, and considered production planning with allocation to satisfy demand. Their model is formulated as a MILP, which considers a robust approach to include the demand uncertainty and is solved using a commercial solver. To date, Izadpanahi et al. [57] conducted the only work on strategic capacity planning that considers environmental issues in their model. They consider capacity expansion through investment, but they do not consider workforce planning to expand capacity. They also consider production planning with allocation to satisfy demand. In their work, they consider that the firm is under a carbon cap, and they pay a fee for any extra emission. Their model is formulated as a MILP, using a robust approach to face the uncertainty, and is solved using a commercial solver. However, they focus on the energy source transition and not on the technological selection of the manufacturing process.

All the listed works, related to strategic capacity planning, illustrate the different characteristics considered. These characteristics can be grouped into (i) problem setting, (ii) decisions, and (iii) formulation. For the problem-setting characteristics, we identify sites (single-site or multi-site), items (single-item or multi-item), and demand (deterministic or uncertain). In the second characteristic group (decisions), we consider the presented by Martínez-Costa et al. [47] and add three more; namely, machines maintenance, sale of discarded equipment, and carbon taxes. Respecting the formulation characteristics, we determine the capacity (discrete or continuous expansion) and type of model.

Table 1. Characteristics of related strategic capacity planning models.

	Setting				Decisions													Formulation
Authors	Scope ${}^a$	Single-site/Multi-site ${}^b$	Single-item/Multi-item ${}^c$	Demand ${}^d$	Capacity size e	Capacity Location	Allocation	Technology Selection	Production planning	Inventory	Backorders	Workforce Planning	New Product Development	Financial Planning	Maintenance	Sale of discarded	Carbon policy	Capacity ${}^f$	Type of model
Turken et al. [38]	S	M	S	D	E	√		√									√	C	MILP
Li et al. [39]	S	M	S	D	E			√									√	C	MILP
Saberi et al. [40]	S	M	S	U	E			√	√								√	C	LP
Chand et al. [50]	M	S	S	D	E/R													D	MILP
Fleischmann et al. [54]	M	M	M	D	E/R	√	√		√			√						C	MILP
Bihlmaier et al. [55]	M	M	M	U	E/R	√	√		√		√	√						C	MILP
Escalona and Ramırez [58]	M	S	M	D	E		√		√			√						D	MINLP
Mitra et al. [51]	M	S	M	U	E/R				√	√								D	MILP
Drake et al. [41]	M	S	S	U	E			√	√								√	C	LP
Benedito et al. [52]	M	S	M	D	E/R				√	√					√	√		C	MILP
Song et al. [42]	M	S	S	U	E			√	√								√	C	LP
Wang and Nguyen [53]	M	S	M	D	E/R		√	√	√	√								C	MILP
Weston et al. [56]	M	S	M	U	E		√		√			√						D	MILP
Izadpanahi et al. [57]	M	S	M	U	E				√						√		√	D	MILP
This paper	M	S	M	D	E/R		√	√	√	√		√			√	√	√	D	MILP

^a S (supply chain), M (manufacturing firm), ^b S (single-site), M (multi-site), ^c S (single-item), M (multi-item), ^d D (deterministic), U (uncertain), ^e E (expansion), R (replacement), ^f C (continuous), D (discrete).

shows the characteristics of the models presented in this section, as well as the model proposed in this work.

From Table 1, it is observed that no work has considered a strategic capacity planning model to explore the technological transition induced by under-carbon prices. Moreover, studies have neglected the crucial aspects of workforce planning and machine maintenance, factors that manufacturing firms need to consider when deciding how best to transition to cleaner technologies. Although the literature has acknowledged the influence of carbon prices on the technology decision of a supply chain, it has not explored the potential of strategic capacity planning to evaluate the effectiveness of carbon taxes in inducing a technological transition.

By incorporating essential factors, such as machine replacement, workforce planning, inventory management, allocation, and maintenance, we can provide valuable insights and practical guidance for firms and environmental authorities to address carbon pricing scenarios.

This work proposes an integrated strategic capacity planning model to assess the impact of carbon taxes to induce a firm’s technological transition. It addresses gaps in the literature by considering factors like machine replacement, workforce planning, and maintenance, contributing to sustainable business practices.

3. Problem Description and Formulation

Consider the strategic capacity planning of a process over a horizon of $T(t=1,\dots ,|T\left|\right)$ periods, where increasing carbon taxes are imposed by the environmental authority. In this process, $I(i=1,\dots ,|I\left|\right)$ items are produced, where an item corresponds to a component or final product. Let $d_{it}$ be the demand of item i satisfied by production or inventory in period t. At each period, the planning of the process capacity can be expanded through the acquisition of machines (expansion through investment) or by modifying the number of work shifts (expansion through operational cost).

Let $K^0$ be the set of operating machines at the beginning of the planning horizon ($t=0$) and $K^c$ be the set of machines that can be acquired over time. Thus, $K=K^0\cup K^c\left(k=1,\dots ,\left|K\right|\right)$ is the set of machines, where $K^0$ can be an empty set. In this paper, machine refers to equipment as well as a production line. The machine capacity is based on available time by period, which depends on the operational conditions of the process and the number of work shifts during the period. Machines wear out on use and are maintained under a preventive policy. Without loss of generality, it is considered that the maintenance policy is a preventive repair [59] and is executed on working hours. We also consider that machines have a remaining useful life that depends on the particular time of operation [60]. Furthermore, as carbon taxes are expected to induce the replacement of older machines with cleaner ones, the discard (and sale) of machines is considered.

Since the number of work shifts can be modified in each period, by opening or closing work shifts, it is possible to hire or fire workers, according to labor requirements. Moreover, at each period, the process can use one, two, or three work shifts with the same length.

The objective of our model is to determine a strategic capacity plan that minimizes costs over the planning horizon considering increasing carbon taxes. In this paper, the strategic capacity plan costs are investment, production, maintenance, labor, hiring–firing, change of work shifts, inventory, and carbon taxes, as well as the salvage value from discarded (sold) machines.

