# On Carbon Taxes Effectiveness to Induce a Clean Technology Transition: An
Evaluation Framework Based on Optimal Strategic Capacity Planning

N. Wolf<EMAIL_ADDRESS>P. Escalona  A. Angulo  J. Weston  Department
of Industrial Engineering, Universidad Técnica Federico Santa María, Avenida
España 1680, Valparaíso, Chile.  Department of Electrical Engineering,
Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso,
Chile.  Friedrich-Alexander-Universität Erlangen-Nürnberg, Department
Mathematik, Cauerstraße 11, 91058 Erlangen, Germany.

###### Abstract

This paper studies carbon taxes effectiveness to induce a transition to
cleaner production when a firm faces different technologies and demands. To
determine carbon taxes effectiveness, we propose a framework based on a
strategic capacity planning under carbon taxes model, that consider proper
perfomance measures. The model, which is formulated as a mixed integer linear
problem (MILP), considers issues that previous work have not studied jointly,
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 carbon taxes by themselves do not necessarily
induce a transition to clean production, since their effectiveness depends on
the available technology relationship and the demand magnitude.

###### keywords:

Carbon taxes effectiveness; strategic capacity planning; clean technology
transition; framework.

††journal: Journal of Sustainability Analytics and Modeling

0.0 pt

##  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, in the
last centuries, human activity has disrupted this balance [11]. Since the
Industrial Revolution, thousands of tons of carbon emissions have been
released into the atmosphere each year. Carbon dioxide is one of the primary
greenhouse gasses, 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 [31]. Therefore, the sustained
increase in carbon emissions is of global concern [61].

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 below 2 °C pre-industrial levels. However, efforts to accomplish
this goal have proven to be insufficient, even more, it was estimated that by
2030 the global emissions will double what should be emitted to fulfill the
controlled rise of temperatures [10].

To control emissions, several policies and mechanisms have been designed to
promote renewable energy and cleaner technologies [63]. Nowadays, carbon
pricing is the main mechanism adopted by environmental authorities [17], as
they have proven to be an effective approach to reduce national emissions
[21]. Two carbon pricing mechanisms can be distinguished. The first ones are
carbon taxes, which set a specific value per ton of carbon emissions,
providing certainty in the cost faced by emitters. Existing carbon taxes
usually increase periodically, such as the French, Swedish, and Canadian
carbon tax [12]. The second ones are emissions trading systems (ETS), which
set a limit on the total tonnes of carbon emissions. ETS distribute the
permitted emissions between emitters, who can later sell or buy to other
emitters [47]. Although both mechanisms promote a reduction of emissions [28],
carbon taxes are more swiftly implementable than ETS, making them the
preferred option for environmental authorities [14].

The effectiveness of carbon taxes on national emissions, along with its effect
on macro indicators has been widely studied, e.g. economic growth, employment
and poverty. However, even when it is vital to reduce the environmental impact
from the industrial sector [30], research on the effectiveness of carbon taxes
in individual firms is still scarce [29]. We consider that an efficient
mechanism to determine the effectiveness of carbon taxes on a firm is using
the strategic capacity planning. A strategic capacity planning under carbon
taxes defines the optimal transition to clean technology of a firm, from which
the effectiveness of carbon taxes can be measure in terms of the cleaning
production level to reached, and the corresponding period of the transition.
To the best of our knowledge, no work has considered a strategic capacity
planning to study the optimal technological transition induced on a individual
firm under carbon taxes.

In this paper, we address the effectiveness of carbon taxes to induce a
transition to cleaner production technologies on a firm that can choose
between different technologies in terms of emissions and investment cost. The
effectiveness of carbon taxes is determined using a framework based on a
strategic capacity planning problem, which is formulated as a single-site and
multi-item strategic capacity model under carbon taxes and deterministic
demand. Our model combine capacity expansion decisions that previous work have
not jointly consider, such as capacity expansion through investment and
replacement, expansion or reduction through operational cost (workforce
planning), demand satisfaction through production and inventory, allocation,
maintenance, sale of discarded machines, and carbon taxes. Furthermore, we
define measures for carbon taxes effectiveness, based on the transition levels
and the periods in which clean production is reach. Thus, a carbon tax is
effective if by the end of the planning horizon a technological transition is
induced on the firm, the greater the transition to clean production and the
sooner it occurs, the higher the effectiveness of the carbon tax.

The main contributions in this paper can be summarized as follows: (i) We
address the effectiveness of carbon taxes to induce a transition to cleaner
technologies on an individual firm perspective; (ii) we proposed a framework
to evaluate the effectiveness of the carbon tax based on a strategic capacity
planning model; (iii) we formulated and solve a strategic capacity planning
model containing decisions not previously addressed jointly; and (iv) we
determined factors that influence the effectiveness of carbon taxes.

The rest of the paper is structured as follows. In Section 2, a literature
review on related work of carbon taxes effectiveness and strategic capacity
planning is presented. In Section 3 we present the proposed framework and
formulate the strategic capacity planning under carbon taxes model. Section 4
reports the computational experiments that explore carbon taxes effectiveness.
Finally, Section 5 presents our conclusions and future lines of research.

##  2 Literature review

There are two main streams of research related to this paper that need to be
explored, (i) the effectiveness of carbon taxes, and (ii) the strategic
capacity planning models.

The effectiveness of carbon taxes has been approached from different
perspectives in literature. In this paper, we classified the effectiveness of
carbon taxes under a macro, supply chain, and individual firm perspective. The
macro perspective focus on quantifying the effect that carbon taxes have on
national or regional indicators [25, 23], the supply chain perspective focus
on the effect that carbon taxes have on the design of the logistic networks
[50], and the individual firm perspective focus on how carbon taxes influence
the operational decisions of firms [64].

From the macro perspective, [32], and [22] analyze qualitatively the
effectiveness of China and Japan’s carbon taxation policy, respectively. Both
authors determine that the value of the carbon tax, to decrease carbon
emissions and transition out of fossil fuels, should be higher than actual
values. [38], and [3] study the consequences that the carbon taxes have on the
gross domestic product (GDP) of China and British Columbia, respectively. They
conclude that carbon taxes do not cause a significant change on the GDP. [51]
and [34] simulated the possible impact of carbon taxes on social welfare and
poverty in Mexico and Finland, showing a negative impact for high carbon
taxes. [6] and [44] study the implications of a carbon tax in employability in
the U.S. and Europe. [6] observed a positive relationship of the carbon tax
and employment in the U.S., while [44] found no robust evidence of a negative
impact on employment.

For the supply chain perspective, a comprehensive review on the reduction of
carbon emissions in supply chains can be found on [13] and [9].

In particular, [62] and [26] research the impact of carbon taxes in the supply
chain cost. [62] compares the effect on costs of carbon taxes and ETS,
observing an increase in costs under a carbon tax policy. [26] observes that
incorporating carbon taxes in the supply chain, results in a minor increase in
the operational costs. [43] and [7] study how carbon taxes affect production.
[43] observes that the cost for manufacturing firms is more sensitive to
carbon taxes than firms that buy the product. [7] concludes that production
quantity decreases proportionally to the carbon tax value. [36] and [24] study
the changes of transportation choices under carbon taxes. [36] observes that
under increasing carbon taxes, the firm outsources more transportation
services. [24] observes that adjusting transportation capacities offers a
better strategy than building extra facilities in response to carbon taxes
uncertainty. [65] and [19] study the effect of carbon taxes on pricing,
showing that final prices and wholesale prices increase as the carbon tax
value increase. [60] and [40] explore the cooperation between the echelons of
the supply chain. [60] observes that depending on the initial level of
contamination and value, a carbon tax policy can promote cooperation in the
supply chain. [40] results shows that,when manufacturers and retailers
cooperate under carbon tax regulations, the emissions reduction and profit
increment can be optimized.

