Enhancing cybersecurity in cloud-assisted Internet of Things environments: A unified approach using evolutionary algorithms and ensemble learning

Mohammed Aljebreen; Hanan Abdullah Mengash; Khalid Mahmood; Asma A. Alhashmi; Ahmed S. Salama; Mohammed Aljebreen; Hanan Abdullah Mengash; Khalid Mahmood; Asma A. Alhashmi; Ahmed S. Salama

doi:10.3934/math.2024763

AIMS Mathematics

2024, Volume 9, Issue 6: 15796-15818. doi: 10.3934/math.2024763

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Research article Special Issues

Enhancing cybersecurity in cloud-assisted Internet of Things environments: A unified approach using evolutionary algorithms and ensemble learning

1.
Department of Computer Science, Community College, King Saud University, P.O. Box 28095, Riyadh 11437, Saudi Arabia
2.
Department of Information Systems, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Information Systems, College of Science & Art at Mahayil, King Khalid University, Saudi Arabia
4.
Department of Computer Science at College of Science, Northern Border University, Arar, Saudi Arabia
5.
Department of Electrical Engineering, Faculty of Engineering & Technology, Future University in Egypt, New Cairo 11845, Egypt

Received: 16 January 2024 Revised: 08 April 2024 Accepted: 10 April 2024 Published: 06 May 2024
MSC : 11Y40

Internet of Things (IoT) security is an umbrella term for the strategies and tools that protect devices connected to the cloud, and the network they use to connect. The IoT connects different objects and devices through the internet to communicate with similarly connected machines or devices. An IoT botnet is a network of infected or cooperated IoT devices that can be remotely organized by cyber attackers for malicious purposes such as spreading malware, stealing data, distributed denial of service (DDoS) attacks, and engaging in other types of cybercrimes. The compromised devices can be included in any device connected to the internet and communicate data with, e.g., cameras, smart home appliances, routers, etc. Millions of devices can include an IoT botnet, making it an attractive tool for cyber attackers to launch attacks. Lately, cyberattack detection using deep learning (DL) includes training neural networks on different datasets to automatically detect patterns indicative of cyber threats, which provides an adaptive and proactive approach to cybersecurity. This study presents an evolutionary algorithm with an ensemble DL-based botnet detection and classification (EAEDL-BDC) approach. The goal of the study is to enhance cybersecurity in the cloud-assisted IoT environment via a botnet detection process. In the EAEDL-BDC technique, the primary stage of data normalization using Z-score normalization is performed. For the feature selection process, the EAEDL-BDC technique uses a binary pendulum search algorithm (BPSA). Moreover, a weighted average ensemble of three models, such as the modified Elman recurrent neural network (MERNN), gated recurrent unit (GRU), and long short-term memory (LSTM), are used. Additionally, the hyperparameter choice of the DL approaches occurs utilizing the reptile search algorithm (RSA). The experimental outcome of the EAEDL-BDC approach can be examined on the N-BaIoT database. The extensive comparison study implied that the EAEDL-BDC technique reaches a superior accuracy value of 99.53% compared to other approaches concerning distinct evaluation metrics.

Keywords:

Citation: Mohammed Aljebreen, Hanan Abdullah Mengash, Khalid Mahmood, Asma A. Alhashmi, Ahmed S. Salama. Enhancing cybersecurity in cloud-assisted Internet of Things environments: A unified approach using evolutionary algorithms and ensemble learning[J]. AIMS Mathematics, 2024, 9(6): 15796-15818. doi: 10.3934/math.2024763

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Abstract

1. Introduction

The models involving entry-exit decisions apply to many situations, as stated in ^[1,2]. Such models may be described simply as follows. A firm decides when to invest in or when to abandon a project that can bring profit. There is plenty of literature on models concerning how to schedule the timing under the assumption that exiting the project does not take time, so we do not list them in detail, but refer to ^{[1,2,3,4,5,6,7,8,9]}. The ideas of entry-exit decisions apply to many concrete issues, for example, relationships of globalization and entrepreneurial entry-exit ^[10], when to invest or expand a start-up firm ^[11], when to enter or exit stock markets ^[12], and how to make a schedule for buying carbon emission rights ^[13].