3.1. Machine State Diagram

The operational status of a machine and its change of state between periods (transitions) can be efficiently represented with a state diagram. Let $S=\{0,1,2,3\}$ be the set of operational states of a machine at any period of the planning horizon, where $s=0$ means that the machine is not operating since it has not been purchased or has already been discarded; $s=1$ means that the machine operates one work shift; $s=2$ means that the machine operates two work shifts; and $s=3$ means that the machine operates three work shifts. Let $E=\{(s,s^{\prime })\in S\times S\}$ be the set of state transitions that can occur at the beginning of any period of the planning horizon. Thus, the state of the machine during a period $t\in T$ is defined by the state transition $e=(s,s^{\prime })\in E$, where s is the state during period $t-1$, and $s^{\prime }$ is the state during period t. Given this definition, the set is defined by the 16 possible state transitions, which we label as follows: $E=\{e_0,\dots ,e_{15}\}$, where $e_0=(0,0)$, $e_1=(0,1)$, $e_2=(0,2)$, $e_3=(0,3)$, $e_4=(1,0)$, $e_5=(1,1)$, $e_6=(1,2)$, $e_7=(1,3)$, $e_8=(2,0)$, $e_9=(2,1)$, $e_{10}=(2,2)$, $e_{11}=(2,3)$, $e_{12}=(3,0)$, $e_{13}=(3,1)$, $e_{14}=(3,2)$ and $e_{15}=(3,3)$.

The set of state transitions E, to make the notation clearer, can be conveniently partitioned into four subsets. Let $E^0=\left\{e_0\right\}$ be the transition set that represents a machine outside the production process since it has not been purchased or has already been discarded; $E^1=\{e_1,$$e_2$, $e_3\}$ be the transitions set that represents the entry into operation of a machine that has not been purchased; $E^2=\{e_5$, $e_6,$$e_7,$$e_9,$$e_{10},$$e_{11},$$e_{13},$$e_{14},$$e_{15}\}$ be the transition set that represents a machine that was operating the previous period and stays in a state different from zero; and $E^3=\{e_4$, $e_8,$$e_{12}\}$ be the transition set that represents the discard of a machine that was operating during the previous period. The state diagrams representing the four partitions of E are illustrated in Figure 1.

Figure 1a shows a machine that during period $t-1$ is in state $s=0$ and in period t remains in state $s^{\prime }=0$. Figure 1b shows a machine that during period $t-1$ is in state $s=0$ and in period t is in a state $s^{\prime }\ne 0$, i.e., the machine begins to operate at the beginning of period t. Figure 1c shows a machine that during period $t-1$ is in a state $s\ne 0$ and in period t is also in a state $s^{\prime }\ne 0$, i.e., the machine stays in the same state or changes to another state different from 0. Figure 1d shows a machine that during $t-1$ is in a state $s\ne 0$ and in period t is in state $s^{\prime }=0$, i.e., the machine is discarded at the beginning of period t.

3.2. The Strategic Capacity Problem with Carbon Taxes

Under a state approach, a machine remains or changes its state at the beginning of a period. Let $X_{ke}^t$ be equal to 1 if the state transition $e\in E$ occurs at the beginning of period t for the machine $k\in K$; 0 otherwise. Furthermore, for any $e=(s,$$s^{\prime })\in E$ in period t we define $H\left(e\right)=s^{\prime }$ as the state of machine k in period t, and $T\left(e\right)=s$ as the state of machine k in period $t-1$.

To establish the relationship between the transitions of any machine throughout the planning horizon, the following constraints are defined:

$$\begin{array}{cccc}\hfill \sum _{e\in E:T\left(e\right)=s}{X}_{ke}^{t+1}-\sum _{e\in E:H\left(e\right)=s}{X}_{ke}^t& =0\hfill & \hfill & \forall k\in K,s\in S,t\in T\setminus \left\{\right|T\left|\right\},\hfill \end{array}$$

(1)

$$\begin{array}{cccc}\hfill \sum _{e\in E}{X}_{ke}^t& =1\hfill & \hfill & \forall k\in K,t\in T\setminus \left\{1\right\},\hfill \end{array}$$

(2)

$$\begin{array}{cccc}\hfill \sum _{t\in T}\sum _{e\in E^1}{X}_{ke}^t& \le 1\hfill & \hfill & \forall k\in K^c,\hfill \end{array}$$

(3)

$$\begin{array}{cccc}\hfill X_{ke}^t& \in \{0,1\}\hfill & \hfill & \forall k\in K,e\in E,t\in T.\hfill \end{array}$$

(4)

Constraint (1) ensures the continuity of transitions for any machine throughout the planning horizon. Constraint (2) forces that for any machine only one transition occurs at the beginning of any period. Constraint (3) ensures that during the planning horizon, a machine $k\in K^c$ can only be bought once, i.e., for any machine $k\in K^c$ a transition $e\in E^1$ can occur, at most, once.

It is considered that operating machines are used the same number of work shifts over the planning horizon, i.e., at the beginning of period t all the operating machines transition to the same state $s^{\prime }$. We assumed that the state s of every machine $k\in K^0$ at the beginning of the planning horizon ($t=0$) is a known parameter, denoted by $s_0$. In the same way, the initial state of a machine $k\in K^c$ is $s=0$. Let $Z_s^t$ be equal to 1 if s work shifts are used during period t; 0 otherwise. To establish the initial state of the different machines and their relation to the work shifts used, the following constraints are defined:

$$\begin{array}{cccc}\hfill \sum _{e\in E^2\cup E^3:T\left(e\right)=s_0}{X}_{ke}^1& =1\hfill & \hfill & \forall k\in K^0,\hfill \end{array}$$

(5)

$$\begin{array}{cccc}\hfill \sum _{e\in E^0\cup E^1}{X}_{ke}^1& =1\hfill & \hfill & \forall k\in K^c,\hfill \end{array}$$