Regarding the effectiveness of carbon taxes to reduce emissions in individual
firms, a comprehensive review on carbon emissions in operational decisions can
be found on [64]. [39] and [53] study the impact of carbon taxes on inventory
decisions. [39] observes that the number of shipments and the lot size depend
on the carbon tax policy, a progressive carbon tax policy offers flexibility
in deciding the number of shipments and the sizes, and a multiple delivery
policy is better than a single delivery policy. [53] shows that emissions from
holding and transportation can be reduced changing the lot size and reorder
point. [41] and [35] study how supplier selection change under carbon taxes.
Both authors observe that carbon taxes can induce manufacturers to choose
lower emitter supplier once carbon tax value is increase enough. [57], [49]
and [37] study the effect of carbon taxes on the optimal path of a vehicle
routing problem. Their results show that a carbon tax policy, under a
minimization costs approach, can reduce carbon dioxide emissions in logistics
network. Furthermore, the effect of the speed and capacity of the vehicles
becomes more relevant as the carbon taxes increase. To the best of our
knowledge, only [15] and [52] study the impact of carbon taxes on the firm
technology and capacity expansion from an individual firm perspective.
However, they use a model based on Stackelber games that only allows them to
analyze the capacity expansion in the short term. In summary, no previous work
explores the long term effectiveness of carbon taxes to induce a transition
toward cleaner technologies in a firm following a strategic planning
perspective.

The second literature stream related to this paper, is strategic capacity
planning. A comprehensive review can be found in Verter and Dincer [55], [54],
[59], [33], and the most recent in [42]. [42] classified 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 described 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.

The capacity expansion problem of Drake et al. [15], studies a single-site and
single-item capacity model under uncertain demand. They considered the
capacity expansion through investment by acquiring units of 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 which maximizes the expected profit. They derived an optimal
closed-form solution that depends on the problem parameters. Song et al. [52],
extend the work of Drake et al. [15], considering capacity at the beginning of
the planning horizon. However, the two related works of Drake et al. [15] and
Song et al. [52] 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. Therefore, it is
essential to consider discarding old machinery when planning capacity under
carbon taxes. A literature review of equipment replacement is presented by
[27]. [8] analyzed a single-site and single-item strategic capacity model
under deterministic demand. They considered capacity expansion through
investment and replacement of machines, where maximum periods of use is
allowed before a machine must be replaced. The model is formulated as a MILP
and solved by a heuristic procedure. [45] studied a single-site and multi-item
strategic capacity model under uncertain demand. They considered capacity
expansion through investment and replacement of machines, 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. [1] formulated a single-site and
multi-item strategic capacity under deterministic demand. They considered
expansion through investment and replacement of machines, 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. [56] analyzed a single-site and multi-item
strategic capacity under deterministic demand. They considered capacity
expansion through investment and replacement of machines, 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 the 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. [20] studied a multi-site and multi-
item strategic capacity model under deterministic demand. They considered
capacity expansion through investment and replacement of machines, and through
operational cost with overtime and change of work shifts. Additionally, they
considered production planning with 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. [4] analyzed a multi-site and
multi-item strategic capacity model under uncertain demand. They considered
expansion through investment by acquiring machines, and through operational
costs by adding new work shifts, such as a late, night, and Saturday work
shift. 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. [58] studied single-site and multi-item under uncertain
demand. They considered 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
presents a robust approach to include the uncertainty and was solved using a
commercial solver.

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 included, and (iii) formulation. For
the problem setting we consider sites (single-site or multi-site), items
(single-item or multi-item) and demand (deterministic or uncertain). In the
decision we include the presented by Martínez-Costa et al. [42], and add three
more, namely, maintenance of machines, sale of discarded equipment, and carbon
taxes. With respect to formulation, we focus on capacity (discrete or
continuous expansion), type of model, and solution approach. Table 1 shows the
characteristics of the models presented in this section, as well as the model
proposed in this work.

Table 1: Characteristics of related strategic capacity planning models.

| Setting | Decisions | Formulation
---|---|---|---
Authors and year |  Single-site/Mult-siteaafootnotemark: a |  Single-item/Multi-itembbfootnotemark: b |  Demandccfootnotemark: c |  Capacity sizeddfootnotemark: d |  Capacity Location |  Allocation |  Technology Selection |  Production planning |  Inventory |  Backordes |  Workforce Planning |  New Product Development |  Financial Planning |  Maintenance |  Sale of discarded |  Carbon tax |  Capacityeefootnotemark: e |  Type of model |  Solution Approach
Chand et al. [8] | S | S | D | E/R |  |  |  |  |  |  |  |  |  |  |  |  | D | MILP | Heuristic
Fleischmann et al. [20] | M | M | D | E/R | ✓ | ✓ |  | ✓ |  |  | ✓ |  |  |  |  |  | C | MILP | Solver
Bihlmaier et al. [4] | M | M | U | E/R | ✓ | ✓ |  | ✓ |  | ✓ | ✓ |  |  |  |  |  | C | MILP | Solver
Escalona and Ramırez [18] | S | M | D | E |  | ✓ |  | ✓ |  |  | ✓ |  |  |  |  |  | D | MINLP | Solver
Mitra et al. [45] | S | M | U | E/R |  |  |  | ✓ | ✓ |  |  |  |  |  |  |  | D | MILP | Solver
Drake et al. [15] | S | S | U | E |  |  | ✓ | ✓ |  |  |  |  |  |  |  | ✓ | C | LP | Closed-form
Benedito et al. [1] | S | M | D | E/R |  |  |  | ✓ | ✓ |  |  |  |  | ✓ | ✓ |  | C | MILP | Solver
Song et al. [52] | S | S | U | E |  |  | ✓ | ✓ |  |  |  |  |  |  |  | ✓ | C | LP | Closed-form
Wang and Nguyen [56] | S | M | D | E/R |  | ✓ | ✓ | ✓ | ✓ |  |  |  |  |  |  |  | C | MILP | Heuristic
Weston et al. [58] | S | M | U | E |  | ✓ |  | ✓ |  |  | ✓ |  |  |  |  |  | D | MILP | Solver
Proposal | S | M | D | E/R |  | ✓ | ✓ | ✓ | ✓ |  | ✓ |  |  | ✓ | ✓ | ✓ | D | MILP | Solver

11footnotetext: S (Single-site), M (Multi-site)22footnotetext: S (Single-
item), M (Multi-item)

33footnotetext: D (Deterministic), U (Uncertain)44footnotetext: E (Expansion),
R (Replacement)

55footnotetext: C (Continuous), D (Discrete)

From Table 1 it can be seen that the strategic capacity planning under carbon
taxes is still narrow, since current works with carbon taxes have not yet
considered relevant decisions to determine the optimal transition to cleaner
technologies, such as replacement, workforce planning, inventory, allocation
or maintenance. All these decisions are addressed in this work, where we
propose an integrated model which considers relevant decisions for a strategic
capacity plan to evaluate the effect of carbon taxes on the clean technology
transition of a firm.