In practice, the exiting process may take a long time, so ignoring it is not reasonable. For example, Brexit took more than 3 years, from June 23, 2016 to January 31, 2020 (https://www.britannica.com/topic/Brexit). The models ^[14,15] take account of the time, but with equal output rates in the regular production and exit periods. It is an apparent feature that the output rate of the project is usually reduced during the exit period. In this paper, we introduce a parameter to describe the output rate during the exit period and completely discuss the effects of output reduction on entry-exit decisions, under the assumption that the commodity price of the project follows a geometric Brownian motion.

We use the optimal stopping theory to carry out the study and obtain the explicit expressions of the optimal activating time and start time of the exit, which is one of the contributions of the paper. With these explicit expressions in hand, we can carefully analyze the effects of output reduction on entry-exit decisions, which is another contribution of the paper.

If the optimal choice is never to exit the project, the output reduction has no effect on the optimal entry-exit decision.

If the optimal exit is in a finite time, the situation becomes complicated. However, we obtain complete criterions, which determine the effects of output reduction on entry-exit decisions (see Section 4).

We outline the structure of this paper. In Section 2, we describe the model in detail. In Section 3, we determine an optimal entry-exit decision. In Section 4, we discuss the effects of output reduction during exit period on entry-exit decisions. Some conclusions are drawn in Section 5.

2. The model

Assume that the price process $P$ of one unit product follows

$\begin{equation} \mathrm{d}P(t) = \mu P(t)\mathrm{d}t+\sigma P(t)\mathrm{d}B(t) \;{\rm{and}}\; P(0) = p , \end{equation}$

(2.1)

where $\mu\in \mathbb{R}$ , $\sigma, \, p > 0$ , and $B$ is a one dimensional standard Brownian motion, which denotes uncertainty. In this paper, all times are stopping times w.r.t. the filtration generated by the Brownian motion $B$ .

Since involving the construction period leads to complicated calculations and distracts us from analyzing the effects of reduction on entry-exit decisions, and the effects of the construction period have been discussed in ^[3,15], we assume that there is no construction period (it may happen when the firm buys a project). The firm activates the project at time $\tau_I$ with the entry cost $K_I$ and completes the abandonment of the project during the time interval $[\tau_O, \tau_O+\delta]$ , with the exit cost valued at $K_O$ at time $\tau_O+\delta$ . Without loss of generality, we assume that the firm produces one unit product per unit time during the period $[\tau_I, \tau_O]$ at the marginal cost $C$ and $\alpha$ ( $0\leq\alpha\leq1$ ) unit products per unit time during the time interval of exiting the project $[\tau_O, \tau_O+\delta]$ .

To answer the two questions, what time is optimal to activate the project and what time is optimal to start the abandonment procedure, we solve the optimization problem

$\begin{equation} \begin{aligned} J(p) = &\sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p\bigg[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t+\alpha\int_{\tau_O}^{\tau_O+\delta}\exp(-r t)(P(t)-C)\mathrm{d}t\\ &-\exp(-r\tau_I)K_I-\exp(-r(\tau_O+\delta))K_O\bigg]. \end{aligned} \end{equation}$

(2.2)

We call stopping times $\tau_I$ and $\tau_O$ the activating times and start times of the exit, respectively, and we call the function $J$ the maximal expected present value of the project.

3. An optimal entry-exit decision

If $r\leq \mu$ , some straight calculations show us that

$\mathbb{E}^p\left[\int_{0}^{+\infty}\exp(-r t)(P(t)-C)\mathrm{d}t\right] = +\infty.$

Thus, we obtain the following result.