(6)

$$\begin{array}{cccc}\hfill \sum _{s\in S}{Z}_s^t& =1\hfill & \hfill & \forall t\in T,\hfill \end{array}$$

(7)

$$\begin{array}{cccc}\hfill \sum _{e\in E^1\cup E^2:H\left(e\right)=s}{X}_{ke}^t& \le Z_s^t\hfill & \hfill & \forall s\in S,k\in K,t\in T,\hfill \end{array}$$

(8)

$$\begin{array}{cccc}\hfill Z_s^t& \in \{0,1\}\hfill & \hfill & s\in S,t\in T.\hfill \end{array}$$

(9)

Constraint (5) ensures that for any machine $k\in K^0$ only one transition $e\in E^2\cup E^3$ can occur at the beginning of period $t=1$, since the state of this kind of machine in $t=0$ is $s=s_0$. Constraint (6) ensures that for any machine $k\in K^c$ only one transition $e\in E^0\cup E^1$ can occur at the beginning of period $t=1$, since the state of this kind of machine in $t=0$ is $s=0$. Constraints (7) and (8) follow that all operating machines are used for the same number of work shifts in period t.

The opening and closing of a work shift at the beginning of period t are defined by the binary variable $O_{st}$ and $C_{st}$, respectively. Let $O_{st}$ ($C_{st}$) be equal to 1 if work shift s is opened (closed) at the beginning of period t; 0 otherwise. Thereby, the opening and closing of a work shift are defined as follows,

$$\begin{array}{cccc}\hfill Z_s^t-Z_s^{t-1}& \le O_s^t\hfill & \hfill & \forall s\in S,t\in T,\hfill \end{array}$$

(10)

$$\begin{array}{cccc}\hfill Z_s^1-Z_{s_0}^0& \le O_s^1\hfill & \hfill & \forall s\in S,\hfill \end{array}$$

(11)

$$\begin{array}{cccc}\hfill Z_s^{t-1}-Z_s^t& \le C_s^t\hfill & \hfill & \forall s\in S,t\in T,\hfill \end{array}$$

(12)

$$\begin{array}{cccc}\hfill Z_{s_0}^0-Z_s^1& \le C_s^1\hfill & \hfill & \forall s\in S,\hfill \end{array}$$

(13)

$$\begin{array}{cccc}\hfill O_s^t,C_s^t& \in \{0,1\}\hfill & \hfill & \forall s\in S,t\in T,\hfill \end{array}$$

(14)

where $Z_{s_0}^0$ is a known parameter that is equal to 1 if $s_0$ work shift are used in period $t=0$; 0 otherwise.

Constraints (10)–(13) ensure that if there is a change of work shift between $t-1$ and t, an opening and closing of work shift will occur.

As mentioned previously, the demand for each period is satisfied with production and/or inventory. Let $Y_{ik}^t$ be the quantity of item i produced by machine k in period t, and $I_i^t$ be the on-hand inventory of item i at the end of period t. Thus, the following constraints ensure that the demand is met:

$$\begin{array}{cccc}\hfill I_i^{t-1}+\sum _{k\in K}{Y}_{ik}^t-I_i^t& \ge d_{it}\hfill & \hfill & \forall i\in I,t\in T\setminus \left\{1\right\},\hfill \end{array}$$

(15)

$$\begin{array}{cccc}\hfill I_i^0+\sum _{k\in K}{Y}_{ik}^1-I_i^1& \ge d_{i1}\hfill & \hfill & \forall i\in I,\hfill \end{array}$$

(16)

$$\begin{array}{cccc}\hfill Y_{ik}^t& \ge 0\hfill & \hfill & \forall i\in I,k\in K,t\in T,\hfill \end{array}$$

(17)

$$\begin{array}{cccc}\hfill I_i^t& \ge 0\hfill & \hfill & \forall i\in I,t\in T,\hfill \end{array}$$

(18)

where $I_i^0$ is the on-hand inventory of item i at the end of period $t=0$, which may be zero. It should be noted that other sources of production used in production planning, such as backorders, outsourcing, or extra time are not considered.

To consider the maintenance of a machine due to its use, we assumed, without loss of generality, a preventive fixed-time maintenance policy, where $FT{M}_k$ is the maximum fixed time between maintenance interventions of machine k. Let W$(w=1,\dots ,$$\left|W\right|)$ be the number of maintenance interventions carried out for each machine during the planning horizon, where the time required for the w-th maintenance of machine k is known and denoted by $RM{T}_{wk}$. Let $M_{wk}^t$ be equal to 1 if the w-th maintenance of the machine k is performed in period t; 0 otherwise, and $T{M}_k^t$ be the accumulated production time of machine k from its last maintenance to the beginning of period t. Consequently, the following sets of constraints define the execution of maintenance.

$$\begin{array}{cccc}\hfill T{M}_k^t& \ge T{M}_k^{t-1}+\sum _{i\in I}\frac{Y_{ik}^{t-1}}{r_{ik}}-FT{M}_k\sum _{w\in W}{M}_{wk}^t\hfill & \hfill & \forall k\in K,t\in T,\hfill \end{array}$$

(19)

$$\begin{array}{cccc}\hfill T{M}_k^t& \le FT{M}_k\hfill & \hfill & \forall k\in K,t\in T,\hfill \end{array}$$

(20)

$$\begin{array}{cccc}\hfill \sum _{\tau =1}^t{M}_{wk}^{\tau }& \ge M_{w+1k}^t\hfill & \hfill & \forall k\in K,t\in T,w\in 1,\dots \left|W\right|-1,\hfill \end{array}$$

(21)

$$\begin{array}{cccc}\hfill \sum _{t\in T}{M}_{wk}^t& \le 1\hfill & \hfill & \forall k\in K,w\in W,\hfill \end{array}$$

(22)

$$\begin{array}{cccc}\hfill M_{wk}^t& \le \sum _{\tau =1}^t\sum _{e\in E^1\cup E^2}{X}_{ke}^{\tau }\hfill & \hfill & \forall k\in K,t\in T,w\in W,\hfill \end{array}$$

(23)

$$\begin{array}{cccc}\hfill T{M}_k^t& \ge 0\hfill & \hfill & \forall k\in K,t\in T,\hfill \end{array}$$

(24)

$$\begin{array}{cccc}\hfill M_{wk}^t& \in \{0,1\}\hfill & \hfill & \forall w\in W,k\in K,t\in T,\hfill \end{array}$$

(25)

where $r_{ik}$ in (19) is the production rate of item i produced with machine k.