##  3 Framework to determine the effectiveness of carbon taxes in a firm

This section presents a framework to determine the effectiveness of carbon
taxes to induce a transition to clean technology on a manufacturing firm. The
framework is based on a strategic capacity planning model that allows
evaluating different demand behavior, planning horizons, and types of
technologies. The model considers relevant decisions on the strategic capacity
planning literature, which have not been considered when evaluating the effect
of carbon taxes on a firm. Furthermore, we propose two measures of
effectiveness, the transition level and transition period. The structure of
the proposed framework is illustrated in Figure 1, and will be further explain
in what follows.

Figure 1: Structure of the framework to determine the effectiveness of carbon
taxes in a firm.

###  3.1 Endogenous and exogenous conditions

This framework allow us to evaluate carbon taxes effectiveness to induce a
clean technology transition on a manufacturing firm, under different
endogenous and exogenous conditions.

The endogenous parameters of the stratetegic capacity planning are those which
are specific to a firm. Different firms and production processes influence
several operational costs, such as production, maintenance, and labor. Thus,
the firm and the scope of its manufacturing process defines the endogenous
parameters of the problem.

On the other hand, the exogenous parameters depend on the external scenario in
which the firm is immersed. In this work, we consider three factors that
define the scenario faced by the firm. The first factor is the carbon tax
policy, i.e., the magnitud and update frequency of the carbon tax implemented.
The second factor is the demand faced by the firm, since the rise of demand is
directly tied with carbon emission growth [16] and the strategic capacity plan
of a firm. The third factor is the technology available in the market, which
differs in terms of the emissions by unit produced and the investment cost.

###  3.2 Strategic capacity planning model

Consider the strategic capacity planning of a process over a horizon of $T\
(t=1,...,|T|)$ periods, where increasing carbon taxes are imposed by the
environmental authority. In this process, $I\ (i=1,...,|I|)$ items are
produced, where an item correspond 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, $K=K^{0}\cup K^{c}\ (k=1,...,|K|)$ is the set of machines, where
$K^{0}$ can be an empty set. In this paper, machine refers to equipment as
well as 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 at the period. Machines wear out on use and are
maintained under a preventive policy. Without loss of generality, it is
considered that the policy is as good as new, and is executed on working
hours. 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 worker, 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 the 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 revenue from discarded (sold) machines.

####  3.2.1 Machine state diagram

The operational state 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 $t$. We label all
elements of the state transition set as follows: $E=\\{e_{0},...,e_{15}\\}$,
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 $E$ can be conveniently partitioned into four
subsets. Let $E^{0}=\\{e_{0}\\}$ be the transition set that represents a
machine that is currently inoperative, 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 2.

(a) $E^{0}$

(b) $E^{1}$

(c) $E^{2}$

(d) $E^{3}$

Figure 2: State diagram partitions.

Figure 2(a) shows a machine that during period $t-1$ is in state $s=0$ and in
period $t$ remains in state $s^{\prime}=0$. Figure 2(b) shows a machine that
during period $t-1$ is in state $s=0$ and in period $t$ is in a state
$s^{\prime}\neq 0$, i.e., the machine begins to operate at the beginning of
period $t$. Figure 2(c) shows a machine that during period $t-1$ is in a state
$s\neq 0$ and in period $t$ is also in a state $s^{\prime}\neq 0$, i.e.,
machine stays in the same state or change to another state different from $0$.
Figure 2(d) shows a machine that during $t-1$ is in a state $s\neq 0$ and in
period $t$ is in state $s^{\prime}=0$, i.e., the machine is discarded at the
beginning of period $t$.

####  3.2.2 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 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 $H(e)=s^{\prime}$ as the state of machine $k$ in period $t$, and
$T(e)=s$ 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 set of constraints are defined:

$\displaystyle\sum_{e\in E:T(e)=s}{X_{ke}^{t+1}}-\sum_{e\in
E:H(e)=s}{X_{ke}^{t}}$ $\displaystyle=0$ $\displaystyle\forall k\in K,s\in
S,t\in T\setminus\\{|T|\\},$ (1) $\displaystyle\sum_{e\in E}{X_{ke}^{t}}$
$\displaystyle=1$ $\displaystyle\forall k\in
K,t\;\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\in
T\setminus\\{1\\}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},$
(2) $\displaystyle\sum_{t\in T}\sum_{e\in E^{1}}X_{ke}^{t}$ $\displaystyle\leq
1$ $\displaystyle\forall k\in K^{c},$ (3) $\displaystyle X_{ke}^{t}$
$\displaystyle\in\\{0,1\\}$ $\displaystyle\forall k\in K,e\in E,t\in T.$ (4)

Constraints (1) ensures the continuity of transitions for any machine
throughout the planning horizon. Constraints (2) forces that for any machine
only one transition occurs at the beginning of any period. Constraints (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:

$\displaystyle\sum_{e\in E^{2}\cup E^{3}:T(e)=s_{0}}{X_{ke}^{1}}$
$\displaystyle=1$ $\displaystyle\forall k\in K^{0},$ (5)
$\displaystyle\sum_{e\in E^{0}\cup E^{1}}{X_{ke}^{1}}$ $\displaystyle=1$
$\displaystyle\forall k\in K^{c},$ (6) $\displaystyle\sum_{s\in S}{Z_{s}^{t}}$
$\displaystyle=1$ $\displaystyle\forall t\in T,$ (7) $\displaystyle\sum_{e\in
E^{1}\cup E^{2}:H(e)=s}{X_{ke}^{t}}$ $\displaystyle\leq Z_{s}^{t}$
$\displaystyle\forall s\in S,k\in K,t\in T,$ (8) $\displaystyle Z_{s}^{t}$
$\displaystyle\in\\{0,1\\}$ $\displaystyle s\in S,t\in T.$ (9)

Constraints (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 machines in $t=0$ is $s=s_{0}$. Constraints (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
machines in $t=0$ is $s=0$. Constraints (7) and (8) follows 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 $t$ is
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 is defined as follows,

$\displaystyle Z_{s}^{t}-Z_{s}^{t-1}$ $\displaystyle\leq O_{s}^{t}$
$\displaystyle\forall s\in S,t\in T,$ (10) $\displaystyle
Z_{s}^{1}-Z_{s_{0}}^{0}$ $\displaystyle\leq O_{s}^{1}$ $\displaystyle\forall
s\in S,$ (11) $\displaystyle Z_{s}^{t-1}-Z_{s}^{t}$ $\displaystyle\leq
C_{s}^{t}$ $\displaystyle\forall s\in S,t\in T,$ (12) $\displaystyle
Z_{s_{0}}^{0}-Z_{s}^{1}$ $\displaystyle\leq C_{s}^{1}$ $\displaystyle\forall
s\in S,$ (13) $\displaystyle O_{s}^{t},C_{s}^{t}$ $\displaystyle\in\\{0,1\\}$
$\displaystyle\forall s\in S,t\in T,$ (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) to (13) ensures 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:

$\displaystyle I_{i}^{t-1}+\sum_{k\in K}{Y_{ik}^{t}}-I_{i}^{t}$
$\displaystyle\geq d_{it}$ $\displaystyle\forall i\in
I,t\;\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\in
T\setminus\\{1\\}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},$
(15) $\displaystyle I_{i}^{0}+\sum_{k\in K}{Y_{ik}^{1}}-I_{i}^{1}$
$\displaystyle\geq d_{i1}$ $\displaystyle\forall i\in I,$ (16) $\displaystyle
Y_{ik}^{t}$ $\displaystyle\geq 0$ $\displaystyle\forall i\in I,k\in K,t\in T,$
(17) $\displaystyle I_{i}^{t}$ $\displaystyle\geq 0$ $\displaystyle\forall
i\in I,t\in T,$ (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 wear of machine due to their use, we assumed, without loss of
generality, a fixed-time maintenance policy, where $FTM_{k}$ is the maximum
fixed time between maintenance of machine $k$. Let $W$ $(w=1,...,$ $|W|)$ be
the numbers of maintenance that can be carried out for each machine, where the
time required for the $w$-th maintenance of machine $k$ is known and denoted
by $RMT_{wk}$. 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 $TM_{k}^{t}$ 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:

$\displaystyle TM_{k}^{t}$ $\displaystyle\geq TM_{k}^{t-1}+\sum_{i\in
I}{\frac{Y_{ik}^{t-1}}{r_{ik}}}-FTM_{k}\sum_{w\in W}{M_{wk}^{t}}$
$\displaystyle\forall k\in K,t\in T,$ (19) $\displaystyle TM_{k}^{t}$
$\displaystyle\leq FTM_{k}$ $\displaystyle\forall k\in K,t\in T,$ (20)
$\displaystyle\sum_{\tau=1}^{t}{M_{wk}^{\tau}}$ $\displaystyle\geq
M_{w+1k}^{t}$ $\displaystyle\forall k\in K,t\in T,w\in 1,...\lvert W\rvert-1,$
(21) $\displaystyle\sum_{t\in T}{M_{wk}^{t}}$ $\displaystyle\leq 1$
$\displaystyle\forall k\in K,w\in W,$ (22) $\displaystyle{M_{wk}^{t}}$
$\displaystyle\leq\sum_{\tau=1}^{t}\sum_{e\in E^{1}\cup E^{2}}{X_{ke}^{\tau}}$
$\displaystyle\forall k\in K,t\in T,w\in W,$ (23) $\displaystyle TM_{k}^{t}$
$\displaystyle\geq 0$ $\displaystyle\forall k\in K,t\in T,$ (24)
$\displaystyle M_{wk}^{t}$ $\displaystyle\in\\{0,1\\}$ $\displaystyle\forall
w\in W,k\in K,t\in T,$ (25)

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

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

Machines have a useful life which wears out according to its production time.
Let $RL_{k}^{t}$ be the remaining useful life of machine $k$ at the beginning
of period $t$. Thus, the production time of a machine cannot exceed its
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:

$\displaystyle\sum_{i\in I}{\frac{Y_{ik}^{t}}{r_{ik}}}+\sum_{w\in
W}{RMT_{wk}M_{wk}^{t}}$ $\displaystyle\leq\mu_{k}\sum_{s\in
S}\sum\limits_{\begin{subarray}{c}\;e\in E^{1}\cup E^{2}:\\\
H(e)=s\end{subarray}}lsX_{ke}^{t}$ $\displaystyle\forall k\in K,t\in T,$ (26)
$\displaystyle\sum_{i\in I}{\frac{Y_{ik}^{t}}{r_{ik}}}$ $\displaystyle\leq
RL_{k}^{t}$ $\displaystyle\forall k\in K,t\in T,$ (27) $\displaystyle
RL_{k}^{t}$ $\displaystyle=RL_{k}^{t-1}+v_{k}\sum_{e\in
E^{1}}{X_{ke}^{t}}-\sum_{i\in I}{\frac{Y_{ik}^{t-1}}{r_{ik}}}$
$\displaystyle\forall k\in
K,t\;\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\in
T\setminus\\{1\\}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},$
(28) $\displaystyle RL_{k}^{1}$ $\displaystyle=RL_{k}^{0}+v_{k}\sum_{e\in
E^{1}}{X_{ke}^{1}}$ $\displaystyle\forall k\in K,$ (29) $\displaystyle
RL_{k}^{t}$ $\displaystyle\geq 0$ $\displaystyle\forall k\in K,t\in T,$ (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. Also,
$RL_{k}^{0}$ represents the remaining useful life of machine $k$ at the end of
period $0$, with $RL_{k}^{0}=0$ for any $k\in K^{c}$.

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

As stated in Section 2, the replacement of machines is an essential decision
to consider in a strategic capacity plan under carbon taxes. The resale
revenue depends on variable $RF_{k}^{t}$, defined as the remaining useful life
of machine $k$ when discarded at the beginning of the period $t$.
Consequently, the following constrains are defined:

$\displaystyle RF_{k}^{t}$ $\displaystyle\leq v_{k}\sum_{e\in
E^{3}}{X_{ke}^{t}}$ $\displaystyle\forall k\in K,t\in T,$ (31) $\displaystyle
RF_{k}^{t}$ $\displaystyle\leq RL_{k}^{t}$ $\displaystyle\forall k\in K,t\in
T,$ (32) $\displaystyle RF_{k}^{t}$ $\displaystyle\geq 0$
$\displaystyle\forall k\in K,t\in T.$ (33)

Constraints (31) and (32) condition the value of the remaining life of a
machine when discarded at the beginning of the 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 the strategic capacity planning. Morever, since the discard of machines
is allowed, the objective function also includes the revenue from selling the
discarded machines. Consequently, we have: $(i)$ the investment cost of
acquiring machine $k$ in period $t$ ($CI_{k}^{t}$); $(ii)$ the production cost
for an item $i$ produced by machine $k$ in period $t$ ($CP_{ik}^{t}$); $(iii)$
the cost of the $w$-th maintenance of machine $k$ in period $t$
($CM_{wk}^{t}$); $(iv)$ the unitary labor cost for machine $k$ in period $t$
($CL_{k}^{t}$); $(v)$ the cost of hiring ($CA_{k}^{t}$) and firing
($CF_{k}^{t}$) workers; $(vi)$ the cost of opening ($CO_{s}^{t}$) and closing
($CC_{s}^{t}$) work shifts $s$; $(vii)$ the holding cost per unit and unit
period of item $i$ in period $t$ ($CH_{i}^{t}$); and, $(viii)$ the value of
carbon taxes in period $t$ ($CT_{t}$), which increases periodically, according
to the carbon tax policy adopted by the environmental authority. Furthermore,
we consider the revenue from sold machines. 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.