Theorem 3.1. Assume that $r\leq \mu$ , then $\tau_I^*$ : $= 0$ is an optimal activating time and $\tau_O^*$ : $= +\infty$ is an optimal start time of the exit, i.e., the firm should never exit the project. In addition, the function $J$ in (2.2) is given by $J\equiv+\infty$ .

In the rest of this section, we assume $r > \mu$ .

Taking $\tau_I = \tau_O$ : $= 0$ in (2.2), we have

$J(p)\geq -K_I+\frac{\alpha(1-\exp((\mu-r)\delta))}{r-\mu}p-\exp(-r\delta)K_O-\alpha\frac{C}{r}(1-\exp(-r\delta)),$

thus, in the remains of this section, we always assume that

$rK_I+\exp(-r\delta)rK_O+\alpha(1-\exp(-r\delta))C\geq0$

to avoid arbitrage opportunities.

Let $\lambda_1$ and $\lambda_2$ be the solutions to the equation

$r-\mu\lambda-\frac{1}{2}\sigma^2\lambda(\lambda-1) = 0$

with $\lambda_1 < \lambda_2$ , then we have $\lambda_1 < 0$ and $\lambda_2 > 1$ .

Theorem 3.2. Assume that $r > \mu$ . The following are true:

(ⅰ) If

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta)\leq0,$

then $(\tau_I^*, \tau_O^*)$ is a solution to (2.2), where

$\tau_I^* = \inf\{t: t > 0, P(t)\geq p_I\}$

and $\tau_O^* = +\infty$ . Here,

$p_I = \frac{\lambda_2}{\lambda_2-1}(r-\mu)\left(\frac{C}{r}+K_I\right).$

In addition,

$J(p) = \left\{ \begin{aligned} &B p^{\lambda_2}, \quad \mathit{\text{if}}\;\,p < p_I,\\ &\frac{p}{r-\mu}-\frac{C}{r}-K_I,\;\;\mathit{\text{if}}\;\, p\geq p_I, \end{aligned} \right.$

where

$B = \frac{{p_I}^{1-\lambda_2}}{\lambda_2(r-\mu)}.$

(ⅱ) If

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta) > 0$

and

$\begin{equation} \alpha(1-\exp((\mu-r)\delta))p_O-\alpha C(1-\exp(-r\delta))-\exp(-r\delta)rK_O-rK_I\leq 0, \end{equation}$

(3.1)

where

$p_O = \frac{\lambda_1}{\lambda_1-1}\frac{r-\mu}{1-\alpha+\alpha\exp((\mu-r)\delta)}\left((1-\alpha)\frac{C}{r}+\exp(-r\delta)\bigg(\alpha\frac{C}{r}-K_O\bigg)\right),$

then $(\tau_I^*, \tau_O^*)$ is a solution to (2.2), where

$\tau_I^* = \inf\{t: t > 0, P(t)\geq p_I\}$

and

$\tau_O^* = \inf\{t: t > \tau_I^*, P(t)\leq p_O\}.$

Here, $p_I$ is the largest solution of the algebraic equation

$A(\lambda_2-\lambda_1)p_I^{\lambda_1}+\frac{(\lambda_2-1)}{r-\mu}p_I-\lambda_2\left(\frac{C}{r}+K_I\right) = 0.$

In addition,

$J(p) = \left\{ \begin{aligned} &B p^{\lambda_2}, \quad \mathit{\text{if}}\;\, p < p_I,\\ &Ap^{\lambda_1}+\frac{p}{r-\mu}-\frac{C}{r}-K_I,\;\;\mathit{\text{if}}\;\, p\geq p_I, \end{aligned} \right.$

where

$A = \frac{1-\alpha+\alpha\exp((\mu-r)\delta)}{\lambda_1(\mu-r)}{p_O}^{1-\lambda_1}$

and

$B = \lambda_1\lambda_2^{-1}Ap_I^{\lambda_1-\lambda_2}+\frac{{p_I}^{1-\lambda_2}}{\lambda_2(r-\mu)}.$

Remark 3.3. We propose condition (3.1) to eliminate the possibility that the firm enters the project at a trigger price lower than the optimal trigger price of the exit, i.e., the firm enters the project and then immediately decides to exit the project.