Constraint (19) updates the time counter from the last maintenance and Constraint (20) ensures that this time does not exceed the maximum fixed time between maintenance of machine k. Constraint (21) implies that the w+1-th maintenance cannot be executed until all previous maintenance has occurred. Constraint (22) ensures that the w-th maintenance can be executed, at most, once. Constraint (23) ensures that maintenance is only applied to operating machines, i.e., only machines that at the beginning of period t stay or transition to a state $s\ne 0$ can be maintained.

Machines have a useful life that is consumed according to their production time. Let $R{L}_k^t$ be the remaining useful life of machine k at the beginning of period t. Thus, the production time of a machine cannot exceed the remaining useful life. On the other hand, it is clear that the production and maintenance time cannot exceed the available time of the period. Consequently:

$$\begin{array}{cccc}\hfill \sum _{i\in I}\frac{Y_{ik}^t}{r_{ik}}+\sum _{w\in W}RM{T}_{wk}{M}_{wk}^t& \le {\mu }_k\sum _{s\in S}\sum _{\begin{array}{c}e\in E^1\cup E^2:\\ H\left(e\right)=s\end{array}}ls{X}_{ke}^t\hfill & \hfill & \forall k\in K,t\in T,\hfill \end{array}$$

(26)

$$\begin{array}{cccc}\hfill \sum _{i\in I}\frac{Y_{ik}^t}{r_{ik}}& \le R{L}_k^t\hfill & \hfill & \forall k\in K,t\in T,\hfill \end{array}$$

(27)

$$\begin{array}{cccc}\hfill R{L}_k^t& =R{L}_k^{t-1}+v_k\sum _{e\in E^1}{X}_{ke}^t-\sum _{i\in I}\frac{Y_{ik}^{t-1}}{r_{ik}}\hfill & \hfill & \forall k\in K,t\in T\setminus \left\{1\right\},\hfill \end{array}$$

(28)

$$\begin{array}{cccc}\hfill R{L}_k^1& =R{L}_k^0+v_k\sum _{e\in E^1}{X}_{ke}^1\hfill & \hfill & \forall k\in K,\hfill \end{array}$$

(29)

$$\begin{array}{cccc}\hfill R{L}_k^t& \ge 0\hfill & \hfill & \forall k\in K,t\in T,\hfill \end{array}$$

(30)

where ${\mu }_k$ is the maximum utilization of machine k, $v_k$ is the useful life of machine k, and l is the work shift length. Moreover, $R{L}_k^0$ represents the remaining useful life of machine k at the end of period 0, with $R{L}_k^0=0$ for any $k\in K^c$.

Constraint (26) ensures that for any machine the production and maintenance time is less or equal that its available time at any period. Constraint (27) ensures that the production time of a machine does not exceed the remaining useful life. Constraints (28) and (29) update and initialize the remaining useful life of a machine, respectively.

As stated in Section 2, the machine’s replacement is an essential decision to consider in a strategic capacity plan under carbon taxes. The salvage value depends on variable $R{F}_k^t$, defined as the remaining useful life of machine k when discarded at the beginning of the period t. Consequently, the following constraints are defined:

$$\begin{array}{cccc}\hfill R{F}_k^t& \le v_k\sum _{e\in E^3}{X}_{ke}^t\hfill & \hfill & \forall k\in K,t\in T,\hfill \end{array}$$

(31)

$$\begin{array}{cccc}\hfill R{F}_k^t& \le R{L}_k^t\hfill & \hfill & \forall k\in K,t\in T,\hfill \end{array}$$

(32)

$$\begin{array}{cccc}\hfill R{F}_k^t& \ge 0\hfill & \hfill & \forall k\in K,t\in T.\hfill \end{array}$$

(33)

Constraints (31) and (32) keep track of the remaining useful life of the machine if discarded at the beginning of period t.

The objective function involves minimizing the total cost of the strategic capacity plan, including the carbon taxes. We consider eight costs associated with strategic capacity planning. Moreover, since the discard of machines is allowed, the objective function also includes the salvage value from selling the discarded machines. Consequently, we have: $\left(i\right)$ the investment cost of acquiring machine k in period t ($C{I}_k^t$); $\left(ii\right)$ the production cost for an item i produced by machine k in period t ($C{P}_{ik}^t$); $\left(iii\right)$ the cost of the w-th maintenance of machine k in period t ($C{M}_{wk}^t$); $\left(iv\right)$ the unitary labor cost for machine k in period t ($C{L}_k^t$); $\left(v\right)$ the cost of hiring ($C{A}_k^t$) and firing ($C{F}_k^t$) workers; $\left(vi\right)$ the cost of opening ($C{O}_s^t$) and closing ($C{C}_s^t$) work shifts s; $\left(vii\right)$ the holding cost per unit and unit period of item i in period t ($C{H}_i^t$); and $\left(viii\right)$ the value of carbon taxes in period t ($C{T}_t$), which increases periodically according to the carbon tax policy adopted by the environmental authority. Let ${\alpha }_k^t$ be the price per hour of useful life remaining of machine k that was discarded in period t. All the aforementioned costs are properly brought to their current value.