Carbon taxes take various forms depending on the regulatory policies of
emission control. Similarly to [2] and [46], we consider that carbon emissions
come from the energy and the resources used in production and inventory. Let
$ep_{ik}$ be the emissions for producing one unit of item $i$ using machine
$k$, and let $eh_{i}$ be the emissions of holding per unit and unit period of
item $i$. Therefore, the carbon emissions in period $t$ are
$\mathcal{E}^{t}=\sum_{i\in I}(\sum_{k\in
K}{ep_{ik}Y_{ik}^{t}}+eh_{i}I_{i}^{t})$. Thus, the strategic capacity planning
under carbon taxes (SPT) is defined as the following MILP:

$\displaystyle\textbf{SPT}:\quad\min$ $\displaystyle\;\;\sum_{k\in K,t\in
T}\left\\{\sum_{e\in E^{1}}{CI_{k}^{t}X_{ke}^{t}}+\sum_{i\in
I}{CP_{ik}^{t}Y_{ik}^{t}}+\sum_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{w\in
w}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}{CM_{wk}^{t}M_{wk}^{t}}+\sum_{e\in
E}{H(e)CL_{k}^{t}{O}_{k}X_{ke}^{t}}\right.$ $\displaystyle\left.+\sum_{e\in
E}{E_{H}(e)CA_{k}^{t}{O}_{k}X_{ke}^{t}}+\sum_{e\in
E}{E_{F}(e)CF_{k}^{t}{O}_{k}X_{ke}^{t}}+\sum_{s\in
S}(CC_{s}^{t}C_{s}^{t}+CO_{st}O_{s}^{t})\right\\}$
$\displaystyle\quad\quad+\sum_{i\in I,t\in T}{CH_{i}^{t}I_{i}^{t}}+\sum_{t\in
T}CT_{t}\mathcal{E}^{t}-\sum_{k\in K,t\in T}{\alpha_{k}^{t}RF_{k}^{t}}$ (34)
$\displaystyle\hskip
5.69054pt\text{s.t:}\quad(\ref{1.1}),(\ref{1.2}),(\ref{1.3}),(\ref{1.4}),(\ref{1.5}),(\ref{1.6}),(\ref{1.7}),(\ref{1.8}),(\ref{1.9}),(\ref{1.10}),(\ref{1.33}),(\ref{1.11}),(\ref{1.34}),$
$\displaystyle\hskip
28.45274pt(\ref{1.12}),(\ref{1.13}),(\ref{1.32}),(\ref{1.14}),(\ref{1.15}),(\ref{1.16}),(\ref{1.17}),(\ref{1.18}),(\ref{1.19}),(\ref{1.20}),$
$\displaystyle\hskip
28.45274pt(\ref{1.21}),(\ref{1.22}),(\ref{1.23}),(\ref{1.24}),(\ref{1.25}),(\ref{1.31}),(\ref{1.26}),(\ref{1.27}),(\ref{1.28}),(\ref{1.29}),$

where $E_{H}(e)=\max\\{0,$ $H(e)-T(e)\\}$ and $E_{F}(e)=\max\\{0,$
$T(e)-H(e)\\}$ are parameters that denote the number of work shifts that are
opened and closed between period $t-1$ and $t$ under state transition $e$,
respectively. The number of workers needed to operate machine $k$ is defined
by the parameter $O_{k}$. In A, a glossary of the used terminology can be
found.

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

###  3.3 Effectiveness carbon taxes measures

To determine carbon tax effectiveness, we defined two effectiveness measures.
The first measure is the transition level, defined as the proportion of
production allocated to each available technology at every period of the
planning horizon. The second measure 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 $N_{j}\cap N_{p}=\emptyset$ for any $j,p\in J:j\neq
p$. Thus, the transition level of technology $j$ reached in period $t$, is
defined as:

$\displaystyle R_{j}^{t}=\sum_{i\in I}\frac{\displaystyle\sum_{k\in
N_{j}}Y^{t}_{ik}}{\displaystyle\sum_{k\in K}Y^{t}_{ik}}\hskip
28.45274pt\forall t\in T,j\in J,$ (35)

where $Y^{t}_{ik}$ is an optimal variable of 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:

$\displaystyle\tau_{\beta}=\min\\{t:\sum_{j\in
J}\gamma_{j}R_{j}^{t}\geq\beta\\},$ (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:

$\displaystyle\gamma_{j}=\frac{\eta_{j}^{-1}-\underset{j\in
J}{\min}(\eta_{j}^{-1})}{\displaystyle\sum_{j\in
J}\left(\eta_{j}^{-1}-\underset{j\in J}{\min}(\eta_{j}^{-1})\right)}\hskip
28.45274pt\forall j\in J,$ (37)

where, $\eta_{j}=|N_{j}|^{-1}\sum_{k\in N_{j}}\sum_{i\in I}ep_{ik}$ for any
$j\in J$. It should be note that the dirtiest technology has a weight equal to
$0$.

###  3.4 Evaluation of carbon taxes effectiveness

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. $\sum_{j\in J}\gamma_{j}R_{j}^{|T|}$. 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 is the
effectiveness of the carbon tax in the firm.

##  4 Computational study

In this section, we present our numerical test and results. The main objective
of the computational study is to explore carbon taxes effectiveness to induce
a clean technology transition when the strategic capacity plan considers
different technologies, in terms of emissions and investment cost using SPT
model. Similarly to [15] and [52], two types of technologies are considered,
more precisely, dirty and clean technology.

We present an illustrative example of a firm that produces eight different
natural juice products, on which carbon taxes are imposed by the environmental
authority. The strategic capacity planning under carbon taxes considers
machines of two technologies, and the clean technology transition is
illustrated in terms of the transition levels ($R_{j}^{t}$). Moreover, besides
the measures proposed in our framework, we compare the total emissions of the
SPT model with the SPWT model. On the other hand, we simulate a firm that
faces several exogenous conditions, keeping the endogenous parameters of the
industrial example. For this we generate 50 instances, varying the magnitude
of the demand. Subsequently, each of these instances were solved considering
different relationships between technologies, in terms of emissions and
investment costs. Thus, to explore carbon taxes effectiveness when the
strategic capacity plan considers different technologies, the transition level
and transition period are computed for each emission and investment
relationship. Each instance considers an eight-year planning horizon. However,
to represent the continuity of the firm we simulated twelve-years and discard
the last four years in the results to avoid the undesirable effect of the
abrupt end of the planning horizon.

All experiments are solved on an Intel Core i5 2.3 GHz x 4 processor with 8 GB
RAM, using Gurobi 9.1.1. We used as termination condition an optimality gap of
$10^{-4}$ within a time limit of $36000$ seconds.

###  4.1 Industrial illustrative example

Consider a firm that requires to determine its strategic capacity expansion
plan for the upcoming 8 years, during which carbon taxes are implemented. The
production considers 8 items, which consisting of the juice from 4 different
fruits packed in glass containers of $1000$ and $300$ $cc$. The juice
production process consists in the grading of 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, it is required a capacity expansion for
the pasteurizing and filling process, which are considered as a production
line. Figure 3 shows the complete juice-making process, where the consider
processes are framed.

Figure 3: Juice packaging production line.

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, behavior that represents the life cycle of most products.
Moreover, increasing carbon taxes are imposed throughout the planning horizon.
Carbon taxes begin at $\$$35 per carbon emission ton, and increase yearly
until reaching to $\$$70 at the end of the planning horizon, following [48].

For simplicity, let $J=\\{1,2\\}$, where $j=1$ and $j=2$ are the dirty and
clean technology, respectively. Let $\widehat{ep}_{ij}$ be the emissions for
producing one unit of item $i$ using technology $j$, where
$ep_{ik}=\widehat{ep}_{ij}$ for any $i\in I$ and $k\in N_{j}$. It is
considered that the investment cost ($CI_{k}^{t}$) 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
$CI_{k}^{t}=\widehat{CI}^{t}_{j}$ for any $k\in N_{j}$ and $t\in T$. Thus, we
consider that carbon emissions and invesment cost 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 industrial example can be found in
B.