In light of [14, Theorems 3.1 and 5.2], we have the following Lemma 3.4, which serves as preparation for the proof of Theorem 3.2.

Lemma 3.4. If $(\tau_1^*, \tau_2^*)$ is a solution to the optimization problem

$\begin{equation} \widetilde{J}(p): = \sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p\bigg[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t -\exp(-r\tau_I)K_I-\exp(-r \tau_O)(l_1P(\tau_O)+l_0)\bigg], \end{equation}$

(3.2)

it is also a solution to (2.2) and $J(x) = \widetilde{J}(x)$ , where

$l_1: = \frac{\alpha(1-\exp((\mu-r)\delta))}{\mu-r}$

and

$l_0: = \alpha\frac{C}{r}(1-\exp(-r\delta))+\exp(-r\delta)K_O.$

Proof. By the strong Markov property of the process $\{(s+t, P(t)), t\geq 0\}$ , where $s\in\mathbb{R}$ , we have

$\begin{aligned} &\mathbb{E}^p\bigg[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t+\alpha\int_{\tau_O}^{\tau_O+\delta}\exp(-r t)(P(t)-C)\mathrm{d}t -\exp(-r\tau_I)K_I-\exp(-r(\tau_O+\delta))K_O\bigg]\\ & = \mathbb{E}^p\bigg[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t+\alpha\exp(-r\tau_O)\mathbb{E}^{P(\tau_O)}\bigg[\int_0^{\delta}\exp(-r t)(P(t)-C)\mathrm{d}t\bigg]\\ &\quad\ -\exp(-r\tau_I)K_I-\exp(-r(\tau_O+\delta))K_O\bigg]. \end{aligned}$

Thus, we need to calculate

$\begin{aligned} &\alpha\mathbb{E}^{P(\tau_O)}\bigg[\int_0^{\delta}\exp(-r t)(P(t)-C)\mathrm{d}t\bigg]-\exp(-r\delta)K_O\\ & = \alpha\int_0^{\delta}\exp(-r t)(\exp(\mu t)P(\tau_O)-C)\mathrm{d}t-\exp(-r\delta)K_O\\ & = -l_1P(\tau_O)-l_0. \end{aligned}$

The proof is complete. □

Remark 3.5. The proof of Lemma 3.4 has an economic meaning as follows. We first discount the benefit during the abandonment period to time $\tau_O$ , then discount this value to time zero.

With the help of Lemma 3.4, we can prove Theorem 3.2.

Proof of Theorem 3.2. (1) By Lemma 3.4, we solve problem (3.2),

$\begin{align} &\sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p\bigg[\int_{\tau_I}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t-\exp(-r\tau_I)K_I-\exp(-r \tau_O)(l_1P(\tau_O)+l_0)\bigg]\\ & = \sup\limits_{\tau_I\leq\tau_O}\mathbb{E}^p\bigg[\exp(-r\tau_I)\int_{0}^{\tau_O-\tau_I}\exp(-r t)(P(t+\tau_I)-C)\mathrm{d}t\\&\quad-\exp(-r\tau_I)K_I-\exp(-r \tau_O)(l_1P(\tau_O)+l_0)\bigg]\\ & = \sup\limits_{\tau_I}\mathbb{E}^p\bigg[\exp(-r\tau_I)(G(P(\tau_I))-K_I)\bigg]\\ & = :H(p), \end{align}$

(3.3)

where

$\begin{equation} G(p): = \sup\limits_{\tau_O}\mathbb{E}^p\bigg[\int_{0}^{\tau_O}\exp(-r t)(P(t)-C)\mathrm{d}t -\exp(-r \tau_O)(l_1P(\tau_O)+l_0)\bigg]. \end{equation}$

(3.4)

(2) Assume

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta)\leq0 .$

We first solve problem (3.4) and then problem (3.3).