Similarly to [61,62], we consider that carbon emissions come from the energy and the resources used in production and inventory. Let $e{p}_{ik}$ be the emissions for producing one unit of item i using machine k, and let $e{h}_i$ be the emissions of holding per unit and unit period of item i. Therefore, the carbon emissions in period t are ${\mathcal{E}}^t={\sum }_{i\in I}({\sum }_{k\in K}e{p}_{ik}{Y}_{ik}^t+e{h}_i{I}_i^t)$. Thus, the strategic capacity planning under carbon taxes (SPT) is defined as the following MILP:

$$\begin{array}{cc}\hfill SPT:min& \sum _{k\in K,t\in T}\left\{\sum _{e\in E^1}C{I}_k^t{X}_{ke}^t+\sum _{i\in I}C{P}_{ik}^t{Y}_{ik}^t+\sum _{w\in w}C{M}_{wk}^t{M}_{wk}^t+\sum _{e\in E}H\left(e\right)C{L}_k^t{OP}_k{X}_{ke}^t\right.\hfill \\ \hfill & \left.+\sum _{e\in E}N{O}_{e}C{A}_k^t{OP}_k{X}_{ke}^t+\sum _{e\in E}N{C}_{e}C{F}_k^t{OP}_k{X}_{ke}^t+\sum _{s\in S}(C{C}_s^t{C}_s^t+C{O}_{st}{O}_s^t)\right\}\hfill \\ \hfill & +\sum _{i\in I,t\in T}C{H}_i^t{I}_i^t+\sum _{t\in T}C{T}_t{\mathcal{E}}^t-\sum _{k\in K,t\in T}{\alpha }_k^{t}R{F}_k^t\hfill \\ \hfill & \mathrm{s}.\mathrm{t}:\left(1\right)–\left(13\right),\hfill \\ \hfill & \left(14\right)–\left(23\right),\hfill \\ \hfill & \left(24\right)–\left(33\right),\hfill \end{array}$$

(34)

where $N{O}_e$ and $N{C}_e$ are the number of work shifts that are opened and closed when a state transition e occurs, respectively. The number of workers needed to operate machine k is defined by the parameter $O{P}_k$. In Appendix A, a glossary of the used terminology can be found.

It is easy to show that the SPT model without carbon taxes, i.e., the objective Function (34) minus ${\sum }_{t\in T}C{T}_t{\mathcal{E}}^t$, is a relaxation of the SPT model. We denote this relaxed model as SPWT. Consequently, $Z_{SPWT}^{*}\le Z_{SPT}^{*}$, where $Z_{SPWT}^{*}$ and $Z_{SPT}^{*}$ are the optimal objective function of SPWT and SPT models, respectively.

4. Effectiveness Carbon Tax Measures

To determine the carbon tax performance, we defined two effectiveness measures. The first one is the transition level, defined as the proportion of production allocated to each available technology at every period of the planning horizon. The second one is the transition period, defined as the period where the weighted proportion of production allocated to clean technologies reaches $\beta$ for the first time, with $\beta \in [0,1]$.

Let J be the set of available technologies. Furthermore, let $N_j$ be an indexed set where $N_j=\{k\in K$: k is of technology $j\}$, with ${\cup }_{j\in J}{N}_j=K$ and $N_j\cap N_p=Ø$ for any $j,p\in J:j\ne p$. Thus, the transition level of technology j reached in period t is defined as:

$$\begin{array}{c}\hfill R_j^t=\sum _{i\in I}\left\{\frac{{\sum }_{k\in N_j}{Y}_{ik}^t}{{\sum }_{k\in K}{Y}_{ik}^t}\right\}\forall t\in T,j\in J,\end{array}$$

(35)

where $Y_{ik}^t$ is an optimal variable of the SPT model. On the other hand, the transition period where the transition level of clean technologies reaches $\beta$ for the first time is defined as:

$$\begin{array}{c}\hfill {\tau }_{\beta }=min\left\{t\in T:\sum _{j\in J}{\gamma }_j{R}_j^t\ge \beta \right\},\end{array}$$

(36)

where ${\gamma }_j$ is the normalized weight for technology $j\in J$, such that ${\sum }_{j\in J}{\gamma }_j=1$ and ${\gamma }_j>{\gamma }_p$ if technology j is consider cleaner than technology p. We propose a criterion based on the possible emissions as follows:

$$\begin{array}{c}\hfill {\gamma }_j=\frac{{\eta }_j^{-1}-{\displaystyle \underset{j\in J}{min}}\left({\eta }_j^{-1}\right)}{{\displaystyle \sum _{j\in J}\left({\eta }_j^{-1}-{\displaystyle \underset{j\in J}{min}}\left({\eta }_j^{-1}\right)\right)}}\forall j\in J,\end{array}$$

(37)

where ${\eta }_j={\left|N_j\right|}^{-1}{\sum }_{k\in N_j}{\sum }_{i\in I}e{p}_{ik}$ for any $j\in J$. It should be noted that the dirtiest technology has a weight equal to zero.

A carbon tax is considered useful if a technological transition towards clean production technologies is induced by the end of the planning horizon. The effectiveness of the carbon tax is determined by the magnitude of change and the time required to reach $\beta$. The weighted average transition level reached by the end of the planning horizon determines the magnitude of the change induced by carbon taxes, i.e., ${\sum }_{j\in J}{\gamma }_j{R}_j^{\left|T\right|}$. The transition period (${\tau }_{\beta }$) measures how long it takes for the carbon tax to induce the desired technological transition. Thus, the higher the transition level and the lower the transition periods, the higher the effectiveness of the carbon tax in the firm.

5. Computational Study and Results

In this section, we numerically evaluate the effectiveness of carbon taxes to induce a transition to clean technologies under an optimal capacity expansion plan determined using the SPT model. We first present an illustrative industrial example based on a real case, denoted as the base case, to show the applicability of the SPT model and illustrate the transition to clean technologies in terms of the level and periods of transition. Then, the base case is perturbed to explore the effect of demand and technologies on carbon tax effectiveness.

The base case considers an eight-year planning horizon, eight products, and two types of technologies (dirty and clean). To avoid the undesired effects of the abrupt termination of the planning horizon, the SPT model considers twelve years and then discards the results for the last four years. The base case perturbation considers 50 demand instances. Each demand instance is solved by considering different relationships between technologies in terms of emissions and investment costs.