To illustrate the technological transition induced by carbon taxes, we
computed the optimal transition level $R_{j}^{t}$ for each $j\in J$ and $t\in
T$ . Figure 4 shows the optimal transition level for both technologies
throughout the planning horizon.

Figure 4: Optimal production allocation by period.

From Figure 4 it can be observed that during the first periods of the planning
horizon ($t=1,$ $2$), the firm only uses dirty technology, since the costs of
carbon taxes are insufficient for the firm to invest in, the more expensive,
clean technology. In the following three periods ($t=3,$ $4,$ $5$), the firm
starts to use clean technology, but still 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 emissions costs makes it
convenient for the strategic plan to invest in clean machines, and the
effectiveness not necessarily will be seen with a 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 model respectively. Figure 5 show the emissions throughout the
planning horizon.

Figure 5: Optimal emissions by period.

From Figure 5 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 in production ($t=3$ according to Figure 4), the growth rate of
emissions decreases considerably.

###  4.2 Test set

In what follows, we explore the impact of different exogenous parameters, such
as the available technology and demand magnitud on carbon taxes effectiveness
to induce a clean technology transition. Carbon taxes effectiveness is
determined by the clean transition level reached at the end of the planning
horizon, i.e., $R_{2}^{8}$. For the test set, we simplify SPT model, such that
the acquisition and replacement of machines become the only variables of
interest. Thus, maintenance is removed of SPT model, the work shifts are set
to one, and the products are aggregated into an equivalent single product. For
each instance of the test set, the aggregated demand of the equivalent product
is determine according to $\bar{d_{t}}=\xi\sum_{i\in I}d_{it}$, where $d_{it}$
is the demand used in the illustrative example and
$\xi\in\left\\{0.05+\frac{U(0.05,25)-0.05}{10}\right\\}$, i.e., $\xi$ varies
uniformly between $0.05$ and $2.5$. Similarly, the emissions for the
equivalent product are denoted by $\widehat{ep}_{j}$ for any $j\in J$. In
summary, we study the strategic capacity plan of a single-product firm under
carbon taxes that increases yearly from $\$$35 to $\$$70 per carbon emission
ton, under different exogenous conditions.

Two emission relationships
($\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}\in\\{0.5,0.7\\}$), and four
investment relationships
($\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}\in\\{1.3,1.4,1.5,1.6\\}$) are
considered. It should be noted that
$\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}<1$ and
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}>1$, since we assume that the less
emitter technology is more expensive.

Therefore, each instance of the test set is evaluated for the eight
combinations of emissions and investment relationships, generating $400$
scenarios. The parameters used for the test set are the aggregated parameters
of the illustrative example.

####  4.2.1 Transition level at the end of the planning horizon

To study the effect of the exogenous conditions considered, we analyze the
clean transition level reached at the end of the planning horizon. Figure 6
shows the proportion of allocated production to clean technology at the end of
the planning horizon ($R_{2}^{8}$) for every scenario.

(a) $\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.5$.

(b) $\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.7$.

Figure 6: $R_{2}^{8}$ for different emissions and investment relationships.

From Figure 6 we observe that carbon taxes effectiveness is highly influenced
by the emission relationship between the clean and dirty technology, and the
demand. More precisely, the greater the emission reduction of the clean
technology, the greater the probability of reaching a technological
transition. In addition, the higher the investment cost of the clean
technology, the lower the probability of reaching a technological transition.
Regarding the demand, the lower the demand size, the lower is the probability
of reaching a technological transition under any technology relationship.
Figure 6(a) shows that the probability of achieving a total technological
transition is greater than zero for every investment relationship considered.
On the other hand, Figure 6(b) shows that the probability of achieving a total
technological transition is greater than zero only for
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}=1.3$, and the probability of
achieving a technological transition is even zero for
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}=1.6$. Table 2 summarizes the
probabilities for achieving different levels of technological transition, and
the expected value of the transition at the end of the planning horizon for
each emission and investment relationship.

| $\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.5$ |  | $\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.7$
---|---|---|---
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}$ | $\mathbb{P}(R_{2}^{8}=0)$ | $\mathbb{P}(R_{2}^{8}\geq 0.50)$ | $\mathbb{P}(R_{2}^{8}\geq 0.75)$ | $\mathbb{P}(R_{2}^{8}\geq 1)$ | $\mathbb{E}(R_{2}^{8})$ |  | $\mathbb{P}(R_{2}^{8}=0)$ | $\mathbb{P}(R_{2}^{8}\geq 0.50)$ | $\mathbb{P}(R_{2}^{8}\geq 0.75)$ | $\mathbb{P}(R_{2}^{8}\geq 1)$ | $\mathbb{E}(R_{2}^{8})$
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

Table 2: Transition expectancy values.

Table 2 shows that for $\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.5$, the
probability to achieve a transition is strictly higher than for
$\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.7$. Moreover, for
$\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.5$ and
$\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.7$, the expected values are
$\mathbb{E}(R^{8}_{2})=0.74$ and $\mathbb{E}(R^{8}_{2})=0$ respectively.
Hence, we infer that the emission relationship has a great impact on carbon
taxes effectiveness to induce a technological transition. Furthermore, when
$\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.5$ the investment relationship
has a lower impact, e.g., $\mathbb{P}(R_{2}^{8}\geq 0.50)=0.96$ for
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}=1.3$ and $\mathbb{P}(R_{2}^{8}\geq
0.50)=0.92$ for $\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}=1.6$. On the
contrary, when $\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.7$ the investment
relationship has a higher impact on the transition, e.g.,
$\mathbb{P}(R_{2}^{8}\geq 0.50)=0.94$ for
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}=1.3$ and $\mathbb{P}(R_{2}^{8}\geq
0.50)=0$ for $\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}=1.6$. Hence, we infer
that the investment relationship has a greater impact on carbon taxes
effectiveness to induce a technological transition when the emission
relationship is higher. In summary, when clean technology is more expensive
and does not reduce emissions enough and the demand is small, the firm has low
or none incentive to change from a dirty technology, since it is more
convenient to pay the carbon tax than to invest in a clean technology.

####  4.2.2 Effect of technology relations on transition periods

For every scenario analized, we computed $\tau_{\beta}$, for
$\beta\in\\{0.50,0.75\\}$. For each $\beta$, only scenarios where
$\tau_{\beta}<\infty$ are considered, e.g., for
$\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.5$ and
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}=1.4$ the $94\%$ of instances reach
or surpass $\beta=0.50$, according to Table 2. Figure 7 shows the box-plots of
$\tau_{\beta}$ for $\beta\in\\{0.50,0.75\\}$ and the different emissions and
investment relationships.

(a) $\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.5$,$\;\tau_{0.50}$.

(b) $\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.5$,$\;\tau_{0.75}$.

(c) $\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.7$,$\;\tau_{0.50}$.

(d) $\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.7$,$\;\tau_{0.75}$.

Figure 7: $\tau_{\beta}$ for different emissions and investment relationships.