For problem (3.4), noting

$\mathbb{E}^p\bigg[\int_0^\infty\exp(-rt)(P(t)-C)\mathrm{d}t\bigg] = \frac{p}{r-\mu}-\frac{C}{r},$

we see that

$\mathbb{E}^p\bigg[\int_0^\infty\exp(-rt)(P(t)-C)\mathrm{d}t\bigg]\geq -l_1p-l_0,$

which implies

$\tau_O^* = +\infty$

and

$G(p) = \frac{p}{r-\mu}-\frac{C}{r}.$

Set

$h(p): = \frac{p}{r-\mu}-\frac{C}{r}-K_I.$

Since

$\{p:p > 0\;\mathrm{and}\;rh(p)-\mu ph'(p)\geq 0\} = \{p:p\geq C+rK_I\},$

the exercise region of problem (3.3) takes the form $[p_I, +\infty)$ for some

$p_I\geq C+rK_I.$

The function $H$ satisfies

$rH-\mu pH'-\frac{1}{2}\sigma^2p^2H'' = 0$

on the continuation region $(0, p_I)$ and is Lipschitz continuous on $(0, +\infty)$ and $C^1$ continuous at $p_I$ . Thus, we get

$H(p) = Bp^{\lambda_2}$

and $B$ and $p_I$ solve

$\left\{ \begin{aligned} &B p_I^{\lambda_2} = \frac{p_I}{r-\mu}-\frac{C}{r}-K_I,\\ &\lambda_2Bp_I^{\lambda_2-1} = \frac{1}{r-\mu}, \end{aligned} \right.$

by which we finish the proof of (ⅰ).

(3) Assume

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta) > 0$

and (3.1) hold. We again first solve problem (3.4) and then problem (3.3).

Set

$g(p): = -l_1p-l_0.$

A straight calculation shows that

$\begin{aligned} \{p:p > 0\;\mathrm{and}\;rg-\mu p g'(p)-p+C\geq0\} = \bigg(0,\frac{(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta)}{1-\alpha(1-\exp((\mu-r)\delta))}\bigg), \end{aligned}$

which means the exercise region of problem (3.4) takes the form $(0, p_O]$ for some

$p_O\leq \frac{(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta)}{1-\alpha(1-\exp((\mu-r)\delta))}.$

The function $G$ satisfies

$rG-\mu pG'-\frac{1}{2}\sigma^2p^2G''-p+C = 0$

on the continuation region $(p_O, +\infty)$ and is Lipschitz continuous on $(0, +\infty)$ and $C^1$ continuous at $p_O$ . Thus, we get

$G(p) = Ap^{\lambda_1}+\frac{p}{r-\mu}-\frac{C}{r},$

and $A$ and $p_O$ solve

$\left\{ \begin{aligned} Ap_O^{\lambda_1}+\frac{p_O}{r-\mu}-\frac{C}{r}& = -l_1p_O-l_0,\\ \lambda_1Ap_O^{\lambda_1-1}+\frac{1}{r-\mu}& = -l_1, \end{aligned} \right.$

which implies coefficient $A$ and the optimal exit trigger pricer of (ⅱ).