The SPT model is solved using Gurobi 9.1.1. The stopping criterion for the SPT model is an optimality gap of ${10}^{-4}$ or 36,000 s of CPU time. All tests were conducted on a PC with an Intel Core i5 2.3 GHz × 4 processor and 8 GB RAM.

5.1. Industrial Illustrative Example: Base Case

Consider a firm that is required to determine its strategic capacity expansion plan for the upcoming 8 years, during which carbon taxes are implemented. The production considers eight items, which consist of the juice from four different fruits packed in glass containers of 1000 and 300 cc. The juice production process consists of grading the fruit, crushing and pressing the fruit, and storing the juice in stainless-steel containers. Once the container reaches a threshold quantity, the juice is pasteurized at 90 °C and then sent to the filling station. In particular, a capacity expansion is required for the pasteurizing and filling processes, which are considered a production line. Figure 2 shows the complete juice-making process, where the production processes subject to the expansion plan are framed in red.

The firm faces a market that grows faster at the beginning of the planning horizon and stabilizes at the end, i.e., it has an increasing demand at a decreasing rate. This demand behavior is representative of the life cycle of most products. Moreover, increasing carbon taxes are imposed throughout the planning horizon. Carbon taxes begin at USD 35 per carbon emission ton and increase yearly until reaching USD 70 at the end of the planning horizon, following [63].

Similarly to Drake et al. [41] and Song et al. [42], two types of technologies are considered (dirty and clean). Let $j=1$ and $j=2$ be the dirty and clean technology, respectively. Let ${\widehat{ep}}_{ij}$ be the emissions for producing one unit of item i using technology j, where $e{p}_{ik}={\widehat{ep}}_{ij}$ for any $i\in I$ and $k\in N_j$. It is considered that the investment cost ($C{I}_k^t$) is higher for the clean technology than for the dirty one. Let ${\widehat{CI}}_j^t$ be the investment cost of technology j in period t, where $C{I}_k^t={\widehat{CI}}_j^t$ for any $k\in N_j$ and $t\in T$. Thus, we consider that carbon emissions and investment costs are the same for all machines of the same technology. It should be noted that ${\gamma }_1=0$ and ${\gamma }_2=1$. The parameters used for the case base are presented in Appendix B.

To illustrate the technological transition induced by carbon taxes, we computed the transition level $R_j^t$, for each $j\in J$ and $t\in T$, according to (35) using the optimal variables resulting from solving the SPT model. Figure 3 shows the transition level for both technologies throughout the planning horizon.

Figure 3 shows that during the first periods of the planning horizon ($t=1,$ 2), the firm only uses dirty technology ($j=1$). This is because the carbon taxes are insufficient for the firm to invest in clean technology, which is more expensive. In the next three periods ($t=3,4,5$), the company starts using clean technology. However, most of its production is allocated to dirty technology. At the beginning of the sixth period ($t=6$), dirty technology is no longer used. Thus, the cost of carbon taxes needed to induce a total technological transition is reached at the beginning of period 6. Therefore, it can be inferred that carbon taxes become a useful mechanism to induce a technological transition once the carbon emission costs make it convenient for the strategic plan to invest in clean machines, and the effectiveness will not necessarily be seen with short-term planning. It should be noted that without carbon taxes, all production is allocated to dirty technology, i.e., without carbon taxes there is no incentive for a clean technology transition.

We computed the total emissions at each period by the optimal solution of SPT and SPWT models, respectively. Figure 4 shows the emissions throughout the planning horizon, where ${\mathcal{E}}_{SPT}^t$ and ${\mathcal{E}}_{SPWT}^t$ are the carbon emissions in period t, resulting from SPT and SPWT, respectively.

From Figure 4, we observe that in the absence of carbon taxes, emissions increase monotonously and grow accordingly with the demand behavior of the firm. On the contrary, under carbon taxes, the emissions decouple from the demand. Furthermore, it can be observed that once the clean technology is incorporated into production ($t=3$ according to Figure 3), the growth rate of emissions decreases considerably.

5.2. Effect of Demand and Technology on Carbon Tax Effectiveness

The effect of demand and technologies on carbon tax effectiveness in inducing a clean technology transition is determined by perturbing the base case. To isolate the effect of demand, the products considered in the base case are aggregated into a single product according to $d_t={\sum }_{i\in I}{d}_{it}$ for any $t=1,\dots ,T$. Furthermore, the emissions when producing the single product with technology j, defined as ${\widehat{ep}}_j$, is the weighted average emissions of the products considered in the base case. Similarly, the production rate, inventory emissions, and production cost of the single product are determined as the weighted average of the products considered in the base case.

The base case perturbation considers 50 demand instances. Let $d_t^{\left(k\right)}=0.05k{d}_t$, with $k=1,\dots ,50$, be the k-th instance demand for period t. Thus, each instance considers more demand than the previous one. It should be noted that instances $k=1$ and $k=50$ consider 0.05 and 2.5 times the base case demand, respectively, and that instance $k=2$0 is the base case.

For each instance k, two emission ratios $\left(\frac{{\widehat{ep}}_2}{{\widehat{ep}}_1}\in \{0.5,0.7\}\right)$, and four investment ratios $\left(\frac{{\widehat{CI}}_2}{{\widehat{CI}}_1}\in \{1.3,1.4,1.5,1.6\}\right)$ are considered. Thus, each instance k is evaluated for the eight combinations of emissions and investment ratios, generating 400 scenarios. It should be noted that emission ratios are strictly less than 1 and the investment ratios are strictly greater than 1. This is because we assume that less emitter technology is more expensive.

5.2.1. Effect on Transition Level

Solving the SPT model for each scenario, we compute the clean transition level reached at the end of the planning horizon. The proportion of production allocated to clean technology at the end of the planning horizon ($R_2^8$) for each scenario, and emission and investment ratios, are shown in Figure 5.