From Figure 7(a), we observe that for
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}\in\\{1.3,1.4\\}$, all instances
that achieved $\beta=0.50$ reach it at the beginning of the planning horizon,
i.e., $\tau_{0.50}=1$. In contrast, for
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}=1.5$ and $1.6$, we observe that
$\tau_{0.50}=1.5$ and $\tau_{0.50}=2.5$, respectively. From Figure 7(b), we
observe that for
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}\in\\{1.3,1.4,1.5\\}$, all instances
which achieved $\beta=0.75$ reach it at $\tau_{0.75}=1$. In contrast, for
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}=1.6$, we observe that
$\tau_{0.75}=4$. On the other hand, Figure 7(c) only contains the box-plots of
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}\in\\{1.3,1.4\\}$, where
$\tau_{0.50}=1.5$ and $\tau_{0.50}=3.5$, respectively. Further, Figure 7(d)
only contains the box-plots of $\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}=1.3$
and $1.4$, where $\tau_{0.75}=1.5$ and $\tau_{0.75}=4$, respectively. It
should be noted that
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}\in\\{1.5,1.6\\}$, no instance
reaches $\beta\in\\{0.5,0.7\\}$. Consequently, there is no box-plot for
$\frac{\widehat{CI}_{2}}{\widehat{CI}_{1}}\in\\{1.5,1.6\\}$ under
$\frac{\widehat{ep}_{2}}{\widehat{ep}_{1}}=0.7$. In summary, we observe that
the firm has incentive to transition earlier to a cleaner technology when the
benefit from reducing emissions is high and the increase of the investment
cost is low, i.e., $\tau_{\beta}$ decreases when the emission and investment
relationships decrease.

##  5 Conclusion

In this paper, we introduce a framework to evaluate carbon taxes effectiveness
to induce a clean technology transition on individual firms. The framework is
composed by three main elements. The first element is a strategic capacity
planning under carbon taxes model, which determines the optimal strategic plan
in terms of acquisition and usage of technologies throughout the planning
horizon. Thus, determines whether there is a technology transition, and if so,
the optimal technological transition strategy. The second element correspond
to the measures to evaluate 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. The third element correspond to the evaluation of the carbon tax
effectiveness itself, using the defined measures.

We apply the framework to an industrial illustrative example where the
strategic capacity plan under carbon taxes faces a clean and a dirty
technology. The technological transition is illustrated in terms of the
transition levels for each technology and total emissions throughout the
planning horizon. Several instances, based on the industrial illustrative
example, were generated to explore the impact of technology relationships, in
terms of emissions and investment, and size of the demand on carbon taxes
effectiveness. From our computational experiments, we observe that a full
clean technology transition is not reached in the planning horizon when the
emissions of the clean technology are a 70$\%$ of the dirty technology, and
the investment cost of clean technologies is 140$\%$ or higher than the dirty
technology. Thus, we infer that for costly clean technology that does not
greatly reduce emissions (compared to dirty technology), carbon taxes are not
effective on incentivizing the firm to change technologies. Furthermore, we
infer that a firm that face small demand has 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 a more expensive technology to reduce
emissions.

There are several issues left for future research. Since carbon taxes
effectiveness is affected by the demand, i.e., the size of the firm, the first
issue to be explored is to determine the optimal carbon tax policy for each
firm size, such that all firms in the industry are encouraged to a clean
technology transition. The environmental authority can choose from a range of
mechanism to reduce emissions and incentivize a transition to clean
technologies. Hence, the second issue is to incorporate into the framework
other common mechanism, 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 is unknown for 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 [5]. Following
this, the third issue is to consider a stochastic strategic capacity planning
model in the framework, such that the model consider uncertainty in the demand
and available technology. The last issue left is the scalability of SPT model,
as it becomes considerably harder to prove the optimality of the solution with
a real case complexity.

## Acknowledgements

Professor Pablo Escalona is grateful for the support of ANID, through grant
FONDECYT 11200287. Jorge Weston has received funding from the European Union’s
Horizon 2020 research and innovation program under the Marie Skłodowska-Curie
grant agreement No 764759 ETN “MINOA”.

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##  Appendix A Glossary of terms

Sets | Definition
---|---
$T$ |  Set of periods indexed by $t$, with $t=1,...,|T|$
$I$ |  Set of items indexed by $i$, with $i=1,...,|I|$
$K^{0}$ | Set of operative machines at the beginning of the planning horizon
$K^{c}$ | Set of machine that can be acquired over the planning horizon
$K$ |  Set of machine indexed by $k=1,...,N$, $K=K^{0}\cup K^{c}$
$S$ |  Set of states in which machine can be situated over the planning horizon, $S=\\{0,1,2,3\\}$ where
|  $s=0$ the machine is inoperative; $s=1$ the machine is operative to be used
one work shift;
|  $s=2$ the machine is available to be used two work shifts; $s=3$ the
machine is available to be used three work shifts;
$E$ |  Set of state transitions, with $E\varsubsetneq\\{(s,s\textprime)\in S\times S\\}$
Operational parameters |
$d_{it}$ |  Demand for item $i$ in period $t$, measure in units.
$s_{0}$ |  State of machine $k$ at the beginning of the planning horizon ($t=0$), for any $k\in K^{0}$
$H_{i}^{0}$ |  On-hand inventory of item $i$ at the beginning of period $1$, measure in units.
$FTM_{k}$ |  Fixed time maintenance of machine $k$, measure in units of time.
$RMT_{wk}$ |  Time required for the $w$-th maintenance of machine $k$, measure in units of time.
$r_{ik}$ |  Production rate of item $i$, produced with machine $k$, measure in unit per unit of time.
$\mu_{k}$ |  Maximum utilization for each machine $k$
$v_{k}$ |  Useful life of machine $k$, measure in units of time.
$l$ |  Available working time for any work shift $s$, measure in units of time.
$ep_{ik}$ |  Emissions for producing one unit of item $i$ using machine $k$, measure in $co_{2}$ ton per unit.
$eh_{i}$ |  Emissions of holding per unit per unit of item $i$, measure $co_{2}$ ton per unit of unit time.
$O_{k}$ |  Number of workers needed to operate machine $k$
Cost parameters |
$CI_{k}^{t}$ |  Investment cost of acquiring a machine $k$ in period $t$, measure in monetary units.
$CP_{ik}^{t}$ |  Production cost for one item $i$ produced by machine $k$ in the period $t$, measure in monetary units per unit.
$CM_{wk}^{t}$ |  Preventive $w$-th maintenance cost of machine $k$ in period $t$, measure in monetary units per maintenance.
$CL_{k}^{t}$ |  Labor cost for machine $k$ in period $t$, measure in monetary units per worker.
$CA_{k}^{t}$ |  Cost of hiring a worker for machine $k$ at the beginning of period $t$ , measure in monetary units per worker.
$CF_{k}^{t}$ |  Cost of firing a worker of machine $k$ at the beginning of period $t$, monetary unit per worker, measure in monetary units per worker.
$CO_{s}^{t}$ |  Cost of opening work shift $s$ in period $t$, measure in monetary units.
$CC_{s}^{t}$ |  Cost of closing work shift $s$ in period $t$, measure in monetary units.
$CH_{i}^{t}$ |  Holding cost per unit and unit time of item $i$ in period $t$, measure in monetary units per unit and unit time.
$CT_{t}$ |  Value of the carbon tax in period $t$, measure in monetary units per $co_{2}$ ton.
$\alpha_{k}^{t}$ |  Price per hour of useful life remaining of used machine $k$ in period $t$, measure in monetary units per unit time.
Parameter functions |
$H(e)$ |  State of machine $k$ in period $t$
$T(e)$ |  State of machine $k$ in period $t-1$
$E_{H}(e)$ |  Increase in work shifts in period $t$, $E_{H}(e)=\max\\{0,H(e)-T(e)\\}$
$E_{F}(e)$ |  Decrease in work shifts in period $t$, $E_{F}(e)=\max\\{0,T(e)-H(e)\\}$
$\mathcal{E}^{t}$ |  Total emissions in period $t$, $\mathcal{E}^{t}=\sum_{i\in I}(\sum_{k\in K}{ep_{ik}Y_{ik}^{t}}+eh_{i}I_{i}^{t})$, measure in $co_{2}$ ton.
Variables |
$X_{ke}^{t}$ |  $1$ if transition state $e$ occurs for machine $k$ at the beginning of period $t$; $0$ otherwise
$Z_{s}^{t}$ |  $1$ if $s$ work shifts are utilized in period $t$; $0$ otherwise
$O_{s}^{t}$ |  $1$ if work shift $s$ is opened in period $t$; $0$ otherwise
$C_{s}^{t}$ |  $1$ if work shift $s$ is closed in period $t$; $0$ otherwise
$Y_{ik}^{t}$ |  Units of item $i$ produce with machine $k$ in period $t$, measure in units.
$I_{i}^{t}$ |  Units of item $i$ in inventory at the end of period $t$, measure in units.
$M_{wk}^{t}$ |  $1$ if maintenance $w$-th is done to machine $k$ in period $t$; $0$ otherwise
$TM_{k}^{t}$ |  Accumulated production time of machine $k$ from its last maintenance at the beginning of period $t$, measure in units of time.
$RL_{k}^{t}$ |  Remaining useful life of machine $k$ at the beginning of period $t$, measure in units of time.
$RF_{k}^{t}$ |  Residual life of machine $k$ when sold in period $t$, measure in units of time.
Subset partitions |
$E^{0}$ |  Transition that represent a machine that remain inoperative, $E^{0}=\\{e_{0}\\}$
$E^{1}$ |  Transitions that represent the entry into operation of a machine, $E^{1}=\\{e_{1},e_{2},e_{3}\\}$
$E^{2}$ |  Transitions that represent a machine that was and stays in operation, $E^{2}=\\{e_{5},e_{6},e_{7},e_{9},e_{10},e_{11},e_{13},e_{14},e_{15}\\}$
$E^{3}$ |  Transitions that represent the discard of a machine, $E^{3}=\\{e_{4},e_{8},e_{12}\\}$
Table 3: Glossary of terms.