To solve problem (3.4), we define

$h(p): = G(p)-K_I.$

In light of (3.1),

$\{p:p > 0\;\mathrm{and}\;rh(p)-\mu ph'(p)\geq 0\} = \{p:p\geq C+rK_I\},$

thus, the exercise region of problem (3.3) takes the form $[p_I, +\infty)$ for some

$p_I\geq C+rK_I.$

The function $H$ satisfies

$rH-\mu pH'-\frac{1}{2}\sigma^2p^2H'' = 0$

on the continuation region $(0, p_I)$ and is Lipschitz continuous on $(0, +\infty)$ and $C^1$ continuous at $p_I$ . Thus, we get

$H(p) = Bp^{\lambda_2},$

and $B$ and $p_I$ solve

$\left\{ \begin{aligned} B p_I^{\lambda_2}& = Ap_I^{\lambda_1}+\frac{p_I}{r-\mu}-\frac{C}{r}-K_I,\\ \lambda_2Bp_I^{\lambda_2-1}& = \lambda_1Ap_I^{\lambda_1-1}+\frac{1}{r-\mu}, \end{aligned} \right.$

by which we finish the proof of (ⅱ). □

4. Discussions

We analyze the effects of output reduction during the exit period on entry-exit decisions.

If $r\leq \mu$ , the firm has an optimal time $\tau_I^* = 0$ to activate the project and should never exit the project. Thus, the reduction does not affect entry-exit decisions.

If $r > \mu$ and

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta)\leq0,$

the optimal time to activate the project is given by

$\tau_I^* = \inf\{t: t > 0, P(t)\geq p_I\},$

where

$p_I = \frac{\lambda_2}{\lambda_2-1}(r-\mu)\left(\frac{C}{r}+K_I\right).$

Thus, the reduction does not affect the optimal activating time. As same as the case of $r\leq\mu$ , the firm should never exit the project and the reduction does not affect exit decisions.

Assume that $r > \mu$ and

$(1-\alpha)C+(\alpha C-rK_O)\exp(-r\delta) > 0.$

We first analyze the effects of reduction on the optimal start time of the exit. By (ⅱ) of Theorem 3.2, the optimal trigger price is an increasing function of $\alpha$ if

$\exp(-\mu\delta)(C-rK_O) > C-\exp(-r\delta)rK_O,$

a decreasing function if

$\exp(-\mu\delta)(C-rK_O) < C-\exp(-r\delta)rK_O,$

and a constant function if

$\exp(-\mu\delta)(C-rK_O) = C-\exp(-r\delta)rK_O.$

We list some examples to illustrate the analysis. Taking

$r = 0.2,\; \; \; \mu = -0.1,\; \; \; \sigma = 0.3,\; \; \; \delta = 2,\; \; \; C = 5,\; \; \; K_I = 20,\; \; \text{and}\; \; K_O = -10,$

we have

$\exp(-\mu\delta)(C-rK_O) > C-\exp(-r\delta)rK_O,$

then the optimal trigger price is an increasing function of $\alpha$ . See . If we replace $\mu = -0.1$ with $\mu = 0.1$ , we have a decreasing function. See Figure 2.

Figure 1.

$\alpha$ -

$p_O$ -increasing.

DownLoad: Full-Size Img PowerPoint

Figure 2.

$\alpha$ -

$p_O$ -decreasing.

DownLoad: Full-Size Img PowerPoint

We conclude that:

(1) If

$\exp(-\mu\delta)(C-rK_O) > C-\exp(-r\delta)rK_O,$

as the reduction increases (i.e., $\alpha$ decreases), the firm will postpone the exit.

(2) If

$\exp(-\mu\delta)(C-rK_O) < C-\exp(-r\delta)rK_O,$

as the reduction increases, the firm will advance the exit.

(3) If

$\exp(-\mu\delta)(C-rK_O) = C-\exp(-r\delta)rK_O,$

the reduction does not affect the exit.

To analyze the effects of reduction on the optimal time to activate the project, we define two functions as follows:

$f(p): = A(\lambda_2-\lambda_1)p^{\lambda_1}$

and

$g(p): = -\frac{(\lambda_2-1)}{r-\mu}p+\lambda_2\left(\frac{C}{r}+K_I\right)$

according to (ⅱ) of Theorem 3.2. Thus, the functions $f$ and $g$ intersect at two points, say, $(p_1, f(p_1))$ and $(p_2, f(p_2))$ with $p_1 < p_2$ , and the optimal trigger price $p_I$ of activating the project is given by $p_I = p_2$ .