Figure 5 shows that the effectiveness of carbon taxes is strongly influenced by the demand size, and by the relationship of clean to dirty technology in terms of emissions and investment costs. More precisely:

• The lower the demand size, the lower the probability of achieving a technology transition under any technology relationship (emissions or cost).
• The higher the clean technology emissions reduction, the higher the probability of achieving a technology transition.
• The higher the clean technology investment cost, the lower the probability of achieving a technology transition.

Figure 5a shows that the probability of achieving a full technological transition ($R_2^8=1$) is strictly greater than zero for every investment ratio. On the other hand, Figure 5b shows that the probability of achieving a full technological transition is strictly greater than zero only when the investment ratio is equal to 1.3, and the probability of achieving a full technological transition is equal to zero when the investment ratio is equal to 1.6.

Table 2. Expected transition level.

	$\frac{{\widehat{\mathit{ep}}}_2}{{\widehat{\mathit{ep}}}_1}=0.5$					$\frac{{\widehat{\mathit{ep}}}_2}{{\widehat{\mathit{ep}}}_1}=0.7$
$\frac{{\widehat{\mathit{CI}}}_{\mathbf{2}}}{{\widehat{\mathit{CI}}}_{\mathbf{1}}}$	$\mathbb{P}({\mathit{R}}_{\mathbf{2}}^{\mathbf{8}}=\mathbf{0})$	$\mathbb{P}({\mathit{R}}_{\mathbf{2}}^{\mathbf{8}}\ge \mathbf{0}.\mathbf{50})$	$\mathbb{P}({\mathit{R}}_{\mathbf{2}}^{\mathbf{8}}\ge \mathbf{0}.\mathbf{75})$	$\mathbb{P}({\mathit{R}}_{\mathbf{2}}^{\mathbf{8}}\ge \mathbf{1})$	$\mathbb{E}\left({\mathit{R}}_{\mathbf{2}}^{\mathbf{8}}\right)$	$\mathbb{P}({\mathit{R}}_{\mathbf{2}}^{\mathbf{8}}=\mathbf{0})$	$\mathbb{P}({\mathit{R}}_{\mathbf{2}}^{\mathbf{8}}\ge \mathbf{0}.\mathbf{50})$	$\mathbb{P}({\mathit{R}}_{\mathbf{2}}^{\mathbf{8}}\ge \mathbf{0}.\mathbf{75})$	$\mathbb{P}({\mathit{R}}_{\mathbf{2}}^{\mathbf{8}}\ge \mathbf{1})$	$\mathbb{E}\left({\mathit{R}}_{\mathbf{2}}^{\mathbf{8}}\right)$
1.3	0.04	0.96	0.90	0.82	0.92	0.06	0.94	0.82	0.56	0.85
1.4	0.06	0.94	0.86	0.76	0.89	0.10	0.90	0.62	0.06	0.70
1.5	0.06	0.94	0.82	0.52	0.84	0.56	0.00	0.00	0.00	0.22
1.6	0.08	0.92	0.74	0.08	0.74	1.00	0.00	0.00	0.00	0.00

summarizes the probabilities for achieving different technological transition levels, and the expected transition value at the end of the planning horizon for each emission and investment ratio.

Under any given investment ratio, Table 2 shows that the probability of achieving a full transition when the ratio of the emission is equal to 0.5 is strictly higher than when the ratio of the emission is equal to 0.7. Similarly, for any given investment ratio, the expected value of achieving a full transition is strictly greater when the emission ratio is equal to 0.5. Thus, we infer that the emission ratio has a great impact on carbon tax effectiveness in inducing a technological transition. Furthermore, when the emission ratio is equal to 0.5, the investment ratio has a lower impact, e.g., $\mathbb{P}(R_2^8\ge 0.50)=0.96$ and $\mathbb{P}(R_2^8\ge 0.50)=0.92$ when the investment ratio is equal to 1.3 and 1.6, respectively. On the contrary, when the emission ratio is equal to 0.7, the investment ratio has a higher impact on the transition, e.g., $\mathbb{P}(R_2^8\ge 0.50)=0.94$ and $\mathbb{P}(R_2^8\ge 0.50)=0$ when the investment ratio is equal to 1.3 and 1.6, respectively. Thus, we infer that the investment ratio has a greater impact on carbon tax effectiveness in inducing a technological transition when the emission ratio is higher. In summary, when demand is low, and clean technology is expensive and does not reduce emissions sufficiently, the company has low or no incentive to change from dirty to clean technology. This is because it is more convenient to pay the carbon tax than to invest in clean technology.

5.2.2. Effect on Transition Periods

For each scenario, we compute the transition period in which the production with clean technology reaches 50% and 75%, i.e., ${\tau }_{0.5}$ and ${\tau }_{0.75}$, respectively. Figure 6 shows the box-plot of ${\tau }_{\beta }$, with $\beta =0.5$ and $\beta =0.75$, under different emission and investment ratios. It should be noted that no scenario reaches 50% or 75% of production with clean technology when the ratio of the emission is equal to 0.7 and the investment ratio is equal to 1.5 or 1.6.

From Figure 6 we observe that:

• As the emissions ratio increases (i.e., the difference in emissions between clean and dirty technology decreases), the longer it takes to reach 50% and 75% production with clean technology.
• As the investment ratio increases (i.e., the difference in cost between clean and dirty technology increases), the longer it takes to reach 50% and 75% production with clean technology.

Thus, we infer that the firm has the incentive to switch to clean technology earlier when the emissions and investment ratio decreases.

6. Discussion and Managerial Insights

In this section, we summarize, interpret, and extrapolate the numerical results obtained in Section 5, their implications, and limitations. From this, we generate some managerial insights aimed at companies and decision-makers of the environmental authority.