##  Appendix B Parameters of the illustrative example

  * Production rate: $\;\;r_{ik}=r_{i}\;\;\forall k\in K$

| i=1 | i=2 | i=3 | i=4 | i=5 | i=6 | i=7 | i=8
---|---|---|---|---|---|---|---|---
$r_{i}$ | 480 | 672 | 576 | 336 | 528 | 624 | 768 | 816

  * Utilization: ${\mu_{k}}=0.85\;\forall k\in K$

  * Useful life: $v_{k}=20000$ $\forall k\in K$

  * Time by shift: $l=2080$

  * Fixed time maintenance: $FTM_{k}=5000$ $\forall k\in K$

  * Time by maintenance: $RMT_{wk}=4$ $\forall w\in W,k\in K$

  * Workers by machine: $O_{k}=2\;\forall k\in K$

  * Shift work at $t=0$: $s_{0}=0$

  * Inventory emissions:

| $i=1$ | $i=2$ | $i=3$ | $i=4$ | $i=5$ | $i=6$ | $i=7$ | $i=8$
---|---|---|---|---|---|---|---|---
$eh_{i}$ | 0.023 | 0.032 | 0.027 | 0.016 | 0.025 | 0.029 | 0.036 | 0.038

  * Production cost: $\;\;CP_{ik}^{t}=CP_{i}^{t}\;\;\forall k\in K$

| $i=1$ | $i=2$ | $i=3$ | $i=4$ | $i=5$ | $i=6$ | $i=7$ | $i=8$
---|---|---|---|---|---|---|---|---
$CP^{1}_{i}$ | 0.075 | 0.105 | 0.09 | 0.053 | 0.083 | 0.098 | 0.12 | 0.128

  * Labor cost: $CL_{k}^{1}=4500$ $\forall k\in K$

  * Cost of hiring and firing: $CA_{k}^{1}=CF_{k}^{1}=5000$ $\forall k\in K$

  * Cost of opening and closing a shift: $CT_{s}^{1}=CU_{s}^{1}=5000$ $\forall s\in S$

  * Holding cost:

| $i=1$ | $i=2$ | $i=3$ | $i=4$ | $i=5$ | $i=6$ | $i=7$ | $i=8$
---|---|---|---|---|---|---|---|---
$CH_{i}^{1}$ | 0.66 | 0.92 | 0.79 | 0.46 | 0.72 | 0.85 | 1.05 | 1.12

  * Selling factor: $\alpha_{k}^{1}=\dfrac{0.8CI_{k}^{1}}{v_{k}}$ $\forall k\in K$

  * Discount factor: The discount factor used is $10\%$. Every costs behave over time as $C_{t}=C_{1}(1+0.1)^{1-t}$

  * Emissions of production:

| $i=1$ | $i=2$ | $i=3$ | $i=4$ | $i=5$ | $i=6$ | $i=7$ | $i=8$
---|---|---|---|---|---|---|---|---
$\widehat{ep}_{i1}$ | 0.30 | 0.21 | 0.25 | 0.42 | 0.27 | 0.23 | 0.18 | 0.17
$\widehat{ep}_{i2}$ | 0.30 | 0.21 | 0.25 | 0.42 | 0.27 | 0.23 | 0.18 | 0.17

  * Investment cost: $\widehat{CI}_{1}^{1}=65000\;$ and $\widehat{CI}_{2}^{1}=104000$

  * Maintenance cost: $\widehat{CM}_{wk}^{1}=600\;$

  * Demand:

$d_{it}$ | $t=1$ | $t=2$ | $t=3$ | $t=4$ | $t=5$ | $t=6$ | $t=7$ | $t=8$ | $t=9$ | $t=10$ | $t=11$ | $t=12$
---|---|---|---|---|---|---|---|---|---|---|---|---
$i=1$ | 20000 | 47726 | 63945 | 75452 | 84378 | 91670 | 97836 | 103178 | 107889 | 112103 | 115916 | 119396
$i=2$ | 28000 | 66816 | 89522 | 105633 | 118129 | 128339 | 136971 | 144449 | 151045 | 156945 | 162282 | 167155
$i=3$ | 24000 | 57271 | 76733 | 90542 | 101253 | 110004 | 117404 | 123813 | 129467 | 134524 | 139099 | 143276
$i=4$ | 14000 | 33408 | 44761 | 52816 | 59064 | 64169 | 68485 | 72224 | 75522 | 78472 | 81141 | 83577
$i=5$ | 22000 | 52498 | 70339 | 82997 | 92815 | 100837 | 107620 | 113495 | 118678 | 123314 | 127507 | 131336
$i=6$ | 26000 | 62044 | 83128 | 98087 | 109691 | 119172 | 127187 | 134131 | 140256 | 145734 | 150691 | 155215
$i=7$ | 32000 | 76361 | 102311 | 120723 | 135004 | 146673 | 156538 | 165084 | 172622 | 179365 | 185465 | 191034
$i=8$ | 34000 | 81134 | 108706 | 128268 | 143442 | 155840 | 166322 | 175402 | 183411 | 190576 | 197057 | 202974