To capture the behavior of $p_I$ as $\alpha$ varies, we only need to examine the behavior of the coefficient $A$ as $\alpha$ varies. Setting

$\begin{align*} M: = &(C-\exp(-r\delta)rK_O)(1-\exp((\mu-r)\delta))\\&+(1-\lambda_1)\exp(-r\delta)(C-rK_O-C\exp(\mu\delta)+\exp((\mu-r)\delta)rK_O) \end{align*}$

and

$N: = -C(1-\exp(-r\delta))(1-\exp((\mu-r)\delta)),$

after some calculations, we find that:

(1) If $M/N\leq0$ , $A$ is increasing on $[0, 1]$ and $p_I$ is decreasing on $[0, 1]$ .

(2) If $M/N\geq1$ , $A$ is decreasing on $[0, 1]$ and $p_I$ is increasing on $[0, 1]$ .

(3) If $0 < M/N < 1$ , $A$ is decreasing on $[0, M/N]$ and increasing on $[M/N, 1]$ and $p_I$ is increasing on $[0, M/N]$ and decreasing on $[M/N, 1]$ .

Numerical examples help us understand the analysis above.

demonstrates the decreasing of $p_I$ ( $r = 0.2$ , $\mu = -0.1$ , $\sigma = 0.3$ , $\delta = 0.6$ , $C = 5$ , $K_I = 20$ , and $K_O = -10$ ). demonstrates the increasing of $p_I$ ( $r = 0.2$ , $\mu = 0.1$ , $\sigma = 0.3$ , $\delta = 2$ , $C = 5$ , $K_I = 20$ , and $K_O = -10$ ), and demonstrates the non-monotonicity of $p_I$ ( $r = 0.2$ , $\mu = -0.1$ , $\sigma = 0.3$ , $\delta = 0.6$ , $C = 5$ , $K_I = 20$ , and $K_O = -10$ ).

Figure 3.

$\alpha$ -

$p_I$ -decreasing.

DownLoad: Full-Size Img PowerPoint

Figure 4.

$\alpha$ -

$p_I$ -increasing.

DownLoad: Full-Size Img PowerPoint

Figure 5.

$\alpha$ -

$p_I$ -non-monotonicity.

DownLoad: Full-Size Img PowerPoint

In other words, here are the effects of the reduction on the optimal time of activating the project:

(1) If $M/N\leq0$ , as the reduction increases (i.e., $\alpha$ decreases), the firm will postpone activating the project.

(2) If $M/N\geq0$ , as the reduction increases, the firm will advance activating the project.

(3) If $0 < M/N < 1$ , as the reduction increases, the firm will postpone activating the project then advance activating the project.

5. Conclusions

There is an apparent phenomenon that firms usually reduce their output rate during stopping the regular production of a project. We intend to analyze the effects of reduction on the optimal entry-exit decision. Since the papers ^[3,15] have investigated the effects of the construction period on entry-exit decisions, we assume that the project has been constructed and the production immediately starts for concentrating on the study of the effects of reduction.

We introduce the output rate into the models ^[14,15] and describe the problem using the optimal stopping theory, obtaining explicit solutions in Theorems 3.1 and 3.2. These explicit solutions help us in discovering the effects of reduction on the optimal entry-exit decision.

We carefully examine the effects of reduction. According the analysis listed in Section 4, we come to the effects of reduction as follows. If the firm should never exit the project to obtain the maximal profit, the reduction does not affect the optimal entry-exit decision. However, if the firm exits the project in finite time, the effects show in a complicated manner. We study the effects analytically and numerically, and provide the relations of the parameters involved in the model that can determine the effects.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Many thanks are due to the editors and reviewers for their constructive suggestions and valuable comments. This work is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. N2023034).

Conflict of interest

No potential conflict of interest exists regarding the publication of this paper.

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