Aspects of interest for companies:

• The model proposed in this paper allows the company to have optimal long-term capacity planning, which guarantees the minimum cost related to the carbon tax when faced with the decision to opt for dirty or clean technologies, at differentiated costs and considering scenarios with different demands. Unlike other models, the model proposed in this paper also considers costs related to machine replacement, workforce planning, and equipment maintenance.
• The model can be applied to firms interested in obtaining strategic capacity planning which contemplates the acquisition of clean technology. This company must also have a prospect of known demand, as well as clarity on its costs related to inventory, production, maintenance, hiring, firing, opening, and closing of shifts, and mainly a way to measure its total carbon dioxide emissions per period.
• In general, demand is a factor that accelerates the technological transition to clean production. Companies with high demand can move more quickly to the majority use of clean technology, regardless of the amount of the carbon tax or even the price of the new technology. If the emissions reductions from clean technology are large, this adoption may be even greater.
• A firm with relatively low demand has virtually no incentive to invest in technology replacement unless the clean technology is extremely cheap and/or the carbon tax is sufficiently high. For these low-demand firms, the achievement of a higher percentage of clean technology adoption is strongly influenced not only by the technology cost but also by its effectiveness in reducing emissions.
• The way that low-demand companies have to ensure their integration and adherence to clean technologies is to be able to count on low-cost and high-efficiency technology in reducing emissions.

Aspects of interest for the environmental authority:

• The overall objective of carbon tax policies is to encourage companies to use clean technology by penalizing the use of dirty technology. The ultimate goal should be that 100 percent of the industrial activities subject to the tax should be carried out using clean technology.
• A challenge is to determine the amount of carbon tax that best encourages the transition of industries to clean technology. That is, to achieve the highest percent of transition in the shortest possible time.
• The proposed model allows the environmental authority to observe the effect of the carbon tax on a company with certain base characteristics. The effectiveness of the tax can be evaluated in terms of the percentage of adherence to new technology (transition level), as well as the speed at which the clean technology is integrated into the company (transition period).
• From the experiments conducted in a base case in a production plant, it is observed that the effectiveness of the carbon tax to encourage technological transition is mainly linked to three factors: the demand, the cost of the new clean technology, and its effectiveness to reduce emissions.
• It should be noted that the effectiveness of the technology in reducing emissions is a factor that, associated with its cost, is relevant regardless of the size of the company. That is, if only an expensive and ineffective technology is available on the market, neither large nor small companies will be motivated to adopt it, regardless of the carbon tax.
• The differentiated impact of carbon taxes on large and small companies can be observed. For large companies, the magnitude of the tax is not a determining factor in achieving technological change. For small companies, the tax is a differentiating factor between undertaking or not the adoption of clean technologies.
• In this sense, decision-makers of the environmental authority could consider alternatives; either to manage a carbon tax differentiated according to business demand, and/or to consider subsidizing small businesses to motivate the adoption of efficient clean technologies.

7. Conclusions

In this paper, we introduce a methodology to evaluate carbon tax effectiveness in inducing a clean technology transition in individual firms. Two main stages compose the methodology. The first corresponds to strategic capacity planning under the carbon taxes model, which determines the optimal strategic plan for an individual firm in terms of acquisition and usage of technologies throughout the planning horizon. The second stage corresponds to the measures to evaluate the effectiveness of the carbon tax. We define the transition level and the transition period, as the proportion of allocated production to each technology and the period where the desired clean transition level is reached, respectively. This second stage may be of interest to decision-makers of the environmental authority.

Our proposed model is applied to an industrial illustrative example where the strategic capacity plan under carbon taxes faces clean and dirty technology. The technological transition is illustrated in terms of the transition levels for each technology and the total emissions throughout the planning horizon. Several instances, based on the industrial illustrative example, were generated to explore the impact of demand size and technology relationships, in terms of emissions and investment, on the carbon taxes’ effectiveness.

We observe that a full clean technology transition is not reached in the planning horizon when the emissions of the clean technology are 70% of the dirty technology, and the investment cost of clean technologies is 140% or higher than the dirty technology. Thus, it can be inferred that carbon taxes are ineffective in incentivizing the firms to adopt costly clean technology that does not significantly reduce emissions (in comparison to dirty technology). Furthermore, we infer that a firm that faces a small demand has a lower incentive to acquire a more costly technology. Moreover, carbon taxes alone do not ensure a clean technology transition because their effectiveness is highly influenced by the incentive of firms to invest in more expensive technology to reduce emissions.

In summary, it is advisable for a firm interested in updating its capacity with clean technology to pay attention to its demand. When a high increase in demand is expected over the planning horizon, it is advisable to consider acquiring clean technology to meet rising carbon taxes. However, the acquisition depends on the technologies available on the market for the industrial process. If the inversion cost that differentiates one technology from another is not high, and the emission reduction is substantial, the sooner the transition can begin, the more cost-effective it will be. Otherwise, it is probably more convenient to not make the transition and to wait for cleaner options to appear in the market or for a decrease in the price. On the other hand, when the demand is expected to stay stable or even decrease, unless the cleaner technology in the market is highly convenient in cost and emissions savings, there is no incentive to start the transition. Based on our approach and the results obtained, we can declare that our model is effective as a capacity planning tool in relation to carbon tax compliance. However, to strengthen planning in companies with special demand conditions, it would be better to apply the proposed methodology or a version of the same to evaluate each particular case.

There are several issues left for future research. Carbon tax effectiveness is affected by demand, which is related to the size of a firm. The first issue to be explored is to determine the optimal carbon tax policy for each firm’s size, such that all firms in the industry are encouraged to adopt it. Environmental authorities have the choice of using a variety of mechanisms to reduce emissions and motivate a transition to clean technologies. Therefore, the second issue is to incorporate into the proposed methodology other common mechanisms, such as the emissions trading system and subsidies, allowing the comparison of the effectiveness of different mechanisms in inducing a clean technology transition in a firm. It is reasonable to consider that demand and available technology are unknown to the firm throughout the planning horizon, due to changes in the market and the incentive promoted by carbon taxes for suppliers to lower technologies emissions [64]. Thus, the third issue is to consider a robust strategic capacity planning model, such that the model considers uncertainty in the demand and available technology. The last remaining issue is the scalability of the SPT model, as it becomes considerably harder to prove the optimality of the solution with a real case complexity. Thus, a solution approach has to be implemented to improve the computational times.